Bibliography

The classical theory of averaging and homogenization, the literature around the regime-switching example, and the wider literature collected on the survey page, grouped by role. Every DOI was checked against Crossref. The literature map draws a selection of them: the regime-switching homogenization papers and the strands nearest to them.

Regime-switching bond prices as linear ODE systems

Because the mean-reversion speed does not switch, the regime-switching model is Markov-modulated exponential-affine, and its bond price reduces to a linear ODE system.

• Cotton, P. (2001). An Analytic Approach to Ornstein–Uhlenbeck Processes with Fluctuating Parameters and Applications in the Modeling of Fixed Income Securities. PhD thesis, Stanford University. pdf · ProQuest.
  Treats Ornstein-Uhlenbeck short rates and hazard rates whose parameters are driven by a Markov chain. It reduces bond prices and survival probabilities to a linear system of ordinary differential equations and expands them for fast switching. Its second half is developed to all orders in the regime-switching pages.
• Elliott, R. J. and Mamon, R. S. (2002). An interest rate model with a Markovian mean reverting level. Quantitative Finance 2(6):454-458. doi:10.1088/1469-7688/2/6/304.
  Takes a Vasicek short rate whose mean-reversion level is driven by a finite-state Markov chain and gives the bond price analytically through a fundamental matrix. It covers the regime-switching Vasicek model in the case of equal volatilities.
• Elliott, R. J. and Siu, T. K. (2009). On Markov-modulated Exponential-affine Bond Price Formulae. Applied Mathematical Finance 16(1):1-15. doi:10.1080/13504860802015744.
  Proves Markov-modulated exponential-affine bond price formulae for regime-switching Hull-White and CIR short rates, with coefficients given by fundamental matrix solutions of linear matrix differential equations, which contains the reduction used in the examples as a special case.
• Landén, C. (2000). Bond pricing in a hidden Markov model of the short rate. Finance and Stochastics 4(4):371-389. doi:10.1007/pl00013526.
  Studies a short rate whose drift and diffusion parameters are modulated by a possibly unobserved Markov process, introduces semi-affine term structures, gives a closed-form bond price when the chain has two states, and studies a Vasicek extension numerically.
• van Beek, M., Mandjes, M., Spreij, P. and Winands, E. (2020). Regime switching affine processes with applications to finance. Finance and Stochastics 24(2):309-333. doi:10.1007/s00780-020-00419-2.
  Shows that an affine process with Markov-switching parameters is affine on the enlarged state space, so transforms reduce to ODE systems, unifying earlier semi-analytic regime-switching bond formulas.
• Rodrigo, M. R. and Mamon, R. S. (2021). Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial and Management Optimization 17(5):2685ff. doi:10.3934/jimo.2020089.
  Gives exact and approximate zero-coupon bond formulas for Markov-modulated affine short rates with time-dependent coefficients and validates them on a regime-switching Vasicek model.

Fast-switching expansions

The nearest published expansions in the inverse switching rate. None treats the regime-switching Vasicek bond price or survival probability.

• Yin, G. (2009). Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process. Asymptotic Analysis 65(3-4):203-222. doi:10.3233/asy-2009-0953.
  Develops uniform asymptotic expansions, with averaged Black-Scholes leading term and full series, for European option prices when the regime chain switches fast, the closest published analogue of the regime-switching expansion but for equity options rather than bond prices.
• Basu, A. and Ghosh, M. K. (2009). Asymptotic analysis of option pricing in a Markov modulated market. Operations Research Letters 37(6):415-419. doi:10.1016/j.orl.2009.06.005.
  Analyzes Markov-modulated option prices in the limits where the chain moves very fast or very slowly relative to the asset.
• Il'in, A. M., Khasminskii, R. Z. and Yin, G. (1999). Asymptotic Expansions of Solutions of Integro-Differential Equations for Transition Densities of Singularly Perturbed Switching Diffusions: Rapid Switchings. Journal of Mathematical Analysis and Applications 238(2):516-539. doi:10.1006/jmaa.1998.6532.
  Obtains asymptotic expansions of the transition densities of diffusions whose coefficients are modulated by a rapidly switching Markov chain, the density-level analogue of the expansion for a Markov-modulated Ornstein-Uhlenbeck process.
• Yin, G. G. and Zhang, Q. (1998). Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach. Springer, Applications of Mathematics 37 (2nd ed. 2013, doi 10.1007/978-1-4614-4346-9). doi:10.1007/978-1-4612-0627-9.
  Constructs asymptotic expansions, with outer and initial-layer terms, for probability distributions of Markov chains with generators of the form Q/epsilon, the general machinery behind expanding in the inverse switching rate.
• Khasminskii, R. Z. and Yin, G. (1996). Asymptotic Series for Singularly Perturbed Kolmogorov-Fokker-Planck Equations. SIAM Journal on Applied Mathematics 56(6):1766-1793. doi:10.1137/s0036139994270085.
  Derives full asymptotic series with error bounds for Kolmogorov-Fokker-Planck equations of diffusions with fast and slow components.
• Berry, M. V. (1984). Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London A 392(1802):45-57. doi:10.1098/rspa.1984.0023.
  Introduces the geometric phase of a cyclically transported eigenspace and its gauge-invariant closed-loop content. The corrector connection here obeys the same exact-differential gauge law, but governs a backward-equation amplitude rather than a quantum phase.
• Sinitsyn, N. A. and Nemenman, I. (2007). Universal Geometric Theory of Mesoscopic Stochastic Pumps and Reversible Ratchets. Physical Review Letters 99:220408. doi:10.1103/PhysRevLett.99.220408.
  Develops geometric phase contributions for slowly driven stochastic kinetics and shows that the periodic contribution depends on the oriented parameter-space protocol rather than its traversal speed; the observable there is a pump current, not the backward-equation amplitude used here.
• Khasminskii, R. Z., Yin, G. and Zhang, Q. (1996). Asymptotic Expansions of Singularly Perturbed Systems Involving Rapidly Fluctuating Markov Chains. SIAM Journal on Applied Mathematics 56(1):277-293. doi:10.1137/S0036139993259933.
  Develops matched outer and boundary-layer expansions for time-varying systems whose singular fast matrix is a finite-state Markov generator, the primary general result underlying the explicit finite-chain composite.
• Meyer, C. D., Jr. (1975). The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains. SIAM Review 17(3):443-464. doi:10.1137/1017044.
  Develops the group inverse for finite Markov chains and its normalization on the stationary and transient subspaces, the algebra used by the explicit finite-chain initial-layer corrector.
• Huang, G., Mandjes, M. and Spreij, P. (2014). Weak convergence of Markov-modulated diffusion processes with rapid switching. Statistics and Probability Letters 86:74-79. doi:10.1016/j.spl.2013.12.013.
  Proves that a diffusion whose drift and diffusion coefficients are modulated by a rapidly switching Markov chain converges weakly to the diffusion with averaged coefficients, which is the leading-order statement underlying the averaged Vasicek limit.
• Huang, G., Jansen, H. M., Mandjes, M., Spreij, P. and Turck, K. D. (2016). Markov-modulated Ornstein-Uhlenbeck processes. Advances in Applied Probability 48(1):235-254. doi:10.1017/apr.2015.15.
  Computes moments and Laplace-transform PDEs of an Ornstein-Uhlenbeck process with Markov-modulated parameters and proves functional central limit theorems when the modulating chain is accelerated, which is the regime-switching state process studied as a process rather than through its exponential functional.
• Huang, G., Mandjes, M. and Spreij, P. (2016). Large deviations for Markov-modulated diffusion processes with rapid switching. Stochastic Processes and their Applications 126(6):1785-1818. doi:10.1016/j.spa.2015.12.005.
  Proves a joint sample-path large deviations principle for a Markov-modulated diffusion and the chain's occupation measure under rapid switching and small noise.

The classics of averaging and homogenization

• Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978). Asymptotic Analysis for Periodic Structures. North-Holland, Studies in Mathematics and its Applications 5 (reprinted AMS Chelsea Publishing, 2011). doi:10.1090/chel/374.
  The standard reference for two-scale expansions and correctors in elliptic and parabolic equations with rapidly varying coefficients, including the probabilistic treatment of diffusions in periodic and random media on which the corrector calculation is modelled; the DOI is for the 2011 AMS Chelsea reprint.
• Papanicolaou, G. C., Stroock, D. W. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. Papers from the Duke Turbulence Conference (Durham, NC, 1976), Duke University Mathematics Series III, Duke University. link.
  Proves diffusion limits for processes driven by fast ergodic Markov noise via the martingale problem, supplying the Poisson-equation and perturbed test function arguments that justify replacing the fast regime by its average.
• Khas'minskii, R. Z. (1966). A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides. Theory of Probability and Its Applications 11(3):390-406. doi:10.1137/1111038.
  Establishes the averaging and diffusion-approximation limit for differential equations driven by rapidly mixing random coefficients, the prototype for letting the regime chain switch at rate 1/epsilon.
• Kurtz, T. G. (1973). A limit theorem for perturbed operator semigroups with applications to random evolutions. Journal of Functional Analysis 12(1):55-67. doi:10.1016/0022-1236(73)90089-x.
  Gives an operator-semigroup limit theorem for singularly perturbed generators of the form (1/epsilon)Q + L, which covers random evolutions such as an Ornstein-Uhlenbeck process modulated by a fast Markov chain.
• Kato, T. (1949). On the Convergence of the Perturbation Method. I. Progress of Theoretical Physics 4(4):514-523. doi:10.1143/ptp/4.4.514.
  Develops convergent perturbation expansions for isolated eigenvalues and their spectral projections, the finite-dimensional spectral tool used to separate the slow Perron mode from the fast complement.
• Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods: Averaging and Homogenization. Springer, Texts in Applied Mathematics 53. doi:10.1007/978-0-387-73829-1.
  A textbook treatment of formal multiscale expansions for Markov chains, ODEs, SDEs and PDEs, including a chapter on averaging for Markov chains that matches the two-state switching setting.
• Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, Grundlehren der mathematischen Wissenschaften 260. doi:10.1007/978-1-4684-0176-9.
  Develops the averaging principle and associated large deviations for systems with fast and slow components, which describe the leading-order behaviour and rare-event corrections beyond a power-series expansion in epsilon.

Fast mean-reverting stochastic volatility

• Cotton, P., Fouque, J.-P., Papanicolaou, G. and Sircar, R. (2004). Stochastic Volatility Corrections for Interest Rate Derivatives. Mathematical Finance 14(2):173-200. doi:10.1111/j.0960-1627.2004.00188.x.
  Derives the corrections to Vasicek and CIR bond prices when the volatility of the short rate is driven by a fast mean-reverting factor. The regime-switching expansion is the same calculation with a two-state chain in place of the diffusion.
• Fouque, J.-P., Papanicolaou, G. and Sircar, K. R. (2000). Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, ISBN 0-521-79163-4. link.
  Introduces fast mean-reverting stochastic volatility asymptotics in which prices are expanded around an effective constant-volatility model with a first-order correction, the template for the fast regime chain.
• Fouque, J.-P., Papanicolaou, G., Sircar, R. and Sølna, K. (2011). Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press. doi:10.1017/cbo9781139020534.
  Extends the fast and slow scale expansions to interest rate and credit models, including Vasicek and CIR bond prices and defaultable bonds with stochastic intensity. It is the closest book-length treatment of the regime-switching problem, with a diffusive fast factor instead of a Markov chain.
• Fouque, J.-P., Papanicolaou, G., Sircar, R. and Sølna, K. (2003). Singular Perturbations in Option Pricing. SIAM Journal on Applied Mathematics 63(5):1648-1665. doi:10.1137/s0036139902401550.
  Gives the error analysis that justifies the first-order fast-scale correction for option prices, including nonsmooth payoffs.
• Fouque, J.-P., Papanicolaou, G., Sircar, R. and Sølna, K. (2003). Multiscale Stochastic Volatility Asymptotics. Multiscale Modeling and Simulation 2(1):22-42. doi:10.1137/030600291.
  Combines a fast and a slow volatility factor in a joint singular and regular perturbation expansion of option prices.
• DeSantiago, R., Fouque, J.-P. and Sølna, K. (2008). Bond markets with stochastic volatility. Advances in Econometrics 22 (Econometrics and Risk Management):215-242. doi:10.1016/s0731-9053(08)22009-8.
  Computes fast and slow stochastic volatility corrections to Vasicek and CIR bond prices, again with a diffusive volatility factor where the regime-switching example uses a two-state chain.

Short rates, affine models and credit

• Schur, I. (1911). Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. Journal für die reine und angewandte Mathematik 140:1-28. doi:10.1515/crll.1911.140.1.
  Proves positivity of the entrywise product of positive semidefinite matrices. The credit-geometry page combines its Gram factorization with rowwise tensor products to obtain the sharp maturity-integrated rank bound.
• Khatri, C. G. and Rao, C. R. (1968). Solutions to Some Functional Equations and Their Applications to Characterization of Probability Distributions. Sankhyā: The Indian Journal of Statistics, Series A 30(2):167-180. JSTOR 25049527.
  Introduces the blockwise Kronecker product now bearing their names. Its rowwise transpose gives the Gram factorization used to separate repeated maturity-loading clusters.
• Gautschi, W. (1978). On inverses of Vandermonde and confluent Vandermonde matrices III. Numerische Mathematik 29:445-450. doi:10.1007/bf01432880.
  Derives inverse-norm bounds for Vandermonde and confluent Vandermonde matrices. The credit-geometry page uses the same coalescing-node structure to quantify how a full-rank maturity-loading Gram matrix becomes singular.
• Hermite, C. (1878). Sur la formule d'interpolation de Lagrange. Journal für die reine und angewandte Mathematik 84:70-79. doi:10.1515/crelle-1878-18788405.
  Introduces the derivative-matching interpolation construction whose confluent Vandermonde structure underlies the clustered-loading collision theorem.
• Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5(2):177-188. doi:10.1016/0304-405x(77)90016-2.
  Introduces the Ornstein-Uhlenbeck short rate and its exponential-affine bond price, which is the model obtained in each regime and in the averaged limit.
• Cox, J. C., Ingersoll Jr., J. E. and Ross, S. A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53(2):385-407. doi:10.2307/1911242.
  Introduces the square-root short rate model with exponential-affine bond prices, the natural alternative regime dynamics when positivity of the hazard rate is required.
• Feller, W. (1951). Two Singular Diffusion Problems. Annals of Mathematics 54(1):173-182. stable link.
  Classifies the attainable and inaccessible boundaries of the square-root diffusion; zero is inaccessible exactly when the Feller ratio is at least one.
• Dambis, K. E. (1965). On the Decomposition of Continuous Submartingales. Theory of Probability and Its Applications 10(3):401-410. doi:10.1137/1110048.
  Represents a continuous local martingale as Brownian motion under its quadratic-variation clock, the time change used to reduce switched vol-of-vol CIR dynamics to a variable-dimension squared-Bessel equation.
• Yamada, T. (1973). On a Comparison Theorem for Solutions of Stochastic Differential Equations and Its Applications. Journal of Mathematics of Kyoto University 13(3):497-512. doi:10.1215/kjm/1250523321.
  Proves the one-dimensional pathwise comparison theorem, giving an alternative positivity proof when the switched square-root model has a common diffusion coefficient.
• Duffie, D. and Kan, R. (1996). A Yield-Factor Model of Interest Rates. Mathematical Finance 6(4):379-406. doi:10.1111/j.1467-9965.1996.tb00123.x.
  Characterizes the multifactor affine term structure models in which bond prices are exponential-affine with coefficients solving Riccati equations.
• Duffie, D., Pan, J. and Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica 68(6):1343-1376. doi:10.1111/1468-0262.00164.
  Gives the general exponential-affine transform formula for affine jump-diffusions via Riccati ODEs, the framework within which regime switching is accommodated by enlarging the state space.
• Cuchiero, C., Keller-Ressel, M. and Teichmann, J. (2012). Polynomial Processes and Their Applications to Mathematical Finance. Finance and Stochastics 16(4):711-740. doi:10.1007/s00780-012-0188-x.
  Shows that moments of polynomial processes are obtained from finite-dimensional matrix exponentials. The augmented CIR state $(v_t,\int_0^t v_sds)$ used for the simultaneous-switching cumulant certificate is polynomial.
• Lando, D. (1998). On Cox Processes and Credit Risky Securities. Review of Derivatives Research 2(2-3):99-120. doi:10.1007/bf01531332.
  Models default as the first jump of a Cox process so that survival probabilities take the form E[exp(-integral of the intensity)], the quantity computed for a regime-switching intensity.
• Duffie, D. and Singleton, K. J. (1999). Modeling Term Structures of Defaultable Bonds. Review of Financial Studies 12(4):687-720. doi:10.1093/rfs/12.4.687.
  Shows that defaultable bonds can be priced by discounting at a default-adjusted short rate, so survival probabilities and bond prices are the same computation.

Regime switching in econometrics and pricing

• Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica 57(2):357-384. doi:10.2307/1912559.
  Introduces the Markov-switching time series model and its filter, the econometric origin of regime-switching interest rate and credit models.
• Hansen, A. T. and Poulsen, R. (2000). A simple regime switching term structure model. Finance and Stochastics 4(4):409-429. doi:10.1007/pl00013523.
  Proposes a short-rate model with Markov regime switching in the parameters and studies its term structure, appearing in the same issue as Landén (2000).
• Ang, A. and Bekaert, G. (2002). Regime Switches in Interest Rates. Journal of Business and Economic Statistics 20(2):163-182. doi:10.1198/073500102317351930.
  Estimates regime-switching short-rate models on US, UK and German data and documents that regimes capture nonlinearities in drift and volatility.
• Bansal, R. and Zhou, H. (2002). Term Structure of Interest Rates with Regime Shifts. Journal of Finance 57(5):1997-2043. doi:10.1111/0022-1082.00487.
  Builds a discrete-time affine term structure model with regime shifts using log-linear approximations and finds that regime shifts improve the fit to yields.
• Wu, S. and Zeng, Y. (2005). A General Equilibrium Model of the Term Structure of Interest Rates under Regime-Switching Risk. International Journal of Theoretical and Applied Finance 8(7):839-869. doi:10.1142/s0219024905003323.
  Develops an equilibrium term structure model with priced regime-shift risk and obtains closed-form affine yields under a log-linear approximation.
• Yao, D. D., Zhang, Q. and Zhou, X. Y. (2006). A Regime-Switching Model for European Options. in Stochastic Processes, Optimization, and Control Theory (H. Yan, G. Yin, Q. Zhang, eds.), Springer, International Series in Operations Research and Management Science, pp. 281-300. doi:10.1007/0-387-33815-2_14.
  Prices European options in a Black-Scholes model with Markov-switching parameters by successive approximation of the coupled pricing system.
• Papanicolaou, A. and Sircar, R. (2014). A regime-switching Heston model for VIX and S&P 500 implied volatilities. Quantitative Finance 14(10):1811-1827. doi:10.1080/14697688.2013.814923.
  Adds Markov regime switching to Heston stochastic volatility to fit VIX and S&P 500 option smiles jointly, an equity counterpart of combining a regime chain with an affine factor.

Models in the examples

• Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies 6(2):327-343. doi:10.1093/rfs/6.2.327.
  Introduces the square-root stochastic variance model and its characteristic function; the Heston example lets the long-run variance switch.
• Lewis, A. L. (2001). A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes. SSRN working paper 282110. doi:10.2139/ssrn.282110.
  Gives the single-integral option price in terms of the characteristic function that the Heston example uses.
• Fischer, W. and Meier-Hellstern, K. (1993). The Markov-modulated Poisson process (MMPP) cookbook. Performance Evaluation 18(2):149-171. doi:10.1016/0166-5316(93)90035-s.
  Collects the generating functions, moments and fitting methods of the Markov-modulated Poisson process, the model in the Poisson counts example.
• Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1-2):125-144. doi:10.1016/0304-405x(76)90022-2.
  Adds lognormal jumps at a Poisson rate to geometric Brownian motion and prices options as a Poisson mixture of Black-Scholes prices; the Merton example lets the volatility and the jump intensity switch.
• Bates, D. S. (1996). Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. Review of Financial Studies 9(1):69-107. doi:10.1093/rfs/9.1.69.
  Combines square-root stochastic variance with lognormal jumps; the Bates example lets the long-run variance and the jump intensity switch together.
• Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The Variance Gamma Process and Option Pricing. European Finance Review 2(1):79-105. doi:10.1023/a:1009703431535.
  Brownian motion with drift run on a gamma clock, with closed-form characteristic function; the variance gamma example lets all three parameters switch.
• Hull, J. and White, A. (1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies 3(4):573-592. doi:10.1093/rfs/3.4.573.
  Extends the Vasicek short rate with a time-dependent drift fitted to the initial term structure; the Hull-White example lets the volatility switch and measures how far each regime moves from the fitted curve.
• Brigo, D. and Mercurio, F. (2006). Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit. Springer, 2nd edition. doi:10.1007/978-3-540-34604-3.
  The standard reference for the two-additive-factor Gaussian model G2++ and its bond, cap and swaption formulas; the G2++ example lets both volatilities and the correlation switch.
• Schrager, D. F. and Pelsser, A. A. J. (2006). Pricing Swaptions and Coupon Bond Options in Affine Term Structure Models. Mathematical Finance 16(4):673-694. doi:10.1111/j.1467-9965.2006.00289.x.
  Develops a general affine approximation for swaptions and coupon-bond options. The exact one-dimensional G2++ conditional-Gaussian quadrature here is narrower and does not claim an exact formula for general affine or finite-rate switching models.
• Choi, J. and Shin, S. (2016). Fast Swaption Pricing in Gaussian Term Structure Models. Mathematical Finance 26(4):962-982. doi:10.1111/mafi.12077.
  Treats the nonlinear exercise boundary in multi-factor Gaussian term-structure models, combining exact lower-dimensional numerical integration with a fast tangent-hyperplane approximation.
• Wolfe, P. (1976). Finding the nearest point in a polytope. Mathematical Programming 11:128-149. doi:10.1007/BF01580381.
  Develops the minimum-Euclidean-norm point problem for a finite convex hull and its equivalent optimal separating-hyperplane formulation.
• Schroder, M. (1989). Computing the Constant Elasticity of Variance Option Pricing Formula. Journal of Finance 44(1):211-219. doi:10.1111/j.1540-6261.1989.tb02414.x.
  Writes the constant elasticity of variance option price in terms of the noncentral chi-square distribution; the CEV example mixes that formula over a random total variance.
• Pyke, R. (1965). Spacings. Journal of the Royal Statistical Society, Series B 27(3):395-449. doi:10.1111/j.2517-6161.1965.tb00602.x.
  Develops the distribution theory of uniform order-statistic spacings; conditional on a fixed jump count, these Dirichlet spacings yield the Beta occupation-time law used in the exact CEV mixture.
• Pedler, P. J. (1971). Occupation Times for Two State Markov Chains. Journal of Applied Probability 8(2):381-390. doi:10.2307/3211908.
  Derives occupation-time distributions for two-state Markov chains; the unequal-rate CEV benchmark uses the corresponding atom-plus-Bessel-density law.
• Reibman, A., Smith, R. and Trivedi, K. (1989). Markov and Markov Reward Model Transient Analysis: An Overview of Numerical Approaches. European Journal of Operational Research 40(2):257-267. doi:10.1016/0377-2217(89)90335-4.
  Develops transient analysis for continuous-time Markov reward models. The nonzero-carry CEV clock is a time-inhomogeneous accumulated reward; its first two moments specialize directly in the two-state case.
• Jamshidian, F. (1989). An Exact Bond Option Formula. Journal of Finance 44(1):205-209. doi:10.1111/j.1540-6261.1989.tb02413.x.
  Prices options on zero-coupon bonds under a Gaussian short rate and decomposes an option on a coupon bond into options on its zero-coupon parts; the decomposition holds regime by regime under switching.
• Kemna, A. G. Z. and Vorst, A. C. F. (1990). A pricing method for options based on average asset values. Journal of Banking and Finance 14(1):113-129. doi:10.1016/0378-4266(90)90039-5.
  Gives the closed-form price of an option on the continuous geometric average under Black-Scholes; the instruments page carries it to a switching volatility.

The wider program

Works behind the sister sites listed on the home page.

• Luce, R. D. (1959). Individual Choice Behavior: A Theoretical Analysis. Wiley, New York (Dover reprint 2005). doi:10.1037/14396-000.
  Axiomatizes probabilistic choice through the choice axiom, yielding the ratio-scale choice model used elsewhere in the author's program.
• Vovk, V., Gammerman, A. and Shafer, G. (2005). Algorithmic Learning in a Random World. Springer (2nd ed. 2022, doi 10.1007/978-3-031-06649-8). doi:10.1007/b106715.
  Develops conformal prediction, giving distribution-free prediction sets with guaranteed coverage under exchangeability.

Diffusion approximation, random evolutions and cumulant expansions

The general theory of equations switched by fast random processes. The first-order rule on the any-equation page appears here as the diffusion approximation, as the limit of a random evolution, and as the second cumulant.

• Curtiss, J. H. (1942). A Note on the Theory of Moment Generating Functions. Annals of Mathematical Statistics 13(4):430-433. doi:10.1214/aoms/1177731541.
  Establishes uniqueness and convergence results for moment-generating functions on a neighborhood of the origin; the switched-CIR note adds an explicit uniform complex zero-free disk and cumulant bounds from a Gamma moment majorant.
• Fejér, L. (1932). Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein möglichst kleines Maximum besitzt. Annali della Scuola Normale Superiore di Pisa 1(3):263–276. EuDML.
  Fejér's primary work on extremal interpolation nodes on an interval. The identifiability note derives the associated Legendre–Gauss–Lobatto/Fekete design directly as the unique maximizer of its leading small-maturity Vandermonde determinant.
• Schoenberg, I. J. (1947). On Totally Positive Functions, Laplace Integrals and Entire Functions of the Laguerre-Polya-Schur Type. Proceedings of the National Academy of Sciences 33(1):11-17. doi:10.1073/pnas.33.1.11.
  Develops the total-positivity mechanism behind ordinary and confluent exponential Chebyshev determinants. On the identifiability page it proves that every ordered positive maturity tuple works for a cyclic scalar chain with real spectrum, including Jordan blocks.
• Gautschi, W. (1962). On Inverses of Vandermonde and Confluent Vandermonde Matrices. Numerische Mathematik 4:117-123. doi:10.1007/BF01386302.
  Derives explicit bounds for inverses of ordinary and confluent Vandermonde matrices. The yield-curve observability note uses the same small-node geometry to obtain the complete singular-value hierarchy and prior-recovery noise amplification.
• Aitken, A. C. (1936). On Least Squares and Linear Combination of Observations. Proceedings of the Royal Society of Edinburgh 55:42–48. doi:10.1017/S0370164600014346.
  The original matrix treatment of generalized least squares. The identifiability note whitens correlated, heteroskedastic Gaussian quote noise and proves matching two-point and projected-GLS recovery bounds in the resulting Mahalanobis geometry.
• Le Cam, L. (1973). Convergence of Estimates Under Dimensionality Restrictions. Annals of Statistics 1(1):38–53. doi:10.1214/aos/1193342380.
  A primary source for statistical lower bounds obtained by reducing estimation to nearby testing problems. The yield-curve identifiability note applies the exact Gaussian two-point experiment to convert long-end singular-value decay into a minimax prior-recovery obstruction.
• Donoho, D. L. (1994). Statistical Estimation and Optimal Recovery. Annals of Statistics 22(1):238–270. doi:10.1214/aos/1176325367.
  Develops the correspondence between minimax statistical estimation and deterministic optimal recovery. The identifiability note places its Gaussian and adversarial quote-error bounds side by side and obtains the exact finite-dimensional threshold from the weakest whitened singular direction.
• Mityagin, B. S. (2020). The Zero Set of a Real Analytic Function. Mathematical Notes 107:529–530. doi:10.1134/S0001434620030189.
  Gives a short proof that the zero set of a nontrivial real-analytic function has Lebesgue measure zero. The identifiability note applies it to a nonzero response-matrix minor, proving that maturity tuples below the maximum attainable rank form a null set.
• Park, S. Y. and Sahai, A. (2013). Intermittent Kalman Filtering: Eigenvalue Cycles and Nonuniform Sampling. arXiv:1308.6107.
  Introduces eigenvalue cycles as a source of sampled-observation instability and proves that randomly dithered nonuniform sampling breaks them. The yield-curve identifiability note derives the exact three-state damped-oscillation aliasing kernel and its universal exceptional maturity lattice.
• Bahadur, R. R. and Ranga Rao, R. (1960). On Deviations of the Sample Mean. The Annals of Mathematical Statistics 31(4):1015-1027. doi:10.1214/aoms/1177705674.
  Derives sharp large-deviation asymptotics beyond the exponential rate, including the lattice tail factor specialized here to binary counts.
• Ney, P. and Nummelin, E. (1987). Markov Additive Processes II: Large Deviations. The Annals of Probability 15(2):593-609. JSTOR 2244063.
  Develops large-deviation theory for Markov-additive processes using positive operators, the general setting of the regular-panel count theorem.
• Cramér, H. (1940). On the theory of stationary random processes. Annals of Mathematics 41(1):215-230. doi:10.2307/1968827.
  Gives the spectral representation of a stationary process. Its vector form uses a nonnegative Hermitian matrix-valued spectral measure, from which the matrix Fejér-kernel crossover follows directly.
• Wiener, N. (1930). Generalized harmonic analysis. Acta Mathematica 55:117-258. doi:10.1007/BF02546511.
  Develops continuous generalized harmonic analysis and the mean-square spectral identities. Its Fourier–Stieltjes lemma identifies the Cesàro mean-square transform with the squared masses of the spectral atoms.
• Wiener, N. and Masani, P. (1958). The prediction theory of multivariate stochastic processes, II: The linear predictor. Acta Mathematica 99:93-137. doi:10.1007/BF02392423.
  Develops the spectral theory of multivariate stationary processes using spectral density matrices, the classical matrix-valued counterpart of scalar stationary-process spectral analysis.
• Deligiannidis, G. and Utev, S. (2013). Variance of partial sums of stationary sequences. Annals of Probability 41(5):3606-3616. doi:10.1214/12-AOP772.
  Characterizes regular variation of partial-sum variance through regular variation of the spectral mass at frequency zero. The low-frequency theorem on the homogenization-scale page is its continuous-time Fejér-kernel analogue, stated directly for cumulative spectral mass with an explicit constant.
• Kubo, R. (1962). Generalized Cumulant Expansion Method. Journal of the Physical Society of Japan 17(7):1100-1120. doi:10.1143/jpsj.17.1100.
  Generalizes cumulant expansions to non-commuting operators with ordered exponentials, the tool behind stochastic line-shape and relaxation theory.
• Griego, R. J. and Hersh, R. (1969). Random Evolutions, Markov Chains, and Systems of Partial Differential Equations. Proceedings of the National Academy of Sciences 62(2):305-308. doi:10.1073/pnas.62.2.305.
  Introduces random evolutions: operators switched by a Markov chain, whose expectations solve coupled systems of partial differential equations. This is the structure of every pricing equation in the examples.
• Papanicolaou, G. C. and Varadhan, S. R. S. (1973). A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations. Communications on Pure and Applied Mathematics 26(4):497-524. doi:10.1002/cpa.3160260405.
  A limit theorem for evolutions driven by strongly mixing noise in a Banach space, applied to stochastic differential equations; an early operator form of the diffusion approximation.
• Papanicolaou, G. C. and Kohler, W. (1974). Asymptotic theory of mixing stochastic ordinary differential equations. Communications on Pure and Applied Mathematics 27(5):641-668. doi:10.1002/cpa.3160270503.
  The asymptotic theory of ordinary differential equations with fast mixing random coefficients, in which the limit is a diffusion whose coefficients are integrated correlation functions.
• Hersh, R. (1974). Random evolutions: A survey of results and problems. Rocky Mountain Journal of Mathematics 4(3):443-477. doi:10.1216/rmj-1974-4-3-443.
  A survey of random evolutions and their limit theorems under rescaling.
• van Kampen, N. G. (1974). A cumulant expansion for stochastic linear differential equations. I. Physica 74(2):215-238. doi:10.1016/0031-8914(74)90121-9.
  A cumulant expansion for linear differential equations with random coefficients. Its second-order term is the time-ordered, correlation-weighted product of the random operators, the physicists' form of the first-order rule for switched operators.
• Phillips, R. G. and Kokotovic, P. V. (1981). A singular perturbation approach to modeling and control of Markov chains. IEEE Transactions on Automatic Control 26(5):1087-1094. doi:10.1109/tac.1981.1102780.
  Singular perturbation for Markov chains with weak and strong interactions, the control-theory origin of aggregating fast states.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley. doi:10.1002/9780470316658.
  The standard reference for convergence of Markov processes through generators and martingale problems, with a chapter on random evolutions and diffusion approximations.
• Aalen, O. O. (1978). Nonparametric Inference for a Family of Counting Processes. Annals of Statistics 6(4):701-726. doi:10.1214/aos/1176344247.
  Develops martingale-based nonparametric inference for counting processes, the framework used to obtain finite-path transition-count confidence bounds.
• Meyer, C. D., Jr. (1980). The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities. SIAM Journal on Algebraic and Discrete Methods 1(3):273-283. doi:10.1137/0601031.
  Derives sharp perturbation bounds for the limiting probabilities of an ergodic finite Markov chain, the finite-chain stability theory used in the plug-in certificate.
• Jerrum, M. R., Valiant, L. G. and Vazirani, V. V. (1986). Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43:169-188. doi:10.1016/0304-3975(86)90174-X.
  Develops polynomial-time approximate counting and sampling, including the repeat-and-take-the-median confidence amplification used for finite-variance probe estimators.
• Avron, H. and Toledo, S. (2011). Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix. Journal of the ACM 58(2), Article 8. doi:10.1145/1944345.1944349.
  Analyzes matrix-free randomized trace estimators and finite sample guarantees for positive semidefinite matrices, the general framework containing the action-probe energy estimator.
• Hutchinson, M. F. (1989). A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines. Communications in Statistics - Simulation and Computation 18(3):1059-1076. doi:10.1080/03610918908812806.
  Introduces the unbiased Rademacher quadratic trace estimator used as the bounded action-probe benchmark.
• Pavliotis, G. A. (2010). Asymptotic analysis of the Green-Kubo formula. IMA Journal of Applied Mathematics 75(6):951-967. doi:10.1093/imamat/hxq039.
  Represents the Green-Kubo formula through a Poisson equation and studies asymptotic limits of the self-diffusion coefficient.
• Duncan, A. B., Lelièvre, T. and Pavliotis, G. A. (2016). Variance Reduction Using Nonreversible Langevin Samplers. Journal of Statistical Physics 163:457-491. doi:10.1007/s10955-016-1491-2.
  Analyzes how antisymmetric perturbations of Langevin dynamics change asymptotic variance, providing a continuous-state comparison for the energy-normalized skew-resolvent factorization.
• Choromanski, K., Rowland, M., Chen, W. and Weller, A. (2019). Unifying Orthogonal Monte Carlo Methods. Proceedings of Machine Learning Research 97:1203-1212. link.
  Studies variance reduction obtained by coupling Monte Carlo samples to be mutually orthogonal; the cycle-smile page gives an exact quadratic Haar-projector specialization.
• Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific. doi:10.1142/1328.
  Lectures on random evolutions, including their averaging and diffusion-approximation limits.
• Kac, M. (1949). On Distributions of Certain Wiener Functionals. Transactions of the American Mathematical Society 65(1):1-13. doi:10.1090/S0002-9947-1949-0027960-X.
  A primary source for the Feynman-Kac representation, used here in its finite-state killed-semigroup form to obtain a switching-rate-uniform contraction bound.
• Pardoux, E. and Veretennikov, A. Yu. (2001). On the Poisson Equation and Diffusion Approximation. I. Annals of Probability 29(3):1061-1085. doi:10.1214/aop/1015345596.
  Solves the Poisson equation for the generator of an ergodic diffusion on the whole space and uses it to prove diffusion approximations; the continuous-state version of solving Q w = f.
• Kontoyiannis, I. and Meyn, S. P. (2003). Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes. Annals of Applied Probability 13(1):304-362. doi:10.1214/aoap/1042765670.
  Develops multiplicative Poisson equations and analytic cumulant limits for geometrically ergodic Markov processes; the switched-CIR calculation here obtains its third-cumulant coefficients directly from ordered correlations and polynomial closure.
• Skorokhod, A. V., Hoppensteadt, F. C. and Salehi, H. (2002). Random Perturbation Methods with Applications in Science and Engineering. Springer, Applied Mathematical Sciences 150. doi:10.1007/b98905.
  Averaging and diffusion approximation for systems perturbed by fast random processes, with applications across science and engineering.
• Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications. Springer, Stochastic Modelling and Applied Probability 35, 2nd ed.. doi:10.1007/b97441.
  Stochastic approximation with correlated and Markovian noise, where the averaged ordinary differential equation and the fluctuation limit play the roles of the averaged law and the Green-Kubo correction.
• Fouque, J.-P., Garnier, J., Papanicolaou, G. and Sølna, K. (2007). Wave Propagation and Time Reversal in Randomly Layered Media. Springer, Stochastic Modelling and Applied Probability 56. doi:10.1007/978-0-387-49808-9.
  Waves in randomly layered media treated by the diffusion approximation for fast random coefficients, including the effective medium and the first corrections.
• Yin, G. G. and Zhang, Q. (2013). Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach. Springer, Stochastic Modelling and Applied Probability 37, 2nd ed.. doi:10.1007/978-1-4614-4346-9.
  The second edition of the two-time-scale Markov chain monograph, with outer expansions, initial layers and applications to control and finance.
• Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Communications in Mathematical Physics 104(1):1-19. doi:10.1007/bf01210789.
  Proves the central limit theorem for time integrals of a function of a reversible Markov process, with the long-run variance given through the resolvent; the reversibility assumption is essential to the proof.
• de Prony, G. R. (1795). Essai expérimental et analytique sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l'eau et de la vapeur de l'alcool, à différentes températures. Journal de l'École Polytechnique 1(2):24-76.
  Introduces the annihilating-polynomial reconstruction now known as Prony's method; the finite-atomic Cox-mixture result uses its Hankel and Vandermonde steps.
• Rebolledo, R. (1980). Central limit theorems for local martingales. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 51:269-286. doi:10.1007/bf00587353.
  Gives central limit criteria for local martingales in terms of predictable quadratic variation and negligible jumps; the weighted fast-chain clock limit is a bounded-jump specialization.
• Hervé, L. and Pène, F. (2010). The Nagaev-Guivarc'h method via the Keller-Liverani theorem. Bulletin de la Société Mathématique de France 138(3):415-489. doi:10.24033/bsmf.2594.
  Develops spectral perturbation methods for strongly ergodic Markov chains, including a first-order Edgeworth expansion; the weighted two-state clock uses the same eigenvalue-versus-boundary-amplitude separation but computes both coefficients directly.
• Kalman, R. E. (1960). Contributions to the Theory of Optimal Control. Boletín de la Sociedad Matemática Mexicana 5:102-119. link.
  Introduces the state-space controllability and observability viewpoint. The yield-curve note turns its finite-horizon observability Gramian into the Fisher information for a continuously observed maturity response.
• Hautus, M. L. J. (1969). Controllability and Observability Conditions of Linear Autonomous Systems. Indagationes Mathematicae (Proceedings) 72(5):443-448. link.
  Gives the spectral observability criterion used here to characterize the null space of the continuous-maturity Fisher Gramian, including nondiagonalizable generators.
• Schmerling, E., Janson, L. and Pavone, M. (2015). Optimal sampling-based motion planning under differential constraints: The drift case with linear affine dynamics. 54th IEEE Conference on Decision and Control:2574-2581. doi:10.1109/cdc.2015.7402604.
  Uses controllability indices to prove the multi-input small-time Gramian eigenvalue orders. The yield-curve note supplies the shifted-response powers and explicit leading block constants for its continuous-maturity Fisher matrix.
• Trélat, E. and Zuazua, E. (2026). Two-term small-time spectral expansions for controllability Gramians. arXiv:2609.37077. link.
  Derives the small-time odd-power eigenvalue hierarchy and refined spectral expansions for controllable single-input Gramians. The yield-curve Fisher response vanishes once more at zero maturity, shifting the hierarchy from powers 1,3,... to 3,5,...; the note gives the corresponding factorial-Hilbert leading constants directly.

Multiscale and regime-switching option pricing and credit

• Hull, J. and White, A. (1987). The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance 42(2):281-300. doi:10.1111/j.1540-6261.1987.tb02568.x.
  Prices an option under uncorrelated stochastic volatility as the Black-Scholes price averaged over integrated variance. Its expansion to second order in the variance of integrated variance is the smile formula on the any-equation page.
• Naik, V. (1993). Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns. Journal of Finance 48(5):1969-1984. doi:10.1111/j.1540-6261.1993.tb05137.x.
  Option valuation when volatility jumps between two levels following a Markov chain, one of the first regime-switching option pricing models.
• Di Masi, G. B., Kabanov, Yu. M. and Runggaldier, W. J. (1995). Mean-Variance Hedging of Options on Stocks with Markov Volatilities. Theory of Probability and Its Applications 39(1):172-182. doi:10.1137/1139008.
  Mean-variance hedging of options on a stock whose volatility is driven by a Markov chain.
• Sircar, K. R. and Papanicolaou, G. C. (1999). Stochastic volatility, smile & asymptotics. Applied Mathematical Finance 6(2):107-145. doi:10.1080/135048699334573.
  Stochastic volatility, the smile and asymptotics: an early account of how fast volatility fluctuations produce a smile around an effective constant volatility.
• Fouque, J.-P., Papanicolaou, G. and Sircar, K. R. (2000). Mean-Reverting Stochastic Volatility. International Journal of Theoretical and Applied Finance 3(1):101-142. doi:10.1142/s0219024900000061.
  The paper that introduced fast mean-reverting stochastic volatility asymptotics for option pricing, with the first correction expressed through a few group parameters that can be fitted to the implied volatility skew.
• Guo, X. (2001). Information and option pricings. Quantitative Finance 1(1):38-44. doi:10.1080/713665550.
  Option pricing with a two-state regime-switching volatility and the role of information about the regime.
• Buffington, J. and Elliott, R. J. (2002). American Options with Regime Switching. International Journal of Theoretical and Applied Finance 5(5):497-514. doi:10.1142/s0219024902001523.
  European and American options when the parameters of the underlying switch with a Markov chain.
• Elliott, R. J., Chan, L. and Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1(4):423-432. doi:10.1007/s10436-005-0013-z.
  Regime-switching option pricing with the equivalent martingale measure chosen by the Esscher transform.
• Fouque, J.-P., Sircar, R. and Sølna, K. (2006). Stochastic Volatility Effects on Defaultable Bonds. Applied Mathematical Finance 13(3):215-244. doi:10.1080/13504860600563127.
  Fast and slow stochastic volatility corrections for defaultable bonds and credit spreads.
• Fouque, J.-P., Wignall, B. C. and Zhou, X. (2008). Modeling correlated defaults: first passage model under stochastic volatility. Journal of Computational Finance 11(3):43-78. doi:10.21314/jcf.2008.192.
  Correlated defaults in a first-passage model under stochastic volatility, with the joint default probabilities corrected by multiscale asymptotics.
• Duffie, D., Eckner, A., Horel, G. and Saita, L. (2009). Frailty Correlated Default. Journal of Finance 64(5):2089-2123. doi:10.1111/j.1540-6261.2009.01495.x.
  Evidence that US corporate defaults cluster beyond what observable covariates explain, modelled with a common latent frailty process driving all default intensities.
• Fouque, J.-P., Lorig, M. and Sircar, R. (2016). Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration. Finance and Stochastics 20(3):543-588. doi:10.1007/s00780-016-0298-y.
  Second-order multiscale stochastic volatility asymptotics, including a stochastic terminal layer, with calibration to implied volatilities.
• Fouque, J.-P., Sircar, R. and Zariphopoulou, T. (2017). Portfolio Optimization and Stochastic Volatility Asymptotics. Mathematical Finance 27(3):704-745. doi:10.1111/mafi.12109.
  Applies fast and slow volatility asymptotics to Merton's portfolio problem, showing the first correction to the value function and the optimal strategy.
• Brigo, D. and Mercurio, F. (2001). A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models. Finance and Stochastics 5(3):369-387. doi:10.1007/PL00013541.
  Develops deterministic-shift extensions of tractable short-rate models and derives analytical formulas for bonds and bond options; the Appendix B erratum isolates the underlying maturity-by-maturity scaling rule.
• Geman, H., El Karoui, N. and Rochet, J.-C. (1995). Changes of numéraire, changes of probability measure and option pricing. Journal of Applied Probability 32(2):443-458. doi:10.2307/3215299.
  Establishes the general change-of-numeraire theorem and forward measures used here to fix the sign and normalization of the futures recursion.
• Birkhoff, G. (1957). Extensions of Jentzsch's theorem. Transactions of the American Mathematical Society 85(1):219-227. doi:10.1090/s0002-9947-1957-0087058-6.
  Proves the projective contraction theorem with its sharp hyperbolic-tangent coefficient; the annuity-measure sensitivity bound has the same tanh(projective-diameter/4) form.

Likelihood and tests for switching

• Carrasco, M., Hu, L. and Ploberger, W. (2014). Optimal Test for Markov Switching Parameters. Econometrica 82(2):765-784. doi:10.3982/ecta8609.
  An information-matrix-type test for Markov switching that uses only the model under the null and is asymptotically equivalent to the likelihood ratio test.
• White, H. (1982). Maximum Likelihood Estimation of Misspecified Models. Econometrica 50(1):1-25. doi:10.2307/1912526.
  Introduces the information-matrix test, based on the difference between the two forms of the information matrix.
• Chesher, A. (1984). Testing for Neglected Heterogeneity. Econometrica 52(4):865-872. doi:10.2307/1911188.
  Shows that the score test for random coefficients is close to the information-matrix test.
• Hansen, B. E. (1992). The likelihood ratio test under nonstandard conditions: Testing the Markov switching model of GNP. Journal of Applied Econometrics 7(S1):S61-S82. doi:10.1002/jae.3950070506.
  A bound for the likelihood ratio test of Markov switching when nuisance parameters are unidentified under the null.
• Garcia, R. (1998). Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models. International Economic Review 39(3):763-788. doi:10.2307/2527399.
  The asymptotic null distribution of the sup likelihood ratio test for Markov switching.
• Cho, J. S. and White, H. (2007). Testing for Regime Switching. Econometrica 75(6):1671-1720. doi:10.1111/j.1468-0262.2007.00809.x.
  Quasi-likelihood ratio tests for regime switching based on mixtures.
• Jarque, C. M. and Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters 6(3):255-259. doi:10.1016/0165-1765(80)90024-5.
  The skewness and kurtosis test of normality of regression residuals.
• Kudô, A. (1963). A multivariate analogue of the one-sided test. Biometrika 50(3/4):403-418. doi:10.1093/biomet/50.3-4.403.
  Develops the multivariate one-sided Gaussian test. With several inequality constraints, the chi-bar-square weights depend on the covariance geometry; the independent orthant gives binomial weights as a special case.
• Sheppard, W. F. (1899). On the application of the theory of error to cases of normal distribution and normal correlation. Philosophical Transactions of the Royal Society of London A 192:101-167. doi:10.1098/rsta.1899.0003.
  Gives the bivariate Gaussian quadrant arcsine identity; combined with central symmetry, it yields the trivariate positive-orthant probability used for the exact three-constraint weights.
• Amelunxen, D. and Lotz, M. (2017). Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective. Discrete & Computational Geometry 58:371-409. doi:10.1007/s00454-017-9904-9.
  Gives a self-contained treatment of conic intrinsic volumes and the polarity identity $v_j(C)=v_{q-j}(C^\circ)$ underlying the complementary-face covariance-inversion formula.
• Clark, P. K. (1973). A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices. Econometrica 41(1):135-155. doi:10.2307/1913889.
  Returns as a normal variance mixture subordinated to a random clock, with fat tails.
• Aït-Sahalia, Y. (2002). Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach. Econometrica 70(1):223-262. doi:10.1111/1468-0262.00274.
  Closed-form Hermite expansions of the transition densities of diffusions for maximum likelihood estimation.
• Lindsay, B. G. (1988). Composite likelihood methods. Contemporary Mathematics 80:221-239. doi:10.1090/conm/080/999014.
  Defines composite likelihoods, sums of log likelihoods of marginal or conditional events, and their estimating equations; the product of pairwise transition densities used on the likelihood page is one.
• Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association 82(398):605-610. doi:10.1080/01621459.1987.10478472.
  The asymptotic law of likelihood ratio and score tests when the true parameter is on the boundary of the parameter space: a mixture of chi-square laws from projecting the Gaussian score onto the tangent cone.
• Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review 56(1):49-62. doi:10.2307/1403361.
  Chi-bar-square distributions for tests of inequality constraints, with the weights given by the probabilities that the projection of a Gaussian vector onto a cone lands on each face.

Correlation regimes and correlation risk

• Margrabe, W. (1978). The value of an option to exchange one asset for another. Journal of Finance 33(1), 177-186. doi:10.1111/j.1540-6261.1978.tb03397.x.
  Prices the option to exchange one lognormal asset for another in closed form, using the second asset as numeraire.
• Carmona, R. and Durrleman, V. (2003). Pricing and hedging spread options. SIAM Review 45(4), 627-685. doi:10.1137/s0036144503424798.
  Surveys the models and numerical methods used to price and hedge spread options.
• Longin, F. and Solnik, B. (2001). Extreme correlation of international equity markets. Journal of Finance 56(2), 649-676. doi:10.1111/0022-1082.00340.
  Uses extreme-value theory to show that international equity correlation rises in bear markets but not in bull markets, and is tied to the market trend rather than to volatility.
• Ang, A. and Chen, J. (2002). Asymmetric correlations of equity portfolios. Journal of Financial Economics 63(3), 443-494. doi:10.1016/s0304-405x(02)00068-5.
  Measures exceedance correlations of US stocks with the market, finds them much larger on the downside, and finds regime-switching models capture the asymmetry better than other models tried.
• Hong, Y. and Tu, J. and Zhou, G. (2007). Asymmetries in stock returns: statistical tests and economic evaluation. Review of Financial Studies 20(5), 1547-1581. doi:10.1093/rfs/hhl037.
  Gives model-free tests for asymmetric correlations, betas and covariances, and measures their value to an investor.
• Fouque, J.-P. and Pun, C. S. and Wong, H. Y. (2016). Portfolio optimization with ambiguous correlation and stochastic volatilities. SIAM Journal on Control and Optimization 54(5), 2309-2338. doi:10.1137/15m1032533.
  Solves worst-case portfolio choice over an ambiguous correlation between two assets, with an asymptotic closed form when their volatilities are fast mean-reverting.
• Driessen, J. and Maenhout, P. J. and Vilkov, G. (2009). The price of correlation risk: evidence from equity options. Journal of Finance 64(3), 1377-1406. doi:10.1111/j.1540-6261.2009.01467.x.
  Finds that market-wide correlation risk is priced, from S&P 100 index options against options on its components.
• Buraschi, A. and Porchia, P. and Trojani, F. (2010). Correlation risk and optimal portfolio choice. Journal of Finance 65(1), 393-420. doi:10.1111/j.1540-6261.2009.01533.x.
  Solves intertemporal portfolio choice with a stochastic (Wishart) covariance matrix and finds a distinct hedging demand against correlation risk.
• Bru, M.-F. (1991). Wishart processes. Journal of Theoretical Probability 4(4), 725-751. doi:10.1007/bf01259552.
  Introduces the Wishart process, a matrix-valued diffusion later used to model stochastic covariance.
• Da Fonseca, J. and Grasselli, M. and Tebaldi, C. (2007). Option pricing when correlations are stochastic: an analytical framework. Review of Derivatives Research 10(2), 151-180. doi:10.1007/s11147-008-9018-x.
  An analytical framework for option pricing with stochastic correlation, built on Wishart covariance dynamics.
• Gourieroux, C. and Sufana, R. (2010). Derivative pricing with Wishart multivariate stochastic volatility. Journal of Business and Economic Statistics 28(3), 438-451. doi:10.1198/jbes.2009.08105.
  Prices derivatives when the covariance matrix of several assets follows a Wishart process.
• Deelstra, G. and Simon, M. (2017). Multivariate European option pricing in a Markov-modulated Levy framework. Journal of Computational and Applied Mathematics 317, 171-187. doi:10.1016/j.cam.2016.11.040.
  Prices multi-asset European options when the parameters of a Levy model are modulated by a Markov chain.
• Kovchegov, Y. and Meredith, N. and Nir, E. (2010). Occupation times and Bessel densities. Statistics & Probability Letters 80(2), 104-110. doi:10.1016/j.spl.2009.09.018.
  Derives occupation-time distributions for continuous-time Markov chains and explains the modified-Bessel form used in the two-state certificate.
• Kato, T. (1995). Perturbation Theory for Linear Operators. Classics in Mathematics, Springer, Berlin and Heidelberg. doi:10.1007/978-3-642-66282-9.
  Develops analytic perturbation of isolated eigenvalues and their spectral projections, used here to separate the principal tilted-generator eigenvalue from the exponentially decaying finite-state modes.
• Brillinger, D. R. (1969). The calculation of cumulants via conditioning. Annals of the Institute of Statistical Mathematics 21(1), 215-218. doi:10.1007/bf02532246.
  Introduces the conditional-cumulant formula used here to organize the exact all-order joint cumulant tensor of the Gaussian occupation-time mixture.
• Coolen-Schrijner, P. and van Doorn, E. A. (2002). The deviation matrix of a continuous-time Markov chain. Probability in the Engineering and Informational Sciences 16(3), 351-366. doi:10.1017/s0269964802163066.
  Defines and studies the continuous-time deviation matrix D = integral_0^infinity (P(t)-Pi) dt, the group-inverse object governing the arbitrary-start initial-layer coefficient used here.
• van Dijk, N. M. and Sladky, K. (2006). On the total reward variance for continuous-time Markov reward chains. Journal of Applied Probability 43(4), 1044-1052. doi:10.1239/jap/1165505206.
  Establishes asymptotically linear variance growth for finite-state continuous-time Markov reward chains and gives formulas for its growth rate; the boundary Hessian here refines the finite-horizon constant term.
• Breuer, L. (2012). Occupation times for Markov-modulated Brownian motion. Journal of Applied Probability 49(2), 549-565. doi:10.1239/jap/1339878804.
  Develops occupation-time transforms for finite-state Markov-modulated Brownian motion, the process class whose exact short-maturity time change supplies the critical sT boundary layer here.

Portfolio choice with regimes

• Merton, R. C. (1969). Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. The Review of Economics and Statistics 51(3), 247-257. doi:10.2307/1926560.
  Solves continuous-time consumption and investment with constant parameters, giving the constant-proportion rule for CRRA utility.
• Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3(4), 373-413. doi:10.1016/0022-0531(71)90038-x.
  Extends the continuous-time portfolio problem to general utilities and changing investment opportunities.
• Zhou, X. Y. and Li, D. (2000). Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework. Applied Mathematics and Optimization 42, 19-33. doi:10.1007/s002450010003.
  Solves the continuous-time Markowitz problem by stochastic linear-quadratic control and gives the efficient frontier in closed form.
• Zhou, X. Y. and Yin, G. (2003). Markowitz's Mean-Variance Portfolio Selection with Regime Switching: A Continuous-Time Model. SIAM Journal on Control and Optimization 42(4), 1466-1482. doi:10.1137/s0363012902405583.
  Solves the continuous-time Markowitz problem when the rate, drifts and volatilities follow an observed Markov chain, with the frontier given by two systems of linear equations.
• Yin, G. and Zhou, X. Y. (2004). Markowitz's Mean-Variance Portfolio Selection With Regime Switching: From Discrete-Time Models to Their Continuous-Time Limits. IEEE Transactions on Automatic Control 49(3), 349-360. doi:10.1109/tac.2004.824479.
  Aggregates a large fast chain into a switching-diffusion limit, restates the continuous-time frontier and shows that strategies built from the limit are asymptotically optimal.
• Bäuerle, N. and Rieder, U. (2004). Portfolio Optimization With Markov-Modulated Stock Prices and Interest Rates. IEEE Transactions on Automatic Control 49(3), 442-447. doi:10.1109/tac.2004.824471.
  Solves utility maximization of terminal wealth with rate, drift and volatility modulated by an observed chain, and compares it with a market of averaged data.
• Sotomayor, L. R. and Cadenillas, A. (2009). Explicit Solutions of Consumption-Investment Problems in Financial Markets with Regime Switching. Mathematical Finance 19(2), 251-279. doi:10.1111/j.1467-9965.2009.00366.x.
  Gives explicit consumption and investment rules for HARA utilities when market parameters follow an observed regime.
• Zhang, Q. and Yin, G. (2004). Nearly-Optimal Asset Allocation in Hybrid Stock Investment Models. Journal of Optimization Theory and Applications 121(2), 419-444. doi:10.1023/b:jota.0000037412.23243.6c.
  Aggregates the classes of a fast, large Markov chain in a stock investment model and builds nearly optimal allocations from the limit problem.
• Yang, Z. and Yin, G. and Wang, L. Y. and Zhang, H. (2013). Near-optimal mean-variance controls under two-time-scale formulations and applications. Stochastics 85(4), 723-741. doi:10.1080/17442508.2013.795567.
  Derives a limit mean-variance problem as a fast chain is aggregated and shows that controls built from it are nearly optimal.
• Rieder, U. and Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. Journal of Applied Probability 42(2), 362-378. doi:10.1239/jap/1118777176.
  Reduces portfolio choice with a hidden Markov-modulated drift to a fully observed problem by filtering and solves it for log and power utility.
• Sass, J. and Haussmann, U. G. (2004). Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain. Finance and Stochastics 8(4). doi:10.1007/s00780-004-0132-9.
  Finds optimal portfolios when the drift is a hidden continuous-time Markov chain filtered from observed prices.
• Honda, T. (2003). Optimal portfolio choice for unobservable and regime-switching mean returns. Journal of Economic Dynamics and Control 28(1), 45-78. doi:10.1016/s0165-1889(02)00106-9.
  Studies portfolio choice when mean returns switch between regimes that the investor must infer from prices.
• Wonham, W. M. (1964). Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering. Journal of the Society for Industrial and Applied Mathematics, Series A: Control 2(3), 347-369. doi:10.1137/0302028.
  Derives the filter for a finite-state Markov chain observed through a diffusion, the basis of regime learning from prices.
• Tu, J. (2010). Is Regime Switching in Stock Returns Important in Portfolio Decisions?. Management Science 56(7), 1198-1215. doi:10.1287/mnsc.1100.1181.
  Finds certainty-equivalent losses above two percent a year from ignoring regime switching in stock returns.

Pulsar timing

• Kramer, M. and Lyne, A. G. and O'Brien, J. T. and Jordan, C. A. and Lorimer, D. R. (2006). A Periodically Active Pulsar Giving Insight into Magnetospheric Physics. Science 312(5773):549-551. doi:10.1126/science.1124060.
  Shows that PSR B1931+24 switches its radio emission on for 5 to 10 days and off for 25 to 35 days, and spins down about 50% faster when on.
• Lyne, A. and Hobbs, G. and Kramer, M. and Stairs, I. and Stappers, B. (2010). Switched Magnetospheric Regulation of Pulsar Spin-Down. Science 329(5990):408-412. doi:10.1126/science.1186683.
  Shows that timing noise in many pulsars comes from abrupt, often quasi-periodic switching between two spin-down rates, correlated with changes in pulse shape.
• Boynton, P. E. and Groth, E. J. and Hutchinson, D. P. and Nanos, G. P. and Partridge, R. B. and Wilkinson, D. T. (1972). Optical Timing of the Crab Pulsar, NP 0532. The Astrophysical Journal 175:217. doi:10.1086/151550.
  Models the timing noise of the Crab pulsar as random walks in phase, frequency or spin-down rate and finds a random walk in frequency.
• Cordes, J. M. and Helfand, D. J. (1980). Pulsar timing. III - Timing noise of 50 pulsars. The Astrophysical Journal 239:640. doi:10.1086/158150.
  Characterizes the timing noise of 50 pulsars in terms of random walks in phase, frequency and spin-down rate.
• Hobbs, G. and Lyne, A. G. and Kramer, M. (2010). An analysis of the timing irregularities for 366 pulsars. Monthly Notices of the Royal Astronomical Society 402(2):1027-1048. doi:10.1111/j.1365-2966.2009.15938.x.
  Analyses the long-term timing irregularities of 366 pulsars observed at Jodrell Bank.
• van Haasteren, R. and Levin, Y. (2012). Understanding and analysing time-correlated stochastic signals in pulsar timing. Monthly Notices of the Royal Astronomical Society 428(2):1147-1159. doi:10.1093/mnras/sts097.
  Treats red timing noise as a Gaussian process and shows how to analyse it together with the timing-model fit.
• Keith, M. J. and Niţu, I. C. (2023). Impact of quasi-periodic and steep-spectrum timing noise on the measurement of pulsar timing parameters. Monthly Notices of the Royal Astronomical Society (2023). doi:10.1093/mnras/stad1713.
  Models the timing noise of 17 Jodrell Bank pulsars with quasi-periodic and steep-spectrum components, and publishes smoothed spin-down rate series for them (Zenodo, doi:10.5281/zenodo.7664166).
• Lee, K. J. and Bassa, C. G. and Janssen, G. H. and Karuppusamy, R. and Kramer, M. and Smits, R. and Stappers, B. W. (2012). The optimal schedule for pulsar timing array observations. Monthly Notices of the Royal Astronomical Society 423(3):2642-2655. doi:10.1111/j.1365-2966.2012.21070.x.
  Appendix A represents a continuous polynomial timing fit as a time-varying projector in the Legendre basis and quantifies how the fit modifies red-noise covariance and power.

Radiative transfer in stochastic media

• Levermore, C. D. and Pomraning, G. C. and Sanzo, D. L. and Wong, J. (1986). Linear transport theory in a random medium. Journal of Mathematical Physics 27, 2526-2536. doi:10.1063/1.527320.
  Derives a formally exact equation for transport through a purely absorbing random mixture and solves it for two materials with Markov statistics, giving the two-exponential transmission law.
• Levermore, C. D. and Wong, J. and Pomraning, G. C. (1988). Renewal theory for transport processes in binary statistical mixtures. Journal of Mathematical Physics 29, 995-1004. doi:10.1063/1.527997.
  Treats purely absorbing binary mixtures with arbitrary chord-length distributions by renewal theory and obtains effective cross sections and sources for a standard transport equation.
• Malvagi, F. and Levermore, C. D. and Pomraning, G. C. (1989). Asymptotic limits of a statistical transport description. Transport Theory and Statistical Physics 18, 287-311. doi:10.1080/00411458908204690.
  Studies limits of the two-equation stochastic transport model, including fluid packets small compared with the mean free path, where the model reduces to a single transport equation.
• Pomraning, G. C. (1991). Linear Kinetic Theory and Particle Transport in Stochastic Mixtures. World Scientific. doi:10.1142/1549.
  The monograph on particle transport through random mixtures of immiscible materials, including the Levermore-Pomraning model.
• Su, B. and Pomraning, G. C. (1994). Limiting correlation length solutions in stochastic radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer 51, 893-912. doi:10.1016/0022-4073(94)90019-1.
  Constructs solutions of stochastic radiative transfer in the limits of small and large correlation length, with scattering, in rod geometry.
• Pomraning, G. C. (1996). Small correlation length solutions for planar symmetry beam transport in a stochastic medium. Annals of Nuclear Energy 23, 843-861. doi:10.1016/0306-4549(95)00080-1.
  Gives small-correlation-length solutions for beam transport through a planar stochastic medium.
• Kiedrowski, B. C. and Vu, E. H. (2025). Transit-Length Distribution for Particle Transport in Binary Markovian Mixed Media. SSRN preprint. doi:10.2139/ssrn.5200776.
  Maps transport in a binary Markovian mixture to the telegraph process and derives the distribution of distance travelled in each material, with a Gaussian form for well-mixed media.
• Adams, M. L. and Larsen, E. W. and Pomraning, G. C. (1989). Benchmark results for particle transport in a binary Markov statistical medium. Journal of Quantitative Spectroscopy and Radiative Transfer 42, 253-266. doi:10.1016/0022-4073(89)90072-1.
  Sets out the standard planar benchmark problems for transport in binary Markovian mixtures.
• Brantley, P. S. (2011). A benchmark comparison of Monte Carlo particle transport algorithms for binary stochastic mixtures. Journal of Quantitative Spectroscopy and Radiative Transfer 112, 599-618. doi:10.1016/j.jqsrt.2010.06.007.
  Compares Monte Carlo algorithms, including chord-length sampling, on the binary-mixture benchmark suite.
• Larmier, C. and Hugot, F.-X. and Malvagi, F. and Mazzolo, A. and Zoia, A. (2017). Benchmark solutions for transport in d-dimensional Markov binary mixtures. Journal of Quantitative Spectroscopy and Radiative Transfer 189, 133-148. doi:10.1016/j.jqsrt.2016.11.015.
  Computes benchmark solutions for transport through binary mixtures generated as coloured Poisson tessellations in one, two and three dimensions.
• Titov, G. A. (1990). Statistical Description of Radiation Transfer in Clouds. Journal of the Atmospheric Sciences 47, 24-38. doi:10.1175/1520-0469(1990)047<0024:sdorti>2.0.co;2.
  Describes radiation transfer in broken cloud fields modelled with Poisson point processes whose inputs are cloud fraction and mean cloud size.
• Malvagi, F. and Byrne, R. N. and Pomraning, G. C. and Somerville, R. C. J. (1993). Stochastic Radiative Transfer in Partially Cloudy Atmosphere. Journal of the Atmospheric Sciences 50, 2146-2158. doi:10.1175/1520-0469(1993)050<2146:srtipc>2.0.co;2.
  Shows that Titov's integral Markovian cloud model is a special case of a low-order differential model of the Levermore-Pomraning type and recommends the latter for climate models.
• Natta, A. and Panagia, N. (1984). Extinction in inhomogeneous clouds. The Astrophysical Journal 287, 228. doi:10.1086/162681.
  Derives analytic expressions for the extinction produced by clumpy dust layers.
• Hobson, M. P. and Scheuer, P. A. G. (1993). Radiative transfer in a clumpy medium - I. Analytical Markov-process solution for an N-phase slab. Monthly Notices of the Royal Astronomical Society 264, 145-160. doi:10.1093/mnras/264.1.145.
  Solves radiative transfer through a slab whose material along a ray is an N-phase Markov process, with absorption and isotropic scattering.
• Kostinski, A. B. (2001). On the extinction of radiation by a homogeneous but spatially correlated random medium. Journal of the Optical Society of America A 18, 1929. doi:10.1364/josaa.18.001929.
  Shows that a statistically homogeneous but spatially correlated medium attenuates radiation more slowly than the exponential law.
• Vasques, R. and Larsen, E. W. (2014). Non-classical particle transport with angular-dependent path-length distributions. II: Application to pebble bed reactor cores. Annals of Nuclear Energy 70, 301-311. doi:10.1016/j.anucene.2013.12.020.
  Applies transport with non-exponential, direction-dependent path-length distributions to pebble-bed reactor cores.
• Flajolet, P. and Odlyzko, A. M. (1990). Singularity Analysis of Generating Functions. SIAM Journal on Discrete Mathematics 3(2):216-240. doi:10.1137/0403019.
  Transfers local algebraic and logarithmic singular expansions into sharp Taylor-coefficient asymptotics; applied on the Clumpy Media page to the conjugate square-root branch points of the exact binary transmission law.

Epidemics in random environments

• Smith, W. L. and Wilkinson, W. E. (1969). On Branching Processes in Random Environments. The Annals of Mathematical Statistics 40, 814-827. doi:10.1214/aoms/1177697589.
  Shows that a Galton-Watson process in an i.i.d. random environment dies out with probability one exactly when the expected log of the mean offspring number is not positive.
• Athreya, K. B. and Karlin, S. (1971). On Branching Processes with Random Environments: I: Extinction Probabilities. The Annals of Mathematical Statistics 42, 1499-1520. doi:10.1214/aoms/1177693150.
  Extends the extinction criterion for branching processes to stationary ergodic environments, where the sign of the typical log growth decides extinction.
• Arnold, L. (1984). A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems. SIAM Journal on Applied Mathematics 44, 793-802. doi:10.1137/0144057.
  Relates the growth rates of the moments of a linear stochastic system to its almost-sure Lyapunov exponent through the convex moment Lyapunov function.
• Wong, E. and Zakai, M. (1965). On the Convergence of Ordinary Integrals to Stochastic Integrals. The Annals of Mathematical Statistics 36, 1560-1564. doi:10.1214/aoms/1177699916.
  Shows that ordinary differential equations driven by smooth approximations of Brownian motion converge to the Stratonovich stochastic differential equation.
• Wang, W. and Zhao, X.-Q. (2008). Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments. Journal of Dynamics and Differential Equations 20, 699-717. doi:10.1007/s10884-008-9111-8.
  Defines the basic reproduction number for compartmental epidemic models with periodic coefficients and proves that it is the invasion threshold.
• Gray, A. and Greenhalgh, D. and Hu, L. and Mao, X. and Pan, J. (2011). A Stochastic Differential Equation SIS Epidemic Model. SIAM Journal on Applied Mathematics 71, 876-902. doi:10.1137/10081856x.
  Perturbs the transmission rate of the SIS model by white noise and shows that the almost-sure extinction threshold falls below the deterministic reproduction number.
• Gray, A. and Greenhalgh, D. and Mao, X. and Pan, J. (2012). The SIS epidemic model with Markovian switching. Journal of Mathematical Analysis and Applications 394, 496-516. doi:10.1016/j.jmaa.2012.05.029.
  Switches the rates of the SIS model by a finite Markov chain and shows that almost-sure extinction or persistence is decided by the stationary average of the per-capita growth rate, not by the reproduction number of the mean system.
• Hernandez-Suarez, C. and Rabinovich, J. and Hernandez, K. (2012). The long-run distribution of births across environments under environmental stochasticity and its use in the calculation of unconditional life-history parameters. Theoretical Population Biology 82, 264-274. doi:10.1016/j.tpb.2012.05.004.
  Computes life-history parameters averaged over a Markovian environment, including a reproduction parameter that governs growth of the expected population.
• Bacaër, N. and Khaladi, M. (2013). On the basic reproduction number in a random environment. Journal of Mathematical Biology 67, 1729-1739. doi:10.1007/s00285-012-0611-0.
  Defines a reproduction number for populations in a Markovian random environment whose position relative to one decides growth or decay of typical populations, and contrasts it with a parameter for the expected population.
• Benaïm, M. and Le Borgne, S. and Malrieu, F. and Zitt, P.-A. (2014). On the stability of planar randomly switched systems. The Annals of Applied Probability 24, 292-311. doi:10.1214/13-aap924.
  Shows that random switching between two stable planar linear systems can produce an unstable system, depending on the switching rate.
• Li, D. and Liu, S. and Cui, J. (2017). Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching. Journal of Differential Equations 263, 8873-8915. doi:10.1016/j.jde.2017.08.066.
  Gives the extinction and persistence threshold of an SIRS model whose rates switch with a Markov chain, as a stationary average over regimes.
• Prodhomme, A. and Strickler, É. (2024). Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switching environment. The Annals of Applied Probability 34. doi:10.1214/23-aap2035.
  Studies a multitype stochastic SIS epidemic in a switching environment and shows that the sign of the top Lyapunov exponent of the linearized switched system governs the time to extinction.
• Monmarché, P. and Strickler, É. (2025). Asymptotic expansion of the invariant measure for Markov-modulated ODEs at high frequency. Journal of Applied Probability 62, 898-924. doi:10.1017/jpr.2024.107.
  Expands the invariant measure of ordinary differential equations modulated by a fast Markov chain in powers of the switching time, with a first-order formula for the top Lyapunov exponent of switched linear systems.
• Monmarché, P. and Schreiber, S. J. and Strickler, É. (2025). Impacts of Tempo and Mode of Environmental Fluctuations on Population Growth: Slow- and Fast-Limit Approximations of Lyapunov Exponents for Periodic and Random Environments. Bulletin of Mathematical Biology 87. doi:10.1007/s11538-025-01443-z.
  Gives first-order slow- and fast-switching approximations of the Lyapunov exponent of structured populations switched periodically or by a Markov chain, the closest prior work to the almost-sure formula here.
• Cotton, P. (2020). Addressing the Herd Immunity Paradox Using Symmetry, Convexity Adjustments and Bond Prices. arXiv:2006.07341 [q-bio.PE]. link.
  Explains why epidemics turn over below the herd-immunity threshold implied by early growth: in a mixture of independent sub-populations the mean early growth is raised by a growth convexity G, from the moment generating function of the transmission rate (Vasicek's bond formula when it follows an Ornstein-Uhlenbeck process), and the susceptible fraction at the peak is raised by a harmonic-mean convexity J, so s(t*) = 1/(F J G R0).
• Cotton, P. (2020). Repeat Contacts and the Spread of Disease: An Agent Model with Compartmental Solution. arXiv:2005.10311 [q-bio.PE]. link.
  Attenuates SIR transmission by the probability that an encounter is novel, derived from agents diffusing in the plane. Epidemics with the same early trajectory and R0 then differ by an order of magnitude in final size, depending on how fast circles of acquaintances go stale.
• Britton, T. and Ball, F. and Trapman, P. (2020). A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2. Science 369(6505):846-849. doi:10.1126/science.abc6810.
  Age and activity heterogeneity lower the disease-induced herd immunity level well below 1 - 1/R0.
• Tkachenko, A. V. and Maslov, S. and Elbanna, A. and Wong, G. N. and Weiner, Z. J. and Goldenfeld, N. (2021). Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity. Proceedings of the National Academy of Sciences 118(17):e2015972118. doi:10.1073/pnas.2015972118.
  Individual social activity that changes over time gives a transient collective immunity below the long-run threshold, with an immunity factor near 4 fitted to the first waves in New York City and Chicago.
• Gomes, M. G. M. and Ferreira, M. U. and Corder, R. M. and King, J. G. and others (2022). Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold. Journal of Theoretical Biology 540:111063. doi:10.1016/j.jtbi.2022.111063.
  Fits models with gamma-distributed susceptibility or exposure to European first waves and finds herd immunity thresholds well below the homogeneous value.

Lumped chains and phase-type durations

• Burke, C. J. and Rosenblatt, M. (1958). A Markovian Function of a Markov Chain. The Annals of Mathematical Statistics 29(4):1112-1122. doi:10.1214/aoms/1177706444.
  Gives conditions under which a function of a Markov chain, a grouping of its states, is itself a Markov chain.
• Gilbert, E. J. (1959). On the Identifiability Problem for Functions of Finite Markov Chains. The Annals of Mathematical Statistics 30(3):688-697. doi:10.1214/aoms/1177706199.
  Asks when two hidden chains produce the same observed process once their states are grouped, the identification question for lumped chains.
• Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton.
  Defines lumpability: grouping states gives a Markov chain for every starting distribution exactly when all states of a group have the same total rate into each other group.
• Heller, A. (1965). On Stochastic Processes Derived From Markov Chains. The Annals of Mathematical Statistics 36(4):1286-1291. doi:10.1214/aoms/1177700000.
  Characterizes the processes that arise as functions of finite Markov chains.
• Colquhoun, D. and Hawkes, A. G. (1981). On the stochastic properties of single ion channels. Proceedings of the Royal Society of London B 211(1183):205-235. doi:10.1098/rspb.1981.0003.
  Derives the open and shut time distributions of a single ion channel whose observed state groups several hidden states. The durations are mixtures of exponentials, one for each hidden state in the group.
• Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore.
  Develops phase-type distributions, the absorption times of finite Markov chains, and the matrix methods built on them. Any distribution of durations on the positive axis can be approximated this way.
• Colquhoun, D. and Hawkes, A. G. (1982). On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Philosophical Transactions of the Royal Society of London B 300(1098):1-59. doi:10.1098/rstb.1982.0156.
  Extends the analysis to bursts of openings and clusters of bursts, whose structure shows how the hidden states connect.

Spatial and nonlinear homogenization

• Tartar, L. (2010). The General Theory of Homogenization: A Personalized Introduction. Springer, Lecture Notes of the Unione Matematica Italiana 7. doi:10.1007/978-3-642-05195-1.
  Develops H-convergence and compensated compactness, the weak-convergence theory of limits of partial differential equations with oscillating coefficients.
• Allaire, G. (1992). Homogenization and Two-Scale Convergence. SIAM Journal on Mathematical Analysis 23(6):1482-1518. doi:10.1137/0523084.
  Introduces two-scale convergence, which passes to the limit in periodically oscillating equations and yields the cell problem and the corrector at once.
• Armstrong, S., Kuusi, T. and Mourrat, J.-C. (2019). Quantitative Stochastic Homogenization and Large-Scale Regularity. Springer, Grundlehren der mathematischen Wissenschaften 352. doi:10.1007/978-3-030-15545-2.
  Gives rates of convergence and the fluctuations of the corrector for divergence-form elliptic equations with stationary random coefficients in space.
• Evans, L. C. (1989). The perturbed test function method for viscosity solutions of nonlinear PDE. Proceedings of the Royal Society of Edinburgh, Section A 111(3-4):359-375. doi:10.1017/s0308210500018631.
  Extends the perturbed test function method to viscosity solutions, giving homogenization for fully nonlinear and Hamilton-Jacobi equations.
• Ziliotto, B. (2017). Stochastic Homogenization of Nonconvex Hamilton-Jacobi Equations: A Counterexample. Communications on Pure and Applied Mathematics 70(9):1798-1809. doi:10.1002/cpa.21674.
  Constructs a stationary ergodic nonconvex Hamilton-Jacobi equation in two dimensions that does not homogenize.
• Cioranescu, D. and Murat, F. (1997). A Strange Term Coming from Nowhere. In Topics in the Mathematical Modelling of Composite Materials, Birkhauser, 45-93. doi:10.1007/978-1-4612-2032-9_4.
  Shows that a Dirichlet problem in a domain with many small holes has a limit with an extra zero-order term that no averaging of the coefficients produces.
• Dal Maso, G. (1993). An Introduction to Gamma-Convergence. Birkhauser, Progress in Nonlinear Differential Equations and Their Applications 8. doi:10.1007/978-1-4612-0327-8.
  The textbook on Gamma-convergence, the notion of convergence of energy functionals under which minimizers converge.
• Hou, T. Y. and Wu, X.-H. (1997). A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media. Journal of Computational Physics 134(1):169-189. doi:10.1006/jcph.1997.5682.
  Builds finite-element basis functions from local solutions of the oscillating equation, so that a coarse grid carries the effective behaviour.

Conformal prediction under dependence

• Daniels, H. E. (1954). Saddlepoint Approximations in Statistics. Annals of Mathematical Statistics 25(4):631-650. doi:10.1214/aoms/1177728652.
  Introduces the statistical saddle-point approximation and its asymptotic expansion; the conformal crossover page applies the same local contour calculation to a finite-state Markov-additive lattice tail, retaining its boundary amplitude.
• Cantelli, F. P. (1928). Sui confini della probabilità. Atti del Congresso Internazionale dei Matematici, Bologna, Tomo VI:47-59. proceedings.
  Introduces the sharp one-sided second-moment inequality used to convert the dependent empirical-CDF variance envelope into a finite-panel lower-coverage guarantee.
• Nettasinghe, B., Chatterjee, S., Tipireddy, R. and Halappanavar, M. (2023). Extending conformal prediction to hidden Markov models with exact validity via de Finetti's theorem for Markov chains. Proceedings of the 40th International Conference on Machine Learning, PMLR 202:25890-25903. link.
  Conformal prediction for observations driven by a hidden Markov chain, with exact marginal validity from a block-randomization that restores exchangeability; the pooled-residual page quantifies instead the conditional error of the ordinary pooled threshold.
• Chernozhukov, V., Wüthrich, K. and Zhu, Y. (2018). Exact and robust conformal inference methods for predictive machine learning with dependent data. Proceedings of the 31st Conference on Learning Theory, PMLR 75:732-749. link.
  Conformal inference for time series through block permutations, exact under a block-exchangeability condition and approximately valid under mixing; the dependence between calibration residuals is the effect it addresses.
• Oliveira, R. I., Orenstein, P., Ramos, T. and Romano, J. V. (2024). Split conformal prediction and non-exchangeable data. Journal of Machine Learning Research 25(225):1-38. link.
  Bounds the marginal coverage gap of split conformal prediction under stationary beta-mixing data in terms of the mixing coefficients and the calibration size; the fast-regime pages give the same gap exactly for a finite chain.
• Esseen, C.-G. (1945). Fourier analysis of distribution functions: A mathematical study of the Laplace-Gaussian law. Acta Mathematica 77:1-125. doi:10.1007/bf02392223.
  Develops the Fourier method behind central-limit refinements for lattice laws, including the discontinuous lattice correction represented here by the half-integer tail coordinate.
• Datta, S. and McCormick, W. P. (1993). On the first-order Edgeworth expansion for a Markov chain. Journal of Multivariate Analysis 44(2):345-359. doi:10.1006/jmva.1993.1020.
  Establishes a first-order Edgeworth expansion for sums related to a homogeneous Markov chain and corrects inaccuracies in earlier Markov-chain formulations.
• Pudlo, P. (2010). Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences. ESAIM: Probability and Statistics 14:435-455. doi:10.1051/ps/2009008.
  Computes local lattice Edgeworth expansions of arbitrary degree for multivariate partial sums of finite-state Markov-chain functionals, including periodic effects.
• Csiszár, I. (1963). Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei 8(1-2):85-108. link.
  Proves that general information divergences cannot increase through a stochastic channel. The joint panel theorem uses this data-processing principle after comparing the hidden Markov path with the iid stationary path.
• Rüschendorf, L. (2009). On the distributional transform, Sklar's theorem, and the empirical copula process. Journal of Statistical Planning and Inference 139(11):3921-3927. doi:10.1016/j.jspi.2009.05.030.
  Develops the randomized distributional transform for arbitrary, possibly discontinuous distributions. It supplies the exact uniformization used to transfer the full beta order-statistic law when pooled conformity scores have ties.
• Rényi, A. (1961). On Measures of Entropy and Information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume I:547-561. Project Euclid.
  Introduces the order-indexed information measures. Monotonicity from order one to order two gives the logarithmic refinement between the exact mutual-information and Hilbert–Schmidt panel bounds.
• Kakutani, S. (1948). On equivalence of infinite product measures. Annals of Mathematics 49(1):214-224. doi:10.2307/1969123.
  Characterizes equivalence versus singularity of product measures through the product of their Hellinger affinities. The irregular two-state panel theorem is its explicit biased-sign specialization.
• Le Cam, L. (1960). Locally asymptotically normal families of distributions. University of California Publications in Statistics 3(2):37-98. link.
  Introduces local asymptotic normality. The irregular-panel theorem uses its heterogeneous Bernoulli triangular-array limit; the joint burn-in/panel theorem retains a separate finite initial-state likelihood factor.
• Bickel, P. J. and Ritov, Y. (1996). Inference in hidden Markov models I: Local asymptotic normality in the stationary case. Bernoulli 2(3):199-228. doi:10.3150/bj/1178291719.
  Establishes local asymptotic normality for stationary hidden Markov models. The overlapping-emission theorem computes the observed score and Fisher information explicitly at hidden independence, then uses this LAN result together with an exact likelihood second moment to obtain the sharp total-variation limit.
• Hardy, G. H. and Littlewood, J. E. (1930). Notes on the Theory of Series (XII): On Certain Inequalities Connected with the Calculus of Variations. Journal of the London Mathematical Society s1-5(1):34-39. doi:10.1112/jlms/s1-5.1.34.
  Introduces the rearrangement inequality used to minimize a deficit integral over all panel subsets of prescribed probability.
• Shevtsova, I. (2011). On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. arXiv:1111.6554. arXiv:1111.6554.
  Proves an iid Berry–Esseen constant below 0.4748; the path-conformal page uses 0.4748 to turn its binomial central crossover into an explicit finite-sample bound.
• Bogachev, L. V. and Ratanov, N. E. (2011). Occupation time distributions for the telegraph process. Stochastic Processes and their Applications 121(8):1816-1844. doi:10.1016/j.spa.2011.03.016.
  Derives occupation-time laws and transforms for the telegraph process using a Feynman–Kac approach; the path-conformal certificate uses the same transform principle with unequal transition rates and Gaussian smoothing.
• Kantorovich, L. V. and Rubinstein, G. S. (1958). On a space of completely additive functions. Vestnik Leningrad University 13(7):52-59.
  Introduces the transport metric and its Lipschitz dual representation used to control the complete random law of calibration-conditional coverage.
• Barber, R. F., Candès, E. J., Ramdas, A. and Tibshirani, R. J. (2021). The limits of distribution-free conditional predictive inference. Information and Inference 10(2):455-482. doi:10.1093/imaiai/iaaa017.
  Shows that exact distribution-free conditional predictive validity is impossible without assumptions; the rare-group calculation on the conformal crossover page gives an elementary finite-sample split-conformal witness while preserving marginal validity.
• Marques F., P. C. (2023). Universal distribution of the empirical coverage in split conformal prediction. arXiv:2303.02770.
  Derives the exact universal finite-batch empirical-coverage law for split conformal prediction under exchangeability using a Polya-urn representation. The repeated-future theorem transfers this iid beta-binomial benchmark to fast hidden Markov scores.

Fluid dynamics and turbulent transport

• Taylor, G. I. (1922). Diffusion by Continuous Movements. Proceedings of the London Mathematical Society s2-20(1):196-212. doi:10.1112/plms/s2-20.1.196.
  Shows that a particle carried by a turbulent velocity spreads, at long times, with a diffusivity equal to the time integral of the velocity autocorrelation. This is the first Green-Kubo formula.
• Kraichnan, R. H. (1968). Small-Scale Structure of a Scalar Field Convected by Turbulence. Physics of Fluids 11(5):945-953. doi:10.1063/1.1692063.
  A scalar advected by a random velocity that is delta-correlated in time, the model in which the eddy diffusivity is exact.
• Hasselmann, K. (1976). Stochastic climate models Part I. Theory. Tellus 28(6):473-485. doi:10.1111/j.2153-3490.1976.tb00696.x.
  Slow climate variables driven by fast weather treated as a diffusion approximation, with the fast variables entering as noise.
• McLaughlin, D. W., Papanicolaou, G. C. and Pironneau, O. R. (1985). Convection of Microstructure and Related Problems. SIAM Journal on Applied Mathematics 45(5):780-797. doi:10.1137/0145046.
  Homogenization of a flow with small-scale structure: the effective large-scale equations acquire an eddy term from a cell problem.
• Frisch, U., She, Z. S. and Sulem, P. L. (1987). Large-scale flow driven by the anisotropic kinetic alpha effect. Physica D 28(3):382-392. doi:10.1016/0167-2789(87)90026-1.
  Multiscale expansion of the Navier-Stokes equations around a small-scale flow that lacks parity invariance, giving a large-scale instability at first order.
• Avellaneda, M. and Majda, A. J. (1991). An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows. Communications in Mathematical Physics 138(2):339-391. doi:10.1007/bf02099496.
  An integral representation and bounds for the effective diffusivity of a passive scalar in periodic and random flows.
• Dubrulle, B. and Frisch, U. (1991). Eddy viscosity of parity-invariant flow. Physical Review A 43(10):5355-5364. doi:10.1103/physreva.43.5355.
  Derives the eddy viscosity of the Navier-Stokes equations by multiscale expansion around a parity-invariant small-scale flow.
• Fannjiang, A. and Papanicolaou, G. (1994). Convection Enhanced Diffusion for Periodic Flows. SIAM Journal on Applied Mathematics 54(2):333-408. doi:10.1137/s0036139992236785.
  Effective diffusivity of a passive scalar in periodic cellular flows, and how it scales with the molecular diffusivity.
• Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press. doi:10.1017/cbo9781139170666.
  A book on turbulence with chapters on multiscale methods, eddy viscosity and the closure problem.
• Majda, A. J. and Kramer, P. R. (1999). Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena. Physics Reports 314(4-5):237-574. doi:10.1016/s0370-1573(98)00083-0.
  A review of turbulent diffusion in simplified random and periodic flows, including homogenization, eddy diffusivity and anomalous regimes.
• Majda, A. J., Timofeyev, I. and Vanden-Eijnden, E. (2001). A mathematical framework for stochastic climate models. Communications on Pure and Applied Mathematics 54(8):891-974. doi:10.1002/cpa.1014.
  Eliminates fast modes of fluid and climate equations by the diffusion approximation, giving reduced stochastic equations for the slow modes.
• Tao, T. (2016). Finite time blowup for an averaged three-dimensional Navier-Stokes equation. Journal of the American Mathematical Society 29(3):601-674. doi:10.1090/jams/838.
  Replaces the bilinear term of the Navier-Stokes equations by an averaged operator that keeps the energy identity, and proves finite-time blowup for the modified equation; the result does not transfer to the original equations.
• OpenAI (2026). Finite time blowup for Navier-Stokes. Preprint, September 2026. link.
  Claims that a smooth, compactly supported force and zero initial velocity give a three-dimensional Navier-Stokes solution with bounded energy whose velocity becomes unbounded in finite time, the forced alternatives of the Clay statement; under evaluation.

Green-Kubo, motional narrowing, relaxation and transport

The same integrated correlations in statistical physics, spectroscopy, chemistry and fluid dynamics.

• Onsager, L. (1931). Reciprocal Relations in Irreversible Processes. I.. Physical Review 37(4):405-426. doi:10.1103/physrev.37.405.
  The reciprocal relations: transport coefficients are symmetric under microscopic reversibility. The same argument makes the Green-Kubo matrix symmetric for a reversible chain.
• Bloembergen, N., Purcell, E. M. and Pound, R. V. (1948). Relaxation Effects in Nuclear Magnetic Resonance Absorption. Physical Review 73(7):679-712. doi:10.1103/physrev.73.679.
  Nuclear magnetic relaxation in liquids, with relaxation rates set by spectral densities of fluctuating local fields.
• Taylor, G. I. (1953). Dispersion of soluble matter in solvent flowing slowly through a tube. Proceedings of the Royal Society A 219(1137):186-203. doi:10.1098/rspa.1953.0139.
  Dispersion of a solute in flow through a tube: shear and cross-stream diffusion combine into an enhanced effective diffusion along the tube.
• Green, M. S. (1954). Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena. II. Irreversible Processes in Fluids. Journal of Chemical Physics 22(3):398-413. doi:10.1063/1.1740082.
  Expresses transport coefficients in fluids as time integrals of equilibrium correlation functions, one half of the Green-Kubo formula.
• Anderson, P. W. (1954). A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion. Journal of the Physical Society of Japan 9(3):316-339. doi:10.1143/jpsj.9.316.
  Exchange narrowing of spectral lines: a frequency that hops between values produces a line whose width falls as the hopping speeds up. The two-site case is the two-state eigenvalue on the yield-curve page with an imaginary forcing.
• Kubo, R. (1954). Note on the Stochastic Theory of Resonance Absorption. Journal of the Physical Society of Japan 9(6):935-944. doi:10.1143/jpsj.9.935.
  The stochastic theory of resonance absorption and motional narrowing, with the line width given by the integrated correlation of the frequency fluctuation.
• Giddings, J. C. and Eyring, H. (1955). A Molecular Dynamic Theory of Chromatography. Journal of Physical Chemistry 59(5):416-421. doi:10.1021/j150527a009.
  A molecule alternating at random between moving and adsorbed phases in chromatography; the band's speed is the time-averaged velocity and its spreading comes from the fluctuations.
• Aris, R. (1956). On the dispersion of a solute in a fluid flowing through a tube. Proceedings of the Royal Society A 235(1200):67-77. doi:10.1098/rspa.1956.0065.
  Method of moments for Taylor dispersion, adding molecular diffusion and giving the effective dispersion coefficient in general geometries.
• Kubo, R. (1957). Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. Journal of the Physical Society of Japan 12(6):570-586. doi:10.1143/jpsj.12.570.
  Linear response theory: transport coefficients as integrated correlation functions, the other half of the Green-Kubo formula.
• Redfield, A. G. (1957). On the Theory of Relaxation Processes. IBM Journal of Research and Development 1(1):19-31. doi:10.1147/rd.11.0019.
  Relaxation theory for spins coupled to fast fluctuations. The operators do not commute, and the rates and frequency shifts come from integrated correlations, as in the any-equation page with commutators.
• Blume, M. (1968). Stochastic Theory of Line Shape: Generalization of the Kubo-Anderson Model. Physical Review 174(2):351-358. doi:10.1103/physrev.174.351.
  Generalizes the Kubo-Anderson stochastic line-shape model to fluctuating Hamiltonians that do not commute at different times.
• Szabo, A., Shoup, D., Northrup, S. H. and McCammon, J. A. (1982). Stochastically gated diffusion-influenced reactions. Journal of Chemical Physics 77(9):4484-4493. doi:10.1063/1.444397.
  Diffusion-influenced reactions whose reactivity is switched on and off by a stochastic gate.
• Zwanzig, R. (1990). Rate processes with dynamical disorder. Accounts of Chemical Research 23(5):148-152. doi:10.1021/ar00173a005.
  Rate processes with dynamical disorder: reaction rates that fluctuate in time, with fast and slow fluctuation limits.
• Jülicher, F., Ajdari, A. and Prost, J. (1997). Modeling molecular motors. Reviews of Modern Physics 69(4):1269-1282. doi:10.1103/revmodphys.69.1269.
  Molecular motors as particles switching between potentials under a chemical cycle that breaks detailed balance.
• Nagasawa, M. (1964). Time Reversions of Markov Processes. Nagoya Mathematical Journal 24:177-204. doi:10.1017/S0027763000011405.
  Develops time reversal for general Markov processes; the stationary reversed semigroup is the adjoint used in the Green-Kubo transpose identity.
• Haussmann, U. G. and Pardoux, E. (1986). Time Reversal of Diffusions. Annals of Probability 14(4):1188-1205. doi:10.1214/aop/1176992362.
  Derives the reverse-time drift and generator of a diffusion; on a stationary circle it flips the probability current.
• Reimann, P. (2002). Brownian motors: noisy transport far from equilibrium. Physics Reports 361(2-4):57-265. doi:10.1016/s0370-1573(01)00081-3.
  A review of Brownian motors, where directed transport needs broken detailed balance and broken symmetry.
• Lawley, S. D. and Keener, J. P. (2015). A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary. SIAM Journal on Applied Dynamical Systems 14(4):1845-1867. doi:10.1137/15m1015182.
  Derives Robin boundary conditions by homogenizing a boundary that switches at random between absorbing and reflecting.

Stochastic switching in biology

• Lewontin, R. C. and Cohen, D. (1969). On Population Growth in a Randomly Varying Environment. Proceedings of the National Academy of Sciences 62(4):1056-1060. doi:10.1073/pnas.62.4.1056.
  Population growth in a randomly varying environment: the expected population and the typical population grow at different rates.
• Tuljapurkar, S. D. (1982). Population dynamics in variable environments. II. Correlated environments, sensitivity analysis and dynamics. Theoretical Population Biology 21(1):114-140. doi:10.1016/0040-5809(82)90009-0.
  The small-noise formula for the stochastic growth rate of structured populations in correlated environments, including the effect of autocorrelation.
• Fredkin, D. R. and Rice, J. A. (1986). On aggregated Markov processes. Journal of Applied Probability 23(1):208-214. doi:10.2307/3214130.
  Aggregated Markov processes, as used for ion channels observed only as open or closed, and what can be identified from them.
• Othmer, H. G., Dunbar, S. R. and Alt, W. (1988). Models of dispersal in biological systems. Journal of Mathematical Biology 26(3):263-298. doi:10.1007/bf00277392.
  Velocity-jump models of dispersal, with the diffusion limit given by the integrated velocity correlation.
• Tuljapurkar, S. (1990). Population Dynamics in Variable Environments. Springer, Lecture Notes in Biomathematics 85. doi:10.1007/978-3-642-51652-8.
  A monograph on population dynamics in variable environments, with products of random matrices and small-noise expansions.
• Schnitzer, M. J. (1993). Theory of continuum random walks and application to chemotaxis. Physical Review E 48(4):2553-2568. doi:10.1103/physreve.48.2553.
  Continuum random walks with run-and-tumble motion and their effective diffusion and drift in chemotaxis.
• Peccoud, J. and Ycart, B. (1995). Markovian Modeling of Gene-Product Synthesis. Theoretical Population Biology 48(2):222-234. doi:10.1006/tpbi.1995.1027.
  The telegraph model of gene expression: a promoter switching between two states modulates the rate of synthesis.
• Paulsson, J. (2004). Summing up the noise in gene networks. Nature 427:415-418. doi:10.1038/nature02257.
  Decomposes noise in gene networks into contributions whose size depends on the time-averaging of fluctuating upstream signals.
• Kussell, E. and Leibler, S. (2005). Phenotypic Diversity, Population Growth, and Information in Fluctuating Environments. Science 309(5743):2075-2078. doi:10.1126/science.1114383.
  Phenotype switching in a fluctuating environment, relating long-run population growth to the information the population has about the environment.
• Raj, A., Peskin, C. S., Tranchina, D., Vargas, D. Y. and Tyagi, S. (2006). Stochastic mRNA Synthesis in Mammalian Cells. PLoS Biology 4(10):e309. doi:10.1371/journal.pbio.0040309.
  Single-molecule counts showing transcriptional bursting in mammalian cells, fitted by a switching promoter model.
• Bressloff, P. C. (2017). Stochastic switching in biology: from genotype to phenotype. Journal of Physics A 50(13):133001. doi:10.1088/1751-8121/aa5db4.
  A review of stochastic switching in biology, from gene expression and ion channels to motors and switching environments, with quasi-steady-state and averaging methods.

Queues and point processes

• He, Q.-M. and Neuts, M. F. (1998). Markov chains with marked transitions. Stochastic Processes and their Applications 74(1):37-52. doi:10.1016/s0304-4149(97)00109-9.
  Markov chains with marked transitions, a general framework for arrival processes driven by a hidden chain.
• Asmussen, S. (2003). Applied Probability and Queues. Springer, Stochastic Modelling and Applied Probability 51, 2nd ed.. doi:10.1007/b97236.
  Applied probability and queues, including Markov-modulated arrivals and their asymptotic variance.
• Dean, J. and Barroso, L. A. (2013). The tail at scale. Communications of the ACM 56(2):74-80. doi:10.1145/2408776.2408794.
  Tail latency in large distributed systems and hedged requests, where the first of several responses wins.

Learning and stochastic approximation

• Duffin, R. J. and Schaeffer, A. C. (1952). A Class of Non-Harmonic Fourier Series. Transactions of the American Mathematical Society 72(2):341-366. doi:10.1090/S0002-9947-1952-0047179-6.
  Introduces Hilbert-space frames and their upper and lower stability bounds; the weighted-race design theorem uses the same frame-operator spectrum to quantify identification conditioning.
• Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Springer. doi:10.1007/978-3-642-75894-2.
  Adaptive algorithms driven by Markovian data, analysed by averaging to an ordinary differential equation and by fluctuation limits.
• Duchi, J. C., Agarwal, A., Johansson, M. and Jordan, M. I. (2012). Ergodic Mirror Descent. SIAM Journal on Optimization 22(4):1549-1578. doi:10.1137/110836043.
  Stochastic mirror descent with data from an ergodic process, with convergence rates depending on the mixing time.
• Gürbüzbalaban, M., Ozdaglar, A. and Parrilo, P. A. (2021). Why random reshuffling beats stochastic gradient descent. Mathematical Programming 186:49-84. doi:10.1007/s10107-019-01440-w.
  Shows that sampling without replacement (random reshuffling) beats independent sampling, so the order in which data is visited matters.

Choice and competing arrivals

• Cox, D. R. (1955). Some Statistical Methods Connected with Series of Events. Journal of the Royal Statistical Society B 17(2):129-164. doi:10.1111/j.2517-6161.1955.tb00188.x.
  Series of events, introducing the doubly stochastic Poisson process now called the Cox process.
• Teicher, H. (1961). Identifiability of Mixtures. Annals of Mathematical Statistics 32(1):244-248. doi:10.1214/aoms/1177705155.
  Proves identifiability for mixtures from one-parameter additively closed families, including the Poisson-mixture setting used to recover the integrated-intensity law from the full count law.
• Stieltjes, T.-J. (1894). Recherches sur les fractions continues. Annales de la Faculté des sciences de Toulouse 8(4):J1-J122; continuation 9(1):A5-A47 (1895). link.
  Introduces the moment problem on the positive half-line. The resulting Hankel and shifted-Hankel positivity criterion gives the exact mixed-Poisson membership test used on the counts page.
• Kuba, M. and Panholzer, A. (2016). On moment sequences and mixed Poisson distributions. Probability Surveys 13:89-155. doi:10.1214/14-PS244.
  Surveys mixed Poisson laws through factorial moments and the Stirling transform, linking nonnegative moment sequences to mixed-Poisson distributions.
• Roueff, F. and Rydén, T. (2005). Nonparametric estimation of mixing densities for discrete distributions. Annals of Statistics 33(5):2066-2108. doi:10.1214/009053605000000381.
  Develops nonparametric estimators and minimax theory for mixing densities in discrete mixture models, including Poisson mixtures, under explicit smoothness restrictions; it supplies the estimation context for the distinction between identifiability and stable inversion.
• Miao, Z., Kong, W., Vinayak, R. K., Sun, W. and Han, F. (2024). Fisher-Pitman permutation tests based on nonparametric Poisson mixtures with application to single cell genomics. Journal of the American Statistical Association 119(545):394-406. doi:10.1080/01621459.2022.2120401.
  Proves that the Poisson-mixture NPMLE is minimax-rate optimal under ordinary Wasserstein-1 loss. On a fixed support [0,B], the upper and lower orders are both B loglog(n)/log(n).
• Jackson, D. (1921). The general theory of approximation by polynomials and trigonometric sums. Bulletin of the American Mathematical Society 27(9-10):415-431. link.
  Establishes the polynomial approximation inequality used to approximate Lipschitz Kantorovich test functions with uniform error of order one over the degree.
• Lim, K. and Han, F. (2024). Smoothed NPMLEs in nonparametric Poisson mixtures and beyond. arXiv:2406.08808. arXiv:2406.08808.
  Proves nearly root-n convergence for Poisson-mixture NPMLE and minimum-distance estimators under Gaussian-smoothed optimal transport, providing a precise contrast with the ordinary unsmoothed Wasserstein obstruction derived on the count-law page.
• Kelly, F. P. (1976). Networks of Queues. Advances in Applied Probability 8(2):416-432. doi:10.2307/1425912.
  Uses stationary time reversal to analyse irreversible queueing networks; the competing-arrivals section uses the same reversed-generator construction to isolate the observable arrow of time.
• Marley, A. A. J. and Colonius, H. (1992). The “horse race” random utility model for choice probabilities and reaction times, and its competing risks interpretation. Journal of Mathematical Psychology 36(1):1-20. doi:10.1016/0022-2496(92)90050-h.
  The horse-race random utility model for choices and response times, and its reading as competing risks.