Bibliography

The classical theory of averaging and homogenization, and the literature around the regime-switching example, grouped by role. Every DOI was checked against Crossref. The same works appear on the literature map.

Regime-switching bond prices as linear ODE systems

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

• 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 model on this site 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 on this site 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 site's 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 site's 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.
• 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 site's 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 exactly the site's 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 site's 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.
• 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 site's 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 the site's 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 on this site 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 the site transfers to a 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 problem on this site, 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 site uses a two-state chain.

Short rates, affine models and credit

• 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.
• 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.
• 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 object the site computes 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 the site's survival probability and bond price calculations 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.

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.