Papers

Research notes developed here

• What a Fast-Switching Yield Curve Identifies (2026). Full note · Interactive example · certificate. For a short rate whose mean level and variance switch with a fast hidden regime: what a bond yield curve can and cannot identify. To first order the curve fixes four coefficients; a fixed-feature Green–Kubo rank $r$ implies at least $r+1$ regimes, a rank from integrated maturity loadings does not count regimes, and the initial regime mixture is recoverable only from very short maturities.

Published

• 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 · errata · Chapter 3 erratum in detail · Appendix B erratum in detail. Vasicek short rates and hazard rates whose mean level and volatility are driven by a fast process, continuous or two-state; the reduction to linear systems and the fast-switching expansions. The errata cover the switching-rate convention, the dynamic-mean kernel of Chapter 3 and the deterministic-shift factors of Appendix B.
• 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 · PDF. Vasicek and CIR short rates with fast mean-reverting stochastic volatility, and the first-order terms that change the shape of the yield curve.
• Cotton, P. (2020). Got Milk? On Homogenization and Survival Probability. Article · code: homogenize. An expository account of regime-switching survival probabilities.
• Cotton, P. (2020). A Fundamental Theorem for Epidemiology. Article, 27 May 2020. Doubling times under a week against peak infection near 20%: de Finetti's theorem reduces the question to mixtures of independent neighborhoods, and the susceptible fraction at the peak to a harmonic mean.
• Cotton, P. (2020). Addressing the Herd Immunity Paradox Using Symmetry, Convexity Adjustments and Bond Prices. Preprint. arXiv:2006.07341. Epidemics that turn over below the herd-immunity threshold implied by their early growth, explained by a growth convexity from Vasicek's bond formula and a harmonic-mean convexity at the peak.
• Cotton, P. (2020). Repeat Contacts and the Spread of Disease: An Agent Model with Compartmental Solution. Preprint. arXiv:2005.10311. SIR with transmission attenuated by the probability that a contact is new; the speed at which acquaintances go stale changes final sizes by an order of magnitude.

Joint default

• Green–Kubo Geometry of Joint Default (2026). Full note · Two-name CIR example · certificate. Several names with CIR intensities whose mean levels follow a shared hidden regime chain. To first order in fast switching, pairwise survival curves identify a fixed Green–Kubo matrix whatever the initial regime prior, and the order of defaults reveals the direction of a cycle. The rank of the maturity-integrated correction does not bound the number of regimes; the note gives the corrected bound.

Working papers and notes

• Cotton, P. (2026). Survival Under Regime-Switching Ornstein–Uhlenbeck Hazard Rates: The Fast-Switching Expansion to All Orders. Note in preparation. Summary · certificate. The survival probability to every order in the inverse switching rate, with the initial layer and unequal switching rates, in closed form.
• Cotton, P. (2026). Fast Switching in Exponential-Affine Models: One Recursion to All Orders. Note in preparation. Summary · code and certificates. Any irreducible finite chain, including defective generators, driving any model whose state factors out of an exponential-affine formula: several regimes, two-factor rates and credit, CIR, jumps, counts, Heston, Bates, Merton and variance gamma. The example pages add what Cox counts identify about the intensity law, exact finite-rate occupation mixtures for CEV, and the frequency window of the Fourier pricing integrals.
• Cotton, P. (2026). Uniform Initial-Layer Bounds for a Fast Finite-State Switch. Note in preparation. Summary · certificate. Explicit first- and second-order composites for unequal two-state rates, including resonant layers, amplitude renormalization and sharp logarithmic crossovers; plus a nonsingular Volterra theorem and an explicit first composite for arbitrary finite chains, signed or complex terminal data and zero stationary mean, with computable a posteriori bounds; and explicit first- and second-order group-inverse composites for regime-independent terminal data, uniform through the initial layer without diagonalizing the generator.
• Cotton, P. (2026). Closed-Form Bond Options Under Fast Regime Switching. Note in preparation. Summary · code · G2++ certificate. One-factor Jamshidian decomposition plus an explicit first-order correction, including the memory of the regime at expiry, in Gaussian integrals. Under G2++ the decomposition stops at zero-coupon claims: coupon-bond exercise boundaries are curved, and Gaussian conditioning reduces the option to a one-dimensional quadrature instead.
• Cotton, P. (2026). Three Numbers in a Yield Curve. Working paper in preparation. Summary · certificate · long maturities. To first order a yield curve sees a fast chain of any size only through a positive semidefinite Green–Kubo matrix, which bounds the curve's shape and leaves different chains indistinguishable; the long end is a convergent eigenvalue series, and keeping it exact gives a formula uniform in maturity.
• Cotton, P. (2026). One Curve from Short to Long Maturities. Note in preparation. Summary · certificate. Matched bond-curve formulas that keep the exact Perron or Floquet growth rate stay accurate at every maturity, for any finite irreducible regime chain and any fixed initial prior, with explicit error orders. A moving invariant law costs one order, and a locally frozen boundary layer is valid only on bounded horizons, where an exact transported layer works at all maturities.
• Cotton, P. (2026). The Likelihood of a Fast-Switching Short Rate. Note in preparation. Summary · transition certificate · filter certificate. The first-order transition density of a discretely observed Vasicek rate with fast hidden regimes is a Hermite correction with Green–Kubo coefficients, and its composite likelihood agrees with the filtered likelihood to first order. Its score test is a Jarque–Bera test with the kurtosis part projected onto positive excess kurtosis; with several one-sided scores the null weights depend on their correlation.
• Cotton, P. (2026). Fast Regimes in Any Pricing Equation. Note in preparation. Summary · certificate. The first-order effective generator for any Markov pricing equation with fast switched parameters; the antisymmetric Green–Kubo sector enters through commutators and is witnessed by a two-channel arrival race exactly when the two hazards have a nonzero antisymmetric pairing. A centered killed-resolvent formula gives all inverse-speed arrival coefficients and an explicit finite-rate truncation bound.
• Cotton, P. (2026). Solvability Beyond the Centered Corrector. Note in preparation. Summary · certificate. The group inverse determines only the centered shape of each corrector; the next Fredholm condition fixes its null-space amplitude. When the fast generator varies slowly a signed geometric transport term appears, distinct from the Green–Kubo term: a parameter-space one-form whose curvature and closed-loop holonomy do not depend on the choice of corrector.
• Cotton, P. (2026). Cumulants of Integrated Variance Under a Fast Regime. Note in preparation. Summary · cumulant certificate · switching-$\kappa$ certificate · stationary moments · boundary certificate. The fourth cumulant of a return under a switched variance is three times the variance of integrated variance only for independent Brownian noise, and that variance is its Green–Kubo term up to a remainder bounded uniformly in maturity. The full note adds leverage, option-price and implied-volatility corrections, a statewise Feller test, and long-maturity cumulant rates when mean reversion and vol-of-vol switch too.
• Cotton, P. (2026). What Sets the Homogenization Scale? Note in preparation. Full note · certificate. For a fast stationary driver the size of the first correction is set by the covariance tail, not by ergodicity. Long-range dependence gives a variance of order $\varepsilon^\alpha$ with $0<\alpha<1$, larger than $\varepsilon$; a vanishing Green–Kubo integral gives a smaller one; and the covariance alone does not determine the limit law.
• Cotton, P. (2026). The Direction of a Regime Cycle in the Heston Smile. Note in preparation. Summary · chain certificate · circle certificate. A chain cycling in one direction through variance levels and vol-of-vol moves the smile, an effect no reversible driver produces. The reversal property extends from finite chains to any stationary Markov driver, with drifted Brownian motion on a circle as a closed-form case.
• Cotton, P. (2026). Switching Correlation: Correlation Smiles and Downside Co-Skewness. Note in preparation. Summary · certificate. A correlation that jumps between calm and crisis regimes: a closed-form correlation volga in exchange and spread options, and a negative third cross-cumulant when correlation rises in down markets. For two regimes the joint return cumulants of every order are exact at finite switching rate.
• Cotton, P. (2026). Portfolio Choice with Fast Regimes. Note in preparation. Summary · certificate. Merton, the Markowitz frontier and a regime-blind investor with bull and bear regimes: each first correction is a contraction of the symmetric Green–Kubo form, even for a nonreversible driver, and observed-regime weights stay exactly myopic.
• Cotton, P. (2026). Pooled Conformal Residuals Under a Fast Hidden Regime. Note in preparation. Summary · certificate. The conditional coverage of a threshold that pools residuals across a fast two-state regime: exponential in the horizon for a terminal score, algebraic for a path average, decaying at the predictor's own rate for a filtered state; calibration dependence at the sampling spacing as a separate, exactly computable effect.
• Cotton, P. (2026). Path-Dependent Conformal Pooling Under Fast Regime Switching. Note in preparation. Summary · certificate. A conformal score accumulated along the forecast path forgets its starting regime only like $\varepsilon/\Delta$, where a terminal score forgets exponentially, so a pooled threshold keeps marginal coverage while its conditional coverage error decays slowly. The note also quantifies how overlapping or dependent calibration windows break iid rank coverage, and what thinning restores.
• Cotton, P. (2026). When Fast Mixing Justifies Pooled Residuals. Note in preparation. Summary · certificate. A finite-state benchmark for when a pooled conformal threshold is accurate conditional on the current regime: the conditional coverage error is an exact function of the horizon relative to the switching time. The calibration spacing separately controls the variance of the empirical threshold; none of this gives distribution-free conditional coverage.
• Cotton, P. (2026). Pulsar Spin-Down Switching and Timing Noise. Note in preparation. Summary · certificate. Timing noise from a spin-down rate that switches between two values: exact two-state formulas, the Green–Kubo strength, exact sampled-design covariances, and an all-degree theorem showing that every polynomial timing fit containing a linear term cancels the first finite-correlation boundary error.
• Cotton, P. (2026). Light Through Clumpy Media to All Orders in the Correlation Length. Note in preparation. Summary · certificate. Transmission through binary and nested Markovian mixtures as a fast-switching expansion around atomic mix, with the Green–Kubo first correction, sharp high-order coefficient asymptotics, an exact exponentially small finite-slab entry-layer bound, and a uniform thin-slab crossover formula when $L/\lambda_c$ remains bounded.
• Cotton, P. (2026). Epidemic Growth in a Fast-Switching Environment. Note in preparation. Summary · certificate. Mean and typical growth of SIR and SEIR epidemics with switching transmission and recovery: the Green–Kubo gap, an exact projective invariant-density formula with sharp endpoints, strict speed monotonicity, and all-speed variance bounds for two-state SEIR, and the effect of the direction of a regime cycle.
• Cotton, P. (2026). An Asymptotic Correction to Luce's Choice Axiom for Competing Correlated Cox Processes. Working draft: PDF. Softmax as the homogenized choice law of fast hidden kinetics, with its Green–Kubo first correction and the arrival-order witness of irreversibility.