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Markov Chain Approximation of One-Dimensional Sticky Diffusions

Christian Meier, Lingfei Li, Gongqiu Zhang

TL;DR

This work develops continuous-time Markov chain (CTMC) approximations for one-dimensional diffusions with a sticky lower boundary, enabling efficient computation of the Feynman-Kac semigroup $\mathcal{P}_t$ and first passage probabilities via matrix exponentials. It introduces two CTMC schemes for discretizing the diffusion generator, with Scheme 2 achieving second-order convergence by matching the first two moments at the sticky boundary, while Scheme 1 is first-order. The approach yields closed-form expressions for FK semigroup actions and first passage probabilities through the CTMC generator $\mathbb{G}_n$, and compares matrix-exponential computation methods, endorsing an extrapolation-based ODE solver for large horizons. Numerical results on a sticky Ornstein–Uhlenbeck short-rate model show accurate bond pricing and robust performance, including effective simulation of sticky paths where Euler discretization fails. The framework is general to drifts, volatilities, and killing rates, with potential extensions to multidimensional sticky diffusions and hybrid simulation schemes.

Abstract

We develop continuous time Markov chain (CTMC) approximation of one-dimensional diffusions with a lower sticky boundary. Approximate solutions to the action of the Feynman-Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short rate model for low interest environment.

Markov Chain Approximation of One-Dimensional Sticky Diffusions

TL;DR

This work develops continuous-time Markov chain (CTMC) approximations for one-dimensional diffusions with a sticky lower boundary, enabling efficient computation of the Feynman-Kac semigroup and first passage probabilities via matrix exponentials. It introduces two CTMC schemes for discretizing the diffusion generator, with Scheme 2 achieving second-order convergence by matching the first two moments at the sticky boundary, while Scheme 1 is first-order. The approach yields closed-form expressions for FK semigroup actions and first passage probabilities through the CTMC generator , and compares matrix-exponential computation methods, endorsing an extrapolation-based ODE solver for large horizons. Numerical results on a sticky Ornstein–Uhlenbeck short-rate model show accurate bond pricing and robust performance, including effective simulation of sticky paths where Euler discretization fails. The framework is general to drifts, volatilities, and killing rates, with potential extensions to multidimensional sticky diffusions and hybrid simulation schemes.

Abstract

We develop continuous time Markov chain (CTMC) approximation of one-dimensional diffusions with a lower sticky boundary. Approximate solutions to the action of the Feynman-Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short rate model for low interest environment.

Paper Structure

This paper contains 19 sections, 15 theorems, 149 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Characterizations of 1D Sticky Diffusions
  3. Existence and Uniqueness of Weak Solutions
  4. Feynman-Kac Semigroup and Infinitesimal Generator
  5. Eigenfunction Expansion Representations
  6. Continuous Time Markov Chain Approximation
  7. Transition Rates of the CTMC
  8. CTMC Approximation of the Feynman-Kac Semigroup and First Passage Probabilities
  9. Eigenfunction Expansions for the CTMC
  10. Convergence Rate Analysis
  11. Numerical Results for the Sticky OU Short Rate Model
  12. Convergence Rates
  13. Some Comparisons
  14. Simulation
  15. Conclusion
  16. ...and 4 more sections

Key Result

Theorem 2.1

Under Assumption assumption:continuous_derivative_coefficients, Equations eq:general_diffusion_sde and eq:general_local_time_sde have a jointly unique weak solution for any $\rho\in(0,\infty)$.

Figures (5)

  • Figure 1: Yield curves for three models.
  • Figure 2: Absolute error vs. $n$ on log-log scale for Model 1 to 3 (from top to bottom) with 6M and 30Y maturities.
  • Figure 3: Comparison of four methods. "CTMC expm" stands for using the scaling and squaring algorithm for computing the matrix exponential in the CTMC method. In the extrapolation approach, extrapolation is only applied once (i.e., $s=2$).
  • Figure 4: Monte Carlo simulation results for zero coupon bond pricing. The blue line indicates the benchmark prices. The left picture shows the result for Model 1 with $\rho=4.0\times 10^{-3}$ and in the right picture the results are obtained by setting $\rho=0.1$ with all other parameters remaining fixed.
  • Figure 5: Sample paths over 10 years generated by the CTMC and Euler scheme.

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.1
  • ...and 13 more