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Adaptive Mesh Construction for the Numerical Solution of Stochastic Differential Equations with Markovian Switching

Cónall Kelly, Kate O'Donovan

TL;DR

This work develops an adaptive mesh framework for solving $d$-dimensional SDEs with Markovian switching (SDEwMS) by extending Jump Adapted-Adaptive Methods to include switching times as meshpoints. A hybrid scheme combining a Milstein-based main map with a backstop is shown to achieve strong mean-square convergence of order $δ$ (and specifically order $δ=1$ with Milstein) under a mean-square consistency bound. Theoretical results are complemented by a practical implementation and application to a nonlinear telomere-length SDEwMS, where switching dynamics influence parameter values and output distributions. The approach yields efficient, robust simulations of regime-switching stochastic systems with direct applicability to biological models and other systems with random mode changes.

Abstract

We demonstrate an approach to the numerical solution of nonlinear stochastic differential equations with Markovian switching. Such equations describe the stochastic dynamics of processes where the drift and diffusion coefficients are subject to random state changes according to a Markov chain with finite state space. We propose a variant of the Jump Adapted-Adaptive approach introduced by K, Lord, \& Sun~(2025) to construct nonuniform meshes for explicit numerical schemes that adjust timesteps locally to rapid changes in the numerical solution and which also incorporate the switching times of an underlying Markov chain as meshpoints. It is shown that a hybrid scheme using such a mesh that combines an efficient explicit method (to be used frequently) and a potentially inefficient backstop method (to be used occasionally) will display strong convergence in mean-square of order $δ$ if both methods satisfy a mean-square consistency condition of the same order in the absence of switching. We demonstrate the construction of an order $δ=1$ method of this type and apply it to generate empirical distributions of a nonlinear SDE model of telomere length in DNA replication.

Adaptive Mesh Construction for the Numerical Solution of Stochastic Differential Equations with Markovian Switching

TL;DR

This work develops an adaptive mesh framework for solving -dimensional SDEs with Markovian switching (SDEwMS) by extending Jump Adapted-Adaptive Methods to include switching times as meshpoints. A hybrid scheme combining a Milstein-based main map with a backstop is shown to achieve strong mean-square convergence of order (and specifically order with Milstein) under a mean-square consistency bound. Theoretical results are complemented by a practical implementation and application to a nonlinear telomere-length SDEwMS, where switching dynamics influence parameter values and output distributions. The approach yields efficient, robust simulations of regime-switching stochastic systems with direct applicability to biological models and other systems with random mode changes.

Abstract

We demonstrate an approach to the numerical solution of nonlinear stochastic differential equations with Markovian switching. Such equations describe the stochastic dynamics of processes where the drift and diffusion coefficients are subject to random state changes according to a Markov chain with finite state space. We propose a variant of the Jump Adapted-Adaptive approach introduced by K, Lord, \& Sun~(2025) to construct nonuniform meshes for explicit numerical schemes that adjust timesteps locally to rapid changes in the numerical solution and which also incorporate the switching times of an underlying Markov chain as meshpoints. It is shown that a hybrid scheme using such a mesh that combines an efficient explicit method (to be used frequently) and a potentially inefficient backstop method (to be used occasionally) will display strong convergence in mean-square of order if both methods satisfy a mean-square consistency condition of the same order in the absence of switching. We demonstrate the construction of an order method of this type and apply it to generate empirical distributions of a nonlinear SDE model of telomere length in DNA replication.
Paper Structure (17 sections, 3 theorems, 56 equations, 4 figures)

This paper contains 17 sections, 3 theorems, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Characterisation of random inputs
  4. Minimal conditions on the drift and diffusion coefficients $f$ and $g$.
  5. Adaptive timestepping for an SDEwMS
  6. Structure of an adaptive mesh
  7. An explicit discretisation scheme for an SDEwMS
  8. The explicit Euler-Maruyama scheme
  9. The Milstein scheme
  10. A hybrid adaptive numerical scheme: general form
  11. A hybrid adaptive numerical scheme: particular form
  12. Main result
  13. Numerical Results
  14. Implementation of an adaptive numerical method for SDEwMS
  15. Example: a stochastic model of telomere length with Markovian switching
  16. ...and 2 more sections

Key Result

Lemma 1

Let $T > 0$ and $c \geq 0$. Let $u$ be a Borel measurable bounded nonnegative function on $[0,T]$, and let $v$ be a nonnegative integrable function on ${0,T}$. If then

Figures (4)

  • Figure 1: Histogram of $L_{30}(\omega)$ for 1000 samples.
  • Figure 2: Histograms of $L_{30}(\omega)$ for 1000 samples for the SDE without switching and choosing $(c,a)=(c_1,a_1)$ (top) and $(c,a)=(c_2,a_2)$ (bottom).
  • Figure 3: Initial telomere lengths ordered (blue), actual final length for a single trajectory (orange), and the mean change in length over all trajectories for a given initial value (green).
  • Figure 4: Density histogram of the mean change in telomere lengths between day $5$ and day $30$.

Theorems & Definitions (8)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Theorem 6
  • proof
  • Remark 7