Numerical Approximations and Convergence Analysis of Piecewise Diffusion Markov Processes, with Application to Glioma Cell Migration

TL;DR

This work develops numerical methods for Piecewise Diffusion Markov Processes ($PDifMPs$) when flow maps are not explicitly solvable, by combining thinning for jump times with Euler–Maruyama approximations of the continuous flow. It provides rigorous mean-square and weak convergence analyses, including a first-order weak error expansion, and introduces a Thinned Euler–Maruyama (TEM) scheme along with a Thinned-Splitting Method (TSM) for PDifMPs. The methods are validated on geometric Brownian motion with jumps and applied to a microscopic glioma cell-migration model, demonstrating accurate trajectory simulations and insights into how environmental cues influence cell motility. Practically, this enables efficient, provably convergent simulations of stochastic hybrid systems with path-dependent jump dynamics in biological and engineering contexts.

Abstract

In this paper, we focus on numerical approximations of Piecewise Diffusion Markov Processes (PDifMPs), particularly when the explicit flow maps are unavailable. Our approach is based on the thinning method for modelling the jump mechanism and combines the Euler-Maruyama scheme to approximate the underlying flow dynamics. For the proposed approximation schemes, we study both the mean-square and weak convergence. Weak convergence of the algorithms is established by a martingale problem formulation. Moreover, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level. Further, we develop and implement a splitting method for this PDifMP model and employ both the Thinned Euler-Maruyama and the splitting scheme in our simulation example, allowing us to compare both methods.

Figures (7)

• Figure 1: Logarithm (base 2) of the (RMSE) of TEM scheme against the logarithm (base 2) of different values of the step size $h$. Parameters: final time $T=1$, initial value $y_0=50$, drift coefficient $\mu=0.001$, volatility coefficient $\sigma=0.002$, and Poisson jump rate $\lambda=0.0001$.
• Figure 2: Logarithm (base 2) of the (RMSE) of the TEM scheme against the logarithm (base 2) of different values of the step size $h$. Parameters for this dynamic model include: final time $T=1$, initial value $y_0=50$, drift coefficient $\mu=0.01$, volatility coefficient $\sigma=0.2$, and thinning Poisson jump rate $\lambda^{*}=0.001$.
• Figure 3: Simulation results of the progression of glioma cell movement under different values of the basal turning rate $\lambda_0$, for fixed time steps $h=10^{-4} \, (\text{s})$ and a constant sensitivity to environmental signals parameter $\lambda_1=0.08 \, (\text{s}^{-1})$.
• Figure 4: Simulation results of the progression of glioma cell movement under different values of the basal turning rate $\lambda_0$, for fixed time steps $h=10^{-4} \, (\text{s})$ and a constant sensitivity to environmental signals parameter $\lambda_1=0.08 \, (\text{s}^{-1})$.
• Figure 5: Simulation results for the glioma cell movement for fixed time steps $h=10^{-4} \, (\text{s})$, with $\lambda_1=0.08 \, (\text{s}^{-1})$ and $\lambda_0=0.7 \, (\text{s}^{-1})$. The values of $a$ and $b$ are taken within the range in Table \ref{['parameter_mod1']}.

Theorems & Definitions (27)

• Theorem 2.1: Mao mao2007stochastic
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Lemma 3.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3