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First- and Half-order Schemes for Regime Switching Stochastic Differential Equation with Non-differentiable Drift Coefficient

Divyanshu Vashistha, Chaman Kumar

TL;DR

The paper develops a first-order randomized Milstein scheme for stochastic differential equations with regime switching (SDEwMS) where the drift is Lipschitz but may be non-differentiable. By randomizing the drift and carefully handling Brownian increments at Markov-switching times, the authors prove a strong convergence rate of $1.0$, substantially improving over classical methods under the weaker drift regularity. They also analyze two variant schemes—the modified randomized and the reduced randomized—that omit key Brownian-increment information and consequently reduce the rate to $1/2$, highlighting the essential role of evaluating Brownian paths at switching times. Numerical experiments on regime-switching mean-reverting and geometric Brownian models confirm the theoretical rates and show that half-order non-randomized/derivative-free schemes can outperform the Euler method in certain scenarios, while full randomized schemes offer higher accuracy at increased cost. The work provides practical, high-order schemes for complex systems with regime-switching dynamics and non-smooth drifts, supported by rigorous analysis and numerical validation.

Abstract

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate $1.0$ of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be $1/2$. Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperforms the classical Euler scheme.

First- and Half-order Schemes for Regime Switching Stochastic Differential Equation with Non-differentiable Drift Coefficient

TL;DR

The paper develops a first-order randomized Milstein scheme for stochastic differential equations with regime switching (SDEwMS) where the drift is Lipschitz but may be non-differentiable. By randomizing the drift and carefully handling Brownian increments at Markov-switching times, the authors prove a strong convergence rate of , substantially improving over classical methods under the weaker drift regularity. They also analyze two variant schemes—the modified randomized and the reduced randomized—that omit key Brownian-increment information and consequently reduce the rate to , highlighting the essential role of evaluating Brownian paths at switching times. Numerical experiments on regime-switching mean-reverting and geometric Brownian models confirm the theoretical rates and show that half-order non-randomized/derivative-free schemes can outperform the Euler method in certain scenarios, while full randomized schemes offer higher accuracy at increased cost. The work provides practical, high-order schemes for complex systems with regime-switching dynamics and non-smooth drifts, supported by rigorous analysis and numerical validation.

Abstract

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be . Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperforms the classical Euler scheme.

Paper Structure

This paper contains 9 sections, 15 theorems, 109 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. General Notation
  3. Main Results
  4. Randomized Milstein Scheme
  5. Importance of the Brownian Increments $\{ B_{\ell}(t_{j})-B_{\ell}(\uptau_{1}^{j-1})\}$
  6. Modified Randomized Scheme
  7. Reduced Randomized Scheme
  8. Rate of Convergence
  9. Numerical Implementation

Key Result

Theorem 2.1

Let Assumptions as:initial to as:diffusion be satisfied. Then, there exists a unique strong solution $X:=\{X(t)\}_{t \in [0,T]}$ of SDEwMS eq:sdems and for all $\tilde{p} \in [0,p]$, we have where $K(d, T, \mathbbm{E}[|X_0|^{p}], C)$ is a positive constant.

Figures (8)

  • Figure 1: Randomized Milstein Scheme vs. Milstein Scheme
  • Figure 2: Randomized and Non-randomized Modified and Reduced Schemes vs. Classical Euler Schemes.
  • Figure 3: Non-Randomized Scheme vs. Euler Scheme
  • Figure 4: Non-Randomized Scheme vs. Euler Scheme
  • Figure 5: Non-Randomized Scheme vs. Euler Scheme
  • ...and 3 more figures

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • ...and 26 more