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Preserving invariant domains and strong approximation of stochastic differential equations

Utku Erdogan, Gabriel Lord

TL;DR

This work develops a general framework for solving stochastic differential equations with solutions confined to a bounded hypercube by constructing domain-preserving integrators from convex combinations of two positivity-preserving flows. The Euler-type scheme achieves strong convergence of order $\tfrac{1}{2}$, with numerical evidence often showing first-order behavior in practice, and a Milstein variant extends higher-order preservation. The approach avoids relying on transformations that are equation-specific and demonstrates robustness across multiple models, including SIS-like dynamics and Nagumo SPDEs. The proposed EM-Weighted variant improves local error and overall accuracy, and the framework is applicable to a broad class of SDEs with diagonal or separable diffusion structures. Overall, the method offers a practical, broadly applicable tool for maintaining invariant domains in stochastic simulations without sacrificing stability or efficiency.

Abstract

In this paper, we develop numerical methods for solving Stochastic Differential Equations (SDEs) with solutions that evolve within a hypercube $D$ in $\mathbb{R}^d$. Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order $\tfrac{1}{2}$, and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that the error constant is in most cases superior.

Preserving invariant domains and strong approximation of stochastic differential equations

TL;DR

This work develops a general framework for solving stochastic differential equations with solutions confined to a bounded hypercube by constructing domain-preserving integrators from convex combinations of two positivity-preserving flows. The Euler-type scheme achieves strong convergence of order , with numerical evidence often showing first-order behavior in practice, and a Milstein variant extends higher-order preservation. The approach avoids relying on transformations that are equation-specific and demonstrates robustness across multiple models, including SIS-like dynamics and Nagumo SPDEs. The proposed EM-Weighted variant improves local error and overall accuracy, and the framework is applicable to a broad class of SDEs with diagonal or separable diffusion structures. Overall, the method offers a practical, broadly applicable tool for maintaining invariant domains in stochastic simulations without sacrificing stability or efficiency.

Abstract

In this paper, we develop numerical methods for solving Stochastic Differential Equations (SDEs) with solutions that evolve within a hypercube in . Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order , and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that the error constant is in most cases superior.

Paper Structure

This paper contains 12 sections, 4 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Domain preservation for the SDE
  3. Derivation of the numerical methods
  4. Positivity preserving integrators
  5. From positivity preservation to domain preservation
  6. Milstein version of the domain preserving method
  7. Convergence analysis of Euler method and parameter selection
  8. Ito-Taylor expansions
  9. Strong convergence
  10. Selection of $\theta$ and local truncation error for the Euler method
  11. Numerical Examples
  12. Example 4: Nagumo SPDE

Key Result

Theorem 1

Let Assumptions ass:fg and ass:X0 hold. Then, the SDE system eq: componentSDE has a unique strong solution $\mathbf{X}(t)$ satisfying for all $i \in \mathbb{N}_{d}$

Figures (4)

  • Figure 1: Sample paths for example 1 (a) and in (b) RMSE against $\Delta t$.
  • Figure 2: RMSE against $\Delta t$ for example 2 with (a) random initial values drawn from $U(0,1)$ and (b) for fixed initial data $X(0)=0.95$.
  • Figure 3: RMSE against $\Delta t$ for example 3 with initial condition $X(0)=9.99$ and $N=10$ (a) and $X(0)=0.95$ with $N=1$ (b). See text for other parameter values.
  • Figure 4: Example 4 (a) $\max_x(X(x,t))$ showing EM-IMP does not preserve the domain and (b) RMSE against $\Delta t$.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof