A numerical method for solving stochastic differential equations with noisy memory

TL;DR

This paper introduces a numerical Euler-Maruyama scheme for stochastic differential equations with noisy memory, demonstrating that it achieves a mean-square error order of .5, similar to standard SDEs, and validates it through an analytical example.

Contribution

The paper develops and proves the effectiveness of a new numerical method for SDEs with noisy memory, matching the convergence order of classical schemes.

Findings

The Euler-Maruyama scheme for noisy memory SDEs has a mean-square error order of .5.

The proposed method performs well on an analytically solvable noisy memory SDE.

The convergence order is comparable to that of standard SDE numerical schemes.

Abstract

Stochastic differential equations with noisy memory are often impossible to solve analytically. Therefore, we derive a numerical Euler-Maruyama scheme for such equations and prove that the mean-square error of this scheme is of order $\sqrt{\Delta t}$. This is, perhaps somewhat surprisingly, the same order as the Euler-Maruyama scheme for regular SDEs, despite the added complexity from the noisy memory. To illustrate this numerical method, we apply it to a noisy memory SDE which can be solved analytically.