A numerical method for solving stochastic differential equations with noisy memory
Provides a numerical method for a class of previously intractable SDEs, with convergence guarantees, benefiting researchers in stochastic analysis and applied fields.
The authors derive a numerical Euler-Maruyama scheme for stochastic differential equations with noisy memory and prove its mean-square error is of order √Δt, matching the rate for regular SDEs. They validate the method on an analytically solvable example.
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 $\sqrt{Δt}$. 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.