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Wong-Zakai approximations with convergence rate for stochastic partial differential equations

Toshiyuki Nakayama, Stefan Tappe

TL;DR

The paper proves a quantitative convergence rate for Wong-Zakai approximations of semilinear SPDEs driven by a finite-dimensional Brownian motion. By first establishing a rate for the Euler-Maruyama approximation and then bounding the distance between Euler-Maruyama and Wong-Zakai schemes, the authors obtain the rate $\\mathbb{E}[\\sup_{t\\in[0,T]} \\|\\xi_m(t) - X(t)\\|^{2p}] \\\le C/m^{p-1}$ under regularity assumptions on the drift and diffusion and without restricting the generator $A$ to a parabolic or analytic type. The framework accommodates arbitrary strongly continuous semigroups, and the results are illustrated with the HJMM equation and additional physical models. This provides rigorous, actionable error estimates for practical simulations of SPDEs in finance and the natural sciences, where Wong-Zakai approximations are frequently used.

Abstract

The goal of this paper is to prove a convergence rate for Wong-Zakai approximations of semilinear stochastic partial differential equations driven by a finite dimensional Brownian motion. Several examples, including the HJMM equation from mathematical finance, illustrate our result.

Wong-Zakai approximations with convergence rate for stochastic partial differential equations

TL;DR

The paper proves a quantitative convergence rate for Wong-Zakai approximations of semilinear SPDEs driven by a finite-dimensional Brownian motion. By first establishing a rate for the Euler-Maruyama approximation and then bounding the distance between Euler-Maruyama and Wong-Zakai schemes, the authors obtain the rate under regularity assumptions on the drift and diffusion and without restricting the generator to a parabolic or analytic type. The framework accommodates arbitrary strongly continuous semigroups, and the results are illustrated with the HJMM equation and additional physical models. This provides rigorous, actionable error estimates for practical simulations of SPDEs in finance and the natural sciences, where Wong-Zakai approximations are frequently used.

Abstract

The goal of this paper is to prove a convergence rate for Wong-Zakai approximations of semilinear stochastic partial differential equations driven by a finite dimensional Brownian motion. Several examples, including the HJMM equation from mathematical finance, illustrate our result.

Paper Structure

This paper contains 6 sections, 28 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. General framework and notation
  3. Convergence rate for the Euler-Maruyama approximations
  4. Distance between the Euler-Maruyama approximations and the Wong-Zakai approximations
  5. An example: The HJMM equation
  6. Further examples

Key Result

Theorem 1.3

Suppose that Assumptions ass-H and ass-D are fulfilled, and let $T > 0$, $p > 1$ and $x_0 \in \mathcal{D}(A)$ be arbitrary. Then there is a constant $C > 0$ such that for each $m \in \mathbb{N}$ we have where $X$ denotes the mild solution to the SPDE (SPDE-Wong-Zakai) with $X(0) = x_0$, and the $(\xi_m)_{m \in \mathbb{N}}$ denote the mild solutions to the PDEs (WZ-PDE-intro) with $\xi_m(0) = x_0$

Theorems & Definitions (55)

  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 45 more