Time-dependent weak rate of convergence for functions of generalized bounded variation

TL;DR

This paper investigates the rate at which approximations of solutions to the backward heat equation, based on simple symmetric random walks, converge for irregular terminal conditions, especially near the terminal time.

Contribution

It provides a detailed analysis of the time-dependent weak rate of convergence for functions of generalized bounded variation, including irregular terminal conditions.

Findings

Convergence rate depends on the regularity of the terminal condition.

Error behavior is characterized as time approaches the terminal time.

Results apply to functions with bounded variation and local Hölder continuity.

Abstract

Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$