A functional limit theorem for irregular SDEs

TL;DR

This paper establishes a limit theorem showing that certain scaled irregular stochastic processes converge to solutions of SDEs, enabling approximation of complex diffusions via simpler random walks.

Contribution

It introduces a method to approximate irregular SDEs by scaled random walks with carefully chosen scaling, extending classical limit theorems to irregular coefficients.

Findings

Convergence of scaled random walks to irregular SDE solutions

Construction of stopping times with expected step size 1/N

Framework for approximating complex diffusions

Abstract

Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{\lfloor N t \rfloor})_{t \in \mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t \in \mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N \to \infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.