A functional limit theorem for irregular SDEs

Abstract

Let $X_1,X_2,\dots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y_{k+1}^N=Y_k^N+a_N(Y_k^N)X_{k+1}$, 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 $(Y_{\lfloor Nt\rfloor }^N{)}_{t\in {\mathbb{R}}_{+}}$ 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 $1/N$.

References in corpus (1)

• Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients