The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes

Abstract

Let $M$ be a symmetric independently scattered random measure on $\lbrack 0, 1\rbrack$ with control measure $m$ which is uniformly in the domain of normal attraction of a stable measure of index $p \in (0, 2\rbrack$. Let $f$ be a non-anticipating process with respect to $X(t) = M\lbrack 0, t\rbrack$ if $m$ is continuous, and a previsible process in general, satisfying $\int^1_0 E|f|^p dm < \infty$. Then the stochastic integral $\int^t_0 f dM$ can be defined as a process in $D\lbrack 0, 1\rbrack$ and is in the domain of normal attraction of a stable process of order $p$ in $D\lbrack 0, 1\rbrack$ in the sense of of weak convergence of probability measures. If $M$ is Gaussian and continuous in probability then the central limit theorem holds in $C\lbrack 0, 1\rbrack$; in particular, Ito and diffusion processes satisfy the CLT. Our main tool is an upper bound for the weak $L^p$ norm of $\sup_{0 \leq t \leq 1} |\int^t_0 f dM|$ in terms of the $L^p(P \times m)$ norm of $f$.