Multiple Integrals of a Homogeneous Process with Independent Increments

Abstract

Let $X(t)$ be a homogeneous process with independent increments having the representation $X(t) = W(t) + \int_{x \neq 0} x\nu^\ast(t, dx)$, where $W(t)$ is a Wiener process with parameter $\sigma^2$ and $\nu^\ast(t, dx) = v(t, dx) - t\mu(dx)$, where $\nu(t, dx)$ is a Poisson random measure with mean measure $t\mu(dx)$. If the $m$th absolute mean of $X(t)$ is finite, then $\int^t_0 dX(t_1) \int^{t_1}_0 dX(t_2) \cdots \int^{t_m - 1}_0 dX(t_m) = \{\partial^m/\partial u^m \exp\{uW(t) + \int_{x \neq 0} \log(1 + ux)\nu^\ast(t, dx) - 1/2tu^2\sigma^2 - t \int_{x \neq 0} \lbrack ux - \log(1 + ux)\rbrack\mu(dx)\}\}_{u = 0}/m!$.