A Local Time for a Storage Process

Abstract

Content process $X$ of a continuous store satisfies $X_t = X_0 + A_t - \int^t_0 r(X_s) ds, t \geqq 0$. Here, $A$ has nonnegative stationary independent increments, and $r$ is a nondecreasing continuous function. The solution $X$ is a Hunt process. Paper considers the local time $L$ of $X$ at $0$. $L$ may be the occupation time of $\{0\}$ if the latter is not zero identically. The more interesting case is where the occupation time of $\{0\}$ is zero but 0 is regular for $\{0\}$; then $L$ is constructed as the limit of a sequence of weighted occupation times of $\{0\}$ for a sequence of Hunt processes $X^n$ approximating $X$. The $\lambda$-potential of $L$ is computed in terms of the Levy measure of $A$ and the function $r$.