Statistics of the occupation time for a random walk in the presence of a moving boundary (Q2766228)

Let \((x_t)_{t\geq 0}\) be a standard Brownian motion on \(\mathbb R\) starting at zero, and fix \(v\in [0,\infty)\). The authors consider the time spent to the right of a moving boundary with velocity \(v\) up to time \(t\), i.e., they study the random variable \(T_t ^+=\int_0^t1\{x_s\geq vs\} ds\). Recent work by T. J. Newman showed that the density of \(T_t^+\) is given by \(\tau\mapsto F^+(\tau)F^-(t-\tau)\) with \(F^\pm(\tau) =(\pi\tau)^{-1/2}\exp\{-v^2\tau/4D\}\pm\) an error term. The paper under review reproves this result using an idea of Kac (1949). The main point is an analysis of the Laplace transform of the probability density of the event \(\{x_t=x, T_t^+=\tau\}\) in \((x,\tau)\). NEWLINENEWLINENEWLINEFurthermore, the analogous problem for an arbitrary random walk on \(\mathbb R\) is studied in the analogous manner using a result by Sparre Andersen (1953). The cases of binomial and Cauchy walks are discussed in some detail.