Hitting times, occupation times, trivariate laws and the forward Kolmogorov equation for a one-dimensional diffusion with memory

Abstract

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d X_t=σ(X_t,X_t)dW_t, where X_t= m ∧ inf_{0 ≤s≤t} X_s. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula for X. We also show that (X_t,X_t) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.