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 dXt=σ(Xt,Xt)dWt, whereXt=m∧ inf0 ≤s≤tXs. In particular, we compute the expected time forXto leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula forX. We also show that (Xt,Xt) 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.