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

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, where Xt = m ∧ inf0≤s≤t Xs. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at zero and we derive a new occupation time formula for X. 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.