We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d Xt=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 0, and 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.
"Hitting times, occupation times, trivariate laws and the forward Kolmogorov equation for a one-dimensional diffusion with memory." Adv. in Appl. Probab. 45 (3) 860 - 875, September 2013. https://doi.org/10.1239/aap/1377868542