Hitting times, occupation times, trivariate laws and the forward Kolmogorov equation for a one-dimensional diffusion with memory (Q2856040)

Apparently motivated by the application in finance, the authors consider one-dimensional diffusions with memory, described by \(dX_t= \sigma(X_t, \underline X_t)\,dW_t\), \(\underline X_t:= m\wedge\text{inf}_{0\leq s\leq t}X_s\), \((W_t)_{t\geq 0}\) the Wiener process, \(X_t\) strictly positive, \(\sigma(x,m)\) continuous with \(0<\sigma(x,m)<\infty\), \(x> 0\) and \(m\geq 0\), \(m< x\). Extending the time-change argument known for ordinary one-dimensional diffusions, they prove weak existence and uniqueness in law of the solution.NEWLINENEWLINE Using Itō's formula and the optimal sampling theorem, the expected time to hit either of the two barriers for \(X\) is calculated. A necessary and sufficient condition is given for the time of hitting \(0\) to be finite, and an extension of the classical occupation time formula is derived.NEWLINENEWLINE The authors further prove the existence of the joint distribution of \(X\) and its minimum and characterize it in terms of two independent tied-down Brownian meanders or equivalently, two independent Bessel-3 bridges. Finally, \(X\) is shown to be a weak solution to a forward Kolmogorov equation, and a new forward equation for down-and-out call options is derived.