First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes

For $a,b >0,$ we consider a temporally homogeneous, one-dimensional diffusion process $X(t)$ defined over $I = (-b, a),$ with infinitesimal parameters depending on the sign of $X(t).$ We suppose that, when $X(t)$ reaches the position $0,$ it is reflected rightward to $\delta$ with probability $p >0$ and leftward to $ - \delta$ with probability $1-p,$ where $\delta >0.$ Closed analytical expressions are found for the mean exit time from the interval $(-b,a),$ and for the probability of exit through the right end $a,$ in the limit $\delta \rightarrow 0 ^+,$ generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.