First-passage problems for degenerate two-dimensional diffusion processes

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d1 <y <d2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.