×

zbMATH — the first resource for mathematics

Hitting place probabilities for two-dimensional diffusion processes. (English) Zbl 1108.60069
The authors consider a 2D difussion process \(X(t):=(X_1(t),X_2(t))\) in a rectangular region, and aim to compute the probability that the process \(X(t)\), starting from a point \((x_1,x_2)\), is hitting first a given side of the rectangle. This problem is solved for three important special cases, viz. when both components of the process (actually, at least one of them) is a 2D Brownian motion, an Ornstein-Uhlenbeck process, or a geometric Brownian motion process. The paper provides the probability functions \(p(x_1,x_2)\) involved by the hitting place problem, for the considered cases, as explicit solutions to variants of a system of stochastic differential equations, expressed as generalized Fourier series. Further extensions of the exposed results are outlined.

MSC:
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J60 Diffusion processes
PDF BibTeX Cite