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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.

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