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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.

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