Strong Solutions of Stochastic Differential Equations with Boundary Conditions

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Abstract
The class B of strong solutions of the boundary problem for a stochastic differential equation with given drift and diffusion coefficients implies here a class of all continuous processes having the given stochastic differential within the interval [y 1,y 2] and not leaving it. It is shown that the class B can be characterized as a class of all solutions of some stochastic integral equation. The instantly reflecting process (IRP) in A. Skorokhod's sense is proved to be the extremal (with respect to some ordering) element of this class and in the case when the drift and diffusion coefficients satisfy the Lipschitz condition it can be constructed by the method of successive approximations. The convergence of discrete approximations, which are constructed by the natural analogy of Euler's formula to the IRP, is studied.

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