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This paper concerns linear random boundary value problems that contain random variables In the boundary conditions and weakly correlated processes in the differential equations. When the correlation length ϵ is small the structure of the solution is pointed out, and the formulas for the density function of the solutions are derived. The discussion is given in terms of second order equations, but extensions to higher order problems are readily apparent.

Abstract: In this paper, we deal with the one-dimensional backward stochastic differential equation (BSDE) driven by Poisson processes. By means of the comparison theorem, we first prove the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies an improved linear growth assumption. Then we extend the result to the case where the coefficient is left or right continuous. [Copyright &y& Elsevier]

Abstract: This paper considers the two-point boundary value problem of stochastic differential equation with the following form:The sufficient and necessary conditions are given for the existence of the adapted solutions. In the simple case that f(t, X t )= f t , the solution can be obtained by introducing a control term f t , extending the solution from X t to (X t , f t ), and constructing a process sequence. All the results obtained are compared with those related to the eigenvalue problems, and the “martingale approach” solution proposed by Pardoux and Peng for backward stochastic differential equations. [Copyright &y& Elsevier]