Existence conditions and variational approach for adapted solutions of the two-point boundary value problem of stochastic differential equations

摘要

This paper considers the two-point boundary value problem of stochastic differential equation with the following form:dXt=f(t,Xt)dt+σ(t,Xt)dWt,AX0+BXT=ξ∗.The sufficient and necessary conditions are given for the existence of the adapted solutions. In the simple case that f(t, Xt) = ft, the solution can be obtained by introducing a control term ft, extending the solution from Xt to (Xt, ft), 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.