Existence conditions and variational approach for adapted solutions of the two-point boundary value problem of stochastic differential equations (Q548350)

The authors give sufficient and necessary conditions for the existence of adapted solutions of the two-point boundary value problem of a stochastic differential equation. The problem considered is in the form: \[ dX_t=f(t,X_t)dt+\sigma (t,X_t)dW_t,\quad AX_0+BX_T=\xi^*, \] where \(T>0\) is a fixed terminal time, \(A\) and \(B\) are two \(d\times d\) matrices, \(\{W(t),\,0\leq t\leq T\}\) is a \(k\)-dimensional standard Brownian motion defined on the probability space \(\{\Omega, F, P\}\) with the natural filtration \(\{F_t:0\leq t\leq T\}\), and \(\xi^*\) is a \(F_t\)-measurable \(R^d\)-valued random variable.