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Existence conditions and variational approach for adapted solutions of the two-point boundary value problem of stochastic differential equations. (English)
Appl. Math. Comput. 217, No. 20, 7900-7912 (2011).
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: $$dX_t=f(t,X_t)dt+σ(t,X_t)dW_t,\quad AX_0+BX_T=ξ^*,$$ 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 $\{Ω, F, P\}$ with the natural filtration $\{F_t:0\leq t\leq T\}$, and $ξ^*$ is a $F_t$-measurable $R^d$-valued random variable.
Reviewer: Jialin Hong (Beijing)