本项目研究代价函数由倒向随机微分方程（BSDE）的解刻画的随机最优控制问题及其在实际中的应用。考虑这类问题中动态规划原理和相应的Hamilton-Jacobi-Bellman (HJB)方程的Sobolev弱解，利用正倒向随机微分方程理论、随机分析理论，以非线性Doob-Meyer分解定理为主要工具，研究最优值函数是HJB方程的唯一Sobolev弱解。在一些金融问题的启发下，重点研究代价函数由带限制的倒向随机微分方程的解刻画的随机最优控制问题中，HJB方程的Sobolev弱解；探讨代价函数由正倒向两个布朗运动驱动的倒向随机微分方程（BDSDE）的解刻画的随机最优控制问题，考虑动态规划原理、对应的非线性Doob-Meyer分解定理和HJB方程的Sobolev弱解。动态规划原理对应的HJB方程中含有上确界，Sobolev弱解使这类方程有更好的概率表示，丰富了HJB方程的弱解理论。

We study the stochastic optimal control problems which described by backward stochastic differential equations (BSDEs) and their applications in real word. In detail, adopting the classical forward-backward stochastic differential equation theory, stochastic analysis theory and the Doob-Meyer decomposition theorem as the main tools, we consider the dynamic programming principles and the Sobolev weak solutions of the corresponding HJB equations, moreover, we will attempt to prove the optimal value functions are the unique Sobolev weak solutions of the HJB equations. Motivated by some problems arising in mathematical finance, we focus on the stochastic recursive optimal control problem with an obstacle constraint for the cost functional which described by the solution of a reflected BSDE, consider the Sobolev weak solution of the corresponding HJB equation. We will dedicate to study stochastic optimal control problem which described by a backward doubly stochastic differential equation, consider the dynamic programming principle, the Doob-Meyer decomposition theorem and the Sobolev weak solution of the corresponding HJB equation. We creatively give the definition of Sobolev weak solution, give better probabilistic interpretation for the HJB equation which has supremum and enrich the weak solution theory of the HJB equation.