受倒向随机微分方程，倒向随机Volterra 积分方程的启示，本项目将引入并研究一类新型倒向随机积分方程，并讨论其在偏微分方程，时间不一致的最优控制问题和金融数学中的应用。主要包括：当倒向随机积分方程生成元分别满足利普希兹条件、平方增长条件时，研究方程适定性及解的比较定理，并利用该解引入几类关于随机过程的连续时间风险度量；受时间不一致性问题启发，引入一类偏微分方程，并利用正向随机微分方程和倒向随机积分方程研究该方程的经典解、粘性解；研究一类以受控随机微分方程为状态方程、倒向随机积分方程为指标泛函的最优控制问题，利用子博弈完备纳什均衡理论和针状变分思想引入时间一致闭环均衡策略，得到均衡值函数满足的非线性偏微分方程并建立相应的粘性解理论。

Inspired by backward stochastic differential equations, backward stochastic Volterra integral equations, in this project we will introduce a new class of backward stochastic integral equations (BSIEs) and study the applications in partial differential equations, time inconsistent optimal control problems and mathematical finance. The main content is as follows. When the generator of the BSIE satisfies the Lipschitz condition and quadratic growth conditions, we will study its well-posedness, the comparison theorem of adapted solutions, and obtain several classes of continuous time risk measures with respect to stochastic processes. Inspired by the time inconsistent problems, we will introduce a class of partial differential equations (PDEs), and study the corresponding classical solutions, viscosity solutions by means of forward stochastic differential equations (SDEs) and the previous BSIEs. We will study a class of optimal control problems when the state equation is controlled SDEs, and the cost functional is controlled BSIEs. Using the subgame perfect Nash equilibrium theory and spike variation, we will introduce the time consistent closed-loop equilibrium strategies, obtain the nonlinear PDEs satisfied by the equilibrium value functions, and establish the corresponding viscosity solutions theory.