与Lévy过程驱动的倒向随机微分方程相关的随机控制和金融问题

This project aims to study the stochastic optimal control and financial applications related to the backward stochastic differential equations driven by Lévy processes. Our research mainly includes the following detailed contents and aims. Firstly, we will study backward stochastic differential equations and backward doubly stochastic differential equations with general terminal value and general random generator, which are driven by Lévy processes, by Malliavin calculus giving the H?lder continuity of the solution and constructing several numerical approximation schemes. Secondly, we will consider the problem of partially observed optimal control for forward and forward-backward stochastic systems (also can be mean-field type), which are driven by Brownian motion and an independent Poisson random measure with a feature that the cost functional is of mean-field type. Malliavin calculus and convex variational technique will be employed to derive a maximum principle for the optimal control of such a system, trying to give the explicit formula of linear-quadratic optimal control. Thirdly, we will deal with a class of backward doubly stochastic differential equations with time delayed coefficients. We will prove the existence and uniqueness of a solution for a sufficient small Lipschitz constant of the coefficients. Moreover, we will give the probabilistic interpretation of the solution for a class of stochastic partial differential equations with delay. Furthermore, we will establish the stochastic maximum principle for the stochastic optimal control related with the backward doubly stochastic differential equations with time delayed coefficients. Fourthly, we also pay attention to some applications in finance by the theory of backward stochastic differential equations that have attract many attentions for its easily explaining specific problems and quick numerical actualization. The first one we concentrate on is the perpetual American style options under model uncertainty. The second one is about Libor Market Model driven by a Lévy process with default risk where we will consider the construction of the model, the calibration and some applications.

本项目旨在研究与Lévy 过程驱动的倒向随机微分方程相关的随机控制和金融问题。主要包括：以 Malliavin 分析为工具，研究一般终端值和随机生成子的 Lévy 过程驱动的倒向随机微分方程、倒向双重随机微分方程解的H?lder 连续性，建立几个数值近似机制；通过Malliavin 分析和凸变分技术研究部分可观测信息下 Lévy 过程驱动的随机系统的均域型最优控制问题，尝试给出线性二次最优控制的显式表达；通过含有时滞算子倒向双重随机微分方程给出一类具有时滞的随机偏微分方程解的概率表示，建立含有时滞算子倒向双重随机微分方程相关的最优控制问题的随机最大值原理。本项目也将借助倒向随机微分方程这一强大理论工具研究一些具体的金融问题，主要包括:1)模型具有不确定情形下永久美式期权的价格及执行边界的刻画；2)构建一类Lévy过程驱动的考虑违约风险的利率市场模型，给出基本指标的定价公式或高效数值算法。

由于应用的广泛性，倒向随机微分方程备受关注，本项目深入研究了与Lévy过程驱动的倒向随机微分方程相关的随机控制和金融问题。主要包括：研究了正倒向双重随机微分方程、双重随机偏微分方程以及反射倒向双重随机偏微分方程解的理论及相关随机最优控制问题；通过Malliavin分析和凸变分技术，研究部分可观测信息下Lévy过程驱动的随机系统的均域型最优控制问题，给出了线性二次最优控制的显式表达；运用带摄动参数粘性解上（下）微分研究正倒向随机次优控制问题，建立伴随方程与值函数摄动上（下）微分之间的联系。通过解的先验估计以及压缩映像原理，利用可料对偶投影性，给出了反射型时滞倒向随机微分方程解的存在唯一性定理；证明了随机流体力学类型方程的解满足中偏差原理和泛函中心极限定理，研究了时空白噪声驱动的带奇异项的反射随机偏微分方程的解以及不变测度的存在唯一性；研究了一类由G-布朗运动驱动的脉冲随机Cohen-Grossberg神经网络的稳定性。本项目还借助倒向随机微分方程这一强大理论工具研究了一些具体的金融问题： 1) 研究市场冲击模型中存在执行风险情形下的最优执行策略问题，且利用二次倒向随机微分方程定义的风险度量引入偏离既定执行速率的风险，在此模型下通过求解最优化期望P&L相关的最优控制问题给出最优执行策略的显式解。2) 给出SABR平面（与随机波动率模型下的定价问题密切相关）中heat kernel的一种新的表示，研究了几种违约证券的定价问题。. 所得结果丰富和发展了倒向随机微分方程、随机（偏）微分方程、Malliavin 分析的研究领域和应用范围，进一步丰富了随机控制以及金融数学理论，得到的成果有交叉应用前景。