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

项目名称： 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

项目编号： No.11471051

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 周清

作者单位： 北京邮电大学

项目金额： 66万元

中文摘要： 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。

中文关键词： Malliavin；分析；倒向随机微分方程；随机控制；金融数学；Lévy；过程

英文摘要： 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.

英文关键词： Malliavin calculus;backward stochastic differential equation;stochastic control;mathematical finance;Lévy process