G-期望下的BSDE和随机最优控制理论及其在金融中的应用

项目名称： G-期望下的BSDE和随机最优控制理论及其在金融中的应用

项目编号： No.11301011

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 范玉莲

作者单位： 北方工业大学

项目金额： 22万元

中文摘要： G-期望这一崭新的理论体系为研究模型不确定情况下的金融问题提供了新的有 力数学工具，G-期望理论本身及其应用都有很多有意义的问题有待解决。本项目将首先寻找合理的范数以得到一般情况下(生成元f,g含有K或eta时)G-BSDE解的先验估计，从而得到G-BSDE解的存在唯一性定理和比较定理；研究G-期望下的随机最优控制理论，值函数满足的动态规划原理和偏微分(HJBI)方程，并将其应用于解决模型不确定情况下投资组合选择问题；应用G-BSDE理论研究模型不确定情况下金融资产定价和风险度量问题，特别是路径依赖型未定权益的定价和风险度量问题，并通过提出最大稳健期望效用无差异定价方法把效用函数纳入模型。项目的研究结果不但能够推动G-期望理论和相关金融理论的研究，而且具有较高的实际应用价值。

中文关键词： G-期望；倒向随机微分方程；随机最优控制；资产定价；资产组合选择

英文摘要： The G-expectation theory can provide powerful instruments to handle contingent claim valuation and portfolio selection problems under model uncertainty. There are many open questions in the G-expectation theory itself as well as in its application. In this project, we will firstly find the proper norms and get the apriori estimates of general G-BSDEs(when the generators f and g contain term K or eta), and prove the corresponding existence and uniqueness theorem of the solutions of the G-BSDEs and the comparison theorem. Secondly, we study the stochastic optimal control theory, the dynamic programming principle and the HJBI equation satisfied by the value function, which will be used to solve portfolio selection problems in financial market with model uncertainty. Finally, we apply the G-BSDE theory to solve the contingent claim valuation and risk measurement problem, especially for path dependent contingent claims, and incoporate the utility function into consideration by proposing the maximum robust expected utility indifference pricing method. The results of this project not only can promote the development of the G-expectation theory and the related fiancial theory, but also have high practical application value.

英文关键词： G-expectation；backward stochastic differential equation；stochastic optimal control；asset pricing；selection