Stochastic Differential Equations and Applications

0204332 Ma主要研究随机微分方程（以下简称SDEs）、随机偏微分方程（SPDEs）及其与偏微分方程（PDEs）的关系。通过对非lipschitz系数倒向随机微分方程（BSDEs）的一般理论的研究，寻求非线性Feynman-Kac公式的一种新的“泛函”形式。本文将研究反射后向SDEs （RBSDEs）解的路径规则性，并着眼于将其应用于此类方程的数值方法。本研究还将通过一些新的解的导数的概率表示公式，得到一类偏微分方程中障碍问题解的正则性。PI建议继续研究非线性SPDEs的随机粘度解的新概念，包括在完全非线性情况下的稳定性和唯一性结果。一些随机控制/随机金融问题，受到涉及投资的风险准备金模型的启发，将受到强烈的关注。以破产概率作为稳定性准则的保险公司最优保留/投资问题为例。费曼-卡茨公式是一个强大的工具，它将概率论与分析联系起来，特别是与偏微分方程理论联系起来。该公式的非线性形式有助于研究许多非线性抛物型偏微分方程，如反应扩散方程，这些方程是由生物学、燃烧理论和化学动力学中有利基因的展开导出的。现代随机金融理论中著名的Black-Scholes公式和随机控制理论中的Hamilton-Jacobi-Bellman方程本质上也是该公式的例子。所提出的研究的主要部分是通过考虑“函数形式”或考虑“随机形式”（全非线性spde的粘度解）来发现将这些公式提升到更高水平的可能性。在金融、保险和许多其他领域中出现的一些随机控制问题自然被认为是该理论的应用。

0204332 Ma The principal investigator proposes to study various issues involving stochastic differential equations (hereafter SDEs), stochastic partial differential equations (SPDEs), and their relations with partial differential equations (PDEs). A new "functional" form of nonlinear Feynman-Kac formula is sought, via a study of the general theory of backward stochastic differential equations (BSDEs) with non-Lipschitz coefficients. The path regularity of the solutions to reflected backward SDEs (RBSDEs) will be studied, with an eye on its application to the numerical method for such equations. The study will also lead to the regularity of solutions to a class of obstacle problems in PDEs, via some new probabilistic representation formulae for the derivatives of such solutions. The PI proposes to continue his research on the new notion of stochastic viscosity solution for nonlinear SPDEs, including the stability and uniqueness results in the fully nonlinear case. Some stochastic control/stochastic finance problems, inspired by a risk reserve model involving investment, will receive strong attention. One example is the optimal retention/investment problem for an insurance company that uses the ruin probability as its stability criterion. The Feynman-Kac Formula is a powerful tool that links probability theory to analysis, especially to the theory of partial differential equations. The nonlinear form of this formula is useful for studying many nonlinear parabolic partial differential equations such as reaction diffusion equations, derived from expansion of advantaged genes in biology, combustion theory, as well as chemical kinetics. The celebrated Black-Scholes formula in modern stochastic finance theory and the Hamilton-Jacobi-Bellman equation in stochastic control theory are also, in essence, examples of the formula. The main part of the proposed research is to discover the possibility of advancing such formulae to a higher level, either by considering the "functional form", or by considering the "stochastic form" (viscosity solution of fully nonlinear SPDEs). Some stochastic control problems arising from finance, insurance, and many other fields are naturally considered as applications of the theory.