Stochastic Differential Equations and Applications

0204332 Ma主要研究者提出研究涉及随机微分方程（以下简称SDEs），随机偏微分方程（SPDE）及其与偏微分方程（PDE）的关系的各种问题。通过对非Lipschitz系数倒向随机微分方程一般理论的研究，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偏微分方程（RBSDEs）解的路径正则性，并着眼于其在数值方法中的应用。研究还将导致一类障碍问题在偏微分方程的解的正则性，通过一些新的概率表示公式，这种解决方案的衍生物。PI建议继续研究非线性SPDE随机粘性解的新概念，包括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，涉及投资的风险准备金模型的启发，将受到强烈的关注。一个例子是保险公司的最优自留/投资问题，使用破产概率作为其稳定性标准。 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对于研究许多非线性抛物型偏微分方程是有用的，例如反应扩散方程，其来源于生物学、燃烧理论以及化学动力学中的扩散方程。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方程实质上也是该公式的例子。所提出的研究的主要部分是发现推进这样的公式到一个更高的水平，无论是通过考虑“功能形式”，或通过考虑“随机形式”（粘性解决方案的完全非线性SPDE）的可能性。金融、保险等领域中的随机控制问题自然也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.