The principal investigator proposes to study six problems in the general area of stochastic differential equations and their applications in finance and insurance. Several long standing problems in the theory of Forward-backward Stochastic Differential Equations (FBSDEs) are investigated, mainly under the non-Markovian framework, allowing less regular coefficients and arbitrary time duration. A ``User's Guide" type result is expected. A class of quasilinear Backward Stochastic PDEs (BSPDE) is studied in the spirit of nonlinear Feynman-Kac formula, and a new type of FBSDEs with coefficients having discontinuity, arising directly from a real application, will be explored for the first time. A class of combined optimal reinsurance and investment problems with random terminal times and possible partial observations is proposed as a direct application of the newly developed results on FBSDEs. Two problems regarding credit risk models with partial information are proposed. One assumes the so-called ``Hypothesis (H)" (or ``immersion property") and focuses on a special BSDE with super-linear growth and exogenous jumps, with an eye on the utility optimization problems involving defaultable assets; and the other tries to understand the relationship between the conditional density and intensity of the defaults in filtering models where the Hypothesis (H) fail. The PI also proposes to investigate optimal execution problems in an ``order-driven" market by first establishing a new model for the dynamics of the Limit Order Book (LOB) using an equilibrium density argument that results in a general type of nonlinear/random shape of the LOB.
The proposed research aims at resolving some of the ``last obstacles" in the theory of FBSDEs and backward SPDEs, and some related topics in finance and insurance. The proposed projects on FBSDEs build on the results initiated by the PI and will lead to a new solution scheme to treat cases that have been open for many years, which will in turn help the proposed research on optimal reinsurance/investment and utility optimization problems involving defautable assets. The study on credit risk models with partial information and optimal liquidation problems are aiming at exploring new modeling aspects in stochastic finance. Most projects in the proposed research have direct or indirect connections to applied fields, especially stochastic control and stochastic finance/insurance. Three problems are directly related to issues in finance and insurance, using the tools developed in the proposed theoretical studies, while one theoretical problem arises from a real project with a local bank in LA. Several parts of the proposed research involve Ph.D students and postdoctoral fellows. The PI will continue strengthening the connections with local financial communities through a regular Math Finance Colloquium series sponsored by the Math Finance Program at USC, for which PI is the director. Ph.D students involved in the proposed research are more likely to obtain internships from the local banks, and some might lead to permanent employment.
The NSF Grant DMS #1106853 resulted in a series of research papers in three areas of theoretical and applied probability: (I) stochastic differential equations; (II) stochastic partial differential equations; and (III) stochastic control and stochastic finance theory. The projects completed under the support of the grant, although different in nature, are often connected and mutually motivating. Several theoretical results in stochastic differential equations are initiated by and applied to respective problems in stochastic finance and insurance, and others are independent problems that are interesting in their own rights. During the past three years, the PI completed 13 papers under the support of the grant. Among them 6 papers have already appeared, 2 papers are in press, 2 papers are conditionally accepted, and 3 papers are under review. Also, 3 papers are in the final stage of editing and will be submitted soon. The main topics that these papers focus on include: 1) some long standing problems in the theory of Forward-backward Stochastic Differential Equations (FBSDEs), in particular the issues regarding the weak solutions, the equations that are non-Markovian (with random coefficients,), and equations allowing less regular (even discontinuous) coefficients; 2) Stochastic Partial Differential Equations (SPDEs), both in forward and backward temporal directions, and under differential notions of solutions. Among others, the Sobolev solutions and Stochastic Viscosity Solutions and their relations with FBSDEs and related Stochastic Taylor Expansions were carefully sought. 3) The SDEs driven by fractional Brownian motions and Poisson point processes, in both additive and multiplicative noise; 4) SDEs of mean-field type and related stochastic control problems with partial observations; 5) Problems in stochastic finance theory, in particular the issues of systemic risk (self-exciting correlated defaults), the liquidity risks (limit order book models and concave price impacts); and 6) Problems in stochastic insurance theory. Intellectual Merit: The research projects under this grant aim at resolving some of the long-standing problems in the theory of FBSDEs and SPDEs, as well as related topics in finance and insurance. The research projects on FBSDEs resulted in new solution scheme to treat cases that have been open for many years, which is then applied in some optimal reinsurance/investment and utility optimization problems. The reserach projects on stochastic viscosity solution resulted in new stochastic Taylor expansion, which will be the building block of the new notions of stochastic viscosity solutions and path-dependent PDEs. The research projects on SDEs driven by fBM with jumps resulted in new theoretical results, in both additive and multiplicative noise cases, and helped the study of optimal dividend problem under Sparre Anderson model, and raised many theoretical questions, besides their original significance in stochastic insurance. Broader Impact: Most projects in the proposed research have direct or indirect connections to applied fields such as stochastic control and stochastic finance/insurance. Some problems either came directly from real project with local bank in LA, or directly related to issues in finance and insurance. The PI has been supervising eight Ph.D students during the past three years, mentoring two visiting scholars, and directing a Master Program that involves a total of about 60 students. Many of the Ph.D students involved in the proposed research have landed on permanent positions in financial industry. The PI will continue strengthening the connections with local academic and financial communities through a regular Math Finance Colloquium series sponsored by the Math Finance Program at USC, for which PI is the director.