Stochastic Differential Equations and Related Topics

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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.