Stochastic Differential Equations and Applications

University Of Southern California, Los Angeles CA

Investigators

Abstract

The principal investigator proposes to study five problems in the general area of stochastic differential equations and their applications in finance. A general framework based on a new type of pathwise stochastic Taylor expansion is proposed to substantially advance the long standing theory of stochastic viscosity solution for fully nonlinear stochastic partial differential equations. The new notion of forward-backward martingale problem (FBMP) and the weak solution to forward-backward SDEs will be further investigated, and the discussion of well-posedness, especially the uniqueness of the solutions will be extended to general cases where the coefficients are allowed to be measurable and/or VMO (Variation Mean Oscillation), reaching the most advanced stage of the theory. A new variant of reflected backward SDEs is proposed with an eye on its applications to various problems in finance where a variant of Skorohod problem and a stochastic representation theorem for optional p rocesses were originated. Two proposed problems are more closely related to finance. The general convex risk measures will be put into the framework of the newly developed theory of filtration consistent nonlinear expectations, and will be investigated using quadratic backward stochastic differential equations and the BMO (Bounded Mean Oscillation) martingale theory. A credit risk model with partial information is proposed, aiming at a general framework where continuous observation, counting process observation, and delayed information can be present at the same time. New types of nonlinear filtering problems are expected to emerge, and some interesting new phenomena exhibited so far have raised new questions for the theory of stochastic differential equations. The proposed research is seeking significant advancement in the field of stochastic differential equations, as well as the related areas such as finance. The proposed projects on stochastic viscosity solution for nonlinear SPDEs and weak solution of FBSDEs will build on the results initiated by the PI to further explore the nature of the respective subjects, and to fill the gaps in the long standing theory. The projects on variant reflected BSDE, on quadratic nonlinear expectations, and on credit risk models with partial information are aiming at developing new tools in stochastic analysis to solve complex but practical problems in finance. Most projects in the proposed research have direct or indirect connections to applied fields, especially stochastic control, stochastic finance, and operations research. Two problems in finance theory will be treated directly, using advanced techniques in stochastic analysis and stochastic differential equations. Several parts of the pro posed research involve Ph.D students and postdoctoral fellows, partly reflecting an educational incentive of this proposal.