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.
During the past four years my research interests have been continuing focusing on three general areas: (I) stochastic differential equations; (II) stochastic control and stochastic finance theory; and (III) stochastic partial differential equations. During 2008 and 2012, ten papers have appeared, two papers are in press, three papers are under review. Intellectual Merit There are seven published papers in categories (I) and (III), plus one in press and two under review. Some of these works are inter-connected, via the so-called (stochastic) Feynman-Kac formula. Many of the equations under investigation are motivated by practical problems in finance, and therefore have high application value. Six papers are devoted to the topic of ``forward-backward stochastic differential equations (FBSDEs)". In the Markovian case our purpose is to establish well-posedness either under a general setting allowing jumps, or with less regular coefficients for which the existing methods do not apply. In the non-Markovian case, we are looking for new framework that unifies and extends all existing solution methods, as well as the connection to stochastic partial differential equations (SPDEs) in the spirit of nonlinear Feynman-Kac formula. Three paper smentioned above are actually from (II), either as a result of a stochastic control problem, or as the underlying model for a utility optimization problem in finance, or even as a fundamental tool to study a certain type of risk measures. These works could certainly be co-listed as the outcome of category (II). Another research focus, which is still on-going is the SDEs driven by fractional Brownian motions. One paper in this area was published and one is under review. These SDEs are useful to deal with models that has long range memory, and is technically more challenging if combined with jumps, as many problems in actuarial science show. We have been successful in the cases either in general form of multiplicative but continuous noise, or additive but jump noises. We are continue our investigation with the help of our study on the fractional Wiener-Poisson spaces with a new norm. Finally, one published papers in (III) concerns the pathwise Taylor expansion which was originally used to define the new notion of stochastic viscosity solution to SPDEs. The recent study shows that such a Taylor expansion, when translated to the new language of "path-derivatives" by Dupire, would be fundamental to the study of the sol-called path-dependent PDEs, initiated recently by S. Peng. We are continue our investigation in the direction. There are three published papers in category (II), plus one in press and one under review. The problems investigated are mostly from either areas of stochastic finance or stochastic actuarial models. Specifically, one paper dealt with Merton's portfolio optimization problem under bounded state-dependent utility functions in a market driven by a L'evy process, one paper studies utility optimization problem for insurance models involving reinsurance, investment, and consumption at the same time, and another work focuses on the correlated default probability for a intensity-based model, and applies it to the pricing of, e.g., the "Universal Variable Insurance" involving married couples. We also study the correlated default models in which the default times of multiple entities depend not only on a common and specific factors, but also on the extent of past defaults in the market, via the average loss process. We show that the Law of Large Numbers holds under certain compatibility conditions. In an on-going project we study an optimal portfolio selection problem under general transaction cost, in which the admissible portfolios are only allowed to have piecewise constant paths, reflecting a strong practical point of view. There are several on-going projects in the pipeline, in all the categories. One of the main focus of the study is path-dependent PDEs and its connection to stochastic PDEs and stochastic control theory. Problems in credit default models, and models with long range memories, along with their applications in actuarial sciences will continue receiving strong attention. Broader Impact Most of the projects under the grant have strong motivations in applied problems in the areas of stochastic control, finance, economics, and actuarial sciences. They also have close relations to other field of mathematics, especially fully nonlinear partial differential equations. During the past three years I have supervised six Ph.D students. Three of them have graduated; two of them are near completion; and one is still in her early stage. I have also supervised one postdoc, one visiting scholar, and one visiting Ph.D student (both for the duration of one year). I have been the director of the Mathematical Finance Master program in the Dana and David Dornsife College of Arts, Letters, and Sciences at USC, and I have been organizing the biweekly {it Mathematical Finance Colloquium} since 2008. I am currently serving as Associate Editor for three Journals, and was the main organizer of two conferences in June of 2011 at USC.
