Recent advances in continuous-time contract theory, some of which have come out from the prior work of the investigators, as well as advances in equilibrium theory and in modeling liquidity, are applied in order to develop more realistic models of interactions of multiple economic agents. In particular, optimal design of joint ventures between two firms is addressed, that is, the optimal time to enter/exit a project, the optimal effort each should invest in it, and how they should share the profits/losses, thus extending the classical real options theory to the case of two agents. New models for finding optimal compensation to managers are developed, in which the manager may have better information than the investors, and in which the investors may acquire better information at a cost. Global properties of prices formed in equilibrium of financial markets with many agents are analyzed, in order to study what factors influence the volatility and the risk premium of the stock market. In addition, the investigators model how prices locally evolve due to changes in liquidity, for example in the presence of fast high-frequency traders. Finally, in a model of a market with many interacting agents, the project explores how the average number of defaults and average loss evolve in the limit when the number of agents grows.
This project is particularly timely given that one of the reasons for the recent crisis in financial markets is the way traders and managers have been compensated. It will help provide insight into what types of compensation schemes are optimal for the parties involved and for the society in general. Another issue that needs better understanding if a similar crisis is to be prevented in the future, is how the markets evolve due to the interaction of many agents, and how lack of liquidity influences the behavior of the market participants and the prices. In particular, this research sheds light on the formation of asset prices, on portfolio strategies, and on the frequency of defaults that can be expected in markets with many agents, as well as how prices are formed in markets with varying degrees of liquidity. Thus, the results of the project will shed light on which issues to focus on when regulating compensation in financial markets and regulating the financial industry in general.
In this project we have developed a viscosity theory for path dependent partial differential euqations. Feasible numerical methods for related high dimenisonal equations are proposed, and its applications on stochastic differential games are investigated. We have also published a research monograph on contract theory and developed some models on credit risk and liquidity risk. Intellectual merit: The path dependent partial differential equations considers continuous paths as its variables, with typical examples include path dependent Hamilton-Jacobi-Bellman equations and path dependent Isaacs equations. It provides a convenient framework for non-Markovian problems. This new type of equations rarely have classical solutions and thus a viscosity theory is desired, both in theory and in applications. The main difficulty is that the state space is not locally compact, a crucial property used in the viscosity theory for standard partial differential equations. Our main innovation is to replace the pointwise maximum in standard theory with an optimal stopping problem under nonlinear expectation. The latter is very technical due to the failure of the dominated convergence theorem under nonlinear expectation. In a series of papers we have established the wellposedness for such equations. Our main technical tool is the quasi-sure stochastic analysis developed in our earlier project on second order backward stochastic differential equations and nonlinear expectation. We have also applied the theory to study stochastic differential games under control against control and obtained the existence of the game value. The theory is also a characterization of the random decoupling field which we used in our study of coupled forward backward stochastic differential equations. We have proposed a feasible numerical scheme for high dimensional fully nonlinear partial differential equations. The standard method works only for problems up to dimesion 3, due to the so called curse of dimensionality. Nuemrical examples show that our scheme works for 12 dimensional problems. Based on our earlier works on the principal agent problems, we publish a research monograph in Springer Finance. The book provides a survey of the literature, carries out a systematic study of the subject, and introduces the related mathematical tools to this important field. We have also proposed three financial models on credit risk and liquidity risk. The first one is a bottom-up self-exciting correlated default model. It provides some insights on systemic risks and is potentially useful for risk management at the level of a country. The second one concerns concave price impact, which has been supported by both theoretical and empirical studies. We show that the optimal portfolio in such a model exists and is piecewise constant. The thrid one is a limit order book model, where the shape of the book is determined endogenously by equilibrium arguments. Broader impacts: The viscosity theory of path dependent partial differntial equations provides a convenient framework fro non-Markovian probelms and a powerful tool for stochastic optimization and game probelms with diffusion control and economic/financial models with volatility uncertainty. The numerical scheme we propose is important for solving equations arising from the above applications. The research monograph we write provides a good reference for both experts and for beginners. The financial models we provide will hopefully give the society and the regulator some insights on the financial system. Six PhD students under my supervision have participated in the project. I have also given numerous conference and seminar talks, invited short courses in international summer schools, and special topics courses and regular graduate courses at USC, mainly based on research findings of this project.
