Optimal Control for Forward-Backward Stochastic Differential Equations and Related Topics

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An optimal control theory framework will be established for forward-backward stochastic differential equations (FBSDEs). A number of fundamental mathematical questions will be addressed. First, well-posedness for FBSDEs with mixed initial and terminal conditions will be established by means of a priori estimates together with the notion of a "bridge", i.e., a map which relates the FBSDE to a Riccati differential inequality with constraints. Secondly, a spike variation technique for FBSDEs will be developed and used to derive a Pontryagin type maximum principle that is satisfied by optimal controls of FBSDEs. Finally, coupled linear FBSDEs with random coefficients and mixed initial and terminal conditions will be studied by introducing decoupling techniques and related linear-quadratic optimal control problems will be solved using stochastic Riccati equations. The theory derived in this project will substantially enrich the existing theory of FBSDEs, and will deeply extend the classical stochastic optimal control theory. Maximizing return and minimizing risks are very common in the world of investment (including stock market, mutual funds, retirement accounts, insurance, social security, etc.). It is well understood that high return is associated with high risks. Careful study shows that people's preferences towards return/risks are usually not linear. The well-known Allais/Ellsberg type paradoxes, which show that decisions made in the presence of high risk are inconsistent with expected utility theory, are excellent counterexamples for this. To compensate for this, some nonlinear preferences (also called nonlinear expectation) can be introduced via the so-called backward stochastic differential equations. Therefore, when such kind of expectation is used, optimal investment problem naturally becomes an optimal control problem for forward-backward stochastic differential equations. A similar situation happens for investment involving contingent claims (such as discount bonds, insurance claims, options, etc.), markets (including financial markets, energy markets, etc.) with large investors (such as some big hedge funds, a main feature of which is the dependence of the price processes on the positions and trading strategies of the large investors), and so on. The theory established in this project will help us to answer, at least in part, the following types of questions: How do the nonlinear preferences affect the optimal trading strategies for an investment? What will be the optimal trading strategies when some contingent claims are allowed to trade? How do the large investors influence the market?