演講訊息 TALKS

摘要：In a series of lectures, I will introduce some useful approaches to study investment problems (or portfolio optimization problems) in finance. In particular, I will consider Merton’s problem and some simple models to show the idea. An approach is the dynamic programming which is a classical powerful method used in control theory. This can be applied to Markovian models. The HJB (Hamilton-Jacobi-Bellman) equation can be derived, which is a nonlinear parabolic or elliptic partial differential equation. A solution of HJB equation can be used to construct an optimal strategy. The second approach is the martingale method. The method uses stochastic calculus and duality idea from convex analysis. In complete markets, this approach can be used to solve the problem without solving (complicated) nonlinear PDE. In incomplete market, the market completion is introduced. The use of BSDE (backward stochastic different equation) is another powerful approach. An adjoint equation is introduced. HJB equation is replaced by a BSDE, a stochastic differential equation, backward in time. The approach is very useful, in particularly, when the investment strategy has constraints. Here are some references for the materials that I will discuss. [1] W. H. Fleming and M. Soner (1993) Controlled Markov Processes and Viscosity Solution. [2] I. Karatzas and S. Shreve(1998), Methods of Mathematical Finance. [3] H. Pham (2002), Continuous Time Stochastic control and Optimization with Financial Applications.