Some Stochastic Control Problems in the Study of Finance

Shuenn-Jyi Sheu

Department of Mathematics and Department of Applied Mathematics

National Central University and National Chengchi University

sheusj@math.ncu.edu.tw

    Merton(1969) studied a continuous time portfolio optimization problem using dynamic programming approach to solve the problem. In Merton (1971) a general Markovian model was discussed and the HJB equation was derived which is a nonlinear PDE with complicated nonlinearity. Solving the HJB equation left open for many years. The development of martingale method in Pliska (1986) and Karatzas-Lehoczky-Sethi (1986) provides a powerful alternative approach to find a solution when the market is complete. For the case of incomplete market, market completion, as a consequence of martingale method, has been studied in Pages(1987), He-Pearson(1991) and Karatzas-Lehoczky-Shreve-Xu(1991), and also some recent works of Haugh-Kogan-Wang(2006), Rogers (2003), Klein-Rogers and Rogers-Zacvkowski(2013).
In this talk, a review is given to the recent developments of the studies of Merton portfolio optimization problems following the dynamic programming approach. They include risk-sensitive portfolio optimization problem, upside chance and downside risk probabilities optimization and optimal consumption problem. The line of developments follow the ideas of Fleming(1995), which suggests to reformulate the risk-sensitive optimization problem as a risk-sensitive stochastic control problem. We will also discuss recent ideas to find a solution for the finite time consumption problem by rewriting the HJB equation as an inf-sup type Isaacs equation, suggested by the idea of market completion. The Merton problem for a model with risk income will be also discussed. When the market is complete, a solution of the HJB equation can be found. When the market is incomplete, the solution of the HJB equation remains a challenge.
We also mention some investment problems with risky income from insurance, and for model with delay. The talk is based on joint works with H. Nagai, H. Hata, L.H Sun and Z. Zhang.

Keyword: Merton problem; dynamic programming; Hamilton-Jacobi-Bellman equationsstochastic control.