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