PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

We consider a financial market with one bond and one stock. The dynamics of the stock price process allow jumps which occur according to a Markov‐modulated Poisson process. We assume that there is an investor who is only able to observe the stock price process and not the driving Markov chain. The investor's aim is to maximize the expected utility of terminal wealth. Using a classical result from filter theory it is possible to reduce this problem with partial observation to one with complete observation. With the help of a generalized Hamilton–Jacobi–Bellman equation where we replace the derivative by Clarke's generalized gradient, we identify an optimal portfolio strategy. Finally, we discuss some special cases of this model and prove several properties of the optimal portfolio strategy. In particular, we derive bounds and discuss the influence of uncertainty on the optimal portfolio strategy.

[1]  Kenneth H. Karlsen,et al.  Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach , 2001, Finance Stochastics.

[2]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[3]  U. Rieder,et al.  Portfolio optimization with unobservable Markov-modulated drift process , 2005, Journal of Applied Probability.

[4]  E. Platen,et al.  A Benchmark Approach to Portfolio Optimization under Partial Information , 2007 .

[5]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[6]  Peter Lakner Utility maximization with partial information , 1995 .

[7]  Ulrich G. Haussmann,et al.  Optimal Terminal Wealth under Partial Information for HMM Stock Returns , 2004 .

[8]  Bernt Øksendal,et al.  Optimal consumption and portfolio in a jump diffusion market , 2001 .

[9]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[10]  Christian Hipp,et al.  Optimal investment for investors with state dependent income, and for insurers , 2003, Finance Stochastics.

[11]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[12]  H. Schmidli On minimizing the ruin probability by investment and reinsurance , 2002 .

[13]  丸山 徹 Convex Analysisの二,三の進展について , 1977 .

[14]  Yoichi Kuwana,et al.  CERTAINTY EQUIVALENCE AND LOGARITHMIC UTILITIES IN CONSUMPTION/INVESTMENT PROBLEMS , 1995 .

[15]  Mark H. Davis Markov Models and Optimization , 1995 .

[16]  Ulrich G. Haussmann,et al.  Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain , 2004, Finance Stochastics.

[17]  P. Lakner Optimal trading strategy for an investor: the case of partial information , 1998 .

[18]  Toshiki Honda,et al.  Optimal portfolio choice for unobservable and regime-switching mean returns , 2003 .

[19]  P. Brémaud Point Processes and Queues , 1981 .