Jump-diffusion risk-sensitive asset management. II: Jump-diffusion factor model. (English) Zbl 1291.35409
Summary: We extend our earlier work on the jump-diffusion risk-sensitive asset management problem in a factor model [M. Davis and S. Lleo, SIAM J. Financ. Math. 2, 22–54 (2011; Zbl 1217.91168)] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential equation (PIDE). We are able to show that finding a viscosity solution to this PIDE is equivalent to finding a viscosity solution to a related PDE, for which classical results give uniqueness. With this in hand, a policy improvement argument and classical results on parabolic PDEs show that the HJB PIDE admits a unique smooth solution. The optimal investment strategy is given by the feedback control that minimizes the Hamiltonian function appearing in the HJB PIDE.
MSC:
35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
35D40 | Viscosity solutions to PDEs |
35R09 | Integro-partial differential equations |
91G10 | Portfolio theory |
91G80 | Financial applications of other theories |
93E20 | Optimal stochastic control |