Utility Maximization With Intermediate Consumption Under Restricted Information for Jump Market Models

The contribution of this paper is twofold: we study power utility maximization problems (with and without intermediate consumption) in a partially observed financial market with jumps and we solve by the innovation method the arising filtering problem. We consider a Markovian model where the risky asset dynamics S t follows a pure jump process whose local characteristics are not observable by investors. More precisely, the stock price process dynamics depends on an unobservable stochastic factor X t described by a jump-diffusion process. We assume that agents' decisions are based on the knowledge of an information flow, [Formula: see text], containing the asset price history, [Formula: see text]. Using projection on the filtration [Formula: see text], the partially observable investment-consumption problem is reduced to a full observable stochastic control problem. The homogeneity of the power utility functions leads to a factorization of the associated value process into a part depending on the current wealth and the so called opportunity process J t . In the case where [Formula: see text], J t and the optimal investment-consumption strategy are represented in terms of solutions to a backward stochastic differential equation (BSDE) driven by the [Formula: see text]-compensated martingale random measure associated to S t , which can be obtained by filtering techniques (Ceci, 2006; Ceci and Gerardi, 2006). Next, we extend the study to the case [Formula: see text], where η t gives observations of X t in additional Gaussian noise. This setup can be viewed as an abstract form of "insider information". The opportunity process J t is now characterized as a solution to a BSDE driven by the [Formula: see text]-compensated martingale random measure and the so called innovation process. Computation of these quantities leads to a filtering problem with mixed type observation and whose solution is discussed via the innovation approach.