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 St 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 Xt described by a jump-diffusion process. We assume that agents' decisions are based on the knowledge of an information flow, $\{{\mathcal G}_t\}_{t \in [0, T]}$, containing the asset price history, $\{\mathcal F}^S_t\}_{t \in [0, T]}$. Using projection on the filtration ${\mathcal G}_t$, 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 Jt. In the case where ${\mathcal G}_t = {\mathcal F}^S_t$, Jt and the optimal investment-consumption strategy are represented in terms of solutions to a backward stochastic differential equation (BSDE) driven by the ${\mathcal F}^S$-compensated martingale random measure associated to St, which can be obtained by filtering techniques (Ceci, 2006; Ceci and Gerardi, 2006). Next, we extend the study to the case ${\mathcal G}_t = {\mathcal F}^S_t \vee {\mathcal F}^\eta_t$, where ηt gives observations of Xt in additional Gaussian noise. This setup can be viewed as an abstract form of "insider information". The opportunity process Jt is now characterized as a solution to a BSDE driven by the ${\mathcal G}_t$-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.