Optimal Investment-Consumption-Insurance with Partial Information and Correlation Between Assets Price and Factor Process

In this research, we present an analysis of the optimal investment, consumption, and life insurance acquisition problem for a wage earner with partial information. Our study considers the non-linear filter case where risky asset prices are correlated to the factor processes under constant relative risk aversion (CRRA) preferences. We introduce a more general framework with an incomplete market, random parameters adapted to the Brownian motion filtration, and a general factor process with a non-linear state estimation and a correlation between the state process (risky asset prices) and the factor process. To address the wage earner's problem, we formulate it as a stochastic control problem with partial information where the risky assets prices are correlated to the factor processes. Our framework is extensive since the non-linear filter applied to the linear case gives a more robust result than the Kalman filter. We obtain the non-linear filter through the Zakai equation and derive a system of the Hamilton-Jacobi-Bellman (HJB) equation and two backward stochastic differential equations (BSDE). We establish the existence and uniqueness of the solution, prove the verification theorem, and construct the optimal strategy.