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

Abstract

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.