An optimal portfolio and consumption problem with a benchmark and partial information

We consider a finite-time optimal portfolio and consumption problem with a benchmark in a continuous time setting. In our model, investors cannot observe the mean return process of risky assets, which is formulated as an Ornstein–Uhlenbeck process. Investors only have access to past prices of risky assets and information about the benchmark, considered observable information. Investors develop their portfolios and consumption strategies using this observable information. The goal in this context is to maximize the expected utility of benchmark-based consumption and benchmark-based terminal wealth. Using Kalman–Bucy filter, we transform the original optimization problem with partial information into an optimization problem with full information. Solving the Hamilton–Jacobi–Bellman equation, we obtain a candidate value function. Using a verification theorem, we prove that the candidate value function is the solution to our model and provide an explicit optimal strategy. Moreover, we run simulations and present few comments to highlight the rationale behind our results.