Asset-liability management with Epstein-Zin utility under stochastic interest rate and unknown market price of risk
We study a continuous-time consumption--investment problem with Epstein--Zin recursive utility under partial information, where the market price of risk is unobservable and the investor faces a terminal liability. The introduction of terminal liabilities fundamentally changes the recursive utility optimisation problem by requiring a new ansatz for the value process. This reformulation leads to a coupled linear forward--backward stochastic differential equation (FBSDE) with unbounded random coefficients, which lies outside the scope of standard well-posedness results for coupled FBSDEs. Exploiting the special structure of the equation, we develop a decoupling reduction method that yields an explicit solution under mild assumptions on the model coefficients. We further establish the Malliavin differentiability of the solution, allowing the optimal investment strategy to be represented in terms of conditional expectations involving Malliavin derivatives, which can be efficiently approximated using Monte Carlo methods. The resulting framework provides explicit expressions for the optimal consumption strategy, portfolio allocation, and value function in a stochastic volatility market with stochastic interest rates and an unobservable market price of risk, extending the classical Kim--Omberg model. Finally, we quantify the utility loss arising from ignoring learning about the market price of risk, thereby illustrating the economic significance of partial information in asset--liability management.