Optimal investment for asset–liability management with delay and partial information under Ornstein–Uhlenbeck process

In this paper, we investigate the optimal investment strategy of asset liability management (ALM) with bounded memory and partial information. Suppose that investors invest their assets in a financial market consisting of a risk-free bond and a risk-free stock, while also taking on liabilities, in which the value of liabilities and the price of risky assets satisfy the Ornstein–Uhlenbeck (O–U) processes whose drift terms are unobserved. By constructing a dynamic portfolio of risk-free bonds, risky stocks and liabilities, a stochastic delay differential equation is obtained to depict the surplus process of investor. The ALM problem is formulated as finding the best strategy to maximize the terminal utility of the sum of terminal surplus and some historical wealth under partial information, and the corresponding full information case is also studied as a supplement. For both cases of partial information and full information, we apply the dynamic programming method to derive HJB equations, verification theorems, and closed-form solutions of optimal strategies and value functions. Moreover the relationship between optimal strategy and value function under full information and partial information is also given. Finally, numerical examples are carried out to illustrate the influence of some important parameters on the obtained results.