Abstract: An optimization problem of maximizing the expected utility from an investor's terminal ratio of asset to liability is studied in a continuous-time incomplete financial market. Suppose that the financial market consists of one risk-free asset and multiple risky assets, whose price processes are modeled by geometric Brownian motions, and that during the investment process, the investor is faced with an uncontrollable exogenous liability, which is also described by a geometric Brownian motion. Corresponding HJB equation and verification theorem are provided, and the closed-form expressions for the optimal investment strategy and optimal value function are derived by adopting the stochastic dynamic programming approach. Furthermore, by employing sensitivity analysis and numerical examples, it can be found that: (1) The optimal investment strategy is independent of the appreciate rate of the exogenous liability and the current ratio of asset and liability;(2) in the case without exogenous liability, the optimal proportions of investing on the risky assets decrease as the volatility of the risky assets or the relative risk aversion coefficient increases, but in the case with exogenous liability, this result only holds when the parameters satisfy some conditions;(3) in the case without exogenous liability, the optimal value function increases as the investment time horizon or the appreciate rate of the risky assets becomes larger, and decreases as the volatility of the risky assets becomes larger, but this result also only holds when the parameters satisfy some conditions.

Key words: CRRA utility, asset-liability ratio, asset-liability management, Hamilton-Jacobi-Bellman equation, optimal strategy