Abstract: In this paper, we study the problem of optimal investment based on the criteria of maximizing probability of survival. On the assumption that investor is faced with a random risk process, the market is incomplete, and the risk of going down-side to the investor cannot be completely eliminated no matter what the investment strategy is. The main contributions are summed up as follows: Based on the maximizing probability of survival, we consider the optimal problem with and without the restriction of borrowing respectively if the risk-free is greater than zero. The closed-form expressions of the optimal strategy and the optimal value function are derived via solving the corresponding HJB equation. In addition, the compares tive static analysis is presented, which explains the quantitative relations among the parameters, survival probability and investment strategy. The results indicate that the proportion invested in the risk asset decreases as the wealth level increases. The more the amount of wealth or the less the given goal level, the greater the survival probability.

Key words: survival probability, stochastic control, borrowing constraints, downside risk