| An insurance company is usually considered to be ruin if its surplus falls below 0 in ruin theory.Assume that the surplus of an insurance company satisfies the classical Cramer-Lundberg model,then if the premium is large enough to cover the interest payment on the debt,the company can still continue to operate even though the surplus is below 0.On the contrary,if the premium is smaller than the interest payment on the debt,the surplus of the company keeps decreasing and never bounces back.So it is called absolute ruin in this case.However,if the company is allowed to invest in a risky asset,it is possible to come back to the non-absolute-ruin situation since the uncertainty of the market.The object of this paper is to find the optimal investment strategy to achieve the maximal probability of coming back to non-absolute-ruin situation in finite time when the company is below the level of absolute ruin.We called it the revival probability.We assume that the amount invest in the risky asset is bounded above by A,which means the company can not borrow as much money as it wants,and also is bounded below by 0,which means short selling is not allowed.To solve the problem,we first establish the Hamilton-Jacobi-Bellman equation,which is a integro-differential equation in our case,and the associate verification theorem by the dynamic programing theory.We prove the existence of the global solution when assuming the existence of non-negative increasing solution near negative infinity.So we can attain the existence of the maximal revival probability by verification theory if we can solve the equation when the independent variable sufficiently small.Moreover,the optimal strategy in this circumstance is investing as much as one can to the risky asset.As an example,we consider the case in which the claim-size is of exponential distribution.We show how to attain the existence of the HJB equation in this example.Finally we present the numerical results of this case. |
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