An optimal investment strategy with maximal risk aversion and its ruin probability

The ruin probability of the reserve of an insurance company, in finite and infinite horizon, when there is the possibility to invest in a risky asset, has recently received a lot of attention. It is well known that for the classical Cramér-Lundberg process (where there is no investment and the claims have exponential moments), the ruin probability decreases exponentially with respect to the initial wealth. Hipp and Plum in 2000, see [HP00], assuming that the price of stock is modeled by the Geometric Brownian Motion, determined the strategy of investment which minimizes the ruin probability using the Hamilton-Jacobi-Bellman equation. In 2003, Gaier, Grandits and Schachermayer, see [GGS03], under the same hypotheses as Hipp and Plum, obtained an exponential bound with a rate that improves the classical Lundberg parameter. The optimal trading strategy they found consists in investing in the stock a constant amount of money, independent of the current level of the reserve. Hipp and Schmidli [HS04] showed that this strategy is asymptotically optimal. In this paper we study the problem from a different point of view. We follow the approach done by Ferguson, (1965) who conjectured that maximizing exponential utility from terminal wealth is intrinsically related to minimizing the probability of ruin. Ferguson studied this problem for a discrete time and discrete space investor. Browne (1995), verified the conjecture in a model without interest rate, where the stock follows a Geometric Brownian Motion, and the Risk Process is a Brownian Motion with drift, see [Fer65], [Bro95], and references therein. We consider the wealth process of the reserve of an insurance company, with claims with exponential moments, when there is investment in a bond and in a stock that follows a Geometric Brownian Motion (see the formulation of the problem in Section 1). We first determine the optimal strategy that maximizes an exponential utility function (− exp−γx) of the wealth process for a finite horizon of time (T ). Then we ask ourselves what the ruin probability is, for this strategy, in the interval [0, T ]. We obtain an exponential bound for the ruin probability that, when applied to the Classical Cramér Risk Process, improves the classical Lundberg parameter for some values of γ. If we take