Ruin Problems For A Risk Model With Stochastic Return On Investment

In this paper, we studied the classical Cramer - Lundberg model by adding a risk investment, which invests a fixed percentage of surplus into a venture capital market (such as stocks), the remaining parts still in the non-risk markets (such as bonds or bank). We gave the differential expression of this model.In this model, using lto formula and the methods for solving stochastic differ-ential equations, we got the integral expression of the surplus process and gained the embedded discrete time process of the surplus process. Then we defined the ruin time and ruin probability and obtained the embedded discrete time process is Markov chain. We got the expression of the ruin probability using the relevant prop-erties of Markov. Defining the ruin probability before the nth claim(n=1.2,…), we obtained the expression of the probability by mathematical induction. Using the monotone convergence theorem, we get another expression of the ruin probability.At last, we use the classical method of ruin theoretical analysis to define the model's adjustment coefficient by the relationship between the adjustment coefficient and the incremental surplus, and obtained an upper bound of the ruin probability. Next, we studied the discounted surplus process and obtained another upper bound of ruin probability of this model by using martingale methods.