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Upper bound for ruin probabilities under optimal investment and proportional reinsurance. (English) Zbl 1199.91088
The authors consider a risk process that is perturbed by a Brownian motion, and it is also supposed that the insurer has the possibility to choose proportional reinsurance with level $$q\in(0,1]$$. The surplus process of the insurer given by $$R^{q}_{t}=u+[q(1+\eta)-(\eta-\theta)]\lambda\mu t+\beta W_{t}-qS_{t}$$, where $$u\geq0$$ is the initial surplus, $$c$$ is the premium rate, $$S_{t}=\sum_{i=1}^{N(t)}Y_{i}$$ is a compound Poisson process, i.e. $$N(t)$$ is a homogeneous Poisson process with intensity $$\lambda$$, $$Y_{i}, i\geq1$$ is a sequence of positive i.i.d. random variables with mean value $$\mu$$. The premium rate for the reinsurance is $$(1+\eta)(1-q)\lambda\mu$$, where $$\eta$$ is the safety loading of the reinsurer, $$\theta=c/\lambda\mu-1$$ is the safety loading of the insurer. $$W_{t}$$ is a standard Brownian motion independent of the claim number process $$N(t)$$ and of $$Y_{i}, i\geq1$$. It is assumed that there is one risky asset with price $$P(t)$$, modelled by geometric Brownian motion, available for the insurer in the financial market. Let $$A$$ be the total amount of money invested in the higher risky asset. The investment strategy $$A$$ and the retention level $$q$$ are constants in time. The wealth of the company at time $$t$$ given by $$X_{t}^{A,q}=u+A(at+\sigma B_{t})+[q(1+\eta)-(\eta-\theta)]\lambda\mu t+\beta W_{t}-qS_{t}$$, where $$B_{t}$$ is another standard Brownian motion independent of the claim number process $$N(t)$$ and of $$Y_{i}, i\geq1$$. The closed-form expressions of the optimal values and an exponential bound for the ruin probability are obtained. It is proved that the case with investment is always better than the case without investment. The authors analyze the optimal values by numerical examples.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60J28 Applications of continuous-time Markov processes on discrete state spaces 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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