## In the insurance business risky investments are dangerous.(English)Zbl 1002.91037

The authors consider the process of insurance company capital $$X_{t}^{u}$$ driven by the equation $X_{t}=u+a\int\limits_{0}^{t}X_{s}ds+ \sigma\int\limits_{0}^{t}X_{s}dw_{s}+ct- \int\limits_{0}^{t} \int x p(ds,dx),$ where $$a,\sigma$$ are arbitrary constants, $$c\geq 0$$; $$w_{t}$$ is a Wiener process independent on the integer-valued random measure $$p(dt,dx)$$ with the compensator $$\tilde p(dt,dx)=\alpha dtF(dx)$$; $$F(dx)$$ is a probability distribution. Let $$\tau^{u}=\inf\{t:\;X_{t}^{u}\leq 0\}$$, $$\Psi(u)=P(\tau^{u}<\infty)$$. The following result is proved.
Assume that $$\sigma>0$$ and let $$F(x)=1-e^{-x/\mu}, x>0$$. If $$\rho=2a/\sigma^2>1$$, then for some $$K>0$$ $$\Psi(u)=Ku^{1-\rho}(1+o(1)),\;u\to\infty$$. If $$\rho<1$$, then $$\Psi(u)=1$$ for all $$u$$.

### MSC:

 91B30 Risk theory, insurance (MSC2010)

### Keywords:

risk process; geometric Brownian motion; ruin probability
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