## On the ruin probability of insurance company functioning on the $$(B,S)$$-market.(Ukrainian, English)Zbl 1150.91421

Teor. Jmovirn. Mat. Stat. 74, 10-22 (2006); translation in Theory Probab. Math. Stat. 74, 11-23 (2007).
The authors study the ruin probability of insurance company, which can invest part of its capital to bank account, driven by equation $$dB_{t}=rB_{t}dt$$ and rest of capital can invest into asset with price, given by $$S_{t}=S_0\exp\left\{\mu t-\sigma^2t/2+\sigma\eta_{t}\right\}$$, where $$\eta_{t}$$ is Ornstein-Uhlenbeck process, given by equation $$d\eta_{t}=-\gamma\eta_{t}dt+\sigma dW_{t}$$, $$\eta_0=0$$. Let $$0\leq u\leq 1$$ be a portion of capital invested in asset, then capital $$\xi_{t}$$ of insurance company is defined by equation $$d\xi_{t}=\xi_{t}(u\mu+(1-u)r-u\gamma\eta_{t})dt+u\xi_{t}\sigma\,dW_{t}+c\,dt-\int y\nu(dy,dt)$$, where $$c>0$$ is a premium rate; $$r$$ is the interest rate; $$W_{t}$$ is the Wiener process; $$\nu(A,t)$$ is the Poisson measure independent on $$W_{t}$$. For different value of $$u$$ power estimates of ruin probability depending on initial capital of insurance company are obtained.

### MSC:

 91B30 Risk theory, insurance (MSC2010)
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