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

Baev, A. V.; Bondarev, B. V.

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.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

Poisson measure; stochastic integral; investor’s portfolio; ruin probability