In the insurance business risky investments are dangerous
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 \(u\).
scientific article; zbMATH DE number 1837545