Limit theorems for generalized risk processes (Q2777832)

Consider the classical risk process \(R_0(t)=ct-\sum_{j=1}^{N_1(t)}X_j, t\geq 0,\) where \(c>0\) is the premium rate, \(\{X_j\}_{j\geq 1}\) are insurance claims (payments) which are assumed to be independent identically distributed random variables (r.v.'s) with mean value \(a\), \(N_1(t)\) is the number of claims up to time \(t\) which is assumed to be the standard Poisson process (homogeneous Poisson process with unit intensity) independent of \(\{X_j\}_{j\geq 1}\). Let \(\Lambda(t),t>0,\) be a random process independent of \(N_1(t)\) with infinitely increasing almost surely finite continuous trajectories starting from the origin. A Cox process controlled by the process \(\Lambda(t)\) is defined as \(N(t)=N_1(\Lambda(t))\). The authors consider a natural generalization of the classical risk process \(R(t)=c\Lambda(t)-\sum_{j=1}^{N(t)}X_j, t\geq 0,\) which is a more flexible mathematical model for the surplus of an insurance company since it takes into account both risk and portfolio fluctuations. In the present paper the authors give a new general criterion for the weak convergence of one-dimensional distributions of the generalized risk processes \(R(t)\) and describe the class of possible limit laws as \(\Lambda(t)\) infinitely grows, which makes it possible to construct asymptotic approximations to distributions of these generalized risk processes.