Explicit expressions for some distributions related to ruin problems (Q1871513)

The authors are concerned with the classical risk process \(R_t=u+ct+\sigma W_t-\sum^{N_t}_{k=1} Z_k\), \(t\geq 0\) which is perturbed by diffusion. Here \(u,c,\sigma > 0\) are constants, \((W_t)_{t\geq 0}\) is a standard Brownian motion, \((N_t)_{t\geq 0}\) is a Poisson process, and \((Z_k)_{k\geq 1}\) is a sequence of non-negative i.i.d. random variables. The last three probabilistic items are assumed to be independent. The authors derive in closed form the ruin probability \(P(\inf_{t\geq 0} R_t < 0)\) as well as the surplus distribution \(P(T < \infty,R_T\geq -y)\), \(y > 0\), at the ruin time \(T\), when the common distribution function of the \(Z_k\), \(k > 1\), is a finite mixture of exponential distributions or a Gamma \((2, \alpha)\) distribution.