The Lokka-Zervos Alternative for a Cramer-Lundberg Process with Exponential Jumps

In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Lokka and Zervos (2008); Lindensjo and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Lokka-Zervos alternative also holds true in the case of a Cramer-Lundberg risk process with exponential claims. More specifically, we show that: if the cost of capital injections is low, then according to a double-barrier strategy, it is optimal to pay dividends and inject capital, meaning ruin never occurs; and if the cost of capital injections is high, then according to a single-barrier strategy, it is optimal to pay dividends and never inject capital, meaning ruin occurs at the first passage below zero.