*Article* **The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps**

### **Florin Avram 1,\*, Dan Goreac 2,3 and Jean-François Renaud 4**


Received: 28 October 2019; Accepted: 2 December 2019; Published: 10 December 2019

**Abstract:** 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 (Løkka and Zervos (2008); Lindensjö 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 Løkka–Zervos alternative also holds true in the case of a Cramér–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.

**Keywords:** stochastic control; optimal dividends; capital injections; bankruptcy; barrier strategies; reflection and absorption; scale functions
