**Jean-François Renaud**

Département de mathématiques, Université du Québec à Montréal (UQAM), Montréal, QC H2X 3Y7, Canada; renaud.jf@uqam.ca

Received: 12 June 2019; Accepted: 26 June 2019; Published: 3 July 2019

**Abstract:** We consider de Finetti's stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strategy is formed by a barrier strategy and that this optimal barrier level is always less than the optimal barrier level when classical ruin is implemented. In addition, we give necessary and sufficient conditions for the barrier strategy at level zero to be optimal.

**Keywords:** stochastic control; spectrally negative Lévy processes; optimal dividends; Parisian ruin; log-convexity; barrier strategies
