**On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function** *W* **and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes**

### **Florin Avram 1,\*, Andras Horváth 2, Serge Provost 3 and Ulyses Solon 1**


Received: 28 October 2019; Accepted: 5 December 2019; Published: 11 December 2019

**Abstract:** This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called "shifted" Padé approximation.

**Keywords:** ruin probability; Pollaczek–Khinchine formula; scale function; optimal dividends; Padé approximations; Laguerre series; Tricomi–Weeks Laplace inversion
