A Novel Approximation Method for Computing the Adjustment Coefficient of a Nonlinear Cramér-Lundberg Risk Model with Gamma Claims

Abstract

This study considers a non-linear Cram & eacute;r-Lundberg risk model and examines the adjustment coefficient (r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{(r)}$$\end{document} when the claims have gamma distribution. The linear models are not always adequate because an insurance company's premium income does not always increase linearly. Therefore, in this study, a more realistic non-linear Cram & eacute;r-Lundberg risk model is mathematically constructed. Then, the ruin probability of this non-linear risk model is studied when the premium function is in the form of square root function, i.e., p(t)=ct\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p}\varvec{(t)}\varvec{=}\varvec{c}\varvec{\sqrt{t}}$$\end{document}. It leads to analyzing the adjustment coefficient (r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{(r)}$$\end{document}, as examining this coefficient is required for finding an upper bound while investigating the ruin probability. However, in general case, it is a challenging procedure to calculate the exact value of r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{r}$$\end{document} from an integral equation. Thus, in this study, the adjustment coefficient r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{r}$$\end{document} is explored by computational methods and a new approximate formula for the practical calculation of the adjustment coefficient is proposed. Moreover, an implementation of the obtained approximate formula, which investigates ruin probability, is included as an example at the end of the paper.