Flow of Dividends under a Constant Force of Interest

Abstract

This study addresses the issue of maximization of dividends of an insurer whose portfolio is exposed to insurance risk. The insurance risk arises from the classical surplus process commonly known as the Cramér-Lundberg model in the insurance literature. To enhance his financial base, the insurer invests in a risk free asset whose price dynamics are governed by a constant force of interest. We derive a linear Volterra integral equation of the second kind and apply an order four Block-byblock method of Paulsen et al. [1] in conjunction with the Simpson rule to solve the Volterra integral equations for each chosen barrier thus generating corresponding dividend value functions. We have obtained the optimal barrier that maximizes the dividends. In the absence of the financial world, the analytical solution has been used to assess the accuracy of our results.

Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

“…To find the survival probabilities φ(u), we took advantage of the fourth-order block-by-block method in conjunction with Simpson's Rule of integration to solve the VIE (18). This method, which produces solutions in blocks of two values, is fully developed in [39] and appears in several papers, e.g., [5,38,49]. Linz [36] has shown that the block-by-block method always converges and has an order of convergence of four (see also [51]).…”

“…In the literature, two-, three-and four-block block-by-block methods have been used to solve Volterra integral equations (e.g., [36] for non-linear VIE-2s, [37] for a system of linear VIE-2s). More recently, Kasozi and Paulsen [38] used the two-block block-by-block method to study the flow of dividends under a constant interest force. They derived a linear VIE-2 and applied a fourth-order block by-block method of Paulsen et al [39] in conjunction with Simpson's rule to solve the Volterra integral equation for the optimal dividend barrier.…”

Dividend Maximization in the Cramer-Lundberg Model using Homotopy Analysis Method

“…However, not many of these address the dividend optimization problem using numerical methods. Kasozi and Paulsen (2005a) used the block-byblock numerical method to study the flow of dividends under a constant force of interest. Their study culminated into a linear Volterra integro equation of the second kind.…”

A Note on the Numerical Solution for Fredholm Integral Equation of the Second Kind with Cauchy kernel

“…The integral equations with Cauchy kernel have been widely used in solving problems associated with aerodynamic, hydrodynamic and elasticity (Lifanov, 1996;Ladopoulos, 2000;Abdou and Naser, 2003;Mohankumar and Natarajan, 2008;Lara and Mariagrazia, 2005;Kasozi and Paulsen, 2005a;Kasozi and Paulsen, 2005b;Ganji et al, 2008;Thukral, 2005).…”