Optimal Dividend Problem With The Dividend Transaction Costs And Transaction Tax

This paper deals with optimal dividend payments in the classical risk model perturbed by a Brownian motion. Since De Finetti(1957) first proposed dividend strategies for insurance risk model, dividend problem has become a important topic and optimal dividend problem has become a most popular topic in the actuarial literature. In fact, the main scope of this paper is to study the relation-ship between the solution of quasi-variational inequality and the optimal return function, then we obtain the solutions of the optimal return function and opti-mal dividend strategy when claims are exponentially distributed by solving the corresponding quasi-variational inequality.This dissertation is divided into four sections.In the first section, we state the background and the hypotheses of the opti-mal dividend problem.In the second section, we give a rigorous mathematical model of the problem and several kinds of important definition.In the third section, we obtain the corresponding quasi-variational inequality and prove the relationship between the optimal return function V(x) and the solution of quasi-variational inequality v(x). If v(x) is continuous on [0,∞) and is continuously differentiate on (0,∞), then V(x)≤v(x).In the forth section, we obtain a closed-form solution of quasi-variational inequality when the claims sizes are exponentially distributed and take the opti-mal return function and optimal dividend strategy. It is shown that there can be essentially two different solutions:(1) Whenever the reserve reaches a barrier u there are reduced to u through a dividend payment u-u, and the reserve process continues. Optimal dividend strategy and the optimal return funtion are as follow:If the initial reserve 0≤x≤u, the first time and amount of dividends defined byτ1*=inf{t≥0:Xt*=u} andξ1*=u-u, the n-th time and amount of dividends defined byτn*=inf{t≥τn-1:Xt*=u} andξn*=u-u, n≥2, where Optimal dividend strategy isπ*={τ1*,τ2*,…,τn*,…;ξ1*,ξ2*,…,ξn*,…},the optimal return funtion is V(x)=γ[(λ1+μ)(λ3-λ2)eλ1x+(λ2+μ)(λ1-λ3)eλ2x+ (λ3+μ)(λ2-λ1)eλ3x].If the initial reserve x≥u,the first time and amount of dividends defined byτ1*=0 andξ1*=x-u,the n-th time and amount of dividends defined by the same above,where Thenπ* is optimal,the optimal return funtion is V(x)=v(u)+k(x-u)-K.(2)Whenever the reserve reaches a barrier u,everything is paid out as divi-dends.If the initial reserve 0≤x