Often The Interest Rate Risk Model Dividends On The Threshold Strategy

The dividend problem is proposed by De Finetti (1957). Some researchers have done some work in this problem, such as Gerber and Shiu (2006) so as.In this paper the threshold strategy is applied with parameter b > 0 andα> 0, whenever the modified surplus is below the level 6, no dividends are paid; when the modified surplus is above b, dividends are paid continuously at a constant rateα.The thesis is divided into two sections according to the contents.In chapter 1, we consider the surplus process of the company is modeled by a Brownian motion, and assume the surplus process does earn interest at the the constant forceρ> 0. If {X(t), t > 0} denote the surplus process, and the pemiums are received at a constant rateμ> 0, then the surplus process follows the following dynamics:dX(t) = (μ+ρX(t))dt +σdW(t), t≥0, (1)whereσ> 0, and {W(t), t≥0} is a standard Brownian motion.Letδ> 0 be the force of interest for valuation, thendenote the present value of all dividends until ruin, where T = min{t≥0|U(t)—D(t) = 0} is the time of ruin.Let v(x; b) denote the expectation of D, in which x is the initial surplus and b is the threshold. Then we obtain that v(x; b) satisfies the following differential equation with conditions v(0; b) = 0 andwith condition v(x; b)�'α/δ, x�'∞.Also the other factors is considered, such as the distribution of T.In chapter 2, let us consider the compound Poisson model perturbed by diffusion and with constant interest. Let {X(t), t > 0} denote the surplus process, then the surplus process is expressed:where {N(t); t≥0} is a Poisson process with parameter A.We obtained that v{x; b) satisfies the following integro-differential equationsin which p(y) be the density function of X_i, i = 1,2,..., and 5 be the discounted interest.In this paper we can consider some special example, such as the individual claim amounts are exponential distributed and mixtured exponential distributed.