Stochastic Interest Rate Risk Model Dividends Discussion

In the thesis, we consider the risk models with stochastic return on investment. Firstly, we get the integro-differential equation satisfied by the value function in the classical risk model with diffusion and stochastic interest, under the barrier strategy. In the case of exponential claim sizes, we get the solutions of the equation. Then, We conside the monment function of the present value of all dividends until ruin. Finally, we consider the optimal dividend strategy in the diffusion models with stochastic interest, and get the equations satisfied by the T-A objective function.The thesis is divided into four chapters according to contents:In Chapter 1, we consider the development of the dividend payment in the risk processes and main results, the conclusions of the classical risk model.In Chapter 2, we present the main risk models of risk theory, and present the contents about the dividend payment. They are all to be ready for the next Chapers.In Chapter 3, we consider the classical risk model with diffusion and stochastic interest, and get the integro-differential equation satisfied by the value function, under the barrier strategy. The equation is following:1/2(σ_R²y²+σ_P²)V"(y;b)+(ry+c)V'(y；b)-(λ+δ)V(y；b) +λ∫₀^yV(y-x;b)p(x)dx=0, 0≤y≤b,with boundary conditionsV(0;b)=0,V'(b;b)=1．In the case of exponential claim sizes, we get the solutions of the equation. Also, we have the equation satisfied by the monment function of the present value of all dividends until ruin:with boundary conditions M(0,α;b)=1,(?)M(y,α;b)/(?)y|y=b=yM(b,α;b).In Chapter 4, we present the equations satisfied by the T-A objective function in the diffusion models with stochastic interest.When the rate of dividend payout is bounded, we get the HJB equation:By solving this equation, we get that the optimal strategy is threshold strategy.When the rate of dividend payout is unbounded, we get the HJB equation:max{1/2-(σ_R²Y² +σ_P²)V"(y) + (c + ry)V'(y) -βV(y) +Λ, 1 - V'(y)} = 0, V(0) = 0.By solving this equation, we get that the optimal strategy is barrier strategy.