O-u Process Model With Investment Dividends

In the thesis we consider the dividends of an O-U insurance risk model with investment and the Laplace transform of the probability density function of the time of ruin.The thesis is divided into two chapters according to contents:In Chapter 1, we consider the question that the difference is immediately paid out as dividends if the initial surplus exceeds a dividend barrier. We assume that if the surplus is positive, the insurance company can invest its surplus in both a risky asset and the risk-free asset according to a fixed proportion. If the surplus is negative, a constant debit interest rate is applied. Dividends are paid to the shareholders according to a barrier strategy. It is shown how the expected discounted value of the dividends and the optimal dividend barrier can be calculated. Rummer's confluent hypergeometric differential equation plays a key role in this chapter. Let X(t) denote the surplus of the company. The surplus before investment isX(t)=x+ct+σ₁B₁{t),where x is the initial surplus, c(>0) is the premium rate, B₁(i) is a standard Brownian motion with varianceσ₁² per unit. We assume that the surplus can be invested in a risk-free bond or a risky asset. If the company invests money in the bank or borrow money from the bank, the rate of return can be describedThat is, we assume that the company's lending rate is r and the borrowing rate is r≥r>0. The price dynamic of the risky asset is given bydS{t)=μS{t)dt+σ₂dB{t). Where B(t) is a standard Brownian motion which is independent of B(t), with varianceσ₂² per unit. Assume that the insurance company invests a fixed proportionαof its surplus in the risky asset when its surplus is positive. When the surplus is negative, the company has to borrow money from the bank using the debt rate r. When no dividends are paid, the surplus process is governed by the following dynamics:Where {W(t),t≥0}is a standard Wiener process. We assume that dividends are paid to the shareholders according to a barrier strategy with parameter b>0. Whenever the surplus is below the lever b, no dividends are paid; When the surplus is about to go above the level b, the excess will be paid as dividends. Let D(t) denote the aggregate dividends by time t and let T = inf{t:X(t)=0} be the time of ruin. Let V(x;b) denote the expected of D(t), in which x is the initial surplus. Then we obtain that V(x;b) satisfies the following differential equation with two conditions. In the chapter, dissolving the equation is the major work.In Chapter 2, we also consider the dividend question of the process as in chapter 1. But, in this chapter, we assume that dividends are paid to the shareholders according to the threshold strategy with parameter b(>0) andβ₁(>0). Whenever the modified surplus is below the lever b, no dividends are paid; when the modified surplus is above 6, dividends are paid continuously at a constant rateβ₁.we also obtain a second-order differential equation of V(x;b) with two boundary conditions. We also consider the distribution of T.