Study Time And Bonus Issues For The First Time By Several Types Of Risk Models

In insurance mathematics, ruin theory is one of the mainly contents in insurance risktheory, and it can supply a very useful early-warning measure for the risk of the insur-ance company. So it has important theoretical and practical significance for the insurancecompany.In this thesis, we pay attention to the jump difusion process: for one hand, with theknowledge of stochastic process and stochastic diferential equation, we research on the firstpassage time which is the first time pass (downward or upward) a flat boundary. Thefirst passage time relate to the classical ruin problem and the expected discounted penaltyfunction or the Gerber-Shiu function, the expected total discounted dividends up to ruin aswell as the pricing options.On the other hand, dividend strategies become an important branch of risk theory.Dividends are payments made to stockholders from a firm’s surplus (or part). Due to itspractical importance, people pay more attentions to it. It is desirable to find a fixed rulewhich produces the largest possible expected sum of discounted dividend, and that is theoptimal dividend problem. There are two common strategies, the barrier dividend strategyand the threshold dividend strategy, and they were proved to be the “optimal” under theircorresponding constraints in their risk models. Therefore, we consider risk models with thepresence of these dividend strategies in Chapter2, Chapter3and Chapter5.In Chapter1, we mainly introduce the basis background, include some basic risk models,dividend strategies, and the basic knowledge of L′evy processes.In Chapter2, we investigate the first passage time to flat boundaries for hyper-exponentialjump (difusion) processes. Explicit solutions of the Laplace transforms of the distribution ofthe first passage time, the joint distribution of the first passage time and undershoot (over-shoot), the joint distribution of the process and running supreme (infima) are obtained. Theprocesses recover many models appearing in the literature such as the compound Poissonrisk models, the difusion perturbed compound Poisson risk models, and their dual models.As applications, we present explicit expressions of the dividend formulae for barrier strategyand threshold strategy.(Published on Journal of Computational and Applied Mathematics.)In Chapter3, we consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative L′evy process. We assume thatdividends are paid to the shareholders according to admissible strategies whose dividend rateis bounded by a constant. We shown that a threshold strategy forms an optimal strategyunder the condition that the L′evy meansure has a completely monotone density.(Publishedon Acta Mathematicae Applicatae Sinica, English Series.)In Chapter4, we consider the first passage time to constant boundaries for mixed-exponential jump difusion processes. Explicit solutions of the Laplace transforms of thedistribution of the first passage time, the joint distribution of the first passage time andundershoot (overshoot) are obtained. As applications, we present explicit expression ofthe Gerber-Shiu functions for surplus processes with two-sided jumps,present the analyticalsolutions for popular path-dependent options such as lookback and barrier options in termsof Laplace transforms and give a closed-form expression on the price of the zero-coupon bondunder a structural credit risk model with jumps.(Submitted.)In Chapter5, we study the optimal dividend problem for a company whose surplusprocess, in the absence of dividend payments, evolves as a generalized compound Poissonmodel in which the counting process is a generalized Poisson process. This model includingthe classical risk model and the P′olya-Aeppli risk model as special cases. The objectiveis to find a dividend policy so as to maximize the expected discounted value of dividendswhich are paid to the shareholders until the company is ruined. We show that under someconditions the optimal dividend strategy is formed by a barrier strategy.(Published on theApplied Mathematics.)...