Application Of Self-exciting Process In Finance And Insurance

The self-exciting process was originally introduced by Hawkes,and it features that each arrival increases the arrival intensity of future arrivals for some period of time.Because of such "self-excite" property,these processes have been widely applied in seismology,neurophysiology,finance and so on.Considering that the ar,rivals of events are also affected by random disturbances,this paper combines the CIR model with the self-exciting process to build a generalized self-exciting model perturbed by diffusion with affine structures.The model consists of two parts,one is a stochastic process with affine structure,self-exciting jump and diffusion term,which can be used to describe stochastic interest rates and insurance company’s asset surplus process.The other is a self-exciting process perturbed by a diffusion with affine structure,which can be used to described the intensity of the jump of the stochastic process in the former part and reflect the self-exciting property of arrivals of events.Firstly,the distribution properties of the model are presented,including the joint Laplace transform,conditional first moment,conditional second moment,con-ditional variance,etc.And the simulation algorithm of the model are given as well,which paves the way for subsequent numerical simulation.Secondly,the model is used in the research of the insurance risk theory.By using the distribution proper,ties,the net profit condition and the pricing formulas of premium principle in the risk model are given.The expression of infinite time ruin probability under the original measure is obtained by Lundberg’s equation and martingale method.Moreover,the upper bound of the infinite time ruin probability is obtained when both jump size and claim size follow exponential distribution.In order to obtain more efficient and accurate simulation results of ruin probability,the process characterizations under the changing measure is presented,and then another expression of ruin probability is derived.Finally,the model is used in the research of stochastic interest rates and mortality model.The pricing of zero-coupon bonds and longevity bonds are solved by the martingale method.Moreover,several numerical examples are given.