Research On The Optimal Dividend With Fines

Classical ruin theory is concerned with the level of insurers' level,considering a simplified version of a real life insurance operation.The theory assumes that the insurance company starts with some non-negative amount of money.The inflow and outflow of cash include the premium income paid by shareholders and the claim expenses incurred by the insurance company.In general,the ruin probability is used to measure the risk of the insurance company.Over the last years,the theory of optimal stochastic control has been applied more and more widely in actuarial mathematics,especially to put the risk management onto a theoretical foundation.A primary goal of solving a stochastic optimal control problem is to characterize the value function,the optimal value of the objective function,and an optimal control about it.A classical and powerful way to attain this goal is by utilizing the dynamic programming principle,which typically leads to an associated HJB equation by heuristically deriving in the sense.The traditional ruin does not seem to conform to the real business operation of the insurance company.As an extension,the insurer will consider injecting capital when ruin occurs and capital injection will generate fixed transaction costs and proportional transaction costs.Meanwhile the probability of bankruptcy will be considered,which is a function of the level of negative surplus.But it is hard to obtain explicit solutions.Vierkotter and Schmidli(2016)studied the optimal dividends with penalty payments for the first time,which allows for a negative surplus and will be effectively regulated based on current surplus levels,correspondingly paying more to avoid bankruptcy.For example,when the surplus level is negative,more penalties will be paid and the small penalties will be paid when the surplus is positive.As a risk measure we consider the difference between the expected discounted dividend and penalty payments.According to the content of this paper,we divide it into the following four chapters:The first chapter is introduction,which introduces the wide application of stochastic control in the optimal dividend problem and many research directions in the ruin time,as well as the background and current situation of dual model research.The second chapter is to introduce models and basic knowledge,focusing on the dual model with diffusion,martingale and stopping time,Brownian motion and Ito formula so on.The third chapter is the main research results of this paper.In Vierkotter and Schmidli(2016),there are two interpretations of these penalty payments:(1)Introducing a preference measure.The value function may be seen as a technical tool to investigate the profitability or risk of a portfolio.Further,with the value function the effect of possible interventions by the risk manager can be measured.In this sense,the dividends measure the profitability,the penalty is a preference measure where large capital is preferred to lower(or negative)capital.(2)Penalty interests for negative capital.The penalty payments can be seen as real costs.For example,if the surplus of an insurance line becomes negative,capital has to be borrowed from other companies.Then this capital cannot be invested anymore leading to a loss of investment return.In this section,we discuss the optimal dividend problem with penalty payments in a dual risk model with diffusion,and show some properties of the value function and derived the HJB equation:max(?)Through the dynamic programming principle and HJB equation,we verify that the optimal dividend strategy is a Barrier strategy.When gains are exponential distribution and penalty payments are exponential,we get the expression of V(x)and the optimal barrier b*And we also get the expressions of V(x)and the optimal barrier b*in linear penalty payments.The fourth chapter is to summarize and forecast this paper.