Optimal Dividend Payment In The Classical Model With Capital Injections Under Power Utility

In this paper,we mainly discuss the optimal dividend payments withcapital injections under power utility in theclassical model And we onlyconsider the dividend payments with both fixed and proportional trans-actioncosts,thereby,the insurancecompany shouldn't pay dividends incontinue times Because we have taken into account the fixed transactioncosts,the problem waschanged into the impulse stochasticcontrol prob-lem And whenever the reserve becomes negative forclaim．capitals shouldbe injected so that thecompany never ruin The final goal is to make share-holders get the most profits,i e maximize the the total expected discountedutility of dividend payments minus the penalised discountedcapital injee-tions,there,we choose the utility function is u(x)=1/γ(kx-K)~γ,whereγ∈(0,1]．We derive the QVI-HJB equation FOr this problem and show theverification theorem In the end,we find the solution and optimal strategyfor exponentially distributedclaim sizes.