Optimal Dividend Strategy Based On The Two-dimensional Dual Model

With the development of economic globalization,the optimal dividend and capital in-jection in the dual model has caused extensive studies in recent years.But most of stud-ies were based on one-dimensional dual model,so the optimal dividend problem of two-dimensional dual model needs to further studies.In the actual investment market,capital injection plays an important role in the operation of assets.In this paper,the optimal control problem of the two-dimensional dual model under the different dividend policies has been studied.Firstly,under the dividend threshold,the optimal dividend problem of the two dimen-sional dual model without capital injection was studied.And we derived two integral dif-ferential equations which satisfies the expected total discounted until ruin.By studying the properties of the value function and the HJB equation satisfied by the value function,we find the optimal strategy,and an the explicit expression of the solution when the profit obeys the exponential distribution.Secondly,under the dividend barrier,the optimal control problem with the capital time has been discussed.The expected total discounted until ruin was studied and derived its integral differential equation.At the same time,the properties of the value function and the HJB equation are given,and the solution of the value function was founded by solving the HJB equation.Finally,according to the correlation between the stochastic control theory,the problem of the optimal dividend and capital injection of a two-dimensional dual model with Brown's motion interference was further studied under the barrier dividend strategy,and the Laplace transform was used to solve the value function V(x),then the analytic expression of the time value function when the profit obeys the exponential distribution was deduced.