Optimal Dividend And Capital Injection Of Dual Model With Diffusion

In recent years, dividend and capital injections optimization problems for financial and insurance corportations have attracted extensive attention. Especially, the economic globalization aggravates, the competition between countries is more and more fierce. The role of independent innovation in the competition is more obvious. And dual model is the biggest role in product research and development process of risk supervision. The dual model with diffusion is appropriate for companies with continuous and constant expenses that are offset by stochastic and irregular gains. Examples include research-based or commision-based companies, they could make profits when they make new breakthrough or new patent for invention. In real life, we could not ignore the impact on the assets control of capital injection or equity issuance. In the real financial market, equity issuance is an important approach for the company to raise capital and reduce risk. The optimal assets control problems by dividend and capital injection became hot topic of research. Under this background, in this paper we study the optimal dividend and capital injection problems for assets control. We further include proportional transactions costs both on dividends and capital injection.In this paper, we use stochastic control theory and optimal dynamic project principle to study the assets control problems for companies. We aim to minimize the risks for the company and maximize the the shareholder’s profit. Firstly, according to the surplus process,for each initial value x and each admissible strategy π to derivation the concrete expression of V(x, π). In order to maximize the expectation of the expected discounted dividend payments minus the expected discounted costs of capital injection. Under the assumption of proportional transaction costs, we identify the value function and the optimal control strategies. To find the optimal strategy π*such that V(x) = V(x, π*). Give some properties of the value function V(x) were examined, then the HJB equations were established with the boundary condition. And the optimal dividend and capital injection strategies were found through the equation.In Chapter 1, we introduce the dual model and give the research status, then we formulate the model for the optimal control problem. The last we will introduce the optimal control theories we may use in this paper.In Chapter 2, we first discuss the optimal control problem without capital injection and some related issues with barrier strategy. Then we obtain the property of the value function V(x) through discuss, and we also give the HJB function that V(x) satistiedmax{(A- δ)V(x), α- V′(x)} = 0.By solving the equations we find the closed form solution to the equation. We use the knowledge of martingale and the verification theorem to confirm the solution.In Chapter 3, we study the assets control problem with dividend and capital injection.That is to say the company managers must seek the best control mode to maximise the expect of the expected discounted dividend payments minus the expected discounted costs of capital injection. We also obtain the property of V(x), and give the HJB functionmax{(A- δ)G(x), α- G′(x), G′(x)- β} = 0.Last, By solving the equations we can obtain the optimal control strategy.