Optimal Portfolio And Consumption Models With Dividen Payment Under Fractional Brownian Motion Environment

Mathematical finance is the one of frontier science which use mathematical tools to research financial and research how to make optimal intertenporal allocation to resources under the uncertain environment. With the rapid development in Financial Mathematics, people pay more and more attention to core questions in Financial Mathematics. When the investor put cash in the financial market, the optimal investment and consumption is to increase the value of the total assets by selecting the portfolio and the consumption of the wealth. The study on the problem of optimal portfolio and consumption can help investors to dominate wealth between investment and consumption reasonably.Many scholars researched on the portfolio and consumption problem:some people have studied under the environment of the standard Brownian Motion, some people have studied under the environment of the Fractional Brownian Motion; at the same time they discussed the situation under a variety of different conditions. Associated with the actual environment, we found that the Fractional Brown motion is the same with people’s intuitive sense. Because that the price of asset in the future not only relate to its existing price,but also relate to the price in the quite a long time in the past.In this paper we assume that the risk assets price is subject to the Fractional Brownian Motion, and we suppose that the utility function is power utility function u{s)=(Sλ)/λ,s>0. We solve the the optimal portfolio with dividend payment and obtain the display expression of price function, the optimal final wealth and the optimal asset portfolio (a*,b*) is a*(t)=C-1[Sη*(t)-b*t)Z(t)], b*(t)=sepTδ-1Z-1(t)[-(φp(t)]exp{‖φ(T)‖2+(ω-q-p)/δ∫0Tφ(x)dx}exp°{-∫0Tφ(x)dBh(x)} Then, we solve the optimal consumption asset portfolio with dividend payment and obtain the display expression of the optimal consumption asset portfolio and the optimal asset portfolio (a*,b*) is a*(t)=e-Pt[S*(t)-b*(t)Z(t)], b*(t)=δ-1eptZ-1(t)Et[DtG]=δ-1eptZ-1(t)(X1+X2).