科研之友
微信
新浪微博
Facebook
分享链接
摘要
We consider a Barenblatt parabolic equation v(t) - (0 <= a <= 1)sup {1/2 a(2)sigma(2)v(xx) + a mu v(x) - cv + x} = 0. This equation comes from finance. In our model, the risk of the insurance company is controllable. The so-called proportional reinsurance means that it is possible for the cedent to divert 1 - a fraction of all premiums to the reinsurance company with the obligation from the latter to pay 1 - a fraction of each claim. The insurance company is willing to maximize the expected dividends (value function) by choosing the control function a(t). First, we proved v is an increasing and convex function with respect to x by the method of stochastic analysis. We can see from the equation above that if -mu/sigma(2) v(x)/v(xx) < 1, then the optimal strategy a*(x, t) = -mu/sigma(2) v(x)/v(xx) < 1, in this situation, the optimal fration tion which must be reinsured is 1 - a*. Otherwise, if -mu/sigma(2) v(x)/v(xx) >= 1 or v(xx) = 0, a* = 1, in this situation, it is optimal to take the maximal risk, using no reinsurance. Thus, we divide the domain into two parts, diverting region 1, and non-diverting region ND. In these two regions, v(x, t) satisfies different types of second-order partial differential equations, the former is fully nonlinear equation, and the latter is a linear equation. The junction of the two regions, i.e., free boundary has particular financial implications. We prove that it can be expressed as a functional form x = h(t). In this paper we not only prove there exists an unique solution v which is three times differentiable, but we also prove x = h(t) is a differentiable curve, and we show its upper and lower bounds and starting point h (0).
