A free boundary problem arising from a stochastic optimal control model under controllable risk

• The proof of the concavity of the value function by the methods of PDE and stochastic analysis are given separately.
• We prove that the equation is degenerate in the left boundary but non-degenerate interior.
• We give high regularity of the solution (value function), we prove it is 3 times differentiable.
• We prove that the free boundary h(t)h(t) is a differentiable curve and show its upper and lower bounds and starting point h(0)=0h(0)=0.

We consider a Barenblatt parabolic equationvt−sup0≤a≤1⁡{12a2σ2vxx+aμvx−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−a1−a fraction of all premiums to the reinsurance company with the obligation from the latter to pay 1−a1−a fraction of each claim. The insurance company is willing to maximize the expected dividends (value function) by choosing the control function a(t)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 −μσ2vxvxx<1, then the optimal strategy a⁎(x,t)=−μσ2vxvxx<1, in this situation, the optimal fraction which must be reinsured is 1−a⁎1−a⁎. Otherwise, if −μσ2vxvxx≥1 or vxx=0vxx=0, a⁎=1a⁎=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 DD and non-diverting region NDND. In these two regions, v(x,t)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)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)x=h(t) is a differentiable curve, and we show its upper and lower bounds and starting point h(0)h(0).