# American Institute of Mathematical Sciences

July  2016, 21(5): 1421-1434. doi: 10.3934/dcdsb.2016003

## Free boundary problem of Barenblatt equation in stochastic control

 1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China 2 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China

Received  January 2014 Revised  March 2014 Published  April 2016

The following type of parabolic Barenblatt equations
min {$\partial_t V - \mathcal{L}_1 V, \partial_t V-\mathcal{L}_2 V$} = 0
is studied, where $\mathcal{L}_1$ and $\mathcal{L}_2$ are different elliptic operators of second order. The (unknown) free boundary of the problem is a divisional curve, which is the optimal insured boundary in our stochastic control problem. It will be proved that the free boundary is a differentiable curve.
To the best of our knowledge, this is the first result on free boundary for Barenblatt Equation. We will establish the model and verification theorem by the use of stochastic analysis. The existence of classical solution to the HJB equation and the differentiability of free boundary are obtained by PDE techniques.
Citation: Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
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