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Free boundary problem of Barenblatt equation in stochastic control

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  • 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.
    Mathematics Subject Classification: 35R35, 60G40, 91B70, 93E20.


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