    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
##### References:
  A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964). Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar  S. Kamin, L. A. Peletier and J. L. Vazquez, On the Barenblatt equation of elasto-plastic filtration,, Indiana Math. J., 40 (1991), 1333.  doi: 10.1512/iumj.1991.40.40060.  Google Scholar  D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation,, Appl Math Optim., 47 (2003), 253.  doi: 10.1007/s00245-003-0764-8.  Google Scholar  A. Kolesnichenko and G. Shopina, Valuation of Portfolios Under Uncertain Volatility: Black-Scholes Barenblatt Equations and the Static Hedging,, {Technical report, (2007).   Google Scholar  N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, Izv. Akad. Nayk SSSR, 47 (1983), 75. Google Scholar  O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1967).   Google Scholar  G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar  H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications,, Springer-Verlag, (2009).  doi: 10.1007/978-3-540-89500-8.  Google Scholar  J. Rochet and S. Villeneuve, Liquidity management and coporate demand for hedging and insurance,, J. Finan. Intermediation, 20 (2011), 303.   Google Scholar  S. E. Shreve, Stochastic Calculus for Finance II,, Springer, (2004). Google Scholar  T. Vargiolu, Existence, Uniqueness and Smoothness for the Black-Scholes-Barenblatt Equation,, Technical Report of the Department of Pure and Appl. Math. of the University of Padava, (2001).   Google Scholar

show all references

##### References:
  A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964). Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar  S. Kamin, L. A. Peletier and J. L. Vazquez, On the Barenblatt equation of elasto-plastic filtration,, Indiana Math. J., 40 (1991), 1333.  doi: 10.1512/iumj.1991.40.40060.  Google Scholar  D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation,, Appl Math Optim., 47 (2003), 253.  doi: 10.1007/s00245-003-0764-8.  Google Scholar  A. Kolesnichenko and G. Shopina, Valuation of Portfolios Under Uncertain Volatility: Black-Scholes Barenblatt Equations and the Static Hedging,, {Technical report, (2007).   Google Scholar  N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, Izv. Akad. Nayk SSSR, 47 (1983), 75. Google Scholar  O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1967).   Google Scholar  G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar  H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications,, Springer-Verlag, (2009).  doi: 10.1007/978-3-540-89500-8.  Google Scholar  J. Rochet and S. Villeneuve, Liquidity management and coporate demand for hedging and insurance,, J. Finan. Intermediation, 20 (2011), 303.   Google Scholar  S. E. Shreve, Stochastic Calculus for Finance II,, Springer, (2004). Google Scholar  T. Vargiolu, Existence, Uniqueness and Smoothness for the Black-Scholes-Barenblatt Equation,, Technical Report of the Department of Pure and Appl. Math. of the University of Padava, (2001).   Google Scholar
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