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 and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
References:
 [1] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [3] S. Kamin, L. A. Peletier and J. L. Vazquez, On the Barenblatt equation of elasto-plastic filtration, Indiana Math. J., 40 (1991), 1333-1362. doi: 10.1512/iumj.1991.40.40060. [4] D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation, Appl Math Optim., 47 (2003), 253-278. doi: 10.1007/s00245-003-0764-8. [5] A. Kolesnichenko and G. Shopina, Valuation of Portfolios Under Uncertain Volatility: Black-Scholes Barenblatt Equations and the Static Hedging, {Technical report, IDE0739}, November 14, 2007. [6] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nayk SSSR, 47 (1983), 75-108. [7] 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, Vol. 23. American Mathematical Society, Providence, R.I., 1967. [8] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302. [9] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [10] J. Rochet and S. Villeneuve, Liquidity management and coporate demand for hedging and insurance, J. Finan. Intermediation, 20 (2011), 303-323. [11] S. E. Shreve, Stochastic Calculus for Finance II, Springer, 2004. [12] 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.

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References:
 [1] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [3] S. Kamin, L. A. Peletier and J. L. Vazquez, On the Barenblatt equation of elasto-plastic filtration, Indiana Math. J., 40 (1991), 1333-1362. doi: 10.1512/iumj.1991.40.40060. [4] D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation, Appl Math Optim., 47 (2003), 253-278. doi: 10.1007/s00245-003-0764-8. [5] A. Kolesnichenko and G. Shopina, Valuation of Portfolios Under Uncertain Volatility: Black-Scholes Barenblatt Equations and the Static Hedging, {Technical report, IDE0739}, November 14, 2007. [6] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nayk SSSR, 47 (1983), 75-108. [7] 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, Vol. 23. American Mathematical Society, Providence, R.I., 1967. [8] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302. [9] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [10] J. Rochet and S. Villeneuve, Liquidity management and coporate demand for hedging and insurance, J. Finan. Intermediation, 20 (2011), 303-323. [11] S. E. Shreve, Stochastic Calculus for Finance II, Springer, 2004. [12] 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.
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