Reflected BSDE with superlinear quadratic coefficient

• Autorzy: Kobylański M. , Lepeltier J. P. , Quenez M. C. , Torres S.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we provide existence of a reflected solution of the one-dimensional backward stochastic differential equation when the coefficient is continuous, has a superlinear growth in y and quadratic growth in z. We also give a characterization of the solution as the value function of an optimal stopping time problem. We also study the links between the solution of the quadratic RBSDE and the corresponding obstacle problem. Then we give an application of quadratic RBSDE’s to the pricing of American contingent claims in an incomplete market.

Słowa kluczowe

EN

backward stochastic differential; reflexion; viscosity solution; American option

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2002
• Tom: Vol. 22, Fasc. 1
• Strony: 51--83
• Opis fizyczny: Biblogr. 13 poz.

Twórcy

autor

Kobylański M.

• Lepeltier Equipe de Mathématiques Universitk du Maine Universitk de Marne la Vallée 5, boulevard Descartes, Champssur-Marne, 77454 Marne-La-VallBe Cedex 2

autor

Lepeltier J. P.

• Université du Maine, Laboratoire Statistiques et Processus, B.P. 535, 72017 Le Mans cedex, France

autor

Quenez M. C.

• Lepeltier Equipe de Mathématiques Universitk du Maine Universitk de Marne la Vallée 5, boulevard Descartes, Champssur-Marne, 77454 Marne-La-Valée Cedex 2

autor

Torres S.

• Universidad de Chile, Facultad de Ciencias Fisicas y Matemáticas Departamento de Ingenieria Matemática Casilla, 170-3 Correo 3, Santiago, Chile

Bibliografia

• [1] N. El Karoui, C. Kapoudjian, E. Pardoux and M. C. Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, Ann. Probab. 25 (1997), pp. 702-737.
• [2] N. El Karoui and M. C. Quenez, Dynamic programming and pricing of a contingent claim in an incomplete market, SIAM J. Control Optim. 33 (1) (1995), pp. 29-66.
• [3] N. El Karoui and M. C. Quenez, Nonlinear Pricing Theory and Backward Stochastic Differential Equations, Financial Mathematics, Lecture Notes in Math. 1656, Bressanone 1996, W. J. Runggaldier (Ed.), Springer, 1997.
• [4] N. El Karoui and R. Rouge, Contingent claim pricing via utility maximization, Mathematical Finance 10 (2) (2000), pp. 259-276.
• [5] S. Hamadane, J. P. Lepeltier and A. Matoussi, Double Barrier Reflected Backward SDE’s with Continuous Coefficient, Pitman Research Notes 364 (1997).
• [6] M. Kobylański, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab. 28 (2000), pp. 558-602.
• [7] D. O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probab. Theory Related Fields 105 (1996), pp. 459-479.
• [8] J. P. Lepeltier and J. San Martin, Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett. 32 (1997), pp. 425-430.
• [9] J. P. Lepeltier and J. San Martin, Existence for BSDE with superlinear-quadratic coefficient, Stochastic and Stochastic Reports 63 (1998), pp. 227-240.
• [10] A. Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficient, Statist, and Probab. Lett 34 (1997), pp. 347-354.
• [11] P. Pardoux and S. Peng, Adapted solution of backward stochastic differential equation, Systems Control Lett. 14 (1990), pp. 55-61.
• [12] P. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci. 176 (1992), pp. 200-217.
• [13] Y. Saisho, Stochastic differential equations for multidimensional domains with reflecting boundary, Probab. Theory Related Fields 74 (1987), pp. 455-477.