Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 395-424.

Dans cet article, nous explicitons la dérivée du flot d'un processus de diffusion réfléchi. Nous obtenons des représentations stochastiques des dérivées des solutions de viscosité d'équations aux dérivées partielles paraboliques semi-linéaires. Nous en déduisons des représentations stochastiques des dérivées des solutions de viscosité d'inégalités variationnelles paraboliques avec conditions au bord de Neumann.

In this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions.

DOI : 10.1214/10-AIHP357
Classification : 60H10, 60H30, 35K55
Mots-clés : forward backward SDEs with refections, Feynman-Kac formulae, derivatives of the flows of reflected SDEs and BSDEs 
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     title = {Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with {Neumann} boundary conditions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {395--424},
     publisher = {Gauthier-Villars},
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Bossy, Mireille; Cissé, Mamadou; Talay, Denis. Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 395-424. doi : 10.1214/10-AIHP357. http://www.numdam.org/articles/10.1214/10-AIHP357/

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 1964. | MR | Zbl

[2] G. Barles. Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differential Equations 106 (1993) 90-106. | MR | Zbl

[3] C. Berthelot, M. Bossy and D. Talay. Numerical analysis and misspecifications in Finance: From model risk to localisation error estimates for nonlinear PDEs. In Stochastic Processes and Applications to Mathematical Finance 1-25. World Sci. Publ., River Edge, NJ, 2004. | MR | Zbl

[4] A. N. Borodin and P. Salminen. Handbook of Brownian Motion Facts and Formulae. Birkhaüser, Basel, 2002. | MR | Zbl

[5] N. Bouleau and F. Hirsch. Propriétés d'absolue continuité dans les espaces de Dirichlet et application aux équations différentielles stochastiques. In Séminaire de Probabilités XX, 1984-1985 131-161. Lectures Notes in Math. 1204. Springer, Berlin, 1986. | Numdam | MR | Zbl

[6] N. Bouleau and F. Hirsch. On the derivability, with respect to the initial data, of solution of a stochastic differential equation with Lipschitz coefficients. In Séminaire de Théorie du Potentiel, Paris, No. 9 39-57. Lecture Notes in Math. 1393. Springer, Berlin, 1989. | Zbl

[7] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.-C. Quenez. Reflected solution of backward SDE's, and related obstacle problem for PDE's. Ann. Probab. 25 (1997) 702-737. | MR | Zbl

[8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Graduated Texts in Mathematics 113. Springer, New York, 1988. | MR | Zbl

[9] L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve. An explicit formula for the Skorokhod map on [0, a]. Ann. Probab. 35 (2007) 1740-1768. | MR | Zbl

[10] D. Lépingle, D. Nualart and M. Sanz. Dérivation stochastique de diffusions réfléchies. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 283-305. | Numdam | MR | Zbl

[11] J. Ma and J. Cvitanić. Reflected forward-backward SDE's and obstacle problems with boundary conditions. J. Appl. Math. Stochastic Anal. 14 (2001) 113-138. | MR | Zbl

[12] J. Ma and J. Zhang. Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 1390-1418. | MR | Zbl

[13] J. Ma and J. Zhang. Representations and regularities for solutions to BSDEs with reflections. Stochastic Process. Appl. 115 (2005) 539-569. | MR | Zbl

[14] J. L. Menaldi. Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983) 733-744. | MR | Zbl

[15] D. Nualart. The Malliavin Calculus and Related Topics. Springer, Berlin, 2006. | MR | Zbl

[16] M. N'Zi, Y. Ouknine and A. Sulem. Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations. Stochastic Process. Appl. 116 (2006) 1319-1339. | MR | Zbl

[17] E. Pardoux. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analysis and Related Topics, VI (Geilo, 1996) 79-127. Progr. Probab. 42. Birkhäuser, Boston, 1998. | MR | Zbl

[18] E. Pardoux and S. G. Peng. Adapted solution of a backward stochastic differential equation. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55-61. | MR | Zbl

[19] E. Pardoux and S. Zhang. Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Related Fields 110 (1998) 535-558. | MR | Zbl

[20] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. A Series of Comprehensive Studies in Mathematics 293. Springer, Berlin, 1999. | MR | Zbl

[21] L. Slominski. Euler's approximations of solutions of SDEs with reflecting boundary. Stochastic Process. Appl. 94 (2001) 317-337. | MR | Zbl

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