Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions

Mireille Bossy, Mamadou Cissé, and Denis Talay

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 395-424.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887275

Digital Object Identifier
doi:10.1214/10-AIHP357

Mathematical Reviews number (MathSciNet)
MR2814416

Zentralblatt MATH identifier
1236.60051

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K55: Nonlinear parabolic equations

Keywords
Forward backward SDEs with refections Feynman–Kac formulae Derivatives of the flows of reflected SDEs and BSDEs

Citation

Bossy, Mireille; Cissé, Mamadou; Talay, Denis. Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 395--424. doi:10.1214/10-AIHP357. https://projecteuclid.org/euclid.aihp/1300887275


Export citation

References

  • [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 1964.
  • [2] G. Barles. Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differential Equations 106 (1993) 90–106.
  • [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.
  • [4] A. N. Borodin and P. Salminen. Handbook of Brownian Motion Facts and Formulae. Birkhaüser, Basel, 2002.
  • [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.
  • [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.
  • [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.
  • [8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Graduated Texts in Mathematics 113. Springer, New York, 1988.
  • [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.
  • [10] D. Lépingle, D. Nualart and M. Sanz. Dérivation stochastique de diffusions réfléchies. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 283–305.
  • [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.
  • [12] J. Ma and J. Zhang. Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 1390–1418.
  • [13] J. Ma and J. Zhang. Representations and regularities for solutions to BSDEs with reflections. Stochastic Process. Appl. 115 (2005) 539–569.
  • [14] J. L. Menaldi. Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32 (1983) 733–744.
  • [15] D. Nualart. The Malliavin Calculus and Related Topics. Springer, Berlin, 2006.
  • [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.
  • [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.
  • [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.
  • [19] E. Pardoux and S. Zhang. Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Related Fields 110 (1998) 535–558.
  • [20] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. A Series of Comprehensive Studies in Mathematics 293. Springer, Berlin, 1999.
  • [21] L. Slominski. Euler’s approximations of solutions of SDEs with reflecting boundary. Stochastic Process. Appl. 94 (2001) 317–337.