African Diaspora Journal of Mathematics

Résolution numérique d'inégalités variationnelles:Localisation avec conditions de Dirichlet

M. Cissé

Full-text: Open access

Abstract

In this paper we present an approximation of viscosity solution of non linear partial differential equation with dirichlet bounded conditions. Our approach used a fully nonlinear PDE in an unbounded domain. To approximate its unique viscosity solution one needs to localize the PDE under consideration and to define artificial boundary conditions. It is known that backward stochastic differential equations (BSDEs) are a useful tool to estimate the error due to misspecified Dirichlet boundary conditions on the artificial boundary [12], but we perfect in this paper their approximation of localization error.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 14, Number 1 (2012), 1-23.

Dates
First available in Project Euclid: 18 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1374153551

Zentralblatt MATH identifier
1285.60055

Subjects
Primary: 60J65; 60H10; 60H05; 60J55; 60J60

Keywords
Forwards Backwards SDE variational inequalities american option viscosity solutions

Citation

Cissé, M. Résolution numérique d'inégalités variationnelles:Localisation avec conditions de Dirichlet. Afr. Diaspora J. Math. (N.S.) 14 (2012), no. 1, 1--23. https://projecteuclid.org/euclid.adjm/1374153551


Export citation

References

  • Bensoussan, A. et Lions, J.L. Applications des inéquations variationnelles en contrôle stochastique. Dunod, 1978.
  • Berthelot C., Bossy M.and Talay D., Numerical analysis and misspecifications in Finance: From model risk to localisation error estimates for nonliear PDEs. in Stochastic processes and applications to mathematical finance, 1-25, World Sci. Publ., River Edge, NJ, 2004.
  • Borodin Andrei N. and Salminen Paavo Handbook of Brownian Motion- Facts and Formulae Probability and Its applications, Birkhauser Second edition (2002)
  • Crandall, M. Ishii, H. and Lions,P-L. User's guide to the viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27, 1-67, (1992).
  • Cvitanic, J. and Ma, J. Reflected forward-Backward SDE's and obstacle problems with boundary conditions, J.Appl. Stochastic anal, 14(2), 113-138, (2001).
  • Darling, R. and Pardoux, E. Backward SDE with Random Terminal Time and Applications to Semilinear Elliptic PDE, Annales of Probability, vol 25, n 3, 1135-1159, (1997).
  • El-Karoui, N. and Hamadène, S. BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastics Processes and their Applications, 107, 145-169, (2003).
  • EL-Karoui, N. Kapoudjian, C. Pardoux, E. Peng, S. and Quenez, MC. Reflected solution of Backward SDE's, and Related Obstacle Problem for PDE's, The Annals of Probability, vol 25, issue 2, pp.702-737, (1997).
  • Elliott, R.J. and Kopp, P.E. Mathematics of Financial Markets, Springer Finance.
  • El-Karoui, N. Peng, S.and Quenez, Mc. Backward Stochastic Differential Equationin Finance, Mathematical Finance, pp 1-71, (1997).
  • Fouque, J.P. Papanicolaou, G. and Sircar, K.R. Derivatives in Financial Markets with Stochastic Volatility,Cambridge u.p, (2000).
  • Lamberton, D. et Lapeyre, B. Introduction au calcul stochastique appliqué la finance, Mathématiques et Applications, Ellipses, (1991).
  • $\emptyset$ksendal B. Stochastic Differential Equations, An introduction with Applications Springer, Sixth Edition, 1998
  • Pardoux, E.; Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Parabolic and Elliptic PDEs of Second Order Stochastic Analysis and Related Topics: The Geilo Workshop, 1996, pp: 79-127 1998
  • Pardoux, E. and Peng, S.; Adapted Solution of Backward Stochastic Differential Equations, Systems and control Letters, 14, pp 55-61, (1990).
  • Pardoux, E. and Peng, S.; Backward Stochastic Differential Equations and Quasilinear Paraboblic Partial Differential Equations, Lecture Notes in Control and Infor. Sci., 176, pp 200-217, (1992). Berlin Springer
  • Pardoux E. and Zhang S.; Generalized BSDEs and nonlinear Neumann boundary value problems, Probability Theory and Related Fields 110, pp. 535-558 (1998)
  • Peng, S.; Probabilistic interpretation for systems of quasilinear parabolic partial differential equation, Stochastics and Stochastics Reports, 37, n1-2, 61-74, (1991).