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Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations

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  • Bouchard, Bruno
  • Touzi, Nizar

Abstract

We suggest a discrete-time approximation for decoupled forward-backward stochastic differential equations. The Lp norm of the error is shown to be of the order of the time step. Given a simulation-based estimator of the conditional expectation operator, we then suggest a backward simulation scheme, and we study the induced Lp error. This estimate is more investigated in the context of the Malliavin approach for the approximation of conditional expectations. Extensions to the reflected case are also considered.

Suggested Citation

  • Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    • Handle: RePEc:eee:spapps:v:111:y:2004:i:2:p:175-206
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    References listed on IDEAS

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    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    2. Arturo Kohatsu & Roger Pettersson, 2002. "Variance reduction methods for simulation of densities on Wiener space," Economics Working Papers 597, Department of Economics and Business, Universitat Pompeu Fabra.
    3. Bruno Bouchard & Ivar Ekeland & Nizar Touzi, 2004. "On the Malliavin approach to Monte Carlo approximation of conditional expectations," Finance and Stochastics, Springer, vol. 8(1), pages 45-71, January.
    4. Coquet, François & Mackevicius, Vigirdas & Mémin, Jean, 1998. "Stability in of martingales and backward equations under discretization of filtration," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 235-248, July.
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    6. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    7. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    8. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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