The Annals of Applied Probability

A regression-based Monte Carlo method to solve backward stochastic differential equations

Emmanuel Gobet, Jean-Philippe Lemor, and Xavier Warin

Full-text: Open access

Abstract

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 3 (2005), 2172-2202.

Dates
First available in Project Euclid: 15 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1121433781

Digital Object Identifier
doi:10.1214/105051605000000412

Mathematical Reviews number (MathSciNet)
MR2152657

Zentralblatt MATH identifier
1083.60047

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Keywords
Backward stochastic differential equations regression on function bases Monte Carlo methods

Citation

Gobet, Emmanuel; Lemor, Jean-Philippe; Warin, Xavier. A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15 (2005), no. 3, 2172--2202. doi:10.1214/105051605000000412. https://projecteuclid.org/euclid.aoap/1121433781


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References

  • Andersen, L. and Brotherton-Ratcliffe, R. (1996). Exact exotics. Risk 9 85–89.
  • Bally, V. (1997). Approximation scheme for solutions of BSDE. In Backward Stochastic Differential Equations (Paris, 1995–1996) (N. El Karoui and L. Mazliak, eds.). Pitman Res. Notes Math. Ser. 364 177–191. Longman, Harlow.
  • Bally, V. and Pagès, G. (2003). Error analysis of the optimal quantization algorithm for obstacle problems. Stochastic Process. Appl. 106 1–40.
  • Bergman, Y. Z. (1995). Option pricing with differential interest rates. Rev. Financial Studies 8 475–500.
  • Bouchaud, J. P., Potters, M. and Sestovic, D. (2001). Hedged Monte Carlo: Low variance derivative pricing with objective probabilities. Physica A 289 517–525.
  • Bouchard, B. and Touzi, N. (2004). Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • Briand, P., Delyon, B. and Mémin, J. (2001). Donsker-type theorem for BSDEs. Electron. Comm. Probab. 6 1–14.
  • Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance (L. C. G. Rogers and D. Talay, eds.) 232–244. Cambridge Univ. Press.
  • Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance Stoch. 6 449–471.
  • Cvitanic, J. and Ma, J. (1996). Hedging options for a large investor and forward–backward SDE's. Ann. Appl. Probab. 6 370–398.
  • Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353–394.
  • El Karoui, N., Peng, S. G. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • El Karoui, N. and Quenez, M. C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29–66.
  • Föllmer, H. and Schweizer, M. (1990). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (M. H. A. Davis and R. J. Elliot, eds.) 389–414. Gordon and Breach, London.
  • Gobet, E., Lemor, J. P. and Warin, X. (2004). A regression-based Monte Carlo method to solve backward stochastic differential equations. Technical Report 533-CMAP, Ecole Polytechnique, France.
  • Golub, G. and Van Loan, C. F. (1996). Matrix Computations, 3rd ed. Johns Hopkins Univ. Press.
  • Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
  • Lapeyre, B. and Temam, E. (2001). Competitive Monte Carlo methods for the pricing of Asian options. Journal of Computational Finance 5 39–59.
  • Lemor, J. P. (2005). Ph.D. thesis, Ecole Polytechnique.
  • Longstaff, F. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple least squares approach. The Review of Financial Studies 14 113–147.
  • Ma, J., Protter, P., San Martín, J. and Soledad, S. (2002). Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 302–316.
  • Ma, J., Protter, P. and Yong, J. M. (1994). Solving forward–backward stochastic differential equations explicitly–-A four step scheme. Probab. Theory Related Fields 98 339–359.
  • Ma, J. and Zhang, J. (2002). Path regularity for solutions of backward stochastic differential equations. Probab. Theory Related Fields 122 163–190.
  • Newton, N. J. (1994). Variance reduction for simulated diffusions. SIAM J. Appl. Math. 54 1780–1805.
  • Pardoux, E. (1998). Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analysis and Related Topics (L. Decreusefond, J. Gjerde, B. Oksendal and A. S. Ustüunel, eds.) 79–127. Birkhäuser, Boston.
  • Pardoux, E. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • Peng, S. (2003). Dynamically consistent evaluations and expectations. Technical report, Institute Mathematics, Shandong Univ.
  • Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures. Stochastic Methods in Finance. Lecture Notes in Math. 1856 165–253. Springer, New York.
  • Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.