Bernoulli

  • Bernoulli
  • Volume 22, Number 1 (2016), 530-562.

Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

Emmanuel Gobet and Plamen Turkedjiev

Full-text: Open access

Abstract

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390–1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.

Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 530-562.

Dates
Received: August 2013
Revised: March 2014
First available in Project Euclid: 30 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1443620859

Digital Object Identifier
doi:10.3150/14-BEJ667

Mathematical Reviews number (MathSciNet)
MR3449792

Zentralblatt MATH identifier
1339.60094

Keywords
backward stochastic differential equations dynamic programming equation empirical regressions Malliavin calculus non-asymptotic error estimates

Citation

Gobet, Emmanuel; Turkedjiev, Plamen. Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression. Bernoulli 22 (2016), no. 1, 530--562. doi:10.3150/14-BEJ667. https://projecteuclid.org/euclid.bj/1443620859


Export citation

References

  • [1] Bally, V., Caramellino, L. and Zanette, A. (2005). Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach. Monte Carlo Methods Appl. 11 97–133.
  • [2] Bender, C. and Steiner, J. (2012). Least-squares Monte-Carlo for BSDEs. In Numerical Methods in Finance (R. Carmona, P. Del Moral, P. Hu and N. Oudjane, eds.). Springer Proceedings in Mathematics 12 257–289. Berlin–Heidelberg: Springer.
  • [3] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [4] Briand, P. and Labart, C. (2014). Simulation of BSDEs by Wiener chaos expansion. Ann. Appl. Probab. 24 1129–1171.
  • [5] Chassagneux, J.-F. and Richou, A. (2013). Numerical simulation of quadratic BSDEs. Preprint. Available at arXiv:1307.5741.
  • [6] Crisan, D. and Delarue, F. (2012). Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations. J. Funct. Anal. 263 3024–3101.
  • [7] Delarue, F. and Guatteri, G. (2006). Weak existence and uniqueness for forward–backward SDEs. Stochastic Process. Appl. 116 1712–1742.
  • [8] Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L. and Touzi, N. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3 391–412.
  • [9] Geiss, C., Geiss, S. and Gobet, E. (2012). Generalized fractional smoothness and $L_{p}$-variation of BSDEs with non-Lipschitz terminal condition. Stochastic Process. Appl. 122 2078–2116.
  • [10] Gobet, E. and Labart, C. (2007). Error expansion for the discretization of backward stochastic differential equations. Stochastic Process. Appl. 117 803–829.
  • [11] Gobet, E. and Makhlouf, A. (2010). $\mathbf{L}_{2}$-time regularity of BSDEs with irregular terminal functions. Stochastic Process. Appl. 120 1105–1132.
  • [12] Gobet, E. and Munos, R. (2005). Sensitivity analysis using Itô–Malliavin calculus and martingales, and application to stochastic optimal control. SIAM J. Control Optim. 43 1676–1713 (electronic).
  • [13] Gobet, E. and Turkedjiev, P. (2014). Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comp. To appear. Available at http://hal.archives-ouvertes.fr/hal-00855760.
  • [14] Golub, G.H. and Van Loan, C.F. (1996). Matrix Computations, 3rd ed. Johns Hopkins Studies in the Mathematical Sciences. Baltimore, MD: Johns Hopkins Univ. Press.
  • [15] Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. New York: Springer.
  • [16] Hu, Y., Nualart, D. and Song, X. (2011). Malliavin calculus for backward stochastic differential equations and application to numerical solutions. Ann. Appl. Probab. 21 2379–2423.
  • [17] Kohatsu-Higa, A. and Pettersson, R. (2002). Variance reduction methods for simulation of densities on Wiener space. SIAM J. Numer. Anal. 40 431–450 (electronic).
  • [18] Ma, J. and Zhang, J. (2002). Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 1390–1418.
  • [19] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
  • [20] Richou, A. (2011). Numerical simulation of BSDEs with drivers of quadratic growth. Ann. Appl. Probab. 21 1933–1964.
  • [21] Turkedjiev, P. (2013). Numerical methods for backward stochastic differential equations of quadratic and locally lipschitz type. Ph.D. thesis, Mathematisch-Naturwissenschaftlichen Fakultät II der Humboldt-Universität zu Berlin.
  • [22] Turkedjiev, P. (2014). Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions. Electron. J. Probab. In revision. Available at http://hal.archives-ouvertes.fr/hal-00862848.