Annals of Applied Probability

Monte Carlo algorithms for optimal stopping and statistical learning

Daniel Egloff

Full-text: Open access


We extend the Longstaff–Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results obtained by Clément, Lamberton and Protter [Finance and Stochastics 6 (2002) 449–471].

Article information

Ann. Appl. Probab., Volume 15, Number 2 (2005), 1396-1432.

First available in Project Euclid: 3 May 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91B28 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control
Secondary: 65C05: Monte Carlo methods 93E24: Least squares and related methods 62G05: Estimation

Optimal stopping American options statistical learning empirical processes uniform law of large numbers concentration inequalities Vapnik–Chervonenkis classes Monte Carlo methods


Egloff, Daniel. Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15 (2005), no. 2, 1396--1432. doi:10.1214/105051605000000043.

Export citation


  • Anthony, M. and Bartlett, P. L. (1999). Neural Network Learning. Cambridge Univ. Press.
  • Barone-Adesi, G. and Whaley, R. (1987). Efficient analytic approximation of American option values. J. Finance 42 301–320.
  • Barron, A., Brigé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301–413.
  • Bartlett, P., Bousquet, O. and Mendelson, S. (2005). Local Rademacher complexities. Ann. Statist. To appear.\goodbreak
  • Bensoussan, A. (1984). On the theory of option pricing. Acta Appl. Math. 2 139–158.
  • Bensoussan, A. and Lion, J. (1982). Application of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam.
  • Benveniste, A., Metiver, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Springer, Berlin.
  • Birman, M. S. and Solomyak, M. Z. (1967). Piecewise polynomial approximation of functions of the classes $w_p^{\alpha}$. Math. USSR Sb. 73 331–355.
  • Boessarts, P. (1989). Simulation estimators of optimal early exercise. Working paper, Carnegie-Mellon Univ.
  • Brigé, L. and Massart, P. (1998). Minimum constrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 329–375.
  • Broadie, M. and Glasserman, P. (1997). Pricing American-style securities using simulation. J. Econom. Dynam. Control 21 1323–1352.
  • Broadie, M. and Glasserman, P. (1998). Monte Carlo methods for pricing high-dimensional American options: An overview. In Monte Carlo: Methodologies and Applications for Pricing and Risk Management (B. Dupire, ed.) 149–161. Risk Books, London.
  • Carl, B. and Stephani, I. (1990). Entropy, Compactness and the Approximation of Operators. Cambridge Univ. Press.
  • Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.
  • Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of the Longstaff–Schwartz algorithm for American option pricing. Finance and Stochastics 6 449–471.
  • Cox, J., Ross, S. and Rubinstein, M. (1979). Option pricing: A simplified approach. J. Financial Economics 7 229–263.
  • Cucker, F. and Smale, S. (2001). On the mathematical foundations of learning. Bull. Amer. Math. Soc. 39 1–49.
  • DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, Berlin.
  • Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, Berlin.
  • Glasserman, P. and Yu, B. (2003). Number of paths versus number of basis functions in American option pricing. Preprint.
  • Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer, New York.
  • Haussler, D. (1995). Sphere packing numbers for subsets of the Boolean $n$-cube with bounded Vapnik–Chervonenkis dimension. J. Combin. Theory Ser. A 69 217–232.
  • Huang, J. and Pang, J. (1998). Option pricing and linear complementarity. J. Comput. Finance 2 31–60.
  • Jaillet, P., Lamberton, D. and Lapeyre, B. (1990). Variational inequalities and the pricing of American options. Acta Appl. Math. 21 263–289.
  • Karatzas, I. (1988). On the pricing of American options. Appl. Math. Optim. 17 37–60.
  • Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.
  • Kohler, M. (2000). Inequalities for uniform deviations of averages from expectations with applications to nonparametric regression. J. Statist. Plann. Inference 89 1–23.
  • Kolmogorov, A. N. and Tikhomirov, V. M. (1959). $\epsilon$-entropy and $\epsilon$-capacity of function spaces. Uspekhi Mat. Nauk. 14 3–86.\goodbreak
  • Kushner, H. J. (1997). Numerical methods for stochastic control in finance. In Mathematics of Derivative Securities (M. A. H. Dempster and S. R. Pliska, eds.) 504–527. Cambridge Univ. Press.
  • Kushner, H. J. and Clark, D. S. (1978). Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer, New York.
  • Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. Springer, Berlin.
  • Laprise, S. B., Su, Y., Wu, R., Fu, M. C. and Madan, D. B. (2001). Pricing American options: A comparision of Monte Carlo simulation approaches. J. Comput. Finance 4 39–88.
  • Ledoux, M. (1996). On Talagrand's deviation inequality for product measures. ESIAM Probab. Statist. 1 63–87.
  • Lee, W., Bartlett, P. and Williamson, R. C. (1996). Efficient agnonstic learning of neural networks with bounded fan-in. IEEE Trans. Inform. Theory 42 2118–2132.
  • Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple least-square approach. Review of Financial Studies 14 113–147.
  • Massart, P. (2000). On the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 863–885.
  • Mendelson, S. (2003). A few notes on statistical learning theory. Advanced Lectures in Machine Learning. Lecture Notes in Comput. Sci. 2600 1–40. Springer, Berlin.
  • Mendelson, S. (2003). Geometric parameters in learning theory. GAFA lecture notes.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
  • Rio, E. (2001). Une ineqalité de Bennet pour les maxima de processus empirques. In Colloque en l'Honneur de J. Bretagnolle (D. D. Castelle and I. Ibragimov, eds.).
  • Rogers, L. C. G. (2002). Monte Carlo valuing of American options. Math. Finance 12 271–286.
  • Shen, X. (1997). On methods of sieves and penalization. Ann. Statist. 25 2555–2591.
  • Shen, X. and Wong, W. H. (1994). On methods of sieves and penalization. Ann. Statist. 22 580–615.
  • Shiryayev, A. N. (1978). Optimal Stopping Rules. Springer, New York.
  • Stone, C. J. (1982). Optimal rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.
  • Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28–76.
  • Tilley, J. A. (1993). Valuing American options in a path simulation model. Transactions of the Society of Actuaries 45 83–104.
  • Tsitsiklis, J. N. and Van Roy, B. (1999). Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Trans. Automat. Control 44 1840–1851.
  • Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York.
  • Van Moerbeke, P. L. J. (1976). On optimal stopping and free boundary problems. Arch. Ration. Mech. Anal. 60 101–148.
  • Vapnik, V. N. (1982). Estimation of Dependences Based on Empirical Data. Springer, New York.
  • Vapnik, V. N. (2000). The Nature of Statistical Learning Theory, 2nd ed. Springer, New York.
  • Vapnik, V. N. and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16 264–280.
  • Vidyasagar, A. (2003). The Theory of Learning and Generalization with Applications to Neural Networks, 2nd ed. Springer, New York.
  • Wald, A. and Wolfowitz, J. (1948). Optimum character of the sequential probability ratio tests. Ann. Math. Statist. 19 326–339.