Bernoulli

  • Bernoulli
  • Volume 18, Number 4 (2012), 1223-1248.

$\varepsilon$-Strong simulation of the Brownian path

Alexandros Beskos, Stefano Peluchetti, and Gareth Roberts

Full-text: Open access

Abstract

We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The dominating processes converge almost surely in the supremum and $L_{1}$ norms. In particular, the rate of converge in $L_{1}$ is of the order $\mathcal{O}(\mathcal{K}^{-1/2})$, $\mathcal{K}$ denoting the computing cost. The a.s. enfolding of the Brownian path can be exploited in Monte Carlo applications involving Brownian paths whence our algorithm (termed the $\varepsilon$-strong algorithm) can deliver unbiased Monte Carlo estimators over path expectations, overcoming discretisation errors characterising standard approaches. We will show analytical results from applications of the $\varepsilon$-strong algorithm for estimating expectations arising in option pricing. We will also illustrate that individual steps of the algorithm can be of separate interest, giving new simulation methods for interesting Brownian distributions.

Article information

Source
Bernoulli, Volume 18, Number 4 (2012), 1223-1248.

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727808

Digital Object Identifier
doi:10.3150/11-BEJ383

Mathematical Reviews number (MathSciNet)
MR2995793

Zentralblatt MATH identifier
1263.65007

Keywords
Brownian bridge intersection layer iterative algorithm option pricing pathwise convergence unbiased sampling

Citation

Beskos, Alexandros; Peluchetti, Stefano; Roberts, Gareth. $\varepsilon$-Strong simulation of the Brownian path. Bernoulli 18 (2012), no. 4, 1223--1248. doi:10.3150/11-BEJ383. https://projecteuclid.org/euclid.bj/1352727808


Export citation

References

  • [1] Anderson, T.W. (1960). A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31 165–197.
  • [2] Bachelier, L. (1900). Théorie de la spéculation. Ann. Sci. École Norm. Sup. (3) 17 21–86.
  • [3] Bertoin, J., Pitman, J. and Ruiz de Chavez, J. (1999). Constructions of a Brownian path with a given minimum. Electron. Commun. Probab. 4 31–37 (electronic).
  • [4] Beskos, A., Papaspiliopoulos, O. and Roberts, G.O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
  • [5] Beskos, A., Papaspiliopoulos, O. and Roberts, G.O. (2008). A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab. 10 85–104.
  • [6] Beskos, A., Papaspiliopoulos, O., Roberts, G.O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382. With discussions and a reply by the authors.
  • [7] Beskos, A. and Roberts, G.O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422–2444.
  • [8] Beskos, A. and Roberts, G.O. (2005). One-shop CFTP; application to a class of truncated Gaussian densities. Methodol. Comput. Appl. Probab. 7 407–437.
  • [9] Casella, B. and Roberts, G.O. (2008). Exact Monte Carlo simulation of killed diffusions. Adv. in Appl. Probab. 40 273–291.
  • [10] Devroye, L. (1986). Nonuniform Random Variate Generation. New York: Springer.
  • [11] Doob, J.L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20 393–403.
  • [12] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Applications of Mathematics (New York) 53. New York: Springer.
  • [13] Jourdain, B. and Sbai, M. (2007). Exact retrospective Monte Carlo computation of arithmetic average Asian options. Monte Carlo Methods Appl. 13 135–171.
  • [14] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
  • [15] Øksendal, B. (2003). Stochastic Differential Equations, 6th ed. Berlin: Springer.
  • [16] Pinsky, R.G. (1985). On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab. 13 363–378.
  • [17] Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Probab. 38 152–164.
  • [18] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
  • [19] Roberts, G. and Shortland, C. (1997). Pricing barrier options with time-dependent coefficients. Math. Finance 7 83–93.
  • [20] Roberts, G.O. and Rosenthal, J.S. (2002). One-shot coupling for certain stochastic recursive sequences. Stochastic Process. Appl. 99 195–208.
  • [21] Rogers, L.C.G. and Shi, Z. (1995). The value of an Asian option. J. Appl. Probab. 32 1077–1088.