Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T23:30:49.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact simulation of multidimensional reflected Brownian motion

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  28 March 2018

Jose Blanchet  and
Karthyek Murthy
Jose Blanchet*
Affiliation:
Columbia University
Karthyek Murthy*
Affiliation:
Columbia University
*
* Current address: Management Science and Engineering, Stanford University, 475 Via Ortega, Stanford, CA 94305, USA. Email address: [email protected]
** Postal address: Department of Industrial Engineering & Operations Research, Columbia University, S. W. Mudd Building, 500 W. 120 Street, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance–rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

Keywords

Unbiased samplingrefine until accept/rejecttolerance enforced simulationacceptance–rejection samplingε-strong simulationiterative algorithmintersection layer

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J60: Diffusion processes 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 137 - 156
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York. CrossRefGoogle Scholar
[2]Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Prob. 15, 2422–2444. CrossRefGoogle Scholar
[3]Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098. CrossRefGoogle Scholar
[4]Beskos, A., Peluchetti, S. and Roberts, G. (2012). ε-strong simulation of the Brownian path. Bernoulli 18, 1223–1248. CrossRefGoogle Scholar
[5]Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Prob. 25, 3209–3250. CrossRefGoogle Scholar
[6]Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization (Appl. Math. (New York) 46). Springer, New York. CrossRefGoogle Scholar
[7]Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Operat. Res. 38, 591–616. CrossRefGoogle Scholar
[8]Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2001). Introduction to Algorithms, 2nd edn. McGraw-Hill, Boston, MA. Google Scholar
[9]Étoré, P. and Martinez, M. (2013). Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process. Monte Carlo Methods Appl. 19, 41–71. CrossRefGoogle Scholar
[10]Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308. CrossRefGoogle Scholar
[11]Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180. CrossRefGoogle Scholar
[12]Pollock, M., Johansen, A. M. and Roberts, G. O. (2016). On the exact and ε-strong simulation of (jump) diffusions. Bernoulli 22, 794–856. CrossRefGoogle Scholar
[13]Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164. CrossRefGoogle Scholar
[14]Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458. CrossRefGoogle Scholar