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Exact simulation of multidimensional reflected Brownian motion

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

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: jose.blanchet@stanford.edu
** Postal address: Department of Industrial Engineering & Operations Research, Columbia University, S. W. Mudd Building, 500 W. 120 Street, New York, NY 10027, USA.
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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 

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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