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A note on integral representations of the Skorokhod map

Published online by Cambridge University Press:  24 March 2016

Patrick Buckingham ,
Brian Fralix  and
Offer Kella
Patrick Buckingham
Affiliation:
Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634-0975, USA. Email address: pbuckin@clemson.edu
Brian Fralix*
Affiliation:
Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634-0975, USA.
Offer Kella
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel. Email address: offer.kella@huji.ac.il
*
*** Email address: bfralix@clemson.edu
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Abstract

We present a very short derivation of the integral representation of the two-sided Skorokhod reflection Z of a continuous function X of bounded variation, which is a generalization of the integral representation of the one-sided map featured in Anantharam and Konstantopoulos (2011) and Konstantopoulos et al. (1996). We also show that Z satisfies a simpler integral representation when additional conditions are imposed on X.

Keywords

Reflection mapregulator mapSkorokhod problem
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 293 - 298
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Anantharam, V. and Konstantopoulos, T. (2011). Integral representation of Skorokhod reflection. Proc. Amer. Math. Soc. 139, 2227–2237. Google Scholar
[2]Andersen, L. N. and Mandjes, M. (2009). Structural properties of reflected L\'evy processes. Queueing Systems 63, 301–322. Google Scholar
[3]Baccelli, F. and Br\'{e}maud, P. (2003). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, 2nd edn. Springer, Berlin. Google Scholar
[4]Cooper, W. L., Schmidt, V. and Serfozo, R. F. (2001). Skorohod--Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233–257. Google Scholar
[5]Halmos, P. R. (1974). Measure Theory. Springer, New York. Google Scholar
[6]Konstantopoulos, T. and Last, G. (2000). On the dynamics and performance of stochastic fluid systems. J. Appl. Prob. 37, 652–667. Google Scholar
[7]Konstantopoulos, T., Zazanis, M. and De Veciana, G. (1996). Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues. Stoch. Process. Appl. 65, 139–146. Google Scholar
[8]Kruk, L., Lehoczky, J., Ramanan, K., and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0,a]. Ann. Prob. 35, 1740–1768. Google Scholar
[9]Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2008). Double Skorokhod map and reneging real-time queues. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4), Institute of Mathematical Statistics, Beachwood, OH, pp. 169–193. CrossRefGoogle Scholar
[10]Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin. Google Scholar
[11]Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163–177. CrossRefGoogle Scholar