Annals of Probability

An explicit formula for the Skorokhod map on [0, a]

Lukasz Kruk, John Lehoczky, Kavita Ramanan, and Steven Shreve

Full-text: Open access


The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map Γ0, a on [0, a] for any a>0 is derived. Specifically, it is shown that on the space $\mathcal{D}[0,\infty)$ of right-continuous functions with left limits taking values in ℝ, Γ0, aa○Γ0, where $\Lambda_{a}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty )$ is defined by $$\Lambda_{a}(\phi)(t)=\phi(t)-\sup_{s\in[0,t]}\biggl[\bigl(\phi(s)-a\bigr)^{+}\wedge\inf_{u\in[s,t]}\phi(u)\biggr]$$ and $\Gamma_{0}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty)$ is the Skorokhod map on [0, ∞), which is given explicitly by $$\Gamma_{0}(\psi)(t)=\psi(t)+\sup_{s\in[0,t]}[-\psi(s)]^{+}.$$ In addition, properties of Λa are developed and comparison properties of Γ0, a are established.

Article information

Ann. Probab., Volume 35, Number 5 (2007), 1740-1768.

First available in Project Euclid: 5 September 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes 60G17: Sample path properties
Secondary: 60J60: Diffusion processes [See also 58J65] 90B05: Inventory, storage, reservoirs 90B22: Queues and service [See also 60K25, 68M20]

Skorokhod map reflection map double-sided reflection map comparison principle reflecting Brownian motion


Kruk, Lukasz; Lehoczky, John; Ramanan, Kavita; Shreve, Steven. An explicit formula for the Skorokhod map on [0, a ]. Ann. Probab. 35 (2007), no. 5, 1740--1768. doi:10.1214/009117906000000890.

Export citation


  • Anderson, R. and Orey, S. (1976). Small random perturbations of dynamical systems with reflecting boundary. Nagoya Math. J. 60 189–216.
  • Anulova, S. V. and Liptser, R. Sh. (1990). Diffusion approximation for processes with normal reflection. Theory Probab. Appl. 35 411–423.
  • Burdzy, K. and Nualart, D. (2002). Brownian motion reflected on Brownian motion. Probab. Theory Related Fields 122 471–493.
  • Chaleyat-Maurel, M., El Karoui, N. and Marchal, B. (1980). Réflexion discontinue et systèmes stochastiques. Ann. Probab. 8 1049–1067.
  • Cooper, W., Schmidt, V. and Serfozo, R. (2001). Skorohod–Loynes characterizations of queueing, fluid, and inventory processes. Queueing Syst. 37 233–257.
  • El Karoui, N. and Chaleyat-Maurel, M. (1976/77). Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur $\mathbb{R}$–-Cas continu. In Temps Locaux. Astérisque 52–53 (J. Azema and M. Yor, eds.) 117–144. Soc. Math. France, Paris.
  • Harrison, M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.
  • Henderson, V. and Hobson, D. (2000). Local time, coupling and the passport option. Finance Stoch. 4 69–80.
  • Janeček, K. and Shreve, S. (2006). Futures trading with transaction costs. To appear.
  • Kruk, \L., Lehoczky, J., Ramanan, K. and Shreve, S. (2006). Diffusion approximation for an earliest-deadline-first queue with reneging. To appear.
  • Kruk, \L., Lehoczky, J., Ramanan, K. and Shreve, S. (2006). Double Skorokhod map and reneging in a real-time queues. In Markov Processes and Related Topics: A Festscrhrift for Thomas G. Kurtz (S. Eithier, J. Feng and Stockbridge, eds.). IMS, Beachwood, OH. To appear.
  • Ramanan, K. and Reiman, M. (2006). The heavy traffic limit of an unbalanced generalized processor sharing model. Preprint.
  • Skorokhod, A. V. (1961). Stochastic equations for diffusions in a bounded region. Theory Probab. Appl. 6 264–274.
  • Soucaliuc, F. and Werner, W. (2002). A note on reflecting Brownian motions. Electron. Comm. Probab. 7 117–122.
  • Tanaka. H. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 163–177.
  • Whitt, W. (2002). An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.