• Bernoulli
  • Volume 19, Number 5B (2013), 2455-2493.

Diffusions with rank-based characteristics and values in the nonnegative quadrant

Tomoyuki Ichiba, Ioannis Karatzas, and Vilmos Prokaj

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We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then “unfolding” the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534–536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited.

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Bernoulli, Volume 19, Number 5B (2013), 2455-2493.

First available in Project Euclid: 3 December 2013

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diffusion with reflection and rank-dependent characteristics semimartingale local time skew representations Skorokhod problem sum-of-exponential stationary densities Tanaka formula and equation unfolding nonnegative semimartingales weak and strong solutions


Ichiba, Tomoyuki; Karatzas, Ioannis; Prokaj, Vilmos. Diffusions with rank-based characteristics and values in the nonnegative quadrant. Bernoulli 19 (2013), no. 5B, 2455--2493. doi:10.3150/12-BEJ459.

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  • [1] Banner, A.D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330.
  • [2] Bass, R.F. and Pardoux, É. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557–572.
  • [3] Bhardwaj, S. and Williams, R.J. (2009). Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation. Queueing Syst. 62 345–382.
  • [4] Burdzy, K. and Marshall, D. (1992). Hitting a boundary point with reflected Brownian motion. In Séminaire de Probabilités, XXVI. Lecture Notes in Math. 1526 81–94. Berlin: Springer.
  • [5] Burdzy, K. and Marshall, D.E. (1993). Nonpolar points for reflected Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 29 199–228.
  • [6] Chen, H. (1996). A sufficient condition for the positive recurrence of a semimartingale reflecting Brownian motion in an orthant. Ann. Appl. Probab. 6 758–765.
  • [7] Dai, J.G. and Harrison, J.M. (1992). Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2 65–86.
  • [8] Dieker, A.B. and Moriarty, J. (2009). Reflected Brownian motion in a wedge: Sum-of-exponential stationary densities. Electron. Commun. Probab. 14 1–16.
  • [9] Dupuis, P. and Williams, R.J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 680–702.
  • [10] Fernholz, E.R. (2002). Stochastic Portfolio Theory: Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 48. New York: Springer.
  • [11] Fernholz, E.R. (2011). Time reversal in an intermediate $n=2$ model. Technical report, INTECH Investment Management LLC, Princeton, NJ.
  • [12] Fernholz, R., Ichiba, T. and Karatzas, I. (2013). A second-order stock market model. Annals of Finance 9 439–454.
  • [13] Fernholz, E.R., Ichiba, T., Karatzas, I. and Prokaj, V. (2013). Planar diffusions with rank-based characteristics and perturbed Tanaka equations. Probab. Theory Related Fields 156 343–374.
  • [14] Foschini, G.J. (1982). Equilibria for diffusion models of pairs of communicating computers—symmetric case. IEEE Trans. Inform. Theory 28 273–284.
  • [15] Harrison, J.M. (1978). The diffusion approximation for tandem queues in heavy traffic. Adv. in Appl. Probab. 10 886–905.
  • [16] Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
  • [17] Harrison, J.M. and Williams, R.J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77–115.
  • [18] Harrison, J.M. and Williams, R.J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.
  • [19] Hobson, D.G. and Rogers, L.C.G. (1993). Recurrence and transience of reflecting Brownian motion in the quadrant. Math. Proc. Cambridge Philos. Soc. 113 387–399.
  • [20] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951–977.
  • [21] Ichiba, T., Karatzas, I. and Shkolnikov, M. (2011). Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 229–248.
  • [22] Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I. and Fernholz, R. (2011). Hybrid atlas models. Ann. Appl. Probab. 21 609–644.
  • [23] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
  • [24] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on $[0,a]$. Ann. Probab. 35 1740–1768.
  • [25] Krylov, N.V. (2004). On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic Process. Appl. 113 37–64.
  • [26] Manabe, S. and Shiga, T. (1973). On one-dimensional stochastic differential equations with non-sticky boundary conditions. J. Math. Kyoto Univ. 13 595–603.
  • [27] Ouknine, Y. (1990). Temps local du produit et du sup de deux semimartingales. In Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Math. 1426 477–479. Berlin: Springer.
  • [28] Ouknine, Y. and Rutkowski, M. (1995). Local times of functions of continuous semimartingales. Stochastic Anal. Appl. 13 211–231.
  • [29] Prokaj, V. (2009). Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79 534–536.
  • [30] Reiman, M.I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441–458.
  • [31] Reiman, M.I. and Williams, R.J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 87–97.
  • [32] Stroock, D.W. and Varadhan, S.R.S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Berlin: Springer.
  • [33] Varadhan, S.R.S. and Williams, R.J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405–443.
  • [34] Williams, R.J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 459–485.
  • [35] Yan, J.A. (1980). Some formulas for the local time of semimartingales. Chinese Ann. Math. 1 545–551.
  • [36] Yan, J.A. (1985). A formula for local times of semimartingales. Northeast. Math. J. 1 138–140.