The Skorokhod problem with two nonlinear constraints

• Autorzy: Li Hanwu

Identyfikatory

DOI

10.37190/0208-4147.00138

• Języki publikacji: EN

Abstrakty

EN

In this paper, we study the Skorokhod problem with two constraints, where both constraints are nonlinear. We prove the existence and uniqueness of a solution and also provide an explicit construction for the solution. In addition, a number of properties of the solution are investigated, including continuity under uniform and J₁ metrics and a comparison principle.

Słowa kluczowe

EN

Skorokhod problem; nonlinear reflecting boundaries; double reflecting boundaries; comparison principle

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2023
• Tom: Vol. 43, Fasc. 2
• Strony: 207--239
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Li Hanwu

• Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, Shandong, China

Bibliografia

• 1] K. Burdzy, W. Kang and K. Ramanan, The Skorokhod problem in a time-dependent interval, Stoch. Process. Appl. 119 (2009), 428-452.
• [2] K. Burdzy and D. Nualart, Brownian motion reflected on Brownian motion, Probab. Theory Related Fields 122 (2002), 471-493.
• [3] M. Chaleyat-Maurel and N. El Karoui, Un problème de réflexion et ses applications au temps local et auxéquations différentielles stochastiques sur R, cas continu, Astérisque 52-53 (1978), 117-144.
• [4] M. Chaleyat-Maurel, N. El Karoui and B. Marchal, Réflexion discontinue et systèmes stochastiques, Ann. Probab. 8 (1980), 1049-1067.
• [5] N. El Karoui and I. Karatzas, A new approach to the Skorokhod problem, and its applications, Stoch. Stoch. Rep. 34 (1991), 57-82.
• [6] A. Falkowski and L. Słomiński, Mean reflected stochastic differential equations with two constraints, Stoch. Process. Appl. 141 (2021), 172-196.
• [7] A. Falkowski and L. Słomi´nski, Backward stochastic differential equations with mean reflection and two constraints, Bull. Sci. Math. 176 (2022), art. 103117, 31 pp..
• [8] A. Hilbert, I. Jarni and Y. Ouknine, Stochastic differential equations with respect to optional semimartingales and two reflecting regulated barriers, arXiv: 2202.12862v1 (2022).
• [9] L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve, An explicit formula for the Skorokhod map on [0, a], Ann. Probab. 35 (2007), 1740-1768.
• [10] A. Mandelbaum and W. Massey, Strong approximations for time-dependent queues, Math. Oper. Res. 20 (1995), 33-63.
• [11] A. V. Skorokhod, Stochastic equations for diffusions in a bounded region, Theory Probab. Appl. 6 (1961), 264-274.
• [12] M. Slaby, Explicit representation of the Skorokhod map with time dependent boundaries, Probab. Math. Statist. 30 (2010), 29-60.
• [13] M. Slaby, An explicit representation of the extended Skorokhod map with two time-dependent boundaries, J. Probab. Statist. 2010, art. 846320, 18 pp.
• [14] L. Słomiński and T. Wojciechowski, Stochastic differential equations with jump reflection at time-dependent barriers, Stoch. Process. Appl. 120 (2010), 1701-1721.
• [15] L. Słomiński and T. Wojciechowski, Stochastic differential equations with time-dependent reflecting barriers, Stochastics 85 (2013), 27-47.
• [16] F. Soucaliuc and W. Werner, A note on reflecting Brownian motions, Electron. Commun. Probab. 7 (2002), 117-122.
• [17] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), 163-177.