A stochastic representation theorem with applications to optimization and obstacle problems. (English) Zbl 1058.60022

Authors’ summary: We study a new type of representation problem for optional processes with connections to singular control, optimal stopping and dynamic allocation problems. As an application, we show how to solve a variant of Skorokhod’s obstacle problem in the context of backward stochastic differential equations.


60G07 General theory of stochastic processes
60G40 Stopping times; optimal stopping problems; gambling theory
60H25 Random operators and equations (aspects of stochastic analysis)
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[1] Bank, P. and Riedel, F. (2001). Optimal consumption choice with intertemporal substitution. Ann. Appl. Probab. 3 750–788. · Zbl 1022.90045
[2] Dellacherie, C. and Lenglart, E. (1982). Sur des problèmes de régularisation, de récollement et d’interpolation en théorie des processus. Séminaire de Probabilités XVI. Lecture Notes in Math. 920 298–313. Springer, Berlin. · Zbl 0538.60035
[3] Dellacherie, C. and Meyer, P. (1975). Probabilités et potentiel , Chapitres I–IV. Hermann, Paris. · Zbl 0323.60039
[4] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. Ecole d’Eté de Probabilités de Saint-Flour IX. Lecture Notes in Math. 876 73–238. Springer, Berlin. · Zbl 0472.60002
[5] El Karoui, N. and Karatzas, I. (1994). Dynamic allocation problems in continuous time. Ann. Appl. Probab. 4 255–286. JSTOR: · Zbl 0831.93069
[6] Hindy, A. and Huang, C.-F. (1993). Optimal consumption and portfolio rules with durability and local substitution. Econometrica 61 85–121. · Zbl 0772.90015
[7] Hindy, A., Huang, C.-F. and Kreps, D. (1992). On intertemporal preferences in continuous time: The case of certainty. J. Math. Econom. 21 401–440. · Zbl 0765.90023
[8] Karatzas, I. (1994). Gittins indices in the dynamic allocation problem for diffusion processes. Ann. Probab. 12 173–192. JSTOR: · Zbl 0536.60058
[9] Morimoto, H. (1991). On average cost stopping time problems. Probab. Theory Related Fields 90 469–490. · Zbl 0729.60037
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