Abstract
A formalism for the optimal stopping of two-parameter processes is developed by analogy with the classical theory. An optimality criterion is established in terms of the conditional pay-off process. The problem of optimal stopping of a process indexed by IN2 is completely solved, in probabilistic terms, by using the notion of tactics. The method consis of searching for an optimal stopping point among the maximal stopping points up to which the Snell envelope is a martingale. On IR 2+ the difficulties arise from the lack of information about the behaviour of two-parameter supermartingales, and particularly the Snell envelope. The optimal stopping of a Brownian sheet is solved and we present the case of the bi-Brownian process. Associated systems of variational inequalities are proposed.
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Bibliography
A. BENSOUSSAN and J.L. LIONS: "Applications des inéquations variationnelles en contrôle stochastique" Dunod, 1978.
J.M. BISMUT and B. SKALLI: "Temps d'arrêt optimal, théorie générale des processus et processus de Markov" Z.f.Wahr.V.Geb. 39, (1977) 301–313.
R. CAIROLI: "Enveloppe de Snell d'un processus à paramètre bidimensionnel" Ann.Inst. H. Poincaré, VolXVIII No1 (1982) 47–53
C. DELLACHERIE and P.A. MEYER: "Probabilités et potentiel" vol. 1 and 2, Hermann, 1975 and 1980.
E.B. DYNKIN: "Markov processes" Springer Verlag, 1965.
N. EL KAROUI: "Les aspects probabilistes du contrôle stochastique" Ecole d'été de St Flour 1979. Lect. Notes in Math. No876, Springer-Verlag (1981) 74–239.
U. KRENGEL and L. SUCHESTON: "Stopping rules and tactics for processes indexed by directed sets" J. of Mult. Anal. Vol 11 No2 (1981) 199–229.
A. MANDELBAUM and R.J. VANDERBEI: "Optimal stopping and supermartingales over partially ordered sets" Z.f.Wahr.V.Geb. 57 (1981) 253–264.
G. MAZZIOTTO and J. SZPIRGLAS: "Arrêt optimal sur le plan" C.R.Acad.Sc.Paris t.293 (1981) 87–90.
P.A. MEYER: "Théorie élémentaire des processus à deux indices" Lect.Notes in Math.No863, Springer-Verlag (1981) 1–39.
A.N. SHIRYAYEV: "Optimal stopping rules" Springer-Verlag, 1978.
J.B. WALSH: "Optional increasing paths" Lect. Notes in Math. No863, Springer-Verlag, (1981) 172–201.
E. WONG and M. ZAKAI: "Martingales and stochastic integrals for processes with a multidimensional parameter" Z.f.Wahr.V. Geb.29 (1974) 109–122.
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Mazziotto, G., Szpirglas, J. (1982). Optimal stopping for two-parameter processes. In: Fleming, W.H., Gorostiza, L.G. (eds) Advances in Filtering and Optimal Stochastic Control. Lecture Notes in Control and Information Sciences, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0004542
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DOI: https://doi.org/10.1007/BFb0004542
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