Abstract
Let (M t )0≤t≤1 be any martingale with initial law M 0 ~ μ0 and terminal law M 1 ~ μ1 and let S≡sup0≤t≤1 M t . Then there is an upper bound, with respect to stochastic ordering of probability measures, on the law of S.
An explicit description of the upper bound is given, along with a martingale whose maximum attains the upper bound.
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References
Azéma, J., Gundy, R.F. and Yor, M.; Sur l'integrabilité uniforme des martingales continues, Séminaire de Probabilités, XIV, 53–61, 1980.
Azéma, J and Yor, M.; Une solution simple au problème de Skorokhod, Séminaire de Probabilités, XIII, 90–115, 1979.
Bertoin, J. and Le Jan, Y.; Representation of measures by balayage from a regular recurrent point, Annals of Probability, 20, 538–548, 1992.
Blackwell, D. and Dubins, L.D.; A Converse to the Dominated Convergence Theorem, Illinois Journal of Mathematics, 7, 508–514, 1963.
Chacon, R. and Walsh, J.B.; One-dimensional potential embedding, Séminaire de Probabilités, X, 19–23, 1976.
Dubins, L.D.; On a theorem of Skorokhod, Annals of Mathematical Statistics, 39, 2094–2097, 1968.
Dubins, L.D. and Gilat, D.; On the distribution of maxima of martingales, Proceedings of the American Mathematical Society, 68, 337–338, 1978.
Hobson, D.G.; Robust hedging of the lookback option, To appear in Finance and Stochastics, 1998.
Kertz, R.P. and Rösler, U.; Martingales with given maxima and terminal distributions, Israel J. Math., 69, 173–192, 1990.
Meyer, P.A.; Probability and Potentials, Blaidsell, Waltham, Mass., 1966.
Perkins, E.; The Cereteli-Davis solution to the H1-embedding problem and an optimal embedding in Brownian motion, Seminar on Stochastic Processes, 1985, Birkhauser, Boston, 173–223, 1986.
Rogers, L.C.G.; Williams' characterisation of the Brownian excursion law: proof and applications, Séminaire de Probabilités, XV, 227–250, 1981.
Rogers, L.C.G.; A guided tour through excursions, Bull. London Math. Soc., 21, 305–341, 1989.
Rogers, L.C.G.; The joint law of the maximum and the terminal value of a martingale, Prob. Th. Rel. Fields, 95, 451–466, 1993.
Rösler, U.; Personal communication.
Strassen, V.; The existence of probability measures with given marginals, Ann. Math. Statist., 36, 423–439, 1965.
Vallois, P.; Sur la loi du maximum et du temps local d'une martingale continue uniformément intégrable, Proc. London Math. Soc., 3, 399–427, 1994.
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Hobson, D.G. (1998). The maximum maximum of a martingale. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101762
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DOI: https://doi.org/10.1007/BFb0101762
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