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 Probability Surveys > Vol. 3 (2006) open journal systems 


The realization of positive random variables via absolutely continuous transformations of measure on Wiener space

D. Feyel
A.S. Ustunel
M. Zakai, Technion - Israel Institute of Technology


Abstract
Let \(\mu\) be a Gaussian measure on some measurable space \(\{W = \{w\}, {\mathcal B} (W)\}\) and let \(\nu\) be a measure on the same space which is absolutely continuous with respect to \(\nu\). The paper surveys results on the problem of constructing a transformation \(T\) on the \(W\) space such that \(Tw = w+u(w)\) where \(u\) takes values in the Cameron-Martin space and the image of \(\mu\) under \(T\) is \(\mu\). In addition we ask for the existence of transformations \(T\) belonging to some particular classes.

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Feyel, D., Ustunel, A.S., Zakai, M., The realization of positive random variables via absolutely continuous transformations of measure on Wiener space, Probability Surveys, 3, (2006), 170-205 (electronic).

References

[1]     V.I. Bogachev, A.V. Kolesnikov, K.V. Medvedev. On triangular transformations of measures. Dokl. Russian Acad. Sci, v. 396, no. 6, pp. 727–732, 2004 (in Russian); English transl.: Dokl. Russian Acad. Sci, v. 69, no. 3, pp. 438–442, 2004. MR2115852

[2]     V.I. Bogachev, A.V. Kolesnikov, K.V. Medvedev. Triangular transformations of measures. To appear. MR2115852

[3]     Y. Brenier: “Polar factorization and monotone rearrangement of vector valued functions”. Comm. pure Appl. Math, 44, 375-417, 1991. MR1100809

[4]     C. Castaing and M. Valadier: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. Vol. 580. Springer, 1977. MR0467310

[5]     C. Dellacherie and P. A. Meyer: Probabilités et Potentiel, Ch. I à IV. Paris, Hermann, 1975. MR0488194

[6]     R.M. Dudley: Real Analysis and Probability. Cambridge University Press, Cambridge, 2003. MR1932358

[7]     X. Fernique: Extension du théorème de Cameron-Martin aux translations aléatoires. Ann. Probab, vol. 31, no. 3, pp. 1296–1304, 2003. MR1988473

[8]     D. Feyel and A.S. Üstünel. The notion of convexity and concavity on Wiener space. Journal of Functional Analysis, vol. 176,pp. 400-428, 2000. MR1784421

[9]     D. Feyel, A.S. Üstünel: Transport of measures on Wiener space and the Girsanov theorem. C.R. Acad. Sci. Paris, vol. 334, no. 1, pp. 1025–1028, 2002. MR1913729

[10]     D. Feyel, A.S. Üstünel: Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theor. Relat. Fields, 128, no. 3, pp. 347–385, 2004. MR2036490

[11]     W. Gangbo and R. J. McCann: “The geometry of optimal transportation”. Acta Mathematica, 177, 113-161, 1996. MR1440931

[12]     H.H. Kuo: Gaussian Measures on Banach Spaces. Lecture Notes in Mathematics, vol. 463, Springer 1975. MR0461643

[13]     H.H. Kuo. White noise distribution theory. CRC Press, 1996. MR1387829

[14]     M. Ledoux and M. Talagrand: Probability in Banach Spaces. Springer, 1991. MR1102015

[15]     K. Ito and M. Nisio: On the convergence of sums of independent random variables. Osaka J. Math., vol. 5, pp. 35–48, 1968. MR0235593

[16]     D. Nualart, M. Zakai. Positive and strongly positive Wiener functionals: Barcelona Seminar on Stochastic Analysis (St. Feliu de Guixols, 1991), 132–146, Progr. Probab., 32 Birkhauser, Basel, 1993. MR1265047

[17]     S.T. Rachev and L. Rüchendorf: “A characterization of random variables with minimum L2-distance”. Journal of Multivariate Analysis, 32, 48-54, 1990. MR1035606

[18]     S.T. Rachev and L. Rüchendorf: Mass Transportation Problems, Vol. I and II. Probability and its Applications, Springer, 1998. MR1619170

[19]     T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, 1972.

[20]     M. Talagrand: “Transportation cost for Gaussian and other product measures”. Geom. Funct. Anal., 6, 587-600, 1996. MR1392331

[21]     A.S. Üstünel: Introduction to Analysis on the Wiener Space. Lecture Notes in Math. Vol. 1610. Springer, 1995. MR1439752

[22]     A.S. Üstünel: Stochastic Analysis on Wiener Space. Electronic version in the cite http://www.finance-research.net/, 2004.

[23]     A.S. Üstünel, M. Zakai: Transformation of Measure on Wiener Space. Springer, Berlin, 2000. MR1736980

[24]     C. Villani: Topics in Optimal Transportation. Amer. Math. Soc., Rhode Island, Providence, 2003. MR1964483

[25]     E. Wong and M. Zakai. A characterization of kernels associated with the multiple integral representation of some functionals of the Wiener process. Systems and Control Lett., vol. 2, pp. 94–98, 1982/3. MR0671862

[26]     M. Zakai. Rotation and Tangent Processes on Wiener Space. Seminaire de Probabilities XXXVIII, Lecture Notes in Mathematics, vol. 1857, pp. 205–225, Springer 2004. MR2126976




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