Probability theory
Conditionally Gaussian stochastic integrals
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1153-1158.

We derive conditional Gaussian type identities of the form

for Brownian stochastic integrals, under conditions on the process (ut)t[0,T] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral 0tABsdBs under the matrix condition AA2=0, using a characterization of Yor [6].

Nous obtenons des identités gaussiennes conditionnelles de la forme

pour les intégrales stochastiques browniennes, sous des conditions sur le processus (ut)t[0,T] exprimées à l'aide du calcul de Malliavin. Ces résultats s'appliquent en particulier à l'intégrale brownienne quadratique 0tABsdBs sous la condition matricielle AA2=0, en utilisant une caractérisation de Yor [6].

Published online:
DOI: 10.1016/j.crma.2015.09.022
Keywords: Quadratic Brownian functionals, Multidimensional Brownian motion, Moment identities, Characteristic functions
Privault, Nicolas 1; She, Qihao 1

1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
     author = {Privault, Nicolas and She, Qihao},
     title = {Conditionally {Gaussian} stochastic integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1153--1158},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.022},
     language = {en},
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Privault, Nicolas; She, Qihao. Conditionally Gaussian stochastic integrals. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1153-1158. doi : 10.1016/j.crma.2015.09.022.

[1] Driver, B.K.; Eldredge, N.; Melcher, T. Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups, 2013 | arXiv

[2] Nualart, D. The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer-Verlag, Berlin, 2006

[3] Privault, N. Cumulant operators and moments of the Itô and Skorohod integrals, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013) no. 9–10, pp. 397-400

[4] Privault, N. Cumulant operators for Lie–Wiener–Itô–Poisson stochastic integrals, J. Theor. Probab., Volume 28 (2015) no. 1, pp. 269-298

[5] Üstünel, A.S.; Zakai, M. Random rotations of the Wiener path, Probab. Theory Relat. Fields, Volume 103 (1995) no. 3, pp. 409-429

[6] Yor, M. Les filtrations de certaines martingales du mouvement brownien dans Rn, Université de Strasbourg, Strasbourg, France, 1977/78 (Lecture Notes in Mathematics), Volume vol. 721, Springer, Berlin (1979), pp. 427-440

[7] Yor, M. Remarques sur une formule de Paul Lévy, Paris, 1978/1979 (Lecture Notes in Mathematics), Volume vol. 784, Springer, Berlin (1980), pp. 343-346 (in French)

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