Comptes Rendus
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

E[exp(i0TutdBt)|0T|ut|2dt]=exp(120T|ut|2dt),
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

E[exp(i0TutdBt)|0T|ut|2dt]=exp(120T|ut|2dt),
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].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.022
Keywords: Quadratic Brownian functionals, Multidimensional Brownian motion, Moment identities, Characteristic functions

Nicolas Privault 1; Qihao She 1

1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
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Nicolas Privault; Qihao She. Conditionally Gaussian stochastic integrals. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1153-1158. doi : 10.1016/j.crma.2015.09.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.022/

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

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

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

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

[6] M. Yor 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] M. Yor 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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