Comptes Rendus
no. 12
Probability theory
Conditionally Gaussian stochastic integrals
Presented by: Jean-François Le Gall
Nicolas Privault1; Qihao She1
1Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1153-1158

We derive conditional Gaussian type identities of the form

E[exp⁡(i∫0TutdBt)|∫0T|ut|2dt]=exp⁡(−12∫0T|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 A†A2=0, using a characterization of Yor [6].

Nous obtenons des identités gaussiennes conditionnelles de la forme

E[exp⁡(i∫0TutdBt)|∫0T|ut|2dt]=exp⁡(−12∫0T|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 A†A2=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 
@article{CRMATH_2015__353_12_1153_0,
     author = {Nicolas Privault and Qihao She},
     title = {Conditionally {Gaussian} stochastic integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1153--1158},
     year = {2015},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     doi = {10.1016/j.crma.2015.09.022},
     language = {en},
}
TY  - JOUR
AU  - Nicolas Privault
AU  - Qihao She
TI  - Conditionally Gaussian stochastic integrals
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1153
EP  - 1158
VL  - 353
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2015.09.022
LA  - en
ID  - CRMATH_2015__353_12_1153_0
ER  - 
%0 Journal Article
%A Nicolas Privault
%A Qihao She
%T Conditionally Gaussian stochastic integrals
%J Comptes Rendus. Mathématique
%D 2015
%P 1153-1158
%V 353
%N 12
%I Elsevier
%R 10.1016/j.crma.2015.09.022
%G en
%F CRMATH_2015__353_12_1153_0
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

[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)

Cited by Sources:

Comments - Policy