Comptes Rendus
no. 7
Probabilités
A backward Itô–Ventzell formula with an application to stochastic interpolation
Pierre Del Moral1  ; Sumeetpal S. Singh2
1 INRIA Bordeaux Research Center, University of Bordeaux, Talance, France
2 Department of Engineering, University of Cambridge, United Kingdom
Comptes Rendus. Mathématique, Volume 358 (2020) no. 7, pp. 881-886.

This Note and its extended version [7] present a novel backward Itô–Ventzell formula and an extension of the Aleeksev–Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.110
Classification : 47D07, 93E15, 60H07

Pierre Del Moral 1 ; Sumeetpal S. Singh 2

1 INRIA Bordeaux Research Center, University of Bordeaux, Talance, France
2 Department of Engineering, University of Cambridge, United Kingdom
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Pierre Del Moral; Sumeetpal S. Singh. A backward Itô–Ventzell formula with an application to stochastic interpolation. Comptes Rendus. Mathématique, Volume 358 (2020) no. 7, pp. 881-886. doi : 10.5802/crmath.110. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.110/

[1] Vladimir Alekseev An estimate for the perturbations of the solution of ordinary differential equations, Vestn. Mosk. Univ. (1961) no. 2, pp. 28-36 | MR

[2] Marc Arnaudon; Pierre Del Moral A duality formula and a particle Gibbs sampler for continuous time Feynman–Kac measures on path spaces (2018) (https://arxiv.org/abs/1805.05044)

[3] Marc Arnaudon; Pierre Del Moral A variational approach to nonlinear and interacting diffusions, Stochastic Anal. Appl., Volume 37 (2019) no. 5, pp. 717-748 | DOI | MR | Zbl

[4] Marc Arnaudon; Pierre Del Moral A second order analysis of McKean–Vlasov semigroups, 2020 (To appear in The Annals of Applied Probability)

[5] Adrian N. Bishop; Pierre Del Moral; Angèle Niclas A perturbation analysis of stochastic matrix Riccati diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 2, pp. 884-916 | DOI | MR | Zbl

[6] Giuseppe Da Prato; Luciano Tubaro Some remarks about backward Itô formula and applications, Stochastic Anal. Appl., Volume 16 (1998) no. 6, pp. 993-1003 | Zbl

[7] Pierre Del Moral; Sumeetpal Sidhu Singh Backward Itô–Ventzell and stochastic interpolation formulae (2019) (https://hal.archives-ouvertes.fr/hal-02161914v4)

[8] David Nualart The Malliavin calculus and related topics, Probability and Its Applications, 1995, Springer, 2006 | Zbl

[9] Daniel Ocone; Etienne Pardoux A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 25 (1989) no. 1, pp. 39-71 | Numdam | Zbl

[10] Etienne Pardoux; Philip E. Protter A two-sided stochastic integral and its calculus, Probab. Theory Relat. Fields, Volume 76 (1987) no. 1, pp. 15-49 | DOI | MR | Zbl

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