Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 11 (2006) > Paper 20 open journal systems 


Convex concentration inequalities and forward-backward stochastic calculus

Thierry Klein, Universite Paul Sabatier
Yutao Ma, Universite de La Rochelle
Nicolas Privault, Universite de La Rochelle


Abstract
Given (Mt)t ∈ R+ and (M*t)t ∈ R+ respectively a forward and a backward martingale with jumps and continuous parts, we prove that E[φ (Mt+M*t)] is non-decreasing in t when φ is a convex function, provided the local characteristics of (Mt)t ∈ R+ and (M*t)t ∈ R+ satisfy some comparison inequalities.
We deduce convex concentration inequalities and deviation bounds for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component.


Full text: PDF

Pages: 486-512

Published on: July 7, 2006
Bibliography
  1. C. Ané and M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs.Probab. Theory Related Fields 116 (2000), 573--602.Math. Review 2001i:60094
  2. V. Bentkus. On Hoeffding's inequalities. Ann. Probab. 32 (2004), 1650--1673. Math. Review 2005e:60041
  3. D.L. Burkholder. Strong differential subordination and stochastic integration. Ann. Probab. 22 (1994), 995--1025. Math. Review 95h:60085
  4. M. Capitaine, E.P. Hsu, and M. Ledoux. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Comm. Probab. 2 (1997), 71--81. Math. Review 99b:60136
  5. C. Dellacherie, B. Maisonneuve, and P.A. Meyer. Probabilités et Potentiel, volume 5. 5 (1992) Hermann. Math. Review number not available.
  6. M. Émery. On the Azéma martingales. Séminaire de Probabilités XXIII, Lecture Notes in Mathematics 1372 (1990), 66-87, Springer Verlag. Math. Review 91e:60140
  7. J. Jacod. Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714 (1979), Springer Verlag. Math. Review 81e:60053
  8. J. Jacod and J. Mémin. Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), 1--37. Math. Review 0418223
  9. O. Kallenberg. Foundations of modern probability. Probability and its Applications (2002), Springer Verlag. Math. Review 2002m:60002
  10. Th. Klein. Inégalités de concentration, martingales et arbres aléatoires. Thèse, Université de Versailles-Saint-Quentin (2003). Math. Review number not available.
  11. J. Ma, Ph. Protter, and J. San Martin. Anticipating integrals for a class of martingales. Bernoulli 4 (1998), 81--114. Math. Review 99j:60071
  12. D. Nualart and J. Vives. Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités XXIV, Lecture Notes in Mathematics 1426 (1990), 154--165, Springer Verlag. Math. Review 92i:60109
  13. B. Øksendal and F. Proske. White noise of Poisson random measures. Potential Anal. 21 (2004), 375--403. Math. Review 2005i:60134
  14. J. Picard. Formules de dualité sur l'espace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), 509--548. Math. Review 98c:60055
  15. I. Pinelis. Optimal tail comparison based on comparison of moments. High dimensional probability (Oberwolfach, 1996), Progr. Probab. 43 (1998), 297--314, Birkhäuser. Math. Review 2000a:60026
  16. N. Privault. An extension of stochastic calculus to certain non-Markovian processes. Prépublication 49 (1997), , Université d'Evry. Math. Review number not available.
  17. N. Privault. Equivalence of gradients on configuration spaces. Random Operators and Stochastic Equations 7 (1999), 241--262. Math. Review 2000c:60087
  18. Ph. Protter. Stochastic integration and differential equations. A new approach. Applications of Mathematics 21 (1990), Springer Verlag. Math. Review 91i:60148
  19. L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000), 427-438.









    Research
    Support Tool
    Capture Cite
    View Metadata
    Printer Friendly
    Context
    Author Address
    Action
    Email Author
    Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489