Stochastic analysis of bernoulli processes

Probability Surveys

Volumn 5, Issue 1, 2008, Pages 435-483

  Privault, Nicolas ^a  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Bernoulli processes; chaotic calculus; discrete time; functional inequalities; Malliavin calculus; option hedging

Indexed keywords

EID: 68749104475     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/08-PS139     Document Type: Article

References (28)

1. C. Ané and M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields, 116(4):573-602, 2000. MR1757600
2. S. Attal. Approximating the Fock space with the toy Fock space. In Séminaire de Probabilités, XXXVI, volume 1801 of Lecture Notes in Math., pages 477-491. Springer, Berlin, 2003. MR1971605
3. S.G. Bobkov. On the Gross and Talagrand inequalities on the discrete cube. Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform., 1:12-19, 1995. MR1717705
4. S.G. Bobkov, C. Houdré, and P. Tetali. The subgaussian constant and concentration inequalities. Israel J. Math., 156:255-283, 2006. MR2282379
5. S.G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal., 156(2):347-365, 1998. MR1636948
6. M. Capitaine, E.P. Hsu, and M. Ledoux. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Comm. Probab., 2:71-81 (electronic), 1997. MR1484557
7. P. Dai Pra, A.M. Paganoni, and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab., 30(4):1959-1976, 2002. MR1944012
8. M. Émery. A discrete approach to the chaotic representation property. In Séminaire de Probabilités, XXXV, volume 1755 of Lecture Notes in Math., pages 123-138. Springer, Berlin, 2001. MR1837280
9. H. Föllmer and A. Schied. Stochastic finance, volume 27 of de Gruyter Stud- ies in Mathematics. Walter de Gruyter & Co., Berlin, 2004. MR2169807
10. F.Q. Gao and N. Privault. Clark formula and logarithmic Sobolev inequalities for Bernoulli measures. C. R. Math. Acad. Sci. Paris, 336(1):51-56, 2003. MR1968902
11. F.Q. Gao and J. Quastel. Exponential decay of entropy in the random transposition and Bernoulli-Laplace models. Ann. Appl. Probab., 13(4):1591-1600, 2003. MR2023890
12. H. Gzyl. An exposé on discrete Wiener chaos expansions. Bol. Asoc. Mat. Venez., XIII(1):3-26, 2006. MR2267626
13. H. Holden, T. Lindstroøm, B. Oøksendal, and J. Uboøe. Discrete Wick calculus and stochastic functional equations. Potential Anal., 1(3):291-306, 1992. MR1245232
14. H. Holden, T. Lindstroøm, B. Oøksendal, and J. Uboøe. Discrete Wick products. In Stochastic analysis and related topics (Oslo, 1992), pages 123-148. Gordon and Breach, Montreux, 1993.
15. C. Houdré and V. Pérez-Abreu. Covariance identities and inequalities for functionals on Wiener and Poisson spaces. Ann. Probab., 23:400-419, 1995. MR1330776
16. C. Houdré and N. Privault. Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli, 8(6):697-720, 2002. MR1962538
17. C. Houdré and P. Tetali. Concentration of measure for products of Markov kernels and graph products via functional inequalities. Combin. Probab. Comput., 10(1):1-28, 2001. MR1827807
18. J. Jacod and Ph. Protter. Probability essentials. Springer-Verlag, Berlin, 2000. MR1736066
19. D. Lamberton and B. Lapeyre. Introduction to stochastic calculus applied to finance. Chapman & Hall, London, 1996. MR1422250
20. M. Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM Probab. Statist., 1:63-87 (electronic), 1995/97. MR1399224
21. M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6), 9:305-366, 2000. MR1813804
22. M. Leitz-Martini. A discrete Clark-Ocone formula. Maphysto Research Report No 29, 2000.
23. N. Privault and W. Schoutens. Discrete chaotic calculus and covariance identities. Stochastics and Stochastics Reports, 72:289-315, 2002. Eurandom Report 006, 2000. MR1897919
24. J. Ruiz de Chávez. Predictable representation of the binomial process and application to options in finance. In XXXIII National Congress of the Mexican Mathematical Society (Spanish) (Saltillo, 2000), volume 29 of Aportaciones Mat. Comun., pages 223-230. Soc. Mat. Mexicana, México, 2001. MR1939446
25. L. Saloff-Coste. Lectures on finite Markov chains. In Lectures on probability theory and statistics (Saint-Flour, 1996), volume 1665 of Lecture Notes in Math., pages 301-413. Springer, Berlin, 1997. MR1490046
26. D. Stroock. Doing analysis by tossing a coin. Math. Intelligencer, 22(2):66-72, 2000. MR1764269
27. D. Williams. Probability with martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. MR1155402
28. L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields, 118(3):427-438, 2000. MR1800540