A new kind of augmentation of filtrations
ESAIM: Probability and Statistics, Tome 15 (2011) , pp. S39-S57.

Let (Ω, , ( t )t≥0, ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to (the σ-algebra generated by ( t )t≥0) a coherent family of probability measures ( t ) indexed by t 0, each of them being defined on t . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual augmentation. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths (typically adapted and continuous or adapted and càdlàg versions) for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. For instance we shall prove that on the Wiener space, equipped with the right-continuous augmentation of the filtration generated by the canonical process, there exists no càdlàg or continuous and adapted version for the local time at level zero of the canonical process. Hence there is an incompatibility between the problem of extending a coherent family of probability measures to and the classical construction of regular versions of paths, requiring the usual assumptions (this situation typically occurs in the problem of penalization of the Brownian paths or in mathematical finance), which to the best of our knowledge, has not been noticed so far. In order to fix this problem, we introduce a new property for filtrations, intermediate between the right continuity and the usual conditions. More precisely, we say that a filtration ( t )t≥0 satisfies the natural assumptions if it is right-continuous and if 0 contains all the sets included in a countable union of negligible sets (Bn)n≥1, such that Bn n for n 1. There is a natural way to obtain, from a given filtration ( t )t≥0, a new filtration which satisfies the natural assumptions: we call it the natural augmentation of ( t )t≥0. We show that most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation, such as the existence of regular version of trajectories or the début theorem, still hold under the natural augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration. For sake of completeness, we also recall (not so well known) Parthasarathy type conditions on the underlying filtration under which the extension problem for a coherent family of probability measures has a solution. In particular, we shall see that this is always the case on the following two fundamental spaces: 𝒞( + , ), the space of continuous functions equipped with the filtration generated by the coordinate process and 𝒟( + , ), the space of càdlàg functions endowed with the filtration generated by the coordinate process.

DOI : https://doi.org/10.1051/ps/2010026
Classification : 60G05,  60G07,  60G44
Mots clés : The usual assumptions, change of probability measure
     author = {Najnudel, Joseph and Nikeghbali, Ashkan},
     title = {A new kind of augmentation of filtrations},
     journal = {ESAIM: Probability and Statistics},
     pages = {S39--S57},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2010026},
     mrnumber = {2817344},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2010026/}
Najnudel, Joseph; Nikeghbali, Ashkan. A new kind of augmentation of filtrations. ESAIM: Probability and Statistics, Tome 15 (2011) , pp. S39-S57. doi : 10.1051/ps/2010026. http://www.numdam.org/articles/10.1051/ps/2010026/

[1] K. Bichteler, Stochastic integration with jumps. Cambridge University Press (2002). | MR 1906715 | Zbl 1189.60004

[2] K. Ciesielski, How good is Lebesgue measure? Math. Intell. 11 (1989) 54-58. | MR 994965 | Zbl 0679.28001

[3] C. Dellacherie and P.-A. Meyer, Un nouveau théorème de projection et de section, in Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526. Springer (1975) 239-245. | Numdam | MR 436311 | Zbl 0346.60023

[4] C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Volume 1. Hermann, Paris (1976). | Zbl 0138.10402

[5] C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Volume 2. Hermann, Paris (1980). | MR 566768 | Zbl 0138.10402

[6] H. Föllmer, The exit measure of a supermartingale. Z. Wahrscheinlichkeitstheorie verw. Geb. 21 (1972) 154-166. | MR 309184 | Zbl 0231.60033

[7] H. Föllmer and P. Imkeller, Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. H. Poincaré 29 (1993) 569-586. | Numdam | MR 1251141 | Zbl 0796.60082

[8] J. Jacod and A. Shiryaev, Limit theorems for stochastic processes. 2nd edition. Springer (2003). | MR 1943877 | Zbl 0635.60021

[9] K.R. Parthasarathy, Probability measures on metric spaces. Academic Press, New York (1967). | MR 226684 | Zbl 0153.19101

[10] D.-W. Stroock and S.-R.-S. Varadhan, Multidimensional diffusion processes, Classics in Mathematics. Springer-Verlag, Berlin (2006). Reprint of the 1997 edition. | MR 2190038 | Zbl 1103.60005