By Jean Picard (auth.), Catherine Donati-Martin, Antoine Lejay, Alain Rouault (eds.)

This is a brand new quantity of the Séminaire de Probabilité which was once begun within the 60's. Following the culture, this quantity comprises as much as 20 unique examine and survey articles on numerous themes concerning stochastic analysisThis quantity comprises J. Picard's complicated direction at the illustration formulae for the fractional Brownian movement. The typical chapters hide quite a lot of topics, comparable to stochastic calculus and stochastic differential equations, stochastic differential geometry, filtrations, research of Wiener area, random matrices and unfastened chance, in addition to mathematical finance. many of the contributions have been awarded on the Journées de Probabilités held in Poitiers in June 2009.

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**Example text**

We prove separately the two parts. e ˛ . t/ ˛ 0 1Cˇ s ;ı ;ı ; sÄuÄt and by applying Theorem 2. The composition rule is evident for smooth functions (use the first equality of (38)), and can be extended by density (the parameter ı is unimportant since we only need the functions on bounded time intervals). Study of T˛ and T˛0 . t s/ˇ ; so T0 f W t 7! 1=t/ is in Hˇ; 2ˇ ı; 2ˇ . The continuity of T˛ and T˛0 is then a e 2˛ . consequence of (41) and of the continuity of ˘ 2˛ and ˘ t u Remark 5. We deduce in particular from Theorem 7 that T˛ and T˛0 are homeomorphisms from H˛ ;0;0C into itself for 0 < ˛ < 1.

T/ is integrable on any Œ0; T , so on Hˇ; ;ı if ˛CˇC > 0. t/ is integrable on any ŒT; 1/, so in particular on Hˇ; ;ı if 2˛ > ˇ C ı. s/ 0 ds s (43) are the inverse transformation of each other. Theorem 7. C . e maps continuously Hˇ; ;ı into Hˇ; C˛;ıC˛ if ˇ C C ˛ > 0 The operator ˘ e ˛2 ˘ e ˛1 D ˘ e ˛1 C˛2 on and ˇ C ı C ˛ > 0. It satisfies the composition rule ˘ Hˇ; ;ı if ˇ C C ˛1 > 0 and ˇ C C ˛1 C ˛2 > 0. ˛ 22 J. ˛ ˇ / . If moreover 2˛ > ˇ C ı and 2˛ > ˇ C , the operator T˛0 satisfies the same property.

T s/H J 1 ds proportional to t H J . This equality can be extended to any function f of HJ , so in particular to W in the case J D 1=2; we deduce the differentiability announced in the theorem. 1. 0/ D 0 and D 1 h is in L2 . 1=2 1 A dt 0 Ä C0 T1 H sup jD 1 f j C T 2 H sup jD 2 f j Á (74) from Theorem 4. 1) and taken from [35]. We use the equivalence of Hilbert spaces (H H0 ) defined in (12). 1, see (100). 0 are equivalent on RC . Theorem 19. For 0 < H < 1, the spaces HH and HH Proof. The proof is divided into the two inclusions; for the second one, we are going to use an analytical result proved in Appendix A.