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Stochastic calculus via regularization in Banach spaces and applications

Russo, F (ENSTA ParisTech)
Monday 10 September 2012, 16:50-17:30

Seminar Room 1, Newton Institute


This talk is based on collaborations with Cristina Di Girolami (Univ. Le Mans) and Giorgio Fabbri (Univ. Evry). Finite dimensional calculus via regularization was first introduced by the speaker and P. Vallois in 1991. One major tool in the framework of that calculus is the notion of covariation [X, Y ] (resp. quadratic variation [X]) of two real processes X, Y (resp. of a real process X). If [X] exists, X is called finite quadratic variation process. Of course when X and Y are semimartingales then [X, Y ] is the classical square bracket. However, also many real non-semimartingales have that property. Particular cases are F¨ollmer-Dirichlet and weak Dirichlet processes, introduced by M. Errami, F. Gozzi and the speaker. Let (Ft, t 2 [0, T]) be a fixed filtration. A weak Dirichlet process is the sum of a local martingale M plus a process A such that [A,N] = 0 with respect to all the local martingales related to the given filtration. The lecture presents the extension of that theory to the case when the integrator process takes values in a Banach space B. In that case very few processes have a finite quadratic variation in the classical sense of M´etivier-Pellaumail. An original concept of quadratic variation (or -quadraticvariation) is introduced, where  is a subspace of the dual of the projective tensor product B ˆ B.

Two main applications are considered.

• Case B = C([-T, 0]). One can express a Clark-Ocone representation formula of a pathdependent random variable with respect to an underlying which is a non-semimartingale withe finite quadratic variation. The representation is linked to the solution of an infinite dimensional PDE on [0, T] × B. • Case when B is a separable Hilbert space H. One investigates quadratic variations of processes which are solutions of an evolution equation, typically a mild solution of SPDEs.


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