2010/12/11 by Cristina Di Girolami, Di Girolami, Cristina, Francesco Russo +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1012.2484
arxiv created 2013/02/27 · arxiv updated 2013/02/28
This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If \X and \Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗πB2)∗), we define the so-called χ-covariation of \X and \Y. If \X=\Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗πB1)∗ then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1=B2=C([-τ,0]) for some τ>0 and \X and \Y being window processes. If X is a real valued process, we call window process associated with X the C([-τ,0])-valued process \X:=X(⋅) defined by Xt(y) = Xt+y, where y ∈ [-τ,0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h=H(XT(⋅)), H:C([-T,0])\longrightarrow\R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u:[0,T]× C([-T,0])\longrightarrow ℝ solving an infinite dimensional partial differential equation.