Infinite dimensional stochastic calculus via regularization

This paper develops some aspects of stochastic calculus via regularization to Banach valued processes. An original concept of χ-quadratic variation is introduced, where χ is a subspace of the dual of a tensor product B ⊗ B where B is the values space of some process X process. Particular interest is devoted to the case when B is the space of real continuous functions defined on [−τ, 0], τ > 0. Itô formulae and stability of finite χ-quadratic variation processes are established. Attention is deserved to a finite real quadratic variation (for instance Dirichlet, weak Dirichlet) process X. The C([−τ, 0])-valued process X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0], is called window process. Let T > 0. If X is a finite quadratic variation process such that [X]t = t and h = H(XT (·)) where H : C([−T, 0]) −→ R is L([−T, 0])-smooth or H non smooth but finitely based it is possible to represent h as a sum of a real H0 plus a forward integral of type ∫ T 0 ξd−X where H0 and ξ are explicitly given. This representation result will be strictly linked with a function u : [0, T ]× C([−T, 0]) −→ R which in general solves an infinite dimensional partial differential equation with the property H0 = u(0, X0(·)), ξt = D0u(t,Xt(·)) := Du(t,Xt(·))({0}). This decomposition generalizes the Clark-Ocone formula which is true when X is the standard Brownian motion W . The financial perspective of this work is related to hedging theory of path dependent options without semimartingales. [2010 Math Subject Classification: ] 60G15, 60G22, 60H05, 60H07, 60H30, 91G10, 91G80