The covariation for Banach space valued processes and applications

Abstract

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 ⊗π B2) * ), 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 ⊗π B1) * 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]) −→ R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u : [0, T ] × C([−T, 0]) −→ R solving an infinite dimensional partial differential equation.

Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion

“…For the particular case of window processes, we also refer to Theorem 6.3 and Sect. 7.2 in [12]. In the present paper, we prove formulae which allow to express functional derivatives in terms of differential operators arising in the Banach space valued stochastic calculus via regularization, with the aim of identifying the building blocks of our functional Itô's formula with the terms appearing in the Itô's formula for window processes.…”

“…This new branch of stochastic calculus has been recently conceived and developed in many directions in [12,[14][15][16]; for more details see [13]. For the particular case of window processes, we also refer to Theorem 6.3 and Sect.…”

“…In particular, unlike (26), since we do not call in the future in our formula (16), we do not have to specify a future time evolution for X , but only a past evolution before time 0. This difference between (16) and (26) is crucial for the proof of the functional Itô's formula. In particular, the adoption of (26) as definition for the horizontal derivative would require an additional regularity condition on u in order to prove an Itô formula for the process t → u(X t ).…”

“…In the present subsection our aim is to make a link between functional Itô calculus, as derived in this paper, and Banach space valued stochastic calculus via regularization for window processes, which has been conceived in [13], see also [12,[14][15][16] …”

“…For example, in [8] a slightly more restrictive definition of strong-viscosity solution was adopted, see Remark 12. Recently, a new branch of stochastic calculus has appeared, known as functional Itô calculus, which results to be an extension of classical Itô calculus to functionals depending on the entire path of a stochastic process and not only on its current value, see Dupire [17], Cont and Fournié [5][6][7]. Independently, Di Girolami and Russo, and more recently Fabbri, Di Girolami, and Russo, have introduced a stochastic calculus via regularizations for processes taking values in a separable Banach space B (see [12][13][14][15][16]), including the case B = C([−T, 0]), which concerns the applications to the path-dependent calculus.…”