Related papers: Causal functional calculus
We derive a functional change of variable formula for {\it non-anticipative} functionals defined on the space of right continuous paths with left limits. The functional is only required to possess certain directional derivatives, which may…
We derive It\^o-type change of variable formulas for smooth functionals of irregular paths with non-zero $p-$th variation along a sequence of partitions where $p \geq 1$ is arbitrary, in terms of fractional derivative operators, extending…
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time…
The core of this article is a general theorem with a large number of specializations. Given a manifold $N$ and a finite number of one-parameter groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots, X_{(d)} $, we…
We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise…
We extend some results about F\"ollmer's pathwise It\^o calculus that have only been derived for continuous paths to c\`adl\`ag paths with quadratic variation. We study some fundamental properties of pathwise It\^o integrals with respect to…
Motivated by questions arising in financial mathematics, Dupire introduced a notion of smoothness for functionals of paths (different from the usual Fr\'echet--Gat\'eaux derivatives) and arrived at a generalization of It\=o's formula…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the…
Path dependence is omnipresent in many disciplines such as engineering, system theory and finance. It reflects the influence of the past on the future, often expressed through functionals. However, non-Markovian problems are often…
Based on an extension of the martingale comparison method some comparison results for path-dependent functions of semimartingales are established. The proof makes essential use of the functional It\^o calculus. A main tool is an extension…
In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration…
We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function.…
Recently, functional It\=o calculus has been introduced and developed in finite dimension for functionals of continuous semimartingales. With different techniques, we develop a functional It\=o calculus for functionals of Hilbert…
We develop a general framework for pathwise stochastic integration that extends F\"ollmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of It\^o, Stratonovich,…
We construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations. We also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function…
We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest,…
In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this formulation and…