Pathwise integration with respect to paths of finite quadratic variation

Abstract

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation

Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

“…Using the concept of the vertical derivative of a functional [8], we extend these results to regular path-dependent functionals of such paths. We obtain an 'isometry' formula in terms of p-th order variations for the pathwise integral and a 'signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation, extending the results of [1] obtained for the case p = 2 to arbitrary even integers p ≥ 2.…”

“…Proof. The proof is similar to the case p = 2 considered in [1]. Indeed, our assumptions allow us to apply [1, Lemma 2.2], which shows that there exists C > 0, only depending on T , F , and S α , such that for all 0…”

“…Using the above result we may derive, as in [1], a pathwise 'signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation. Let…”

“…This result has many interesting ramifications and applications in the pathwise approach to stochastic analysis, and has been extended in different ways, to less regular functions using the notion of pathwise local time [2,10,24], as well as to path-dependent functionals and integrands [1,7,8,25]. The central role played by the concept of quadratic variation has led to the assumption that they do not extend to less regular paths with infinite quadratic variation.…”

On the support of solutions to stochastic differential equations with path-dependent coefficients

“…We construct Hölder-continuous approximations of the solution using an adapted linear interpolation of Brownian motion and show that this approximation converges in probability to the solution in Hölder norm. A key ingredient is the use of functional estimates derived in [3] using the Functional Ito calculus, combined with interpolation error estimates in Hölder norm for stochastic processes.…”