Searcharxiv⌕ Search

arXiv subjects

Anna Ananova

Publications and source records attributed to Anna Ananova.

5 recordsLinked to original sources
arXiv2026

Functional representation and functional calculus for controlled paths

We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along H\"older controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for $\gamma-$H\"older controls with $\tfrac{1}{2}\geq \gamma>\sqrt{2}-1$, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals $G_0,\ldots,G_p$ which satisfy compatible higher-order controlled Taylor estimates along $\gamma$-H\"older paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_\omega^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional It\^o calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.

math.FA↗
arXiv2026

A pathwise Ito formula for weakly differentiable functions

We prove a version of F\"ollmer's pathwise It\^o formula for weakly differentiable functions of continuous paths with finite quadratic variation along a sequence of partitions. For each such path $\omega$, we introduce a path-dependent Sobolev space $W_{\omega,\pi}^{2}$ defined through smooth approximation of the weak Hessian in a seminorm generated by discrete weighted occupation measures of the path $\omega$. For $F\in W_{\omega,\pi}^{2}$, we construct the pathwise integral $\int \nabla F(\omega)\,d^{\pi}\omega$ and the covariation $[\nabla F(\omega),\omega]_{\pi}$, and prove the change-of-variable formula $$ F(\omega(t))-F(\omega(0)) = \int_{0}^{t}\nabla F(\omega(s))\,d^{\pi}\omega(s) + \frac12[\nabla F(\omega),\omega]_{\pi}(t). $$ Our result does not require any assumption on the existence of local time for the path; the Ito term appears as a quadratic covariation. For Brownian motion, we show that functions in $W^{2+,p}(\mathbb{R}^{d})\cap W^{2,1}(\mathbb{R}^{d})$ belong almost surely to the corresponding path-dependent space, outside a polar exceptional set of starting points. If, in addition, $F\in W_{\mathrm{loc}}^{1,2}(\mathbb{R}^{d})$, the pathwise integral agrees with the stochastic It\^o integral, yielding a pathwise version of the multidimensional F\"ollmer--Protter formula.

math.PR↗
arXiv2020

Model-free Analysis of Dynamic Trading Strategies

We introduce a model-free approach for analyzing the risk and return for a broad class of dynamic trading strategies, including pairs trading, mean-reversion trading and other statistical arbitrage strategies, in terms of excursions of a trading signal away from a reference level. Our results are derived in a pathwise setting, without any probabilistic assumptions. We introduce the notion of {\delta}-excursion, defined as a path which deviates by {\delta} from a reference level before returning to this level. We show that every continuous path has a unique decomposition into {\delta}-excursions. This decomposition is useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss, and drawdown. We show that the high-frequency limit of mean-reversion strategies may be described in terms of the (p-th order) local time of the signal. In particular, our results yield a financial interpretation of the local time of an irregular path. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

q-fin.MF↗
arXiv2016

Pathwise integration with respect to paths of finite quadratic variation

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.

math.PR↗