arXiv subjects

Anna P. Kwossek

Publications and source records attributed to Anna P. Kwossek.

7 recordsLinked to original sources

arXiv2026

Universal approximation with signatures of non-geometric rough paths

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 the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Itô integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.

math.PR↗

arXiv2026

Stochastic differential equations driven by fractional Brownian motion: dependence on the Hurst parameter

Stochastic models with fractional Brownian motion as source of randomness have become popular since the early 2000s. Fractional Brownian motion (fBm) is a Gaussian process, whose covariance depends on the so-called Hurst parameter $H\in (0,1)$. Consequently, stochastic models with fBm also depend on the Hurst parameter $H$, and the stability of these models with respect to $H$ is an interesting and important question. In recent years, the continuous (or even smoother) dependence on the Hurst parameter has been studied for several stochastic models, including stochastic integrals with respect to fBm, stochastic differential equations (SDEs) driven by fBm and also stochastic partial differential equations with fractional noise, for different topologies, e.g., in law or almost surely, and for finite and infinite time horizons. In this manuscript, we give an overview of these results with a particular focus on SDE models.

math.PR↗

arXiv2025

Pathwise analysis of log-optimal portfolios

Based on the theory of càdlàg rough paths, we develop a pathwise approach to analyze stability and approximation properties of portfolios along individual price trajectories generated by standard models of financial markets. As a prototypical example from portfolio theory, we study the log-optimal portfolio in a classical investment-consumption optimization problem on a frictionless financial market modelled by an Itô diffusion process. We identify a fully deterministic framework that enables a pathwise construction of the log-optimal portfolio, for which we then establish pathwise stability estimates with respect to the underlying model parameters. We also derive pathwise error estimates arising from the time-discretization of the log-optimal portfolio and its associated capital process.

q-fin.MF↗

arXiv2025

A rough path approach to pathwise stochastic integration à la Föllmer

We develop a general framework for pathwise stochastic integration that extends Föllmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of Itô, Stratonovich, and backward Itô integration. More precisely, for a continuous path admitting both quadratic variation and Lévy area along a fixed sequence of partitions, we define pathwise stochastic integrals as limits of general Riemann sums and prove that they coincide with integrals defined with respect to suitable rough paths. Furthermore, we identify necessary and sufficient conditions under which the quadratic variation and the Lévy area of a continuous path are invariant with respect to the choice of partition sequences.

math.PR↗

arXiv2025

Functional differential equations driven by càdlàg rough paths

The existence of unique solutions is established for rough differential equations (RDEs) with path-dependent coefficients and driven by càdlàg rough paths. Moreover, it is shown that the associated solution map, also known as Itô-Lyons map, is locally Lipschitz continuous. These results are then applied to various classes of rough differential equations, such as controlled RDEs and RDEs with delay, as well as stochastic differential equations with delay. To that end, a joint rough path is constructed for a càdlàg martingale and its delayed version, that corresponds to stochastic Itô integration.

math.PR↗

arXiv2025

Pathwise convergence of the Euler scheme for rough and stochastic differential equations

The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by càdlàg paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, Itô processes, Lévy processes and general càdlàg semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes.

math.PR↗

arXiv2025

Universal approximation property of neural stochastic differential equations

We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of Itô diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.

math.PR↗