arXiv subjects

Fabrice Wunderlich

Publications and source records attributed to Fabrice Wunderlich.

5 recordsLinked to original sources

arXiv2025

Relative compactness of It\^o integrals on the M1 Skorokhod space with identification of limits

We establish general results for weak relative compactness of sequences of It\^o integrals with respect to Skorohod's functional M1 topology, under sharp general conditions. Moreover, we are able to explicitly characterise the form of the limit points of all convergent subsequences. This result closes a longstanding gap in the literature, where weak relative compactness had previously only been shown under the significantly more restrictive J1 topology, or the S topology which is too coarse to preserve continuity for most common functionals.

math.PR↗

arXiv2024

Functional CLTs for subordinated Lévy models in physics, finance, and econometrics

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) Lévy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.

math.PR↗

arXiv2024

Functional weak convergence of stochastic integrals for moving averages and continuous-time random walks

There is an extensive theory of weak convergence for moving averages and continuous-time random walks (CTRWs) with respect to Skorokhod's M1 and J1 topologies. Here we address the fundamental question of how this translates into functional limit theorems in the M1 or J1 topology for stochastic integrals driven by these processes. As an important application, we provide weak approximation results for general SDEs driven by time-changed L\'evy processes. Such SDEs and their associated fractional Fokker--Planck--Kolmogorov equations are central to models of anomalous diffusion in statistical physics. Our results yield a rigorous functional characterisation of these as continuum limits of the underlying models driven by CTRWs. With regard to strictly M1 convergent moving averages and correlated CTRWs, it turns out that the convergence of stochastic integrals can fail decidedly and fundamental new challenges arise compared to the J1 setting. Nevertheless, we identify natural classes of integrand processes for which there is M1 convergence of the stochastic integrals. We also show that these results are flexible enough to yield functional limit theorems in the M1 topology for certain stochastic delay differential equations driven by moving averages.

math.PR↗

arXiv2023

Weak Convergence of Stochastic Integrals on Skorokhod Space in Skorokhod's J1 and M1 Topologies

We provide criteria for It\^o integration to behave continuously with respect to Skorokhod's J1 and M1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.

math.PR↗

arXiv2020

Asymptotic strong Feller property and local weak irreducibility via generalized couplings

In this short note we show how the asymptotic strong Feller property (ASF) and local weak irreducibility can be established via generalized couplings. We also prove that a stronger form of ASF together with local weak irreducibility implies uniqueness of an invariant measure. The latter result is optimal in a certain sense and complements some of the corresponding results of Hairer, Mattingly (2008).

math.PR↗