Conference Agenda

Based on discrete observations within the nonparametric Gaussian white noise model $dY_t = sigma(t)dW_t$, we develop a test to infer if the volatility function $sigma(cdot)$ is constant. In particular, at prescribed significance, we simultaneously identify those time intervals where a violation of the constancy hypothesis occurs without a priori knowledge of their number and size. The testing procedure is shown to be minimax-optimal and adaptive for infill asymptotics and these results entail that a deviation from the null hypothesis of constancy is best measured in terms of $sup_{tin [0,1]}|sigma(t)^2 /|sigma|_{L^2}^2 - 1|$. The derivation of the optimal constants requires to build hypotheses with height solving $F_n(x)=0$ for given functions $F_n$ and to understand the asymptotic behavior of their solution, which is done using the implicit function theorem.

We study the Langevin stochastic differential equation and its discrete approximations: the Euler–Maruyama scheme, commonly referred to as the Unadjusted Langevin Algorithm (ULA), and direct sampling from the continuous-time process. We show that the ULA process is geometrically ergodic in $mathbb{R}^d$ under suitable conditions and derive a corresponding drift condition using a Foster–Lyapunov test function. We then analyze time-inhomogeneous approximations with diminishing step sizes and establish geometric recurrence for both chains—the ULA and the directly sampled chain.

Statistical analysis of stochastic processes increasingly relies on functional limit theorems for path-dependent estimators, particularly in the presence of jumps. Many estimators in econometrics and time series analysis, such as statistics used for cointegration testing, self-normalized inference, or high-frequency volatility estimation, can be expressed as functionals of stochastic integrals with random, data-dependent integrands, or as continuous-time limits thereof. Their asymptotic validity therefore hinges on weak convergence results that remain stable beyond the classical continuous-path regime. In particular, Skorokhod’s M1 topology becomes increasingly relevant, since it captures convergence in situations where large discontinuities are approximated by clusters of smaller jumps, a behavior that is typically not captured in the classical framework of the J1. Such phenomena arise naturally in econometrics and high-frequency data settings.

This talk develops a weak limit theory for stochastic integrals on the space of càdlàg paths under Skorokhod’s M1 topology. I present a new, self-contained approach based on good decompositions of semimartingale integrators, yielding tractable conditions under which Itô integration is continuous jointly in the integrator and integrand. The results unify classical J1 continuity theorems and provide new conclusions in M1. I also show that for families of local martingales, M1-tightness implies J1-tightness under a mild localised uniform integrability condition. I conclude with a discussion of applications, including anomalous diffusion models represented as stochastic integrals with respect to continuous-time random walks.