IS01: Statistics for stochastic processes

date: 7/17/2025, time: 14:00-15:30, room: IM WS

Organizer: Fabienne Comte (Université Paris Cité, MAP5)

Chair: Fabienne Comte (Université Paris Cité, MAP5)

Fractional interacting particle system: drift parameter estimation via Malliavin calculus

Chiara Amorino (UNIVERSITAT POMPEU FABRA)

We address the problem of estimating the drift parameter in a system of \(N\) interacting particles driven by additive fractional Brownian motion of Hurst index \(H \geq 1/2\). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \(N \to \infty\). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \(H \in (0,1)\). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators.

This is based on a joint work with I. Nourdin and R. Shevchenko.

Adaptive minimax estimation for discretely observed Lévy processes

Céline Duval (Sorbonne Université - LPSM)

We study the nonparametric estimation of the density \(f_\Delta\) of an increment of a Lévy process based on observations with a sampling rate \(\Delta\). The class of Lévy processes considered is broad, including both processes with a Gaussian component and pure jump processes. A key focus is on processes where \(f_\Delta\) is smooth for all \(\Delta\). We use a spectral estimator of \(f_\Delta\) and derive both upper and lower bounds, showing that the estimator is minimax optimal in both low- and high-frequency regimes. In low-frequency settings, we recover parametric convergence rates, while in high-frequency settings, we identify two regimes based on whether the Gaussian or jump components dominate. The rates of convergence are closely tied to the jump activity, with continuity between the Gaussian case and more general jump processes.

This is based on a joint work with T. Jalal and E. Mariucci.

Estimation for SPDEs from noisy observations

Markus Reiß (Humboldt-Universität zu Berlin)

We consider stochastic evolution equations of the form \(dX(t)=A_\theta X(t)dt+BdW_t\) with a generator \(A_\theta\) on a Hilbert space, involving an unknown real or functional parameter \(\theta\). We consider observations \(dY(t)=X(t)dt+\varepsilon dV_t\), \(t\in[0,T]\), in space-time white noise \(dV\) and ask about optimal estimation of \(\theta\). Minimax lower bounds reveal a rich picture, which we shall describe in detail for second-order elliptic operators \(A_\theta=\nabla\cdot(\theta_2\nabla+\theta_1)+\theta_0\). Optimal rates depend on the order of the coefficient \(\theta_i\), the dimension and the asymptotics taken. An even richer structure appears for nonparametric estimation. The lower bound proofs rely on Hellinger bounds for cylindrical Gaussian measures and functional calculus for non-commuting, unbounded normal operators. A rate-optimal parametric estimator is obtained by a subtle preaveraging approach. Finally, a nonparametric diffusivity estimator and several open problems are presented.

Bibliography

\([1]\) Pasemann, G. and Reiß, M. (2024) Nonparametric Diffusivity Estimation for the Stochastic Heat Equation from Noisy Observations, arXiv, https://arxiv.org/abs/2410.00677

\([2]\) Pasemann, G. and Reiß, M. (2025), Information bounds for inference in stochastic evolution equations observed under noise, arXiv, https://arxiv.org/abs/2505.14051