CS26: Inference for stochastic equations

date: 7/17/2025, time: 16:00-17:30, room: IM WS

Organizer: Ciprian Tudor (University of Lille, France)

Chair: Ciprian Tudor (University of Lille, France)

Parameter estimation for SDEs with Rosenblatt noise

Petr Čoupek (Charles University, Faculty of Mathematics and Physics)

The talk is devoted to parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes. The talk is based on the recent article \([1]\).

Bibliography

\([1]\) Čoupek, P., Kříž, P., Maslowski, B. “Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes", Stochastic Processes and their Applications, vol. 179, 2025, art. no. 104499.

Statistical inference for semi-linear SPDEs using spatial information

Sascha Gaudlitz (Humboldt-Universität zu Berlin)

We consider the Bayesian non-parametric estimation of the reaction term in a semi-linear parabolic SPDE. Posterior contraction rates are proven by making use of the spatial ergodicity of the SPDE while the time horizon is fixed. We additionally prove a non-parametric Bernstein- von Mises Theorem for the posterior distribution. The analysis of the posterior requires new concentration results for spatial averages of transformation of the SPDE, which are based the combination of the Clark-Ocone formula with bounds on the marginal densities.

Inference for the nonlinear stochastic heat equation

Ciprian Tudor (University of Lille)

We consider the nonlinear stochastic heat equation with fractional Laplacian, driven by the Gaussian space-time white noise and we analyse the asymptotic behavior of the quadratic and higher order variations of its mild solution. The idea is to approximate the increments of the solution to the nonlinear heat equation with those of the solution to the linear heat equation (which is related to the fractional Brownian motion). Based on these variations, we construct estimators for several parameters that may appear in such a model: the drift and diffusion parameters or the parameter associated with the order of the fractional Laplacian.