# Parameter estimation for SPDEs based on discrete observation in time and space

#### 17.12.2018, 10:15  –  Golm, Haus 9, Raum 2.22 Forschungsseminar Wahrscheinlichkeitstheorie

Florian Hildebrandt (Hamburg)

Stochastic partial differential equations (SPDEs) are becoming increasingly popular for modeling phenomena from the natural sciences and finance and thus require statistical methods for their calibration. We study parameter estimation for the parabolic, linear, second order SPDE

<tex>dX_t = (\theta_2 \Delta X_t +\theta_1 \nabla X_t + \theta_0 X_t)\,dt + \sigma dW_t</tex>

with Dirichlet boundary condition in dimension one. Assuming a discrete observation pattern given by a grid in time and space, our aim is to understand the interplay between temporal and spatial sampling frequencies. Focusing on volatility estimation, recent works study a high frequency regime in time with only few spatial observations. Instead, we first consider a larger space than time sampling frequency. Using space increments, we are able to construct an asymptotically normal and efficient estimator. Next, we present a volatility estimator based on space-time increments. It turns out that although the analysis of this estimator works quite differently across different sampling regimes it always satisfies a central limit theorem with parametric rate of convergence. Finally, we discuss first results about estimation of (subsets of) the whole parameter vector .

The talk is based on joint work with Mathias Trabs.

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