One Day Workshop on SPDEs
Date: December 2, 2022
Location: Radboud University, Nijmegen
Organizers: Gabriel Lord, Stefanie Sonner
10:30-11:15 Sonja Cox (University of Amsterdam)
11:15-12:00 Peter Kloeden (University of Tübingen)
13:30-14:15 Christian Hamster (Wageningen University)
14:15-15:00 Oliver Tse (TU Eindhoven)
Titles and abstracts
Sonja Cox Affine infinite-dimensional stochastic covariance models
Infinite-dimensional stochastic covariance models are used e.g. for forward prices (of currencies or commodities). Such models require the construction of a flexible yet tractable class of stochastic process taking values in the cone of positive operators. In the finite-dimensional (i.e., matrix valued) setting, so-called `affine covariance models' are popular -- examples include the Wishart processes (1991) and the Barndorff-Nielsen and Stelzer model (2007). A full characterization of the class of all affine processes taking values in the cone of positive semi-definite matrices was given by Cuchiero et al. (2011).
We have constructed a class of affine infinite-dimensional stochastic processes taking values in the cone of positive Hilbert-Schmidt operators that allows for state-dependent jumps. Diffusion causes difficulties: we have shown that an infinite-dimensional Wishart process practically only exist if the initial value of the process is of finite rank -- in this case, the process remains of finite rank.
As mentioned above, these affine models are popular due to their tractablility: the characteristic funtion is given in terms of (infinite-dimensional) Riccati equations. Nevertheless, simulations pose some challenges that I will also briefly discuss.
Joint work with Christa Cuchiero, Sven Karbach, and Asma Khedher.
Peter Kloeden RODEs and their numerical approximation
Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion, but when it is a diffusuion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors. They are also useful in biology.
Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the
solutions of RODES can be obtained when the stochastic process has H\"older continuous sample paths and then used to derive pathwise convergent numerical schemes of arbitrarily high order. RODEs with Ito noise will also be considered as well as RODEs with affine structure and Poisson noise.
- Xiaoying Han and E. Kloeden, Random Ordinary Differential Equations and their Numerical Solution, Springer Nature Singapore, 2017.
Christian Hamster Understanding Stochastic Waves in Cell Movement Models, from Modelling to (S)PDEs
Single-cell organisms are remarkably good at sensing food, especially if you consider that they lack our sensing organs and have to measure a gradient in the food supply over the length of a single cell. The precise mechanisms behind this gradient sensing are not fully understood yet. Still, scientists have determined many molecules that are relevant to the motion of the cell and we can see how these molecules are activated in wavelike patterns. These processes can be used to build Gillespie-type stochastic models for cell movement. These models are complex, both numerically and analytically, so we often summarise everything into 'simpler' PDEs. In this talk, I would like to advocate an in-between option, so-called Chemical Langevin Equations, effectively an SPDE approximation of the Gillespie algorithms. This approach allows us to use all the insights from the underlying deterministic PDE, without throwing away the stochastic nature of the models.
Oliver Tse Large deviations for singularly interacting diffusions
The large deviations of interacting diffusions have been a field of interest since the seminal work of D. A. Dawson and J. Gärtner in 1987. The large deviation principle not only provides the almost sure convergence of these interacting diffusions to their corresponding mean-field limits—so-called McKean-Vlasov equations—but also provides a variational formulation for the limiting distribution.
In this talk, I will give a brief introduction to the basic concepts of large-deviation theory, particularly Sanov’s theorem and Varadhan’s integral lemma. I will then discuss how these concepts can be used to prove large-deviations principles for interacting diffusions, mentioning the technical difficulties in applying standard theory and illuminating the need for extensions of the standard theory. The last part of the talk will focus on insights into an extension that is applicable for interacting diffusions with singular interacting kernels.