Many reallife phenomena, like weather patterns or prices of stocks exhibit jumptype behavior.
In fact, from a certain mathematical perspective, the jumptype phenomena–understood as Lévytype stochastic processes–may be considered more general than the continuous phenomena–understood as diffusion processes.
The theory of Lévytype processes belongs to the field of stochastic processes.
However, because of numerous and deep links with other areas of mathematics, it is also significant for potential theory, the theory of nonlocal partial differential equations, as well as statistics and financial mathematics.
The first workshop "Nonlocal Operators and Markov Processes" on
2630 October 2020 focused on the interplay of Lévytype processes with nonlocal partial differential equations and potential theory. The second workshop "Nonlocal Operators and Markov Processes II" on 2226 March 2021 is devoted to the statistics of stochastic processes, especially jump processes in continuous time. We will discuss probabilistic models and statistical methods  theory, applications and simulation techniques. The workshop is addressed to people with different backgrounds, including students and PhD students.
The workshops are organised as part of the
Polish  German Beethoven Classic 3 grant "Sensitivity Analysis of Nonlocal Operators with Applications to Jump Processes" from National Science Center (Poland) 2018/31/G/ST1/02252 and German Research Foundation SCHI491/111,
carried out under the supervision of René Schilling and Krzysztof Bogdan.
Agenda:
Date: Mon 22March2021  Fri 26March2021
Format: Virtual Zoom Conference
Time: 3 talks a day: 10:0011:00, 11:0012.00 and 14:0015:00 (GMT+1h)
In this workshop we focus on:
 parametrix constructions in stochastic processes
 statistics of stochastic processes
 numerical aspects of the above
Schedule:
Times are given in DresdenWroclaw (GMT+1h). The contents of courses and individual talks are given below the timetable.
Monday, 22 Mar:
10:0011:00
11:0012:00
14:0015:00
Tuesday, 23 Mar:
10:0011:00
11:0012:00
14:0015:00
Wednesday, 24 Mar:
10:0011:00
11:0012:00
14:0015:00
Thursday, 25 Mar:
10:0011:00
11:0012:00
14:0015:00
Friday, 26 Mar:
10:0011:00
11:0012:00
14:0015:00
Contents of lectures:

Noufel Frikha (talk): Wellposedness of McKeanVlasov SDEs, related PDE on the Wasserstein space and some new quantitative estimates for propagation of chaos
 Summary: I will present some recent results on the wellposedness in the weak and strong sense of some nonlinear stochastic differential equations (in the sense of McKeanVlasov) driven by Brownian and/or jump processes which go beyond those derived from the standard CauchyLipschitz theory (see e.g. the monograph of Sznitman). Then, in the Brownian setting, I will show how the underlying noise regularizes the equation and allows to prove that the transition density of the dynamics exists and is smooth, especially in the measure direction, under the uniform ellipticity assumption. Such smoothing effects then in turn allow to establish the existence and uniqueness for the Cauchy problem associated to a Kolmogorov PDE stated on the Wasserstein space (the space of probability measures with finite second order moment) with irregular terminal condition and source term. This PDE stated on an infinite dimensional space plays a key role in deriving new quantitative estimates of propagation of chaos for the meanfield approximation by systems of interacting particles. This presentation is based on several recent works in collaboration with: P.E. Chaudru de Raynal (Université Savoie Mont Blanc), V. Konakov (HSE Moscow), L. Li (UNSW Sydney) and S. Menozzi (Université d'Evry Val d'Essone).
 Reinhard Höpfner (course): Point process models and local asymptotics in statistics
 Likelihood ratio processes for point processes (purely probabilistic)
 Main results of the statistical theory of local asymptotic normality (LAN) in the sense of LeCam (purely statistical)
 Detailed treatment of one example where probability and statistics together allow to specify asymptotically optimal estimators for parameters in a multivariate point process model
 Arnaud Gloter (course): Malliavin calculus for pure jump processes and statistical applications
 Malliavin calculus and integration by parts setting
 Malliavin calculus on the Poisson space and application to S.D.E driven by jump processes
 Statistical applications for S.D.E driven by stable like processes: approximation of the score function and LAMN properties
 Arturo KohatsuHiga (course): Interaction between simulation and theory
 Simulation of Poisson random measures
 Exact simulation methods for jump driven SDE's
 Simulation methods for infinite activity Lévy processes and their associated functionals
 Summary: I will mostly concentrate on simulation methodology with associated theoretical results such as convergence results and their rate. I also plan to discuss the interactions between theory and applications. Some interesting and feasible open problems will be discussed, too.
 Alex Kulik (course): Lectures on the parametrix method
 Basic constructions: diffusion processes and parabolic PDEs
 Sensitivities and approximations
 Nonlocal PDEs: new effects and methods

Summary:
I will provide a selfcontained introduction to the parametrix method, which is a powerful analytical tool for construction and investigation of heat kernels of diffusion and Lévytype processes.
 Andrea Pascucci (talk): A class of Kolmogorov SPDEs and applications to stochastic filtering
 René Schilling (talk): Detecting Independence with Distance Multivariance based on Lévy Metrics

Summary: We introduce two new measures for the dependence of n>1 random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted L^2distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to ntuplets of random variables. We show that total distance multivariance can be used to detect the independence of n random variables and has a simple finitesample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.
This is joint work with B. Böttcher and M. KellerRessel.