The Lamperti representation of multitype branching processes
Löic Chaumont, University of Angers
Local vs. global growth for spatial branching processes
Janos Englander, University of Colorado, Boulder
We will consider spatial branching processes (discrete particle systems and superprocesses) and review some recent results and examples that concern the large time growth (or decay) of these processes -- both locally and globally. Exponential as well as super-exponential growth will be studied. These results are related to spine techniques and also to the spectral and gauge theories of elliptic operators with potentials.
The talk is based on a joint project with Z-Q. Chen (Seattle) and on another one with Y. Ren (Beijing) and R. Song (Urbana).
Mixed zero-sum controller stopper games
Daniel Hernández-Hernández, CIMAT
In this talk we consider a stochastic differential equation that is controlled by means of an additive finite-variation process. A singular stochastic controller, who is a minimizer, determines this finite variation process while a discretionary stopper, who is a maximizer, chooses a stopping time at which the game terminates. We consider two closely related games that are differentiated by whether the controller or the stopper has a first-move advantage. The games' performance indices involve a running payoff as well as a terminal payoff and penalize control effort expenditure. We derive a set of variational inequalities that can fully characterize the games' value functions as well as yield Markovian optimal strategies. In particular, we derive the explicit solutions to two special cases and we show that, in general, the games' value functions fail to be continuously differentiable. We shall also provide some recent results for controlled Lévy processes in this context.
Limit theorems for ambit processes
Mark Podolskij, Heidelberg University
In this talk we will present some limit theorems for ambit processes. Ambit processes are stochastic models of moving average structure with additional stochastic component. We concentrate on the asymptotic behavior of power variation of ambit processes. We will see that the limit theory strongly depends on the driving Levy motion. When the driver is a Brownian motion the weak convergence is towards a mixed Gaussian law. When the driving motion is a pure jump Levy process some non-standard limits appear.
Strong laws of large numbers for supercritical superdiffusions
Andreas E. Kyprianou, Universy of Bath
One of the most fundamental questions regarding spatial branching particle systems is how they distribute mass through space. In this respect, there is a long history in the literature looking at laws of large numbers on compact domains. Getting strong limit theorems has proved to be a challenge, particularly for superprocesses. In this talk we outline a new approach to this using a skeletal decomposition of superprocesses to bootstrap results concerning particle processes into the superprocess setting. This is joint work with Maren Eckhoff and Matthias Winkel.