- 27 March 2012, The University of Melbourne
- Organiser: Kostya Borovkov
Speakers and talks
- Andrew Barbour (Melbourne & Monash Unis) . The asymptotics of the Aldous gossip process.
- Daniel Dufresne (Melbourne Uni). Gram-Charlier Distributions.
- Fima Klebaner (Monash Uni). Law of Large Numbers for the age distribution in population dependent branching process.
- Owen Jones (Melbourne Uni). Exact simulation of diffusions.
Abstracts
Andrew Barbour. The asymptotics of the Aldous gossip process.
As a model for the spread of gossip, Aldous used the (discrete) 2-D torus to represent space, with gossip spreading between neighbours, but also occasionally at long range. The development of a continuous space version was shown by Durrett and Chatterjee to have some randomness at the start, but thereafter to run an almost deterministic course, described by a mysterious function $h$, that also appears in Aldous’s paper. Here, we explain the asymptotics, and identify $h$, entirely in terms of branching processes. Our arguments remain valid for the spread on quite general, locally homogeneous manifolds, in any number of dimensions.
Daniel Dufresne. Gram-Charlier Distributions.
Not the greatest model for stock returns, Gram-Charlier distributions still have some interest. Will talk about their history and properties, maybe option pricing.
Fima Klebaner. Law of Large Numbers for the age distribution in population dependent branching process.
We consider a population of particles evolving in continuous time. If the lifespans are not exponential such process (the Belman-Harris process) is not Markov. However, considered as a collection of ages, as a finite counting measure $A=\sum_a \delta_a,$ it is Markov. We give the generator of such process, and its semimartingale decomposition. Assuming further that the parameters (lifespans, offspring distributions) depend on the population composition, in such a way that it is supercritical below some threshold K and subcritical above it. We prove Law of Large Numbers for the measure valued process $\frac{1}{K}A_t^K$ as K ⭢ ∞, and describe the measure-valued limit $A_t$. Convergence is shown in the space of trajectories, the Skorohod space $D(R^+,M(R^+))$, where $M(R^+))$ is the space of measures on $R^+$ metrizable by weak convergence. This is joint work with K. Hamza (Monash) and P. Jagers (Chalmers).
Owen Jones . Exact simulation of diffusions.
Abstract: Some recent results by Nan Chen (Hong Kong) and Giesecke and Smelovy (Stanford), on the exact simulation of diffusion, are described. Unlike the approach of Beskos and Roberts (2005), the diffusion is simulated on a regular spatial lattice, rather than at regular points in time.
