Three lectures on Stochastic Processes


Organised jointly by UvA (Colloquium on Probability, Statistics and Financial Mathematics) and CWI (Spatial Stochastics Seminar)


Date and Location

October 23, 2001
Korteweg-de Vries Institute for Mathematics
Universiteit van Amsterdam
Plantage Muidergracht 24
Room P.016


Speakers


Program

14.30-15.15 Jean Mémin On the robustness of backward stochastic differential equations
15.15-16.00 Harry van Zanten On Donsker Theorems for Additive Functionals of Ergodic Diffusion Processes
16.00-16.15 coffee break
16.15-17.00 Marc Yor On subordinators, self-similar Markov processes and some factorizations of the exponential variable

There is ps and a pdf file with program and abstracts



Abstracts
Jean Mémin: On the robustness of backward stochastic differential equations
In this talk we study the robustness of backward stochastic differential equations (BSDE in short) with respect to the Brownian motion; more precisely we will show that if $W^n$ is a martingale approximation of a Brownian motion $W$ then the solution of the BSDE driven by the martingale $W^n$ converges to the solution of the classical BSDE, namely the BSDE driven by $W$. Here we will not assume that $W^n$ has the predictable representation property. As a byproduct of the result we obtain the convergence of the "Euler scheme" for BSDEs corresponding to the case where $W^n$ is a time discretization of $W$.
Harry van Zanten: On Donsker Theorems for Additive Functionals of Ergodic Diffusion Processes
In this talk we discuss the uniform central limit problem for additive functionals of an ergodic, $1$-dimensional diffusion process. We consider a regular diffusion $X$ on an open interval $I$, with finite speed measure $m$ and diffusion local time $(l_t(x): t \ge 0, x \in I)$. If $\Lambda$ is a collection of signed measure on $I$ and the total variations of these signed measures are uniformly bounded, we give a sufficient condition on $\Lambda$ under which the random map

\begin{displaymath} \lambda \mapsto \sqrt{t}\int_I \left(\frac{1}{t}l_t(x)-\frac{1}{m(I)}\right)\,\lambda(dx) \end{displaymath}

converges weakly, as $t \to \infty$, to a tight weak limit in the space $\ell^\infty(\Lambda)$ of bounded functions on $\Lambda$. The condition on $\Lambda$ is formulated in terms of the metric entropy of the class with respect to a suitable metric. We also discuss a number of applications of the abstract result.
Marc Yor: On subordinators, self-similar Markov processes and some factorizations of the exponential variable
In this lecture, I shall prove that if $I=\int_0^\infty ds\, \exp(-\xi_s)$ is the `exponential functional' associated to $(\xi_s,s\geq 0)$, a subordinator, then it is always a factor in a multiplicative decomposition of the exponential variable. I shall illustrate this result with several examples.