This paper is devoted to the proof of Donsker's theorem for backward stochastic differential equations (BSDEs for short). The main objective is to give a simple method to discretize in time a BSDE. Our approach is based upon the notion of ``convergence of filtrations'' and covers the case of a $(y,z)$-dependent generator.

Equivalent upper and lower bounds for the $L^1$ norm of Hilbert space valued infinitely divisible random variables are obtained and used to find bounds for different types of stochastic integrals.

We consider ballistic diffusion limited aggregation on a finite strip $[0, L-1]$ times $\mathbb{Z}_+$ in $\mathbb{Z}^2$ for some $L$ in $\mathbb{Z}_+$. We provide numerical bounds on the growth in the height process.

We consider a continuous time random walk on the $d$-dimensional lattice $\mathbb{Z}^d$: the jump rates are time dependent, but symmetric and strongly elliptic with ellipticity constants independent of time. We investigate the implications of heat kernel estimates on recurrence-transience properties of the walk and we give conditions for recurrence as well as for transience: we give applications of these conditions and discuss them in relation with the (optimal) Wiener test available in the time independent context. Our approach relies on estimates on the time spent by the walk in a set and on a 0-1 law. We show also that, still via heat kernel estimates, one can avoid using a 0-1 law, achieving this way quantitative estimates on more general hitting probabilities.

In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different proof of the main technical lemma that guarantees the existence of a certain kernel. Moreover, we generalize the result of Talagrand to a family of kernels which in one particular case allows us to produce the Poissonian bound without using the truncation argument. We also give some examples of applications of the abstract concentration inequality to empirical processes that demonstrate some interesting properties of Talagrand's kernel method.

In 1971, Yamada and Watanabe showed that pathwise uniqueness holds for the SDE $dX= \sigma (X)dB$ when sigma takes values in the n-by-m matrices and satisfies $|\sigma (x)- \sigma (y)| < |x-y|\log(1/|x-y|)^{1/2}$. When $n=m=2$ and $\sigma$ is of the form $\sigma _{ij}(x)= \delta_{ij}s(x)$, they showed that this condition can be relaxed to $| \sigma(x)-\sigma(y)| < |x-y|\log(1/|x-y|)$, leaving open the question whether this is true for general $ 2\times m$ matrices. We construct a $2\times 1$ matrix-valued function which negatively answers this question. The construction demonstrates an unexpected effect, namely, that fluctuations in the radial direction may stabilize a particle in the origin.

Let $(\xi_k, k\ge 0)$ be a Markov chain on ${-1,+1}$ with $\xi_0=1$ and transition probabilities $P(\xi_{k+1}=1| \xi_k=1)=a>b=P(\xi_{k+1}=-1| \xi_k=-1)$. Set $X_0=0$, $X_n=\xi_1+\cdots +\xi_n$ and $M_n=\max_{0\le k\le n}X_k$. We prove that the process $2M-X$ has the same law as that of $X$ conditioned to stay non-negative.

Given a probability law $\pi$ on a set $S$ and a function $g : S \rightarrow R$, suppose one wants to estimate the mean $\bar{g} = \int g d\pi$. The Markov Chain Monte Carlo method consists of inventing and simulating a Markov chain with stationary distribution $\pi$. Typically one has no a priori bounds on the chain's mixing time, so even if simulations suggest rapid mixing one cannot infer rigorous confidence intervals for $\bar{g}$. But suppose there is also a separate method which (slowly) gives samples exactly from $\pi$. Using $n$ exact samples, one could immediately get a confidence interval of length $O(n^{-1/2})$. But one can do better. Use each exact sample as the initial state of a Markov chain, and run each of these $n$ chains for $m$ steps. We show how to construct confidence intervals which are always valid, and which, if the (unknown) relaxation time of the chain is sufficiently small relative to $m/n$, have length $O(n^{-1} \log n)$ with high probability.

We give a new relatively compact proof of the famous identity for the distribution of the first hitting time of a linear boundary by a skip-free process with stationary independent increments. The proof uses martingale identities and change of measure.

Let $\xi$ be a subordinator with Laplace exponent $\Phi$, $I=\int_{0}^{\infty}\exp(-\xi_s)ds$ the so-called exponential functional, and $X$ (respectively, $\hat X$) the self-similar Markov process obtained from $\xi$ (respectively, from $\hat{\xi}=-\xi$) by Lamperti's transformation. We establish the existence of a unique probability measure $\rho$ on $]0,\infty[$ with $k$-th moment given for every $k\in N$ by the product $\Phi(1)\cdots\Phi(k)$, and which bears some remarkable connections with the preceding variables. In particular we show that if $R$ is an independent random variable with law $\rho$ then $IR$ is a standard exponential variable, that the function $t\to E(1/X_t)$ coincides with the Laplace transform of $\rho$, and that $\rho$ is the $1$-invariant distribution of the sub-markovian process $\hat X$. A number of known factorizations of an exponential variable are shown to be of the preceding form $IR$ for various subordinators $\xi$.

Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$. This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process;. The purpose of this note is to remark that, assuming $n > p$, the eigenvalues of $M(t)$ evolve like $p$ independent squared Bessel processes of dimension $2(n-p+1)$, conditioned (in the sense of Doob) never to collide. More precisely, the function $h(x)=\prod_{i < j}(x_i-x_j)$ is harmonic with respect to $p$ independent squared Bessel processes of dimension $2(n-p+1)$, and the eigenvalue process has the same law as the corresponding Doob $h$-transform. In the case where the entries of $A(t)$ are real Brownian motions, $(M(t))_{t > 0}$ is the Wishart process considered by Bru (1991). There it is shown that the eigenvalues of $M(t)$ evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same $h$-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time.

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that $0$ is surrounded by the union of $K$ and $A$.