We define a sequence of mappings $\Gamma_k:D_0(R_+)^k\to D_0(R_+)^k$ and prove the following result: Let $N_1,\ldots,N_n$ be the counting functions of independent Poisson processes on $R_+$ with respective intensities $\mu_1 \lt \mu_2 \lt \cdots \lt \mu_n$. The conditional law of $N_1,\ldots,N_n$, given that $$N_1(t)\le\cdots\le N_n(t), \mbox{ for all }t\ge 0,$$ is the same as the unconditional law of $\Gamma_n(N)$. From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman's representation for the 3-dimensional Bessel process.

The large deviation principle is established for the Fleming-Viot process with neutral mutation when the process starts from a point on the boundary. Since the diffusion coefficient is degenerate on the boundary, the boundary behavior of the process is investigated in detail. This leads to the explicit identification of the rate function, the quasi-potential, and the structure of the effective domain of the rate function.

We define a generalized Brownian bridge and we provide some information about its filtration. Two decompositions of this process as a semi-martingale are given. The first one is a Volterra decomposition and the second one is its canonical decomposition in its own filtration.

We present a class of processes which enjoy an exponential analogue of Pitman's $2M-X$ theorem, improving hence some works of H. Matsumoto and M. Yor.

The product of subsequent partial sums of independent, identically distributed, square integrable, positive random variables is asymptotically lognormal. The result extends in a rather routine way to non-degenerate U-statistics.

The inequality of Vapnik and Chervonenkis controls the expectation of the function by its sample average uniformly over a VC-major class of functions taking into account the size of the expectation.

Using Talagrand's kernel method we prove a similar result for the classes of functions for which Dudley's uniform entropy integral or bracketing entropy integral is finite.

For each $d\geq 2$, we give examples of $d$-dimensional periodic lattices on which the hard-core and Widom-Rowlinson models exhibit a phase transition which is monotonic, in the sense that there exists a critical value $\lambda_c$ for the activity parameter $\lambda$, such that there is a unique Gibbs measure (resp. multiple Gibbs measures) whenever $\lambda$ is less than $\lambda_c$ (resp. $\lambda$ greater than $\lambda_c$). This contrasts with earlier examples of such lattices, where the phase transition failed to be monotonic. The case of the cubic lattice $Z^d$ remains an open problem.

The purpose of this note is to give a PDE satisfied by a call option when the price process is a semimartingale. The main result generalizes the PDE in the case when the stock price is a diffusion. Its proof uses Meyer-Tanaka and occupation density formulae. Presented approach also gives a new insight into the classical Black-Scholes formula. Rigorous proofs of some known results are also given.

We present a reordered version of the inclusion-exclusion principle, which is useful when computing the probability of a union of events which are close to independent. The advantages of this formulation are demonstrated in the context of 3 classic problems in combinatorics.

A distribution (function) $F$ on $[0,1]$ with $F(t)$ less or equal $t$ for all $t$ is called subdiagonal. The extreme subdiagonal distributions are identified as those whose distribution functions are almost surely the identity, or equivalently for which $F \circ F = F$. There exists a close connection to exchangeable random orders on $\{1,2,3,...\}$.

In a first-passage percolation model on the square lattice $Z^2$, if the passage times are independent then the number of geodesics is either $0$ or $+\infty$. If the passage times are stationary, ergodic and have a finite moment of order $\alpha \gt 1/2$, then the number of geodesics is either $0$ or $+\infty$. We construct a model with stationary passage times such that $E\lbrack t(e)^\alpha\rbrack \lt \infty$, for every $0 \lt \alpha \lt 1/2$, and with a unique geodesic. The recurrence/transience properties of reversible random walks in a random environment with stationary conductances $( a(e);e$ is an edge of $\mathbb{Z}^2)$ are considered.

We give another proof of the following result from a joint paper with Bálint Tóth: A Brownian motion reflected on an independent time-reversed Brownian motion is a Brownian motion.

We state and prove a simple quantitative bound on the total variation distance after k iterations between two Markov chains with different initial distributions but identical transition probabilities. The result is a simplified and improved version of the result in Rosenthal (1995), which also takes into account the $\epsilon$-improvement of Roberts and Tweedie (1999), and which follows as a special case of the more complicated time-inhomogeneous results of Douc et al. (2002). However, the proof we present is very short and simple; and we feel that it is worthwhile to boil the proof down to its essence. This paper is purely expository; no new results are presented.

We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain.

Fill, Machida, Murdoch, and Rosenthal (2000) presented their algorithm and its variants to extend the perfect sampling algorithm of Fill (1998) to chains on continuous state spaces. We consider their algorithm for absolutely continuous stochastically monotone kernels, and show the correctness of the algorithm under a set of certain regularity conditions. These conditions succeed in relaxing the previously known hypotheses sufficient for their algorithm to apply.

The spatial structure of a class of superprocesses which arise as limits in distribution of a class of interacting particle systems with location dependent branching is investigated. The criterion of their state classification is obtained. Their effective state space is contained in the set of purely-atomic measures or the set of absolutely continuous measures according as one diffusive coefficient $c(x) \equiv 0$ or $|c(x)| \geq \epsilon \gt 0$ while another diffusive coefficient $h \in C^{2}_{b}(R)$.

We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables $ y,z $ and an only square integrable terminal data. Let $ L_N $ be the Lipschitz constant of the coefficient on the ball $ B(0,N) $ of $ R^d\times R^{dr} $. We prove that if $ L_N = O (\sqrt {\log N }) $, then the corresponding BSDE has a unique solution. Moreover, the stability of the solution is established under the same assumptions. In the case where the terminal data is bounded, we establish the existence and uniqueness of the solution also when the coefficient has an arbitrary growth (in $ y $) and without restriction on the behaviour of the Lipschitz constant $ L_N $.

Let $X, X_1, X_2, \dots $ be a sequence of nondegenerate, independent and identically distributed random variables and set $S_n=X_1+\dots +X_n$, $V_n^2=X_1^2+\dots +X_n^2$. We answer a question of Gotze, Gine and Mason by providing a simple necessary and sufficient condition for tightness of $S_n/V_n$.

We prove that random walks in i.i.d. random environments which oscillate in a given direction have velocity zero with respect to that direction. This complements existing results thus giving a general law of large numbers under the only assumption of a certain zero-one law, which is known to hold if the dimension is two.

We study the optimal control problem for $\mathbb{R}^d$-valued absolutely continuous stochastic processes with given marginal distributions at every time. When $d=1$, we show the existence and the uniqueness of a minimizer which is a function of a time and an initial point. When $d \gt 1$, we show that a minimizer exists and that minimizers satisfy the same ordinary differential equation.

Consider a martingale $M$ with bounded jumps and two sequences $a_n, b_n \to \infty$. We show that if the rescaled martingales $$ M^n_t =\frac{1}{\sqrt{a_n}}M_{b_n t}$$ converge weakly, then the limit is necessarily a continous Ocone martingale. Necessary and sufficient conditions for the weak convergence of the rescaled martingales are also given.

In this paper we study the a.s. convergence of all solutions of the Itô-Volterra equation \[ dX(t) = (AX(t) + \int_{0}^{t} K(t-s)X(s),ds)\,dt + \Sigma(t)\,dW(t) \] to zero. $A$ is a constant $d\times d$ matrix, $K$ is a $d\times d$ continuous and integrable matrix function, $\Sigma$ is a continuous $d\times r$ matrix function, and $W$ is an $r$-dimensional Brownian motion. We show that when \[ x'(t) = Ax(t) + \int_{0}^{t} K(t-s)x(s)\,ds \] has a uniformly asymptotically stable zero solution, and the resolvent has a polynomial upper bound, then $X$ converges to 0 with probability 1, provided \[ \lim_{t \rightarrow \infty} |\Sigma(t)|^{2}\log t= 0. \] A converse result under a monotonicity restriction on $|\Sigma|$ establishes that the rate of decay for $|\Sigma|$ above is necessary. Equations with bounded delay and neutral equations are also considered.