Let X be a locally bounded semimartingale. Using the theory of \textit{BMO}-martingales we give a sufficient criterion for a martingale measure for X to minimize relative entropy among all martingale measures. This is applied to prove convergence of the q-optimal martingale measure to the minimal entropy martingale measure in entropy for $q\downarrow 1$ under the assumption that X is continuous and that the density process of some equivalent martingale measure satisfies a reverse $\mathit{LLogL}$\small -inequality.

We consider the one-dimensional stochastic equation \[ X_t=X_0+\int^t_0 b(X_s)\,dM_s \] where M is a continuous local martingale and b a measurable real function. Suppose that $b^{-2}$ is locally integrable. D. N. Hoover asserted that, on a saturated probability space, there exists a solution X of the above equation with $X_0=0$ having no occupation time in the zeros of b and, moreover, the pair (X, M) is unique in law for all such X. We will give an example which shows that the uniqueness assertion fails, in general.

Following Lindeberg's approach, we obtain a new condition for a stationary sequence of square-integrable and real-valued random variables to satisfy the central limit theorem. In the adapted case, this condition is weaker than any projective criterion derived from Gordin's theorem [\textit{Dokl. Akad. Nauk SSSR} \textbf{188} (1969) 739--741] about approximating martingales. Moreover, our criterion is equivalent to the {\it conditional central limit theorem}, which implies stable convergence (in the sense of Rényi) to a mixture of normal distributions. We also establish functional and triangular versions of this theorem. From these general results, we derive sufficient conditions which are easier to verify and may be compared to other results in the literature. To be complete, we present an application to kernel density estimators for some classes of discrete ime processes.

The product Bernoulli measures $\rho_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process with irreducible random walk kernel $p(\cdot)$. In $d=1$, stationary measures that are not translation invariant are known to exist for specific $p(\cdot)$ satisfying $\sum_xxp(x)>0$. These measures are concentrated on configurations that are completely occupied by particles far enough to the right and are completely empty far enough to the left; that is, they are blocking measures. Here, we show stationary blocking measures exist for all exclusion processes in $d=1$, with $p(\cdot)$ having finite range and $\sum_x xp(x)>0$.

In this paper we study a new type of "Taylor expansion" for Itô-type random fields, up to the second order. We show that an Itô-type random field with reasonably regular "integrands" can be expanded, up to the second order, to the linear combination of increments of temporal and spatial variables, as well as the driven Brownian motion, around even a random (t,x)-point. Also, the remainder can be estimated in a "pathwise" manner. We then show that such a Taylor expansion is valid for the solutions to a fairly large class of stochastic differential equations with parameters, or even fully-nonlinear stochastic partial differential equations, whenever they exist. Using such analysis we then propose a new definition of stochastic viscosity solution for fully nonlinear stochastic PDEs, in the spirit of its deterministic counterpart. We prove that this new definition is actually equivalent to the one proposed in our previous works, at least for a class of quasilinear SPDEs.

Weak convergence of the Euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of Lebesgue measure zero. The rate of convergence is presented when coefficients are Hölder continuous. Monte Carlo simulations are also discussed.

We develop a nonlinear martingale theory for time discrete processes $(Y_n)_{n\in \NN_0}$. These processes are defined on any filtered probability space $(\O,\F,\F_n,\P)_{n\in\NN_0}$ and have values in a metric space (N,d) of nonpositive curvature (in the sense of A. D. Alexandrov). The defining martingale property for such processes is \[ \E(Y_{n+1}|\F_n)=Y_n, \qquad \P\mbox{-a.s.,} \] where the conditional expectation on the left-hand side is defined as the minimizer of the functional \[ Z\mapsto\E d^2(Z,Y_{n+1}) \] within the space of $\F_n$-measurable maps $Z\dvtx \O\to N$. We give equivalent characterization of N-valued martingales (using merely the usual linear conditional expectations) and derive fundamental properties of these martingales, for example, a martingale convergence theorem. Finally, we exploit the relation with harmonic maps. It turns out that a map $f\dvtx M\to N$ is harmonic w.r.t. a given Markov kernel p on M if and only if it maps Markov chains $(X_n)_{n\in\NN}$ (with transition kernel p) on M onto martingales $(f(X_n))_{n\in\NN}$ with values in $N$. The nonlinear heat flow $f\dvtx \N_0\times M\to N$ of a given initial map $f(0,\cdot)\dvtx M\to N$ at time n is obtained as the "filtered expectation," \[ f(n,x) := \E_x [ f(X_n) |\!|\!| (\F_k)_{k\ge 0}] \] of the random map $f(X_n)$. Similarly, the unique solution to the Dirichlet problem for a given map $g\dvtx M\to N$ and a subset $D\subset M$ is obtained as \[ f(x) := \E_x [ g(X_{\tau(D)})|\!|\!| (\F_k)_{k\ge 0}]. \] In both cases, a crucial role is played by the notion of filtered expectation $\E_x [\cdot |\!|\!| (\F_k)_{k\ge 0}]$ which will be discussed in detail. Moreover, we prove Jensen's inequality for expectations and filtered expectations and we prove (weak and strong) laws of large numbers for sequences of i.i.d. random variables with values in N. Our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed, for example, in Emery and Kendall. The goal is to extend the previous framework towards processes with values in metric spaces. This will lead to a stochastic approach to the theory of (generalized) harmonic maps with values in such "singular" spaces as developed by Jost and Korevaar and Schoen.

We obtain deviation inequalities for some classes of functions of infinitely divisible random vectors having finite exponential moments.

Let T be a symmetric statistic based on sample of size n drawn without replacement from a finite population of size N, where $N>n$. Assuming that the linear part of Hoeffding's decomposition of T is nondegenerate we construct a one term Edgeworth expansion for the distribution function of T and prove the validity of the expansion with the remainder $O(1/n^*)$ as $n^*\to \infty$, where $n^*=\min\{n,N-n\}$.

We consider random walks on $\Z$ in a stationary random medium, defined by an ergodic dynamical system, in the case when the possible jumps are $\{-L,\ldots,-1,0,+1\}$ for some fixed integer L. We provide a recurrence criterion expressed in terms of the sign of the maximal Liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent. Next, we characterize the existence of the absolutely continuous invariant measure for the Markov chain of "the environments viewed from the particle" and also characterize, in the transient cases, the existence of a nonzero drift. To study the validity of the central limit theorem, we consider the notion of harmonic coordinates introduced by Kozlov. We characterize the existence of both the invariant measure and the harmonic coordinates and show in the recurrent case that the existence of those two objects is equivalent to the validity of an invariance principle. We give sufficient conditions for the validity of the central limit theorem in the transient cases. Finally, we consider the previous results in the context of a random medium defined by an irrational rotation on the circle and study their realization in terms of regularity and Diophantine approximation.

General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge and conditional gauge theorems hold. These classes are larger than the classical Kato class when the process is Brownian motion in a bounded $C^{1,1}$ domain.

We consider a model of interface growth in two dimensions, given by a height function on the sites of the one-dimensional integer lattice. According to the discrete time update rule, the height above the site x increases to the height above $x-1$, if the latter height is larger; otherwise the height above x increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution F and that the initial state is such that the origin is far above the other sites. We explicitly identify the asymptotic shape and prove that, in the pure regime, the fluctuations about that shape, normalized by the square root of time, are asymptotically normal. This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory.

Let $S_n$ be a random walk in $\bz^d$ and let $R_n$ be the range of $S_n$. We prove an almost sure invariance principle for $R_n$ when $d=3$ and a law of the iterated logarithm for $R_n$ when $d=2$.

Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.

Let $X,X_1,X_2,\ldots$ be a sequence of independent and identically distributed random variables. Let $ ^{(1)}X_n,\ldots,{^{(n)}X}_n$ be an arrangement of $X_1$, $X_2,\ldots,X_n $ in decreasing order of magnitude, and set ${^{(r_n)}S}_n= {}^{(r_{n}+1)}X_n+\cdots + {^{(n)}X}_{n}$. This is known as the modulus trimmed sum. We obtain a complete characterization of the class of limit laws of the normalized modulus trimmed sum when the underlying distribution is symmetric and $ r_n \to \infty$, $r_nn^{-1}\to 0$.

The aim of this note is to reduce a number of assumptions in the recent paper of W. Bryc by showing that some of them imply the others and to give alternative, simpler proofs of some of Bryc's results.

Compound Poisson processes are often useful as approximate models, when describing the occurrence of rare events. In this paper, we develop a method for showing how close such approximations are. Our approach is to use Stein's method directly, rather than by way of declumping and a marked Poisson process; this has conceptual advantages, but entails technical difficulties. Several applications are given to illustrate the procedure.