We show that for all p > pc(Z) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. Thi...

This material was used for a course on stochastic analysis at UW–Madison in fall 2003. The text covers the development of the stochastic integral of predictable processes with respect to cadlag semimartingale integrators, Itô’s formula in an open domain in R, and an existence and uniqueness theorem for an equation of the type dX = dH + F (t, X) dY where Y is a cadlag semimartingale. The text is...

It was shown by the authors that two one-dimensional probability measures in convex order admit a martingale coupling with respect to which integral of |x?y| is smaller than twice their W1-distance (Wasserstein distance index 1). We showed replacing and W1 respectively |x?y|? W?? does not lead finite multiplicative constant. show here constant recovered when product W? times centred ?-th moment...

Solution to the optimal stopping problem V (x) = sup τ Eeg(x+Xτ ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K) or gp(x) = (K−x) Results, interpreted as option prices of perpetual options in Bachelier’s model are expressed in terms of the distribution of the overall supremum in cas...

Let X be the unique normal martingale such that X0 = 0 and d[X]t = (1− t − Xt−) dXt + dt and let Yt := Xt + t for all t > 0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set {t > 0 : Yt = 1} is shown to be non-...

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipsch...

We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. These are weak in probabilistic sense i.e., probability space and driving Wiener process an integral part solution. derive relative energy inequality for equations and, as corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use inequality, i...