Several situations when one can prove weak uniqueness of solutions of Itô equations with discontinuous or/and degenerate coefficients are presented. In particular, the cases are considered in which the set of discontinuity is a cone, or a straight line, or else a discrete set of points.

We consider a nearest neighbors random walk on . The jump rate from site x to site x+1 is equal to the jump rate from x+1 to x and is a bounded, strictly positive random variable [eta](x). We assume that is distributed by a locally ergodic probability measure. We prove that, under diffusive...

Let (Xn)n[greater-or-equal, slanted]0 be a Harris ergodic Markov chain and f be a real function on its state space. Consider the block sums [zeta](i) for f, i[greater-or-equal, slanted]1, between consecutive visits to the atom given by the splitting technique of Nummelin. A regularity condition...

We observe (Yt) at times i/n, i=0,...,n, in the parametric stochastic volatility modeldYt=[Phi]([theta],WtH) dWt,where (Wt) is a Brownian motion, independent of the fractional Brownian motion (WtH) with Hurst parameter . The sample size n increases not because of a longer observation period, but...

We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian...

Consider the American put and Russian option (Ann. Appl. Probab. 3 (1993) 603; Theory Probab. Appl. 39 (1994) 103; Ann. Appl. Probab. 3 (1993) 641) with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with...

A process which we call symbiotic branching, is suggested covering three well-known interacting models: mutually catalytic branching, the stepping stone model, and the Anderson model. Basic tools such as self-duality, particle system moment duality, measure case moment duality, and moment...

The corrected diffusion effects caused by a noncentered stochastic system are studied in this paper. A diffusion limit theorem or CLT of the system is derived with the convergence error estimate. The estimate is obtained for large t (on the interval (0,t*), t* of the order of [var epsilon]-1)....

We use the concept of time-space chaos (see Peccati (Ann. Inst. Poincaré 37(5) (2001) 607; Prépublication n. 648 du Laboratoire de Probabilités et Modèles Aléatoires de l'Université Paris VI; Chaos Brownien d'espace-temps, décompositions de Hoeffding et problèmes de convergence associés,...

A supercritical bisexual Galton-Watson branching process with population-size-dependent mating is considered and some necessary and sufficient conditions for the almost sure and L1-convergence of the suitably normalized process are given. Similar results for the underlying sequences of females...