In Lochowski (2008) [9] we defined truncated variation of Brownian motion with drift, Wt=Bt+[mu]t,t=0, where (Bt) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c0. We prove that truncated variation is a random...

In this article, we consider the properties of hitting times for G-martingales and the stopped processes. We prove that the stopped processes for G-martingales are still G-martingales and that the hitting times for a class of G-martingales including one-dimensional G-Brownian motion are...

Let {Xn,n=1} be a strictly stationary sequence of random variables and Mn=max{X1,X2,...,Xn}. Assume that some random variables X1,X2,... can be observed and the sequence of random variables indicate which X1,X2,... are observed, thus . In paper (Mladenovic and Piterbarg, 2006 [3]), the limiting...

This paper considers the nonlinear theory of G-martingales as introduced by Peng (2007) in [16] and [17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in Soner et al. (2009) [20] for the second-order stochastic target...

For a given bivariate Lévy process (Ut,Lt)t=0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed...

We consider iid Brownian motions, Bj(t), where Bj(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by Bj:n(t), and we consider a sequence Qn(t)=Bj(n):n(t), where j(n)/n--[alpha][set membership, variant](0,1). This...

This paper focuses on the degree sequence of a random graph process with copying and vertex deletion. A phase transition is revealed as the following: when copying strictly dominates deletion, the model possesses a power law degree sequence; and when deletion strictly dominates copying, it...

We prove the hydrostatics of boundary driven gradient exclusion processes, Fick's law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.

We consider a random walk with a negative drift and with a jump distribution which under Cramér's change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a...