Abstract. This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on 3 closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L modulus of continuity for a 1-dimensional Brownian motion. (c) Ext...

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, the obtention of their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest by themselves. 1. Motivation, introduction 1.1. To (Bt, t ...

For a random process X consider the random vector deened by the values of X at times 0 < U n;1 < ::: < U n;n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n;i are the order statistics of n independent uniform (0; 1) variables, independent of X. The joint law of this random vector is explicitly described when X is a Brownian motion...

For a random process X consider the random vector defined by the values of X at times 0 < Un,1 < ... < Un,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the Un,i are the order statistics of n independent uniform (0, 1) variables, independent of X. The joint law of this random vector is explicitly described when X is a Brownian motion. ...

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in Rn that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D ∩ 2−kZn moving at the rate 2−2k with stationary init...

We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on E = (0,∞)×R, which is not invariant by translation. We then prove that the hitting probability of a Borel set A ⊂ E for the graph of the Brownian snake starting at (0, 0) is comparable, up to multiplicative constants, to its capacity. This implies that super-Brown...

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to rst hit > 0 at tim...

In this paper we construct a measure on pairs of Brownian motions starting at the same point conditioned so their paths do not intersect. The construction of this measure is a start towards the rigorous understanding of nonintersecting Brownian motions as a conformal eld. Let B 1 ; B 2 be independent Brownian motions in R 2 starting at distinct points on the unit circle. Let T j r be the rst ti...

Minimal area regions are constructed for Brownian paths and perturbed Brownian paths. While the theoretical optimal region cannot be obtained in closed form, we provide practical confidence regions based on numerical approximations and local time arguments. These regions are used to provide informal convergence assessments for both Monte Carlo and Markov Chain Monte Carlo experiments, via the B...

In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is...