Arratia, and later Tóth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we call the Brown-ian Web as a random variable taking values in an appropriate (metric) space whose points are (compact) sets of paths. This leads to general con...

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L 2 norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm.

Itô’s semimartingale driven by a Brownian motion is typically used in modeling the asset prices, interest rates and exchange rates, and so on. However, the assumption of Brownian motion as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropriate. In the paper we propose a statisti...

We define a local time flow of skew Brownian motions, i.e., a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the RayKnight theorem on local times. In our case, however, t...

Brownian motions in the innnite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent one-dimensional Brownian motions. The proof involves continuous tensor products and continuous quantum measurements. A by-product: a Brownian motion in a separab...

In this paper scientific uncertainty is defined as the impossibility to choose the correct stochastic process for the value of a public policy. The real option value of waiting under scientific uncertainty is derived using the difference between the geometric Brownian motion and the mean reverting process by applying contingent claim analysis. The results are compared with those generated by ei...

The detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the subspace spanned by linear functions. Karhunen–Loeve expansion for the process is obtained, together with the explicit formula for the Laplace transform of the squared L 2 norm. Distribution identities are established in connection with the second order Brownian bridge dev...

We simulate several models of random curves in the half plane and numerically compute their stochastic driving process (as given by the Loewner equation). Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion. We find that just testing the normality of the proc...

The super-Brownian motion X % in a super-Brownian medium % constructed in DF96a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ((DF96a]) and three ((DF96b]). Here we ll the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d 3); given the catalyst, the variance of the proc...

The set of zeros of a Brownian motion gives rise to a product system in the sense of William Arveson (that is, a continuous tensor product system of Hilbert spaces). Replacing the Brownian motion with a Bessel process we get a continuum of non-isomorphic product systems.