Path Regularity of the Brownian Motion and the Brownian Sheet

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates for Brownian motion and the Brownian sheet are obtained when the balls are given by certain H61der norms. As an application, we establish some lira inf results for Gaussian samples of these processes, as well as Chung's LIL in this setting, but now we introduce some notation. Let I = [0, 1 ], with Co(I) the continuous real-valued function f ( . ) on I such that f ( 0 ) = 0....

Brownian Motion: The Link Between Probability and Mathematical Analysis

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Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [7] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

Images of the Brownian Sheet

An N-parameter Brownian sheet in Rd maps a non-random compact set F in R+ to the random compact set B(F ) in R d. We prove two results on the image-set B(F ): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive d 2 -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985), and Khoshnevisan (1999). (2) If dimH F > d 2 ...