Abstract
For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.
Highlights
Introduction and main resultsIn recent work of Assaf et al [4], a novel path transform, called the quantile transform Q has been studied both in discrete and continuous settings
We focus on studying the Vervaat transform of Brownian motion
And PT0,λ be the distribution of Brownian bridge of length T from 0 to λ, and PTxy be the distribution of Brownian motion starting from x until the first time at which it hits y for y < x
Summary
In recent work of Assaf et al [4], a novel path transform, called the quantile transform Q has been studied both in discrete and continuous settings. As shown in Assaf et al [4, Theorem 8.16], the scaling limit of this transformation of simple random walks is the quantile transform of Brownian motion B := (Bt; 0 ≤ t ≤ 1): Q(B)t. Vervaat [50] showed that if B is conditioned to both start and end at 0, V (B) is normalized Brownian excursion: Theorem 1.1. We use the path decomposition to derive a collection of properties of the Vervaat transform of Brownian bridges and Brownian motion. We prove that these processes are not Markov (Subsection 3.2). We provide explicit formulae for the first two moments of the Vervaat transform of Brownian motion (Subsection 4.4)
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