scholarly journals Isomorphisms of β-Dyson’s Brownian motion with Brownian local time

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Titus Lupu
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Brownian Local Time

scholarly journals A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME INCREMENTS

2011 ◽  
Vol 11 (01) ◽  
pp. 5-48
Author(s):  
JAY ROSEN
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Random Variable ◽  
Normal Random Variable ◽  
Higher Moments ◽  
Second Moment ◽  
The Third ◽  
Brownian Local Time

Let [Formula: see text] denote the local time of Brownian motion. Our main result is to show that for each fixed t[Formula: see text] as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of [Formula: see text]. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.

Hilbert transform of G-Brownian local time

2014 ◽  
Vol 14 (04) ◽  
pp. 1450006 ◽  
Author(s):  
Litan Yan ◽  
Qinghua Zhang ◽  
Bo Gao
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Hilbert Transform ◽  
Natural Extension ◽  
Itô Integral ◽  
Brownian Local Time ◽  
Ito Integral ◽  
Natural Result ◽  
The Hilbert Transform

Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals [Formula: see text] We show that the integral diverges and the convergence [Formula: see text] exists in 𝕃2 for all a ∈ ℝ, t > 0. This shows that [Formula: see text] coincides with the Hilbert transform of the local time [Formula: see text] of G-Brownian motion B for every t. The functional is a natural extension to classical cases. As a natural result we get a sublinear version of Yamada's formula [Formula: see text] where the integral is the Itô integral under the G-expectation.

scholarly journals Some limit theorems connected with Brownian local time

2006 ◽  
Vol 2006 ◽  
pp. 1-5
Author(s):  
Raouf Ghomrasni
Keyword(s):  
Brownian Motion ◽  
Large Class ◽  
Local Time ◽  
Limit Theorems ◽  
Standard Brownian Motion ◽  
Time Process ◽  
Continuous Version ◽  
Brownian Local Time ◽  
Class Of Functions

Let B=(Bt)t≥0 be a standard Brownian motion and let (Ltx;t≥0,x∈ℝ) be a continuous version of its local time process. We show that the following limitlim⁡ε↓0(1/2ε)∫0t{F(s,Bs−ε)−F(s,Bs+ε)}ds is well defined for a large class of functions F(t,x), and moreover we connect it with the integration with respect to local time Ltx . We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.

Some eigenvalue identities for Brownian motion

1989 ◽  
Vol 105 (3) ◽  
pp. 587-596 ◽  
Author(s):  
Paul McGill
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
General Problem ◽  
Convolution Equation ◽  
Additive Functional ◽  
Brownian Local Time ◽  
Integral Convolution ◽  
Explicit Evaluation ◽  
The Law

The general problem can be stated as follows. Take a Brownian motion Bt started at − x < 0, and consider the additive functional At = ∫L(a,t)m(da), where L(a, t) is the Brownian local time. We suppose that m = m+ — m−, where these are positive measures supported respectively on (0, ∞) and (— ∞, 0). Then, with the equalization time defined by T = inf {t > 0: At = 0}, we ask for an explicit evaluation of the law π (x,dy) = P−x[BT∈dy]. In [8, 9] we showed how π (x,dy) can be obtained by solving an integral convolution equation of Wiener-Hopf type. The method used there exploits a technique of Ray [10].

Self-avoiding random walk: A Brownian motion model with local time drift

1987 ◽  
Vol 74 (2) ◽  
pp. 271-287 ◽  
Author(s):  
J. R. Norris ◽  
L. C. G. Rogers ◽  
David Williams
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Local Time ◽  
Motion Model ◽  
Time Drift ◽  
Brownian Motion Model

Pricing path-dependent options in a Black-Scholes market from the distribution of homogeneous Brownian functionals

2004 ◽  
Vol 41 (01) ◽  
pp. 1-18
Author(s):  
T. Fujita ◽  
F. Petit ◽  
M. Yor
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Black Scholes ◽  
Explicit Formulae ◽  
Path Dependent

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, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

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