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[in Journal: Stochastics and Dynamics] AND [Keyword: Local Time] (8)	28 Mar 2025	Run
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articleNo Access
ON ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING THE MAXIMUM PROCESS
- R. BELFADLI,
- S. HAMADÈNE, and
- Y. OUKNINE
Stochastics and Dynamics01 Jun 2009
Preview Abstract
We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation
The second type is the equation
The third type is the equation
We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE
articleNo Access
INFINITELY MANY BROWNIAN GLOBULES WITH BROWNIAN RADII
- MYRIAM FRADON and
- SYLVIE RŒLLY
Stochastics and Dynamics01 Dec 2010
Preview Abstract
We consider an infinite system of non-overlapping globules undergoing Brownian motions in ℝ³. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional stochastic differential equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.
articleNo Access
A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME INCREMENTS
- JAY ROSEN
Stochastics and Dynamics01 Mar 2011
Preview Abstract
Let denote the local time of Brownian motion. Our main result is to show that for each fixed t
as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of . This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.
articleNo Access
Hilbert transform of G-Brownian local time
- Litan Yan,
- Qinghua Zhang, and
- Bo Gao
Stochastics and Dynamics22 Sep 2014
Preview Abstract
Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals
We show that the integral diverges and the convergence
exists in 𝕃² for all a ∈ ℝ, t > 0. This shows that coincides with the Hilbert transform of the local time 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
where the integral is the Itô integral under the G-expectation.
articleNo Access
On limit theorems of some extensions of fractional Brownian motion and their additive functionals
- M. Ait Ouahra,
- S. Moussaten, and
- A. Sghir
Stochastics and Dynamics26 Mar 2017
Preview Abstract
This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.
articleNo Access
The quadratic covariation for a weighted fractional Brownian motion
- Xichao Sun,
- Litan Yan, and
- Qinghua Zhang
Stochastics and Dynamics04 May 2017
Preview Abstract
Let $B^{a, b}$ be a weighted fractional Brownian motion with indices $a$ and $b$ satisfying $a > - 1,$ $- 1 < b < 0,$ $| b | < 1 + a$ . In this paper, motivated by the asymptotic property
$E [{(B_{s + 𝜀}^{a, b} - B_{s}^{a, b})}^{2}] = O (𝜀^{1 + b}) ≁ 𝜀^{1 + a + b} = E [{(B_{𝜀}^{a, b})}^{2}], 𝜀 \to 0$
for all $s > 0$ , we consider the generalized quadratic covariation ${[f (B^{a, b}), B^{a, b}]}^{(a, b)}$ defined by
${[f (B^{a, b}), B^{a, b}]}_{t}^{(a, b)} = lim_{𝜀 ↓ 0} \frac{1 + a + b}{𝜀^{1 + b}} \int_{𝜀}^{t + 𝜀} {f (B_{s + 𝜀}^{a, b}) - f (B_{s}^{a, b})} (B_{s + 𝜀}^{a, b} - B_{s}^{a, b}) s^{b} d s,$
provided the limit exists uniformly in probability. We construct a Banach space $ℋ$ of measurable functions such that the generalized quadratic covariation exists in $L^{2} (Ω)$ and the generalized Bouleau–Yor identity
${[f (B^{a, b}), B^{a, b}]}_{t}^{(a, b)} = - \frac{1}{(1 + b) 𝔹 (a + 1, b + 1)} \int_{ℝ} f (x) ℒ^{a, b} (d x, t)$
holds for all $f \in ℋ$ , where $ℒ^{a, b} (x, t) = \int_{0}^{t} δ (B_{s}^{a, b} - x) d s^{1 + a + b}$ is the weighted local time of $B^{a, b}$ and $𝔹 (\cdot, \cdot)$ is the Beta function.
articleNo Access
The functional Meyer–Tanaka formula
- Yuri F. Saporito
Stochastics and Dynamics01 Aug 2018
Preview Abstract
The functional Itô formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer–Tanaka formula. Following the idea of the proof of the classical time-dependent Meyer–Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obłój.
articleNo Access
Occupation times of discrete-time fractional Brownian motion
- Manfred Denker and
- Xiaofei Zheng
Stochastics and Dynamics27 Jan 2019
Preview Abstract
We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter $\frac{3}{4} < H < 1$ . Using results from infinite ergodic theory, it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.