1/fnoise and one-dimensional Brownian motion in a singular potential

1994 ◽  
Vol 50 (4) ◽  
pp. 2491-2495 ◽  
Author(s):  
H. F. Ouyang ◽  
Z. Q. Huang ◽  
E. J. Ding
Keyword(s):  
Brownian Motion ◽  
Singular Potential ◽  
One Dimensional

scholarly journals BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid Oufdil
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Random Measure ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Stochastic Optimal Control Problem ◽  
Uniqueness Of The Solution ◽  
Logarithmic Growth

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

One-Dimensional Brownian Motion and Related Processes

2018 ◽  
pp. 369-409
Author(s):  
Giuseppe Campolieti ◽  
Roman N. Makarov
Keyword(s):  
Brownian Motion ◽  
One Dimensional

4. Gelification and soapiness

2020 ◽  
pp. 63-90
Author(s):  
Tom McLeish
Keyword(s):  
Brownian Motion ◽  
Self Assembly ◽  
Soft Matter ◽  
Intermolecular Forces ◽  
Two Dimensional ◽  
Important Distinction ◽  
One Dimensional ◽  
The Third ◽  
Self Assembled ◽  
Weak Intermolecular Forces

‘Gelification and soapiness’ looks at the third class of soft matter: ‘self-assembly’. Like the colloids of inks and clays, and the polymers of plastics and rubbers, ‘self-assembled’ soft matter also emerges as a surprising consequence of Brownian motion combined with weak intermolecular forces. Like them, it also leads to explanations of a very rich world of materials and phenomena, such as gels, foams, soaps, and ultimately to many of the structures of biological life. There is an important distinction that needs to be made between one-dimensional and two-dimensional self-assembly.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (04) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝ p -valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

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