scholarly journals Exact sampling of diffusions with a discontinuity in the drift

2016 ◽  
Vol 48 (A) ◽  
pp. 249-259 ◽  
Author(s):  
Omiros Papaspiliopoulos ◽  
Gareth O. Roberts ◽  
Kasia B. Taylor
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Rejection Sampling ◽  
Exact Methods ◽  
Exact Sampling ◽  
One Dimensional ◽  
Sample Paths ◽  
Finite Dimensional

AbstractWe introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time, and are based on the principle of retrospective rejection sampling. A simple illustration is provided.

The local time of a two-dimensional diffusion

1994 ◽  
Vol 445 (1925) ◽  
pp. 657-667
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Dimensional Subspace ◽  
Two Dimensions ◽  
Local Times ◽  
Two Dimensional ◽  
One Dimensional ◽  
Cantor Function ◽  
Continuous State ◽  
Markovian Random Process

When a markovian random process taking values in a continuous state-space, such as R, visits a particular point repeatedly, it is natural to seek some quantity which records how long it spends there. Typically, however, the number of visits made to the point is uncountably infinite, and the (Lebesgue) length of time spent there is zero. One interesting object to consider is the local time, sometimes thought of as the occupation density of the process, which at each point is a random Cantor function that increases only when the process visits the point. The review article by Rogers (1989) contains a good introduction to the local time of a one-dimensional brownian motion and its relevance to the excursions of brownian motion from zero. In two dimensions, a typical diffusion, such as brownian motion in the plane, never revisits a point, so it does not have a local time. In this paper we shall construct the local times of some particular two-dimensional diffusions on a one-dimensional subspace, and show that they are jointly continuous in both time and space.

scholarly journals An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers

2016 ◽  
Vol 22 (1) ◽  
pp. 1-23 ◽  
Author(s):  
David Dereudre ◽  
Sara Mazzonetto ◽  
Sylvie Roelly
Keyword(s):  
Brownian Motion ◽  
Sampling Method ◽  
Transition Density ◽  
Explicit Representation ◽  
Rejection Sampling ◽  
Skew Brownian Motion ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Transition Densities ◽  
The One

AbstractIn this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover, we propose a rejection sampling method to simulate this density in an

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

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