scholarly journals A boundary local time for one-dimensional super-Brownian motion and applications

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Thomas Hughes
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
One Dimensional ◽  
Super Brownian Motion

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.

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