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Local time (mathematics)



In the mathematical theory of stochastic processes, local time is a property of diffusion processes like Brownian motion that characterizes the time a particle has spent at a given level. Local time is very useful and often appears in various stochastic integration formulas if the integrand is not sufficiently smooth, for example in Tanaka's formula.

Strict definition

Formally, the definition of the local time is

\ell(t,x)=\int_0^t \delta(x-b(s))\,ds

where b(s) is the diffusion process and δ is the Dirac delta function. It is a notion invented by P. Lévy. The basic idea is that \ell(t,x) is a (rescaled) measure of how much time b(s) has spent at x up to time t. It may be written as

\ell(t,x)=\lim_{\epsilon\downarrow 0} \frac{1}{2\epsilon} \int_0^t 1\{ x- \epsilon < b(s) < x+\epsilon \} ds,

which explains why it is called the local time of b at x.

See also

  • Tanaka's formula
  • Brownian motion
  • Red noise, also known as brown noise (Martin Gardner proposed this name for sound generated with random intervals. It is a pun on Brownian motion and white noise.)
  • Diffusion equation

References

  • K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, 1990, Birkhäuser, ISBN 978-0817633868 .
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Local_time_(mathematics)". A list of authors is available in Wikipedia.
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