Local time (mathematics)

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

[edit] Strict definition

Formally, 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 $x$ .

[edit] 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