Local time (mathematics)

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A sample path of an Itō process together with its surface of local times.

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.

[edit] 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 $x$ .

[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