Local_time_(mathematics)

Local time (mathematics)

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.

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 $x$ .

References

K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, 1990, Birkhäuser, ISBN 978-0817633868 .

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Strict definition

See also

References