Definitions

# 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

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

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

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

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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