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.

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Strict definition

Formally, the definition of the local time is

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 is a (rescaled) measure of how much time b(s) has spent at x up to time t. It may be written as

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 .