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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.
Additional recommended knowledge
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.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Local_time_(mathematics)". A list of authors is available in Wikipedia.|