local time

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(t,x)=int_0^t delta(x-b(s)),ds

where b(s) 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(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_{epsilondownarrow 0} frac{1}{2epsilon} int_0^t 1{ x- epsilon < b(s) < x+epsilon } ds,

which explains why it is called the local time of b at x.

See also

References

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

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