Formal Definition
Mathematically, 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 Paul Lévy. The basic idea is that ℓ(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
which explains why it is called the local time of b at x.
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