In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields.
Formal definition For a continuous real-valued semimartingale ( B s ) s ≥ 0 {\displaystyle (B_{s})_{s\geq 0}} , the local time of B {\displaystyle B} at the point x {\displaystyle x} is the stochastic process which is informally defined by
L x ( t ) = ∫ 0 t δ ( x − B s ) d [ B ] s , {\displaystyle L^{x}(t)=\int _{0}^{t}\delta (x-B_{s})\,d[B]_{s},}
where δ {\displaystyle \delta } is the Dirac delta function and [ B ] {\displaystyle [B]} is the quadratic variation. It was introduced by Paul Lévy. The basic idea is that L x ( t ) {\displaystyle L^{x}(t)} is an (appropriately rescaled and time-parametrized) measure of how much time B s {\displaystyle B_{s}} has spent at x {\displaystyle x} up to time t {\displaystyle t} . More rigorously, it may be written as the almost sure limit
L x ( t ) = lim ε ↓ 0 1 2 ε ∫ 0 t 1 { x − ε < B s < x + ε } d [ B ] s , {\displaystyle L^{x}(t)=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}\int _{0}^{t}1_{\{x-\varepsilon <B_{s}<x+\varepsilon \}}\,d[B]_{s},}
which may be shown to always exist. Note that in the special case of Brownian motion (or more generally a real-valued diffusion of the form d B = b ( t , B ) d t + d W {\displaystyle dB=b(t,B)\,dt+dW} where W {\displaystyle W} is a Brownian motion), the term d [ B ] s {\displaystyle d[B]_{s}} simply reduces to d s {\displaystyle ds} , which explains why it is called the local time of B {\displaystyle B} at x {\displaystyle x} . For a discrete state-space process ( X s ) s ≥ 0 {\displaystyle (X_{s})_{s\geq 0}} , the local time can be expressed more simply as
L x ( t ) = ∫ 0 t 1 { x } ( X s ) d s . {\displaystyle L^{x}(t)=\int _{0}^{t}1_{\{x\}}(X_{s})\,ds.}
Tanaka's formula Tanaka's formula also provides a definition of local time for an arbitrary continuous semimartingale ( X s ) s ≥ 0 {\displaystyle (X_{s})_{s\geq 0}} on R : {\displaystyle \mathbb {R} :}
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