Local time (mathematics)

Formal definition

For a real valued diffusion process ( B s ) s ≥ 0 , the local time of B at the point x is the stochastic process

where δ is the Dirac delta function. It is a notion invented by Paul Lévy. The basic idea is that L x ( t ) 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 . For a discrete state-space process ( X s ) s ≥ 0 , the local time can be expressed more simply as

Tanaka's formula

Tanaka's formula provides a definition of local time for an arbitrary continuous semimartingale ( X s ) s ≥ 0 on R :

A more general form was proven independently by Meyer and Wang; the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If F : R → R is absolutely continuous with derivative F ′ , which is of bounded variation, then

where F − ′ is the left derivative.

If X is a Brownian motion, then for any α ∈ ( 0 , 1 / 2 ) the field of local times L = ( L x ( t ) ) x ∈ R , t ≥ 0 has a modification which is a.s. Hölder continuous in x with exponent α , uniformly for bounded x and t . In general, L has a modification that is a.s. continuous in t and cadlag in x .

Tanaka's formula provides the explicit Doob–Meyer decomposition for the one-dimensional reflecting Brownian motion, ( | B s | ) s ≥ 0 .

Ray–Knight theorems

The field of local times L t = ( L t x ) x ∈ E associated to a stochastic process on a space E is a well studied topic in the area of random fields. Ray–Knight type theorems relate the field L_t to an associated Gaussian process.

In general Ray–Knight type theorems of the first kind consider the field L_t at a hitting time of the underlying process, whilst theorems of the second kind are in terms of a stopping time at which the field of local times first exceeds a given value.

First Ray–Knight theorem

Let (B_t)_{t ≥ 0} be a one-dimensional Brownian motion started from B₀ = a > 0, and (W_t)_t≥0 be a standard two-dimensional Brownian motion W₀ = 0 ∈ R². Define the stopping time at which B first hits the origin, T = inf { t ≥ 0 : B t = 0 } . Ray and Knight (independently) showed that

where (L_t)_{t ≥ 0} is the field of local times of (B_t)_{t ≥ 0}, and equality is in distribution on C[0, a]. The process |W_x|² is known as the squared Bessel process.

Second Ray–Knight theorem

Let (B_t)_{t ≥ 0} be a standard one-dimensional Brownian motion B₀ = 0 ∈ R, and let (L_t)_{t ≥ 0} be the associated field of local times. Let T_a be the first time at which the local time at zero exceeds a > 0

Let (W_t)_{t ≥ 0} be an independent one-dimensional Brownian motion started from W₀ = 0, then

Equivalently, the process ( L T a x ) x ≥ 0 (which is a process in the spatial variable x ) is equal in distribution to the square of a 0-dimensional Bessel process, and as such is Markovian.

Generalized Ray–Knight theorems

Results of Ray–Knight type for more general stochastic processes have been intensively studied, and analogue statements of both (1) and (2) are known for strongly symmetric Markov processes.