Local time (mathematics)

A sample path of an Itō process together with its surface of local times.

In the mathematical theory of stochastic processes, local time is a stochastic process associated with diffusion 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 real valued diffusion process $(B_s)_{s\ge 0}$ , the local time of $B$ at the point $x$ is the stochastic process

L^x(t) =\int_0^t \delta(x-B_s)\,ds,

where $\delta$ 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

L^x(t) =\lim_{\varepsilon\downarrow 0} \frac{1}{2\varepsilon} \int_0^t 1_{\{ x- \varepsilon < B_s < x+\varepsilon \}} \, ds,

which explains why it is called the local time of $B$ at $x$ . For a discrete state-space process $(X_s)_{s\ge 0}$ , the local time can be expressed more simply as^[1]

L^x(t) =\int_0^t 1_{\{x\}}(X_s) \, ds.

Tanaka's formula

Tanaka's formula provides a definition of local time for an arbitrary continuous semimartingale $(X_s)_{s\ge 0}$ on $\mathbb R:$ ^[2]

L^x(t) = |X_t - x| - |X_0 - x| - \int_{0}^t \left( 1_{(0,\infty)}(X_s - x) - 1_{(-\infty, 0]}(X_s-x) \right) dX_s, \qquad t \geq 0.

A more general form was proven independently by Meyer^[3] and Wang;^[4] the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If $F:\mathbb R \rightarrow \mathbb R$ is absolutely continuous with derivative $F',$ which is of bounded variation, then

F(X_t) = F(X_0) + \int_{0}^t F'_{-}(X_s) dX_s + \frac12 \int_{-\infty}^\infty L^x(t) dF'(x),

where $F'_{-}$ is the left derivative.

If $X$ is a Brownian motion, then for any $\alpha\in(0,1/2)$ the field of local times $L = (L^x(t))_{x \in \mathbb R, t \geq 0}$ has a modification which is a.s. Hölder continuous in $x$ with exponent $\alpha$ , uniformly for bounded $x$ and $t$ .^[5] In general, $L$ has a modification that is 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, $(|B_s|)_{s \geq 0}$ .

Ray–Knight theorems

The field of local times $L_t = (L^x_t)_{x \in E}$ 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, $T = \inf\{t \geq 0 \colon B_t = 0\}$ . Ray^[6] and Knight^[7] (independently) showed that

$\left\{ L^x(T) \colon x \in [0,a] \right\} \stackrel{\mathcal{D}}{=} \left\{ |W_x|^2 \colon x \in [0,a] \right\} \,$

(1)

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

T_a = \inf \{ t \geq 0 \colon L^0_t > a \}.

Let (W_t)_{t ≥ 0} be an independent one-dimensional Brownian motion started from W₀ = 0, then^[8]

$\left \{ L^x_{T_a} + W_x^2 \colon x \geq 0 \right \} \stackrel{\mathcal{D}}{=} \left\{ (W_x + \sqrt a )^2 \colon x \geq 0 \right \}. \,$

(2)

Equivalently, the process $(L^x_{T_a})_{x \geq 0}$ (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.

Notes

↑ Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus. Springer.
↑ Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 428–449. ISBN 0387949577.
↑ Meyer, P. A. (2002) [1976]. "Un cours sur les intégrales stochastiques". Séminaire de probabilités 1967–1980. Lect. Notes in Math. 1771. pp. 174–329. doi:10.1007/978-3-540-45530-1_11.
↑ Wang (1977). "Generalized Itô's formula and additive functionals of Brownian motion". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 41: 153–159.
↑ Kallenberg (1997). Foundations of Modern Probability. New York: Springer. p. 370. ISBN 0387949577.
↑ Ray, D. (1963). "Sojourn times of a diffusion process". Illinois Journal of Mathematics 7 (4): 615–630. MR 0156383. Zbl 0118.13403.
↑ Knight, F. B. (1963). "Random walks and a sojourn density process of Brownian motion". Transactions of the American Mathematical Society 109 (1): 56–86. JSTOR 1993647.
↑ Marcus; Rosen (2006). Markov Processes, Gaussian Processes and Local Times. New York: Cambridge University Press. pp. 53–56. ISBN 0521863007.

References

K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, 1990, Birkhäuser, ISBN 978-0-8176-3386-8.
M. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, 1st edition, 2006, Cambridge University Press ISBN 978-0-521-86300-1
P.Morters and Y.Peres, Brownian Motion, 1st edition, 2010, Cambridge University Press, ISBN 978-0-521-76018-8.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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