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 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 $(B_{s})_{s\geq 0}$ , the local time of $B$ at the point $x$ is the stochastic process which is informally defined by

L^{x}(t)=\int _{0}^{t}\delta (x-B_{s})\,d[B]_{s},

where $\delta$ is the Dirac delta function and $[B]$ is the quadratic variation. It is a notion invented by Paul Lévy. The basic idea is that $L^{x}(t)$ is an (appropriately rescaled and time-parametrized) measure of how much time $B_{s}$ has spent at $x$ up to time $t$ . More rigorously, it may be written as the almost sure limit

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 $dB=b(t,B)\,dt+dW$ where $W$ is a Brownian motion), the term $d[B]_{s}$ simply reduces to $ds$ , which explains why it is called the local time of $B$ at $x$ . For a discrete state-space process $(X_{s})_{s\geq 0}$ , 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 also provides a definition of local time for an arbitrary continuous semimartingale $(X_{s})_{s\geq 0}$ 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}+{\frac {1}{2}}\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 a.s. continuous in $t$ and càdlàg 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 $L_{t}=(L_{t}^{x})_{x\in E}$ 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 started from 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_{t}^{0}>a\}.

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

\left\{L_{T_{a}}^{x}+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 $(L_{T_{a}}^{x})_{x\geq 0}$ (which is a process in the spatial variable $x$ ) is equal in distribution to the square of a 0-dimensional Bessel process started at $a$ , 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, Paul-Andre (2002) [1976]. "Un cours sur les intégrales stochastiques". Séminaire de probabilités 1967–1980. Lect. Notes in Math. Vol. 1771. pp. 174–329. doi:10.1007/978-3-540-45530-1_11. ISBN 978-3-540-42813-8.
↑ Wang (1977). "Generalized Itô's formula and additive functionals of Brownian motion". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 41 (2): 153–159. doi:10.1007/bf00538419. S2CID 123101077.
↑ Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 370. ISBN 0387949577.
↑ Ray, D. (1963). "Sojourn times of a diffusion process". Illinois Journal of Mathematics. 7 (4): 615–630. doi:10.1215/ijm/1255645099. 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. doi:10.2307/1993647. 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. Mörters and Y. Peres, Brownian Motion, 1st edition, 2010, Cambridge University Press, ISBN 978-0-521-76018-8.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus. Springer.

[Kallenberg-2] Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 428–449. ISBN 0387949577.

[3] Meyer, Paul-Andre (2002) [1976]. "Un cours sur les intégrales stochastiques". Séminaire de probabilités 1967–1980. Lect. Notes in Math. Vol. 1771. pp. 174–329. doi:10.1007/978-3-540-45530-1_11. ISBN 978-3-540-42813-8.

[4] Wang (1977). "Generalized Itô's formula and additive functionals of Brownian motion". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 41 (2): 153–159. doi:10.1007/bf00538419. S2CID 123101077.

[Kallenberg2-5] Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 370. ISBN 0387949577.

[6] Ray, D. (1963). "Sojourn times of a diffusion process". Illinois Journal of Mathematics. 7 (4): 615–630. doi:10.1215/ijm/1255645099. MR 0156383. Zbl 0118.13403.

[7] Knight, F. B. (1963). "Random walks and a sojourn density process of Brownian motion". Transactions of the American Mathematical Society. 109 (1): 56–86. doi:10.2307/1993647. JSTOR 1993647.

[8] Marcus; Rosen (2006). Markov Processes, Gaussian Processes and Local Times. New York: Cambridge University Press. pp. 53–56. ISBN 0521863007.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[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 Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes 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 Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length 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	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) 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 Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
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 Karhunen–Loève theorem 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 Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category