Local time (mathematics)

<h2 id="formal-definition">Formal definition</h2>
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 <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> and 
 
 
 
 [
 B
 ]
 
 
 {\displaystyle [B]}
 
 is the <a href="/facts/Quadratic_variation/15jHz4xU">quadratic variation</a>. It is a notion invented by <a href="/facts/Paul_L%25C3%25A9vy_(mathematician)/RKFMXMkL">Paul Lévy</a>. 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 <a href="/facts/Ito_diffusion/vlwAqWSU"> diffusion</a> 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<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

L
          
            x
          
        
        (
        t
        )
        =
        
          ∫
          
            0
          
          
            t
          
        
        
          1
          
            {
            x
            }
          
        
        (
        
          X
          
            s
          
        
        )
        
        d
        s
        .
      
    
    {\displaystyle L^{x}(t)=\int _{0}^{t}1_{\{x\}}(X_{s})\,ds.}

<h2 id="tanakas-formula">Tanaka's formula</h2>
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} :}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

L
          
            x
          
        
        (
        t
        )
        =
        
          |
        
        
          X
          
            t
          
        
        −
        x
        
          |
        
        −
        
          |
        
        
          X
          
            0
          
        
        −
        x
        
          |
        
        −
        
          ∫
          
            0
          
          
            t
          
        
        
          (
          
            
              1
              
                (
                0
                ,
                ∞
                )
              
            
            (
            
              X
              
                s
              
            
            −
            x
            )
            −
            
              1
              
                (
                −
                ∞
                ,
                0
                ]
              
            
            (
            
              X
              
                s
              
            
            −
            x
            )
          
          )
        
        
        d
        
          X
          
            s
          
        
        ,
        
        t
        ≥
        0.
      
    
    {\displaystyle 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<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and Wang;<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If 
 
 
 
 F
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} }
 
 is absolutely continuous with derivative 
 
 
 
 
 F
 ′
 
 ,
 
 
 {\displaystyle F',}
 
 which is of bounded variation, then

F
        (
        
          X
          
            t
          
        
        )
        =
        F
        (
        
          X
          
            0
          
        
        )
        +
        
          ∫
          
            0
          
          
            t
          
        
        
          F
          
            −
          
          ′
        
        (
        
          X
          
            s
          
        
        )
        
        d
        
          X
          
            s
          
        
        +
        
          
            1
            2
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          L
          
            x
          
        
        (
        t
        )
        
        d
        
          F
          
            −
          
          ′
        
        (
        x
        )
        ,
      
    
    {\displaystyle 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
 
 −
 
 ′
 
 
 
 {\displaystyle F'_{-}}
 
 is the left derivative.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Brownian motion, then for any 
 
 
 
 α
 ∈
 (
 0
 ,
 1
 
 /
 
 2
 )
 
 
 {\displaystyle \alpha \in (0,1/2)}
 
 the field of local times 
 
 
 
 L
 =
 (
 
 L
 
 x
 
 
 (
 t
 )
 
 )
 
 x
 ∈
 
 R
 
 ,
 t
 ≥
 0
 
 
 
 
 {\displaystyle L=(L^{x}(t))_{x\in \mathbb {R} ,t\geq 0}}
 
 has a modification which is a.s. Hölder continuous in 
 
 
 
 x
 
 
 {\displaystyle x}
 
 with exponent 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
, uniformly for bounded 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 t
 
 
 {\displaystyle t}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> In general, 
 
 
 
 L
 
 
 {\displaystyle L}
 
 has a modification that is a.s. continuous in 
 
 
 
 t
 
 
 {\displaystyle t}
 
 and <a href="/facts/C%25C3%25A0dl%25C3%25A0g/R9Jz9QLh">càdlàg</a> in 
 
 
 
 x
 
 
 {\displaystyle x}
 
.
Tanaka's formula provides the explicit <a href="/facts/Doob%25E2%2580%2593Meyer_decomposition_theorem/UD44QwPD">Doob–Meyer decomposition</a> for the one-dimensional <a href="/facts/Reflecting_Brownian_motion/Wb4Knnzv">reflecting Brownian motion</a>, 
 
 
 
 (
 
 |
 
 
 B
 
 s
 
 
 
 |
 
 
 )
 
 s
 ≥
 0
 
 
 
 
 {\displaystyle (|B_{s}|)_{s\geq 0}}
 
.

<h2 id="rayknight-theorems">Ray–Knight theorems</h2>
The field of local times 
 
 
 
 
 L
 
 t
 
 
 =
 (
 
 L
 
 t
 
 
 x
 
 
 
 )
 
 x
 ∈
 E
 
 
 
 
 {\displaystyle L_{t}=(L_{t}^{x})_{x\in E}}
 
 associated to a stochastic process on a space 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a well studied topic in the area of random fields. Ray–Knight type theorems relate the field Lt to an associated <a href="/facts/Gaussian_process/MrBq7kYW">Gaussian process</a>.
In general Ray–Knight type theorems of the first kind consider the field Lt 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.

<h3>First Ray–Knight theorem</h3>
Let (Bt)t ≥ 0 be a one-dimensional Brownian motion started from B0 = a > 0, and (Wt)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
 ≥
 0
 :
 
 B
 
 t
 
 
 =
 0
 }
 
 
 {\displaystyle T=\inf\{t\geq 0\colon B_{t}=0\}}
 
. Ray<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and Knight<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> (independently) showed that

<table><tbody><tr><td> { L x ( T ) : x ∈ [ 0 , a ] } = D { | W x | 2 : x ∈ [ 0 , a ] } {\displaystyle \left\{L^{x}(T)\colon x\in [0,a]\right\}{\stackrel {\mathcal {D}}{=}}\left\{|W_{x}|^{2}\colon x\in [0,a]\right\}\,} </td> <td></td> <td>1</td></tr></tbody></table>
where (Lt)t ≥ 0 is the field of local times of (Bt)t ≥ 0, and equality is in distribution on C[0, a]. The process |Wx|2 is known as the squared <a href="/facts/Bessel_process/RKVXhbc8">Bessel process</a>.

<h3>Second Ray–Knight theorem</h3>
Let (Bt)t ≥ 0 be a standard one-dimensional Brownian motion B0 = 0 ∈ R, and let (Lt)t ≥ 0 be the associated field of local times. Let Ta be the first time at which the local time at zero exceeds a > 0

T
          
            a
          
        
        =
        inf
        {
        t
        ≥
        0
        :
        
          L
          
            t
          
          
            0
          
        
        >
        a
        }
        .
      
    
    {\displaystyle T_{a}=\inf\{t\geq 0\colon L_{t}^{0}>a\}.}

Let (Wt)t ≥ 0 be an independent one-dimensional Brownian motion started from W0 = 0, then<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<table><tbody><tr><td> { L T a x + W x 2 : x ≥ 0 } = D { ( W x + a ) 2 : x ≥ 0 } . {\displaystyle \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\}.\,} </td> <td></td> <td>2</td></tr></tbody></table>
Equivalently, the process 
 
 
 
 (
 
 L
 
 
 T
 
 a
 
 
 
 
 x
 
 
 
 )
 
 x
 ≥
 0
 
 
 
 
 {\displaystyle (L_{T_{a}}^{x})_{x\geq 0}}
 
 (which is a process in the spatial variable 
 
 
 
 x
 
 
 {\displaystyle x}
 
) is equal in distribution to the square of a 0-dimensional <a href="/facts/Bessel_process/RKVXhbc8">Bessel process</a> started at 
 
 
 
 a
 
 
 {\displaystyle a}
 
, and as such is Markovian.

<h3>Generalized Ray–Knight theorems</h3>
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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Tanaka%2527s_formula/wJU9BmfL">Tanaka's formula</a></li>
<li><a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a></li>
<li><a href="/facts/Random_field/8N7Id48z">Random field</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, 1990, Birkhäuser, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3386-8.</li>
<li>M. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, 1st edition, 2006, Cambridge University Press <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-86300-1</li>
<li>P. Mörters and Y. Peres, Brownian Motion, 1st edition, 2010, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-76018-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus. Springer. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 428–449. ISBN 0387949577. <a href="0387949577" target="_blank">0387949577</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-3-540-42813-8" target="_blank">978-3-540-42813-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.1007%2Fbf00538419" target="_blank">https://doi.org/10.1007%2Fbf00538419</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Kallenberg (1997). Foundations of Modern Probability. New York: Springer. pp. 370. ISBN 0387949577. <a href="0387949577" target="_blank">0387949577</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.1215%2Fijm%2F1255645099" target="_blank">https://doi.org/10.1215%2Fijm%2F1255645099</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.2307%2F1993647" target="_blank">https://doi.org/10.2307%2F1993647</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Marcus; Rosen (2006). Markov Processes, Gaussian Processes and Local Times. New York: Cambridge University Press. pp. 53–56. ISBN 0521863007. <a href="0521863007" target="_blank">0521863007</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Local time (mathematics) open-in-new

Local time (mathematics)