BROWNIAN LOCAL TIME

BROWNIAN LOCAL TIME

RAOUF GHOMRASNI

Received 26 October 2004; Revised 11 April 2005; Accepted 12 April 2005

LetB=(Bt)t≥0be a standard Brownian motion and let (L^x_t;t≥0,x∈R) be a continuous version of its local time process. We show that the following limit limε↓0(1/2ε)₀^t{F(s,Bs− ε)₋F(s,B_s+ε)_}dsis well defined for a large class of functionsF(t,x), and moreover we connect it with the integration with respect to local timeL^x_t. We give an illustrative ex- ample of the nonlinearity of the integration with respect to local time in the random case.

Copyright © 2006 Raouf Ghomrasni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

1.1. The local time of the Brownian motionBat the pointais defined as follows:

L^a_t =Plim

ε↓0

1 2ε

_t

01(_|Bs−a|≤ε)ds, (1.1) which equivalently could be written as follows:

L^a_{t =P}lim

ε↓0

1 2ε

_t

0

1(Bs−ε≤a)−1(Bs+ε≤a)

ds. (1.2)

Here we are, more generally, interested in the limit inL¹: limε↓0

1 2ε

_t

0

Fs,B_s−ε−Fs,B_s+εds (1.3)

for some functionF.

Our motivation comes from the desire to connect Chitashvili and Mania results [1]

with those of Eisenbaum [2].

Hindawi Publishing Corporation

Journal of Applied Mathematics and Stochastic Analysis Volume 2006, Article ID 26961, Pages1–5

DOI10.1155/JAMSA/2006/26961

1.2. We give an example which illustrates that the integration with respect to (L^x_t; 0≤t≤ 1,x∈R) does not admit a linear extension in the random case (seeSection 3.2for details) and in particular local time is not a 1-integrator, which is also proved by Eisenbaum [2].

2. Notation and preliminaries

LetB=(B_t)_t≥0be a standard Brownian motion and let (L^x_t;t≥0,x∈R) be a continuous version of its local time process. Let (Ᏺt)_t≥0denote the natural filtration generated byB.

Without loss of generality, we restrict our attention to functions defined on [0, 1]×R. For a measurable function f from [0, 1]×RintoR, define the norm · by

f =2 1

0

Rf²(s,x)e^−x2/2s_√ds dx 2πs

1/2

+ 1

0

R

x f(s,x)e^−x2/2sds dx

s^√2πs. (2.1) LetᏴbe the set of functions f such thatf<∞.

In Eisenbaum [2], it is shown that the integration with respect toLis possible in the following sense. Let f_Δbe an elementary function on [0, 1]×R, meaning that

f_Δ(t,x)=

(si,xj)_∈Δ

f_i,j1(si,si+1](t)1_(xj,xj+1](x), (2.2) whereΔ= {(si,xj), 1≤i≤n, 1≤j≤m}is an [0, 1]×Rgrid, and, for every (i,j), fi j is in R. For such a function, integration with respect toLis defined by

1 0

Rf_Δ(s,x)dL^x_s=

(si,xj)∈Δ

f_i,jL^xsi+1^j+1−L^xsi^j+1−L^xsi+1^j +L^xsi^j

. (2.3)

Let f be an element ofᏴ. For any sequence of elementary functions (f_Δk)k∈N con- verging to f in Ᏼ, the sequence (₀¹_Rf_Δk(s,x)dL^x_s)_k∈N converges inL¹. The limit ob- tained does not depend on the choice of the sequence (f_Δk) and represents the integral ₁

0

Rf(s,x)dL^x_s.

Theorem 2.1 (see [2]). Let (A(x,t); x∈R, 0≤t≤1) be a continuous random process taking values inR, such that for anytin [0, 1] and any ω,A(·,t) is absolutely continu- ous with respect todx. Note∂A/∂x its derivative and ask∂A/∂x to be continuous. Then 1

0

RA(x,s)dL^x_sexists and the following hold:

(i) for any couple (a,b) inR²witha < b _t

0

_a

bA(x,s)dL^x_s= − _t

0

∂A

∂x

Bs,sds+ _t

0A(b,s)dsL^b_s− _t

0A(a,s)dsL^a_s; (2.4) (ii)

1 0

RA(x,s)dL^x_s = − 1

0

∂A

∂x

Bs,sds; (2.5)

(iii)

t 0

_a

b A(x,s)dL^x_s (ω)= _t

0

_a

bA(x,s)(ω)dL^x_s(ω). (2.6)

3. Main results 3.1. Deterministic case

Theorem 3.1. LetFbe a bounded element ofᏴ. The following equalities hold inL¹: limε↓0

1 ε

_t

0

Fs,B_s−Fs,B_s−εds= − _t

0

RF(s,x)dL^x_s; (3.1) limε↓0

1 ε

_t

0

Fs,Bs+ε−Fs,Bs

ds= − _t

0

RF(s,x)dL^x_s; (3.2) limε↓0

1 2ε

_t

0

Fs,B_s−ε−Fs,B_s+εds= _t

0

RF(s,x)dL^x_s. (3.3) Remark 3.2. (1) If we takeF(t,x)=1_(x≤a)in (3.1), we have the very definition ofL^a_t.

(2) Eisenbaum [2] has shown that for any Borelian functionb(t), limε↓0

1 2ε

_t

01(|Bs−b(s)|<ε)ds= _t

0

R1(−∞,b(s))(x)dL^x_s inL¹, (3.4) which corresponds to (3.3) withF(t,x)=1(x≤b(t)).

Proof. DefineHε(t,x)=(1/ε)_x^x_−εF(t,y)d y. ThenHε→FinᏴasε↓0. On the one hand, (∂/∂x)Hε(t,x)=(1/ε){F(t,x)−F(t,x−ε)}. It follows that (see Eisenbaum [2, Theorem 5.1(ii)])₀^t_RH_ε(s,x)dL^x_s=−(1/ε)₀^t{F(s,B_s)−F(s,B_s−ε)}ds. On the other hand,₀^t_RH_ε(s, x)dL^x_s→_t

0

RF(s,x)dL^x_sinL¹.

Corollary 3.3 (see [3]). The following relation holds inL¹:

limε↓0

1 2ε

_t

0g(s)I(b(s)−ε < B_s< b(s) +ε)ds= _t

0g(s)dL^b_s (3.5) for a continuous functiong: [0,t]_→Rand a continuous curveb(_·) with bounded variation on [0,t].

Proof. We applyTheorem 3.1to the functionF(t,x)=g(t)I(x < b(t)). It follows that (1/

2ε)₀^tg(s)I(b(s)−ε < Bs< b(s) +ε)ds→_t

0

Rg(s)I(x < b(s))dL^x_sinL¹asε↓0. We conclude using (see [4, Corollary 2.9]) that for the continuous functiong, we have₀^tg(s)∂sL^b(s)s = _t

0g(s)dL^b_s.

3.2. Random function case. Leta,bbe inRwitha < b. Letᏹbe the set of elementary processesAsuch that

A(s,x)=

(si,xj)_∈Δ

Ai j1si,si+1](s)1(xj,xj+1](x), (3.6) where (s_i)1≤i≤nis a subdivision of (0, 1], (x_j)1≤j≤mis a finite sequence of real numbers in (a,b],Δ= {(si,xj), 1≤i≤n, 1≤j≤m}, and, isAi janᏲsj-measurable random variable such that|Ai j| ≤1 for every (i,j).

Eisenbaum [2] asked the following question: does integration with respect to (L^x_t; 0≤ t≤1,x∈R) admit a linear extension toᏼthe field generated byᏹ, verifying the follow- ing property?

If (A_n)_n≥0 converges a.e. to A(t,x), then (₀¹_a^bA_n(s,x)dL^x_s)_n≥0 converges in L¹ to ₁

0

_b

aA(s,x)dL^x_s.

She only obtained a negative answer to the following weaker question:

Is the set 1

0

_b

a A(s,x)dL^x_s,A∈ᏹbounded inL¹? (3.7) Consequently, integration with respect to (L^x_t; 0≤t≤1, x∈R) does not admit a continuous extension inL¹.

Here we give an illustrative example, thanks to a result obtained by Walsh, which shows the lack of a linear extension.

Let us defineA_ε(t,x)=(1/ε)_x^x_−εL_t^yd yandA_ε(t,x)=(1/ε)_x^x+εL_t^yd y. We see easily that A_ε(t,x) (resp.,A_ε(t,x)) converges a.e. toL^x_t, nevertheless we have

limε↓0

_t

0

RA_ε(s,x)dL^x_s=lim

ε↓0

_t

0

RA_ε(s,x)dL^x_s. (3.8) Remark 3.4. The integrals₀^t_RA_ε(s,x)dL^x_sand₀^t_RA_ε(s,x)dL^x_sare well defined thanks to Theorem 2.1, however, one does not know whether₀^t_RL^x_sdL^x_s is well defined or not.

Let us recall, for the convenience of the reader, Walsh’s theorem about the decomposi- tion ofA(t,Bt) :=_t

01_{Bs≤Bt}ds.

Theorem 3.5 (see [5]). A(t,Bt) has the decomposition At,Bt

= _t

0L^B_s^sdBs+Xt, (3.9)

where

Xt=lim

ε↓0

1 2ε

_t

0

L^B_s^s−L^B_s^s−εds=t+ lim

ε↓0

1 2ε

_t

0

L^B_s^s+ε−L^B_s^sds. (3.10)

The limits exist in probability, uniformly fortin compact sets.

Our example follows by recalling the following property:

_t

0

RAε(s,x)dL^x_s= −1 ε

_t

0

L^B_s^s−L^B_s^s−εds. (3.11)

References

[1] R. J. Chitashvili and M. G. Mania, Decomposition of the maximum of semimartingales and gen- eralized Itˆo’s formula, New Trends in Probability and Statistics, Vol. 1 (Bakuriani, 1990), VSP, Utrecht, 1991, pp. 301–350.

[2] N. Eisenbaum, Integration with respect to local time, Potential Analysis 13 (2000), no. 4, 303–

328.

[3] G. Peskir, A change-of-variable formula with local time on curves, Research Report 428, Depart- ment of Theoretical Statistics, University of Aarhus, Aarhus, 2002, 17 pp.

[4] Ph. Protter and J. San Mart´ın, General change of variable formulas for semimartingales in one and finite dimensions, Probability Theory and Related Fields 97 (1993), no. 3, 363–381.

[5] J. B. Walsh, Some remarks onA(t,Bt), S´eminaire de Probabilit´es 27, Lecture Notes in Mathemat- ics, vol. 1557, Springer, Berlin, 1993, pp. 173–176.

Raouf Ghomrasni: Fakult¨at II–Mathematik und Naturwissenschaften, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

E-mail address:[email protected]