Intersection local time for two independent fractional Brownian motions
Intersection Local Time for two Independent
Fractional Brownian Motions
David Nualart
Department of Mathematics, University of Kansas
405 Snow Hall, Lawrence, 66045 KS, USA
nualart@math.ku.edu, http://www.math.ku.edu/˜nualart/
Salvador Ortiz-Latorre
Facultat de Matemàtiques, Universitat de Barcelona
Gran Via 585 08007 Barcelona, Spain
sortiz@ub.edu
Abstract
Let BH and eBH be two independent, d-dimensional fractional Brow-
nian motions with Hurst parameter H ∈ (0, 1) . Assume d ≥ 2. We prove
that the intersection local time of BH and eBH
s )dsdt
exists in L2 if and only if Hd < 2.
Keywords: Fractional Brownian motion. Intersection local time.
Mathematics Subject Classification MSC2000: 60G15, 60F25,
60G18, 60J55.
1 Introduction
We consider two independent fractional Brownian motions onRd, d ≥ 2, with the
same Hurst parameter H ∈ (0, 1) . This means that we have two d-dimensional
independent centered Gaussian processesBH =
BHt , t ≥ 0
and B̃H = {B̃Ht , t ≥
0} with covariance structure given by
s ] = E[B̃
s ] = δijRH (s, t) ,
where i, j = 1, ..., d, s, t ≥ 0 and
RH (s, t) ≡
t2H + s2H − |t− s|
http://arxiv.org/abs/0704.1259v1
http://www.math.ku.edu/~nualart/
The object of study in this paper will be the intersection local time of BH and
B̃H , which is formally defined as
I(BH , B̃H) ≡
t − B̃
s )dsdt,
where δ0 (x) is the Dirac delta function. It is a measure of the amount of time
that the trajectories of the two processes, BH and B̃H , intersect on the time
interval [0, T ] . As we pointed out before, this definition is only formal. In order
to give a rigorous meaning to I(BH , B̃H) we approximate the Dirac function by
the heat kernel
pε (x) = (2πε)
exp(− |x|
/2ε),
in Rd. Then, we can consider the following family of random variables indexed
by ε > 0
H , B̃H) ≡
t − B̃
s )dsdt,
that we will call the approximated intersection local time of BH and B̃H . We
are interested in the L2 (Ω) convergence of Iε(B
H , B̃H) as ε tends to zero.
For H = 1/2, the processes BH and B̃H are classical Brownian motions.
The intersection local time of independent Brownian motions has been studied
by several authors (see Wolpert [9] and Geman, Horowitz and Rosen [2]). The
approach of these papers rely on the fact that the intersection local time of
independent Brownian motions can be seen as the local time at zero of some
Gaussian vector field. This approach easily allows to consider the intersection
of k independent Wiener processes, k ≥ 2. The applications of the intersection
local time theory for Brownian motions range from the construction of relativis-
tic quantum fields, see Wolpert [10], to the construction of the self-intersection
local time for the Brownian motion, see LeGall [4]. Further research has been
done in order to study such problems for other types of stochastic processes,
mainly Lévy processes with a particular structure (strongly symmetric), see
Marcus and Rosen [6].
In the general case, that is H 6= 1/2, only the self-intersection local time has
been studied. Rosen studied in [11] the planar case and a recent paper by Hu
and Nualart [3] gives a complete picture for the multidimensional case. On the
other hand, Nualart et al. [8] used a weighted version of the 3-dimensional self-
intersection local time for the study of probabilistic models for vortex filaments
based on the fractional Brownian motion . In recent years the fBm has become
an object of intense study. A stochastic calculus with respect to this process
has been developed by many authors, see Nualart [7] for an extensive account
on this subject. Because of its interesting properties, such as short/long range
dependence and selfsimilarity, the fBm it’s being widely used in a variety of areas
such finance, hydrology and telecommunications engineering, see [8]. Therefore,
it seems interesting to study the intersection local time for this kind of processes.
The aim of this paper is to prove the existence of the intersection local time
of BH and B̃H , for an H 6= 1/2 and d ≥ 2. We have obtained the following
result.
Theorem 1 (i) If Hd < 2, then the family of random variables Iε(B
H , B̃H)
converges in L2 (Ω). We will denote this limit by I(BH , B̃H).
(ii) If Hd ≥ 2, then
E[Iε(B
H , B̃H)] = +∞
Var[Iε(B
H , B̃H)] = +∞.
If {B
t , t ≥ 0} is a planar Brownian motion, then
B1/2s −B
diverges almost sure, when ε tends to zero. Varadhan, in [12], proved that the
renormalized self-intersection local time defined as limε→0(Iε − E[Iε]), exists in
L2 (Ω). Condition (ii) implies that Varadhan renormalization does not converge
in this case.
For Hd ≥ 2, according to the previous theorem, Iε(B
H , B̃H) doesn’t con-
verge in L2 (Ω) and therefore I(BH , B̃H), the intersection local time of BH and
B̃H , doesn’t exist. The proof of Theorem 1.1 rest on Lemma 4, which deals with
the integral of a negative power of the determinant of some covariance matrix.
The paper is organized as follows. In Section 2 we prove Theorem 1.1. In
order to clarify the exposition, some technical lemmas needed in the proof are
stated and proved in the Appendix.
2 Intersection Local Time of BH and B̃H , Case
Hd < 2
Let BH and B̃H two independent fractional Brownian motions on Rd with the
same Hurst parameter H ∈ (0, 1) .
Using the following classical equality
pε (x) =
ei〈ξ,x〉e−ε
from Fourier analysis, and the definition of Iε(B
H , B̃H), we obtain
H , B̃H) =
ei〈ξ,B
t − eB
s 〉e−ε
2 dξdsdt. (1)
Therefore,
E[Iε(B
H , B̃H)] =
E[ei〈ξ,B
t − eB
s 〉]e−ε
2 dξdsdt
e−(ε+s
2H+t2H)
2 dξdsdt
(ε+ s2H + t2H)−d/2dsdt, (2)
where we have used that 〈ξ, BHt − B̃
s 〉 ∼ N(0, |ξ|
s2H + t2H
), so
E[ei〈ξ,B
t − eB
s 〉] = e−(s
2H+t2H )
and the fact that
e−(ε+s
2H+t2H )
2 dξ =
ε+ s2H + t2H
According to the representation (1) for Iε(B
H , B̃H), we have that
E[I2ε (B
H , B̃H)] =
[0,T ]4
E[ei(〈ξ,B
t − eB
s 〉+〈η,B
v − eB
u 〉)]
× e−ε
|ξ|2+|η|2
2 dξdηdsdtdudv. (3)
Let introduce some notation that we will use throughout this paper,
λ = λ (s, t) = s2H + t2H ,
ρ = ρ (u, v) = u2H + v2H ,
µ = µ (s, t, u, v) =
s2H + t2H + u2H + v2H − |t− v|
− |s− u|
Notice that λ is the variance of B
t − B
s , ρ is the variance of B
v − B
and µ is the covariance between B
s and B
u , where B
H,1 and
BH,2 are independent one-dimensional fractional Brownian motions with Hurst
parameter H.
Using that 〈ξ, BHt − B̃
s 〉 + 〈η,B
v − B̃
u 〉 ∼ N(0, λ |ξ|
+ ρ |η|
+ 2µ〈ξ, η〉)
and (3) we can write for all ε > 0
E[I2ε (B
H , B̃H)]
[0,T ]4
{(λ+ε)|ξ|2+(ρ+ε)|η|2+2µ〈ξ,η〉}dξdηdsdtdudv
[0,T ]4
((λ + ε)(ρ+ ε)− µ2)−d/2dsdtdudv. (4)
The last equality follows from the well known fact that
〈x,Ax〉dx =
(detA)
A = Idd ⊗
λ+ ε µ
µ ρ+ ε
where Idd is the d-dimensional identity matrix and ⊗ denotes the Kronecker
product of matrices. We also have that
detA = det
Idd ⊗
λ+ ε µ
µ ρ+ ε
= (det (Idd))
λ+ ε µ
µ ρ+ ε
= ((λ + ε)(ρ+ ε)− µ2)d.
Proof of Theorem 1. Suppose first that Hd < 2. A slight extension of (4)
yields
E[Iε(B
H , B̃H)Iη(B
H , B̃H)] =
[0,T ]4
((λ + ε)(ρ+ η)− µ2)−d/2dsdtdudv.
Consequently, a necessary and sufficient condition for the convergence in L2 (Ω)
of Iε(B
H , B̃H) is that
[0,T ]4
(λρ− µ2)−d/2dsdtdudv < +∞.
Then the result follows from Lemma 4.
Now suppose that Hd ≥ 2, then from (2) and using monotone convergence
theorem
E[Iε(B
H , B̃H)] =
s2H + t2H
)−d/2
dsdt,
and this integral is divergent by Lemma 3. According to the expression (2) for
E[Iε(B
H , B̃H)] and the expression (4) for E[I2ε (B
H , B̃H)] we obtain
Var[Iε(B
H , B̃H)] = lim
E[Iε(B
H , B̃H)2]−
E[Iε(B
H , B̃H)]
[0,T ]4
(λρ− µ2)−d/2 − (λρ)
dsdtdudv.
Dε := {(s, t, u, v) ∈ R
+ | s
2 + t2 + u2 + v2 ≤ ε2}. (5)
We can find ε > 0 such that Dε ⊂ [0, T ]
. Making a change to spherical coor-
dinates, as the integrand is always positive, we have
[0,T ]4
(λρ− µ2)−d/2 − (λρ)
dsdtdudv
(λρ− µ2)−d/2 − (λρ)
dsdtdudv =
r3−2Hddr
Ψ(θ) dθ,
where the integral in r is convergent if and only if Hd < 2, and the angular in-
tegral is different from zero thanks to the positivity of the integrand. Therefore,
if Hd ≥ 2, then
Var[Iε(B
H , B̃H)] = +∞.
3 Appendix
For clarity of exposition, we state and prove some technical lemmas in this
appendix.
Lemma 2 Let α > 0, and let
γ (α, x) ≡
e−yyα−1dy (6)
be the lower incomplete gamma function. Then for all ε < α and x > 0,
γ (α, x) ≤ K (α)xε,
where K (α) ≡ 1
∨ Γ (α) and Γ (α) = γ (α,+∞).
Proof. If x ≥ 1,
γ (α, x) ≤ Γ (α) xε,
for all ε > 0. On the other hand, if x < 1,
γ (α, x) ≤
yα−1dy =
if ε < α.
Lemma 3 The following integral
s2H + t2H
)−d/2
dsdt,
is finite if and only if Hd < 2.
Proof. It easily follows from a polar change of coordinates.
Lemma 4 Let
[0,T ]4
(λρ− µ2)−d/2dsdtdudv,
then AT < +∞ if and only if Hd < 2.
Proof. The necessary condition follows from a spherical change of coordinates.
We can find ε > 0 such that Dε ⊂ [0, T ]
, where Dε is given in (5) . As the
integrand in AT is always positive we have
(λρ− µ2)−d/2dsdtdudv =
r3−2Hddr
φ (θ) dθ,
where the integral in r is convergent if and only if Hd < 2, and the angular in-
tegral is different from zero thanks to the positivity of the integrand. Therefore,
if Hd ≥ 2, then AT = +∞.
Suppose now that Hd < 2. By symmetry we have that
AT = 4
λρ− µ2
)−d/2
dsdtdudv,
where
T ≡ {(s, t, u, v) : 0 < v < t, 0 < t ≤ T, 0 < u < s, 0 < s ≤ T )}.
Notice that
λρ− µ2 = detVar (Z) ,
where Z ≡ (B
t − B̃
s , B
v − B̃
u ). Due to the independence of B
H and
B̃H , we have that
Var (Z) = Var(B
t , B
v ) + Var(B̃
s , B̃
λρ− µ2 ≥ det(Var(B
t , B
v )) + det(Var(B̃
s , B̃
u )),
because the matrices Var(B
t , B
v ) and Var(B̃
s , B̃
u ) are strictly positive
definite (see A8, (viii) in [5]). Then
AT ≤ 4
(ϕ (t, v) + ϕ (s, u))−d/2dsdtdudv,
where
ϕ (t, v) ≡ det(Var(B
t , B
v )) = t
2Hv2H −
t2H + v2H − |t− v|
Using Fubini’s Theorem and
λ−α =
Γ (α)
e−λzzα−1dz,
for all λ, α > 0, we obtain
(ϕ (t, v) + ϕ (s, u))−d/2dsdtdudv
e−(ϕ(t,v)+ϕ(s,u))zz
−1dzdsdtdudv
−1A2 (z) dz, (7)
where
A (z) ≡
e−ϕ(t,v)zdvdt.
As A (z) < +∞, for all z ∈ [0, 1], the integral (7) is convergent in a neighborhood
of zero. Hence, we have to study the convergence of
−1A2 (z) dz.
Due to the homogeneity of order 4H of ϕ (t, v) , if we make the change of coor-
dinates t = z−
4H x, v = z−
4H y, we obtain
−1A2 (z) dz =
−1− 1
∫ Tz 14H
e−ϕ(x,y)dydx
Now, using that {(x, y) : 0 < x < Tz
4H , 0 < y < x} ⊂ {(x, y) : x2 + y2 ≤
2T 2z
2H } ≡ S, and making a polar change of coordinates we have
∫ Tz 14H
e−ϕ(x,y)dydx ≤
e−ϕ(x,y)dydx =
∫ π/4
∫ √2Tz 14H
4Hϕ(θ)drdθ,
where ϕ (θ) ≡ ϕ (cos θ, sin θ) . After the new change of variable x = r4Hϕ (θ) ,
the last integral is equal to
∫ π/4
ϕ (θ)
∫ 22HT 4Hzϕ(θ)
−1dxdθ
∫ π/4
ϕ (θ)
(2H)−1, 22HT 4Hzϕ (θ)
where γ (α, x) is given by (6) . Applying Lemma 2,
−1A2 (z)dz
−1− 1
(∫ π/4
ϕ (θ)
(2H)−1, 22HT 4Hzϕ (θ)
24HεT 8Hε
)∫ +∞
−1− 1
+2εdz
(∫ π/4
ϕ (θ)
2H dθ
The integral in z is convergent provided ε < 2−Hd
. It’s an exercise of compu-
tation of limits to prove that ϕ (θ) ∼ θ2H as θ ↓ 0 and ϕ (θ) ∼ (π/4 − θ)2H as
θ ↑ π/4, the main tool is to substitute the trigonometric functions by their first
order approximations at the respective points. As a consequence, the integral
∫ π/4
ϕ (θ)
2H dθ
is always convergent.
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	Introduction
	Intersection Local Time of BH and B"0365BH, Case Hd<2
	Appendix
