One version of the clark representation theorem for arratia flows
The article contains the description of functionals from a family of coalescing Brownian particles. A new type of the stochastic integral is introduced and used.
Gespeichert in:
| Datum: | 2005 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2005
|
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/4426 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | One version of the clark representation theorem for arratia flows / A.A. Dorogovtsev // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 63–70. — Бібліогр.: 4 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4426 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-44262009-11-10T12:00:31Z One version of the clark representation theorem for arratia flows Dorogovtsev, A.A. The article contains the description of functionals from a family of coalescing Brownian particles. A new type of the stochastic integral is introduced and used. 2005 Article One version of the clark representation theorem for arratia flows / A.A. Dorogovtsev // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 63–70. — Бібліогр.: 4 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4426 519.21 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The article contains the description of functionals from a family of coalescing Brownian particles. A new type of the stochastic integral is introduced and used. |
| format |
Article |
| author |
Dorogovtsev, A.A. |
| spellingShingle |
Dorogovtsev, A.A. One version of the clark representation theorem for arratia flows |
| author_facet |
Dorogovtsev, A.A. |
| author_sort |
Dorogovtsev, A.A. |
| title |
One version of the clark representation theorem for arratia flows |
| title_short |
One version of the clark representation theorem for arratia flows |
| title_full |
One version of the clark representation theorem for arratia flows |
| title_fullStr |
One version of the clark representation theorem for arratia flows |
| title_full_unstemmed |
One version of the clark representation theorem for arratia flows |
| title_sort |
one version of the clark representation theorem for arratia flows |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4426 |
| citation_txt |
One version of the clark representation theorem for arratia flows / A.A. Dorogovtsev // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 63–70. — Бібліогр.: 4 назв.— англ. |
| work_keys_str_mv |
AT dorogovtsevaa oneversionoftheclarkrepresentationtheoremforarratiaflows |
| first_indexed |
2025-07-02T07:40:30Z |
| last_indexed |
2025-07-02T07:40:30Z |
| _version_ |
1836520078303035392 |
| fulltext |
Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 63–70
UDC 519.21
ANDREY A. DOROGOVTSEV
ONE VERSION OF THE CLARK REPRESENTATION
THEOREM FOR ARRATIA FLOWS
The article contains the description of functionals from a family of coalescing Brown-
ian particles. A new type of the stochastic integral is introduced and used.
Introduction
The aim of this article is to establish the Clark representation for functionals from an
Arratia flow of coalescing Brownian particles [1-4]. The following description of this flow
will be used. We consider the random process {x(u); u ∈ R} with values in C([0; 1]) such
that, for every u1 < · · · < un,
1) x(uk, ·) is the standard Wiener process starting at the point uk,
2) ∀t ∈ [0; 1]
x(u1, t) ≤ · · · ≤ x(un, t),
3) The distribution of (x(u1, ·), . . . , x(un, ·)) coincides with the distribution of the
standard n-dimensional Wiener process starting at (u1, . . . , un) on the set
{f ∈ C([0; 1], Rn) : fk(0) = uk, k = 1, . . . , n, f1(t) < · · · < fn(t), t ∈ [0; 1]}.
Roughly speaking, the process x can be described as a family of Wiener particles which
start from every point of R, move independently up to the moment of the meeting, and
then coalesce and move together.
The following fact is well known. If {w(t); t ∈ [0; 1]} is the standard Wiener process
and a square-integrable random variable α is measurable with respect to w, then α can
be represented as a sum
α = Eα +
∫ 1
0
f(t)dw(t)
with the usual Itô stochastic integral on the right-hand side. Our aim is to establish a
variant of this theorem in the case where α is measurable with respect to the Arratia flow
{x(u, t); u ∈ [0; U ], t ∈ [0; 1]}. It follows from the description above that the Brownian
motions {x(u, ·); u ∈ [0; U ]} are not jointly Gaussian. So, the original Clark theorem
cannot be used in this situation. The article is divided into three parts. In the first part,
the construction of the stochastic integral with respect to the Arratia flow is presented.
The next part is devoted to the variants of the Clark theorem for a finite number of the
Brownian motions stopped at random times. In the last part, the modification of the
construction from the first part is applied to the representation of the functionals from
the flow.
2000 AMS Mathematics Subject Classification. Primary 60H05, 60H40, 60J65, 60K35.
Key words and phrases. Brownian motion, coalescence, Clark representation, stochastic integral.
63
64 ANDREY A. DOROGOVTSEV
1. Spatial stochastic integral with respect to the Arratia flow
Let {x(u); u ∈ R} be the Arratia flow, i.e. the flow of Brownian particles with coa-
lescence described above. For U > 0, consider a partition π of the interval [0;U ], π =
{u0 = 0, . . . , un = U}. For k = 1, . . . , n, we define
τ(uk) = inf{1, t ∈ [0; 1] : x(uk, t) = x(uk−1, t)}.
Note that τ(uk), k = 1, . . . , n are the stopping moments with respect to the flow
Fπ
t = σ(x(uk, s), k = 1, . . . , n, s ≤ t).
For a bounded measurable function a : R → R, let us consider the sum
(1.1) Sπ =
n∑
k=1
∫ τ(uk)
0
a(x(uk, s))dx(uk, s).
Our aim is to investigate the limit of Sπ under
|π| = max
k=0,...,n−1
(uk+1 − uk) → 0
and its properties depending on the function a and the spatial variable U.
Let us begin with the moments of Sπ. It follows from the standard properties of Itô
stochastic integrals that
ESπ = 0
and
(1.2) ES2
π = E
n∑
k=1
∫ τ(uk)
0
a2(x(uk, s))ds.
Let us denote
Sπ =
n∑
k=1
∫ τ(uk)
0
a2(x(uk, s))ds.
Consider the sequence of increasing partitions {πn; n ≥ 1} of the interval [0;U ] with
|πn| → 0, n → ∞.
Lemma 1.1. There exists a limit
(1.3) lim
n→∞ Sπn a.s.
Proof. To prove the lemma, we will check two properties of the sequence {Sπn ; n ≥ 1} :
(1.4) ∀n ≥ 1 : Sπn ≤ Sπn+1,
and
(1.5) sup
n≥1
ESπn < +∞.
Note that it is enough to prove (1.4) in the case where πn+1 contains only one additional
point v0 comparing with πn. Suppose that πn = {u0 = 0, . . . , un = U} and πn+1 = {u0 =
0, . . . , uk, v0, uk+1, . . . , un = U}. Denote
τ́ (uk+1) = inf{1, t ∈ [0; 1] : x(uk+1, t) = x(v0, t)}.
Now
Sπn+1 − Sπn =
∫ τ́(uk+1)
0
a2(x(uk+1, s))ds+
+
∫ τ(v0)
0
a2(x(v0, s))ds −
∫ τ(uk+1)
0
a2(x(uk+1, s))ds.
ONE VERSION OF THE CLARK REPRESENTATION THEOREM 65
There are two possibilities. In the first one, τ(v0) < τ(uk+1). Now τ́ (uk+1) = τ(uk+1).
So in this case
Sπn+1 − Sπn =
∫ τ(v0)
0
a2(x(v0, s))ds ≥ 0.
The next case is τ(v0) ≥ τ(uk+1). This possibility can be realized only if τ(v0) = τ(uk+1).
Now τ́ (uk+1) ≤ τ(v0) and∫ τ(uk+1)
0
a2(x(uk+1, s))ds =
∫ τ́(uk+1)
0
a2(x(uk+1, s))ds +
∫ τ(uk+1)
τ́(uk+1)
a2(x(v0, s))ds.
So, in this case,
Sπn+1 − Sπn =
=
∫ τ(v0)
0
a2(x(v0, s))ds −
∫ τ(uk+1)
τ́(uk+1)
a2(x(v0, s))ds =
=
∫ τ(v0)
0
a2(x(v0, s))ds −
∫ τ(v0)
τ́(uk+1)
a2(x(v0, s))ds =
=
∫ τ́(uk+1)
0
a2(x(v0, s))ds ≥ 0.
Hence, (1.4) is true. Let us estimate the expectation of Sπn . Consider two independent
standard Wiener processes w1, w2 which start from 0 and u > 0, correspondingly. Denote
τ = inf{1, t : w1(t) = w2(t)}.
Then
(1.6) Eτ =
∫ 1
0
∫ u
−u
p2t(v)dv +
∫ u
−u
p2(v)dv,
where pt is the density of the normal distribution with zero mean and covariance t. It
follows from (1.6) that
(1.7) Eτ ∼ 3u
2
√
π
, u → 0 + .
Consequently,
lim
n→∞ESπn ≤ U
3
2
√
π
sup
R
a2.
Now the statement of the lemma follows from (1.4) and (1.7).
Remark 1. It follows from the proof of the lemma that there exists a limit
lim
n→∞ESπn .
Lemma 1.2. There exists a limit
m(U) = L2 − lim
n→∞Sπn .
Proof. Let the partitions πn, πn+1 be the same as in the proof of the previous lemma.
Then
ESπnSπn+1 =E
n∑
j1=1
∫ τ(uj1)
0
a(x(uj1 , s))dx(uj1 , s)·
· (
∑
j2 �=k+1
∫ τ(uj2)
0
a(x(uj2 , s))dx(uj2 , s) +
∫ τ(v0)
0
a(x(v0, s))dx(v0, s)+
+
∫ τ́(uk+1)
0
a2(x(uk+1, s))dx(uk+1, s)) =
66 ANDREY A. DOROGOVTSEV
=E
∑
j �=k+1
∫ τ(uj)
0
a2(x(uj , s))ds + E
∫ τ(uk+1)
0
a(x(uk+1, s))dx(uk+1, s)·
· (
∫ τ(v0)
0
a(x(v0, s))dx(v0, s) +
∫ τ́(uk+1)
0
a(x(uk+1, s))dx(uk+1, s))) =
=E
∑
j �=k+1
∫ τ(uj)
0
a2(x(uj , s))ds + E
∫ τ(uk+1)∧τ́(uk+1)
0
a2(x(uk+1, s))ds+
+ E
∫ τ(uk+1)∧τ(v0)
τ́(uk+1)∧τ(v0)
a2(x(uk+1, s))ds =
=E
∑
j �=k+1
∫ τ(uj)
0
a2(x(uj , s))ds + E
∫ τ(uk+1)
0
a2(x(uk+1, s))ds =
=ES2
πn
.
Consequently, for all n ≤ m,
ESπnSπm = ESπn .
Now the statement of the lemma follows from Remark 1.
Remark 2. Note that the limit m(U) does not depend on the choice of the sequence of
partitions {πn; n ≥ 1}.
To prove this, we need in the estimation of the rate of convergence ES2
πn
to its limit.
Lemma 1.3. There exists a constant C such that, for every partition π of the interval
[0; U ],
(1.8) |ES2
π − Em2(U)| ≤ C|π| sup
R
a2.
Proof. First consider the partitions π′, π′′, where π′′ is obtained from π′ by adding one
point on the interval [uk, uk+1]. As was mentioned in the proof of Lemma 1,
(1.9) Sπ′′ − Sπ′ =
∫ ζ(v0)
0
a2(x(v0, s))ds.
Here,
ζ(v0) = inf{1; t : (x(v0, t) − x(uk, t))(x(v0, t) − x(uk+1, t)) = 0}.
Let us estimate Eζ(v0). Consider the standard Wiener process −→w on the plane, which is
starting from the point −→r . Suppose that this point lies inside the angle with the vertex
in the origin. Let the value of the angle be less than π
2 , and let the angle lie in the
part of the plane where the both coordinate are nonnegative. Define ζ́ as the first exit
time of −→w from the angle. Then, enlarging the angle up to π
2 and using one-dimensional
expressions like (1.6), we can check that there exists C > 0 such that
(1.10) Eζ́ ∧ 1 ≤ Cr1r2,
where −→r = (r1, r2).
From these remarks, we can conclude that there exists C1 > 0 such that
(1.11) Eζ(v0) ≤ C1(uk+1 − v0)(v0 − uk).
This conclusion can be obtained if we note that ζ(v0) is the minimum of 1 and the first
exit time of the 3-dimensional Wiener process from the space angle with the value π
3 . It
follows from (1.9) and (1.11) that
(1.12) E(Sπ′′ − Sπ′) ≤ C1 sup
R
a2(uk+1 − v0)(v0 − uk).
Now let us consider the general case where π′′ is obtained from π′ by the adding of a few
new points. Denote, by v1 < · · · < vm, the new points on the interval [uk, uk+1]. Then
ONE VERSION OF THE CLARK REPRESENTATION THEOREM 67
the new amount which is obtained in E(Sπ′′ − Sπ′) from these points can be estimated
due to (1.12) by the sum
C1 sup
R
a2
m∑
j=1
(vj − vj−1)(uk+1 − vj),
where we suppose that v0 = uk. Consequently
E(Sπ′′ − Sπ′) ≤ C1 sup
R
a2
n−1∑
k=0
(uk+1 − uk)2 ≤ C1 sup
R
a2U |π′|.
This inequality leads to the existence of the limit
lim
|π|→0
ESπ.
It follows from the proof of Lemma 2 that
Em2(U) = lim
|π|→0
ESπ.
It is clear now that (1.8) holds.
The independence m(U) on the choice of the sequence {πn; n ≥ 1} now follows in
standard way.
For U ≥ 0, we define the σ−field
F̃U = σ(x(u, ·); 0 ≤ u ≤ U).
Lemma 1.4. The process {m(U); U ≥ 0} is (F̃U )−martingale.
Proof. The measurability of m(U) with respect to F̃U is evident. Let 0 ≤ U1 < U2.
Consider the partition |π| of [0; U2] which contains the point U1. Then
E(Sπ/F̃U1) =
∑
uk≤U1
∫ τ(uk)
0
a(x(uk, s))dx(uk, s)+
+ E(
∑
uk>U1
∫ τ(uk)
0
a(x(uk, s))dx(uk, s)/F̃U1).
To prove that the last summand is equal to zero, it is enough to consider the expression
E(
∫ τ(u)
0
a(x(u, s))dx(u, s)/F̃U1 )
for u > U1. Take 0 ≤ u1 < · · · < un = U1. For a bounded Borel function f : C([0; 1])n →
R, the expectation
(1.13) E
∫ τ(u)
0
a(x(u, s))dx(u, s)f(x(u1, ·), . . . , x(un, ·))
can be rewritten as
E
∫ τ́
0
a(w(s))dw(s)f́ (w1, . . . , wn),
where w and w1, . . . , wn are independent standard Wiener processes starting from the
points u and u1, . . . , un, correspondingly, τ́ is a stopping time for (w, w1, . . . , wn), andf́
is a bounded Borel function on C([0; 1])n. Denote, by Γ, the σ−field corresponding to τ́ .
Then
E
∫ τ́
0
a(w(s))dw(s)f́ (w1, . . . , wn) = E
∫ τ́
0
a(w(s))dw(s)E(f́ (w1, . . . , wn)/Γ)
= E
∫ τ́
0
a(w(s))dw(s)f̃ (w̃1, . . . , w̃n),
68 ANDREY A. DOROGOVTSEV
where w̃k(s) = wk(s ∧ τ́ ), k = 1, . . . , n, and f̃ is a new bounded Borel function. Due to
the Clark representation theorem,
f̃(w̃1, . . . , w̃n) = c +
n∑
k=1
∫ τ́
0
ηk(s)dwk(s),
where, for k = 1, . . . , n, ηk is the square-integrable random function adapted to the flow
Γt = σ(w(s), w1(s), . . . , wn(s), s ≤ t).
Consequently,
E
∫ τ́
0
a(w(s))dw(s)f̃ (w̃1, . . . , w̃n) = E
∫ τ́
0
a(w(s))dw(s)·
·
(
c +
n∑
k=1
∫ τ́
0
ηk(s)dwk(s)
)
= 0.
Hence, (1.13) is also equal to zero. Finally,
E(
∑
k>U1
∫ τ(uk)
0
a(x(uk, s))dx(uk, s)/F̃U1) = 0.
Taking the limit where the diameter of the partition tends to 0, we get the statement of
the lemma.
2. Clark representation for a finite
family of coalescing Brownian motions
This section is devoted to the integral representation of the functionals from
x(u1, ·), . . . , x(un, ·), u1 < u2 < . . . < un.
Let us start with the following simple lemma, which was already used in the previous
section.
Lemma 2.1. Let w be the standard Wiener process on [0; 1], and let 0 ≤ τ ≤ 1 be the
stopping time for w. Suppose that the square-integrable random variable α is measurable
with respect to {w(τ ∧ t); t ∈ [0; 1]}. Then α can be represented as
α = Eα +
∫ τ
0
f(t)dw(t)
with the certain adapted square-integrable random function f.
Proof. Note that
σ(w(τ ∧ t); t ∈ [0; 1]) = Fτ ,
where Fτ is the σ-field corresponding to the stopping moment τ. Now, due to the original
Clark theorem,
α = Eα +
∫ 1
0
f(t)dw(t).
It remains now to apply the conditional expectation with respect to Fτ to the both sides
of this equality. Lemma 2.1 is proved.
Consider the following situation. Let w1, w2 be independent standard Wiener pro-
cesses on [0; 1], and let τ be a stopping time with respect to its join flow of σ-fields.
The processes w1, w2 and the random variable τ can be considered on the product of
probability spaces Ω1 × Ω2. Here, Ω1 is related to w1 and Ω2 is related to w2.
Lemma 2.2. For every fixed ω1 ∈ Ω1, the random variable τ(ω1, ·) on Ω2 is the stopping
moment for w2 on Ω2.
ONE VERSION OF THE CLARK REPRESENTATION THEOREM 69
Proof. The set {ω2 : τ(ω1, ω2) < t} is the cross section of {τ < t} in Ω1 × Ω2. Hence,
its measurability with respect to σ(w2(s); s ≤ t) follows from the usual arguments of
measure theory.
The previous two lemmas lead to the following result.
Theorem 2.1. Let w0, w1, . . . , wn be independent standard Wiener processes on [0; 1]
and, for every k = 1, . . . , n, τk is the stopping time for the process (w0, w1, . . . , wk).
Suppose that the square-integrable random variable α is measurable with respect to the
set (w0(·), w1(τ1 ∧ ·), . . . , wn(τn ∧ ·)). Then α can be represented as
α = Eα +
n∑
k=0
∫ τk
0
fk(t)dwk(t),
where τ0 = 1 and fk is adapted to the flow generated by wk under fixed wj , j �= k.
Proof. Denote w̃k(t) = wk(τk ∧ t), k = 1, . . . , n. Consider the random variable α −
E(α/w̃0, . . . , w̃n−1). It is measurable with respect to w̃n under fixed w̃0, . . . , w̃n−1 and
has zero mean. Due to the previous lemma, it can be written as
α − E(α/w̃0, . . . , w̃n−1) =
∫ τn
0
fn(t)dwn(t),
where the random function fn under fixed w̃0, . . . , w̃n−1 is adapted to the flow generated
by wn. Repeat the same procedure for the random variable E(α/w̃0, . . . , w̃n−1). Then
E(α/w̃0, . . . , w̃n−1) − E(α/w̃0, . . . , w̃n−2) =
∫ τn−1
0
fn−1(t)dwn−1(t).
After n steps, we get the statement of the theorem.
Remark 3. Note that the representation from the theorem has the property
Eα2 = (Eα)2 +
n∑
k=1
E
∫ τk
0
fk(t)2dt.
Consider an example of the application of Theorem 2.1.
Example 2.1. Let α be the square-integrable random variable measurable with respect
to x(u0, ·), . . . , x(un, ·), where u0, . . . , un are different points. Define the random mo-
ments
τ0 = 1, τk = inf{1, t : x(uk, t) ∈ {x(u0, t), . . . , x(uk−1, t)}}, k = 1, . . . , n.
Then α can be represented as
α = Eα +
n∑
k=0
∫ τk
0
fk(t)dx(uk, t),
where, for every k, the random function fk is measurable with respect x(u0, ·), . . . , x(uk, ·)
and (under fixed x(u0, ·), . . . , x(uk−1, ·)) is adapted to the flow x(uk, τk ∧ ·). In this
representation,
Eα2 = (Eα)2 +
n∑
k=0
E
∫ τk
0
fk(t)2dt.
3. Clark representation
Let α be a square-integrable random variable measurable with respect to {x(u, ·); u ∈
[0; U ]}. Suppose that {un; n ≥ 0} is a dense set in [0; U ] containing 0 and U. Define
the random moments {τk; k ≥ 0} as in Theorem 2.1. The following analog of the Clark
representation holds.
70 ANDREY A. DOROGOVTSEV
Theorem 3.1. The random variable α can be represented as an infinite sum
(3.1) α = Eα +
∞∑
n=0
∫ τk
0
fk(t)dx(uk, t),
where {fk} satisfy the same conditions as those in Theorem 2.1, and the series converges
in the square mean. Moreover,
Eα2 = (Eα)2 +
∞∑
n=0
E
∫ τk
0
fk(t)dt.
Proof. As was mentioned in the first section, x has cadlág trajectories as a random
process in C([0; 1]). Consequently,
σ(x(u, ·); u ∈ [0; U ]) = σ(x(un; ·); n ≥ 0) =
∞∨
n=0
σ(x(u0, ·), . . . , x(un, ·)).
Hence, due to the Lévy theorem,
α = L2- lim
n→∞E(α/x(u0, ·), . . . , x(un, ·)).
Due to Theorem 2.1,
E(α/x(u0, ·), . . . , x(un, ·)) = Eα +
n∑
k=0
∫ τk
0
fk(t)dx(uk, t),
where τk and fk do not change with n. So, taking the limit as n → ∞, we get the
statement of the theorem.
Note that the sum in (3.1) is closely related to the spatial stochastic integral which
was built in the first section. Really, suppose that a is a bounded measurable function
on R and the set {un; n ≥ 0} is dense in [0; U ] with u0 = 0, u1 = U.
Lemma 3.1.
∞∑
n=0
∫ τn
0
a(x(un, t))dx(un, t) = m(U) +
∫ 1
0
a(x(0, t))dx(0, t),
where m(U) was defined in the first section.
Proof. Note that, for every n ≥ 1, the points u0, . . . , un if ordered in the growing order
form a partition of [0;U ]. Under n → ∞, these partitions increase, and their diameters
tend to zero. To prove the lemma, it remains to note that, for every n ≥ 1, the sum∑n
k=0
∫ τk
0
a(x(uk, t))dx(uk, t) coincides with the sum Sπ for the corresponding partition.
Lemma 3.1 is proved.
Bibliography
1. Arratia, R. A., Brownian motion on the line (1979), PhD dissertation, Univ. Wisconsin, Madi-
son.
2. Le Jan, Yves, Raimond, Oliver, Flows, coalescence and noise, The Annals of Probability 32
(2004), no. 2, 1247-1315.
3. Dorogovtsev, A.A., One Brownian stochastic flow, Theory of Stochastic Processes 10(26)
(2004), no. 3-4, 21-25.
4. Dorogovtsev, A.A., Some remarks on the Wiener flow with coalescence, Ukrainian Math. Journ.
57 (2005), no. 10, 1327-1333.
E-mail : adoro@imath.kiev.ua
|