Elect. Comm. in Probab. 15 (2010), 213–226
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
STOCHASTIC FLOWS OF DIFFEOMORPHISMS FOR ONE-DIMENSIONAL
SDE WITH DISCONTINUOUS DRIFT
STEFANO ATTANASIO
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
email: s.attanasio@sns.it
Submitted July 2,2009, accepted in final form May 24,2010
AMS 2000 Subject classification: 60H10
Keywords: Stochastic flows, Local time
Abstract
The existence of a stochastic flow of class C 1,α , for α < 21 , for a 1-dimensional SDE will be proved
under mild conditions on the regularity of the drift. The diffusion coefficient is assumed constant
for simplicity, while the drift is an autonomous BV function with distributional derivative bounded
from above or from below. To reach this result the continuity of the local time with respect to the
initial datum will also be proved.
1
Introduction
The problem of the existence and smoothness of the stochastic flow under conditions of low regularity of the coefficients has been much studied. Apart from the intrinsic interest of the problem,
there is an interest due to the range of possible applications of these results to PDE theory. For
example, in [3] the uniqueness of the stochastic linear transport equation with Hölder continuous
drift was proved, through new results about stochastic flows of class C 1,α . Because of the greater
regularity of stochastic flows compared to deterministic flows, there are cases in which a PDE admits infinitely many solutions in the deterministic case, but it becomes well posed if it is perturbed
by a stochastic noise. In addition in [2] it is proved that in some cases, through a zero-noise limit,
it is possible to find a criterion to select one particular solution.
We consider an equation of the form
d X tx = b(X tx )d t + dWt
X x (0) = x
(1)
A complete proof of existence of the stochastic flows of class C 1,α is known only when b is Hölder
continuous and bounded, see [3]. In the 1-dimensional case, an important example that deals
with discontinuous b has been studied in [5]. Moreover there are preliminary results in [6]. The
class of bounded variation (BV) fields b emerges from these works as a natural candidate for the
flow property, although only a few partial properties have been proved. Moreover, BV fields are
the most general class considered also in the deterministic literature, see [1]: in any dimension,
213
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Electronic Communications in Probability
when b ∈ BV and the negative part of the distributional divergence of b is bounded, a generalized
notion of flow exists and is unique.
The aim of this work is to give a complete proof of existence of the stochastic flows of class C 1,α , for
α < 12 , in dimension one, when b ∈ BV and the positive or the negative part of the distributional
derivative of b is bounded. This result, although restricted to the 1-dimensional case, is stronger
than the deterministic one both because we accept a bound on b0 from any side, and because we
construct a flow of class C 1,α , not only a generalized flow.
The partial results of the paper [6] suggest the problem whether b ∈ BV is sufficient. We cannot
reach this result without a one-side control on b0 . The fact that a similar assumption is imposed
in [1] is maybe an indication that it is not possible to avoid it.
2
Flow of homeomorphisms and known results
All results contained in this paper will be proved under the following hypothesis:
1. b ∈ BV (R) and b = b1 − b2 , with b1 and b2 increasing and bounded functions
2. b1 ∈ W 1,∞ or b2 ∈ W 1,∞
We will first suppose b1 ∈ W 1,∞ . Under this hypothesis we will prove that the local time of the
stochastic differential equations (SDE) solutions is Hölder continuous with respect to the initial
data. Thanks to this result we will prove the existence of the stochastic flow of class C 1,α , for
α < 21 . At the end of the paper, using standard facts on the backward equations, we will show that
the results proved hold if we replace the hypothesis b1 ∈ W 1,∞ with the hypothesis b2 ∈ W 1,∞ .
A result about one-dimension stochastic flows, under the hypothesis b ∈ BV, was given in [6].
There it was first proved the non-coalescence property through an elegant proof different from the
one proposed in these notes. Then, the flow continuity was proved, and this property, together
with the continuity of the flow of the backward equation, implies the homeomorphic property of
the flow. However the proof of the continuity of the flow appears to be incomplete. Indeed, in
order to apply Kolmogorov’s lemma, the following inequality is shown:
–
™
E
y
x
sup (X s∧τ
− X s∧τ )2
0≤s≤t
≤ n(x − y)2
where τ is a stopping time depending on x and y. This doesn’t appear sufficient to apply Kolmogorov’s lemma.
Note that, as a standard consequence of the pathwise uniqueness we have that ∀h > 0, a.s.,
X tx+h ≥ X tx . Moreover, assuming b1 ∈ W 1,∞ , we have
(X tx+h
−
X tx ) − (X sx+h
−
X sx )
=
Z
t
b(X rx+h ) −
b(X rx )d r
s
Z
≤
t
kDb1 k∞ (X rx+h − X rx )d r
s
From this inequality, using Gronwall’s lemma we obtain:
(X tx+h − X tx ) ≤ e(t−s)kDb
1
k∞
(X sx+h − X sx )
This fact, together with the proof of the non-coalescence property, contained in [6], is sufficient
to prove the existence of a stochastic flow of homeomorphisms. Therefore the proof of the homeomorphic property contained in [6] can be corrected easily under our hypothesis 1 and 2.
Stochastic flows of diffeomorphisms
215
However we are interested in stronger results about smoothness of the flow. In particular we
are interested in the smoothness of the inverse flow, which is a basic ingredient, for instance, in
the analysis of stochastic transport equations. While the homeomorphic property implies only the
continuity of the inverse flow, we will prove that the inverse flow is of class C 1,α for α < 12 .
Notation
Throughout the paper we will assume given a stochastic basis with a 1-dimensional Brownian
motion (Ω, (F t ) t≥0 , F , P, Wt ). Moreover, for each 0 ≤ s < t we denote by Fs,t the completed σalgebra generated by Wu − Wr for s ≤ r ≤ u ≤ t. We will use the following notation: L = kbk∞ ,
K = kDb1 k∞ , b∗ = b1 + b2 , L ∗ = kb∗ k∞ .
We will denote by X tx the unique solution of the stochastic differential equation (1).
3
Local-time continuity with respect to the initial data
Definition 3.1. Let X tx be the unique solution of equation (1), and let a ∈ R. We will denote by
L ta (X x ) its local time at a, i.e. the continuous and increasing process such that
|X tx
− a| = |x − a| +
Z
t
sgn(X sx − a)dX sx + L ta (X x )
0
Further details about local time can be found in [8].
Remark 1. Recall the following inequality which is used, for example, to prove the continuity with
respect to (a, t) of the local time: Let X t = X 0 + A t + M t be a continuous semimartingale, where
M t is a continuous local martingale, vanishing in 0 and A t is a continuous process with bounded
variation, vanishing in 0. Suppose that sup t≤T |M t | ∨ sup t≤T |A t | ≤ K. Then it holds:
Z
p
t
1{a<X s ≤b} d⟨M ⟩s  ≤ C p,K |a − b| p
E
0
Theorem 3.2. There exists a modification of L ta (X x ) which is jointly continuous in (a, t, x), and it is
Hölder continuous in (a, x), of order α, for α < 12 .
Proof. Define an increasing sequence of stopping times as follows:
Tn := inf t ≥ 0 : |Wt | ∨ t L ≥ n
We have Tn ↑ ∞ a.s. Denote by X Tn the unique stopped solution of equation (1). It is sufficient to
a
prove the theorem for L t∧T
(X x ). Note that ∀t ≥ 0 |X t∧Tn − x| ≤ 2n, and X Tn satisfies the hypothesis
n
of remark 1. By definition of L ta (X x ) it follows:
a
x
L t∧T
(X x ) = |X t∧T
− a| − |x − a| −
n
n
y
Xt |
Z
0
t∧Tn
sgn(X sx − a)b(X sx )ds −
Z
t∧Tn
sgn(X sx − a)dWs
0
Thanks to the inequality |X tx −
≤ e K t |x − y|, which holds a.s. the first term on the right
hand admits a modification jointly continuous in (a, t, x), and lipschitz continuous in (a, x). In
particular it is Hölder continuous in (a, x), of order α, for α < 12 . The second term is obviously
continuous in (a, t, x) and Hölder continuous in (a, x), of order α, for α < 12 . We now prove that
the third one admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x),
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Electronic Communications in Probability
of order α, for α < 12 . We will apply Kolmogorov’s lemma to the space C([0, ∞)), endowed with
the sup norm. It will be useful to apply remark 1.
Let (a, x) ∈ R2 and (b, y) ∈ R2 . We will consider only the case b > a and y > x. In the other cases
the following estimates are similar. It holds:

p
Z
t∧Tn
sgn(X sx − a)b(X sx ) − sgn(X sy − b)b(X sy )ds 
E sup
t≥0
0

p
t∧Tn
Z
sgn(X sx
≤ C p E sup
t≥0
t≥0
p
sgn(X sx − b)b(X sx ) − sgn(X sy − b)b(X sy )ds 
0
p
Z
1a≤X sx <b ds  + C p E 
Tn
≤ C p E  2L
0
Z
+C p E 

t∧Tn
Z
+C p E sup
Z
−
b)b(X sx )ds
0


−
a)b(X sx ) − sgn(X sx
p
Tn
|b(X sx )(sgn(X sx − b) − sgn(X sy − b))|ds 
0
p
Tn

| sgn(X sy − b)(b(X sx ) − b(X sy ))|ds  ≤ C p E  2L
p
Tn
Z
1a≤X sx <b ds 
0
0

Tn
Z
+C p E  2L
p
Z
1X sx ≤b<X sy ds  + C p E 
0

Z
|b(X sx ) − b(X sy )|ds 
0
p
Tn
≤ C p E  2L
p
Tn
x
1a≤X s∧T
0
n

 + C p E  2L
<b ds
p
Tn
Z
x
1 b−eK T ( y−x)<X s∧T
0
Z
+C p E 
≤b ds
n

p
Tn
b∗ (X sy ) − b∗ (X sx )ds 
0
Z
≤ C p,n L p |a − b| p + C p,n L p |x − y| p + C p E 
p
Tn
b∗ (X sy ) − b∗ (X sx )ds 
0
We need to estimate the last term to apply Kolmogorov’s lemma:
define h = e K T ( y − x); let f be
Rr
00
0
00
00
∗
∗
∗
such that f (r) = b (hr + h) − b (hr), and f (r) = −L + −∞ f (s)ds. Note that f (r) ≥ 0 ∀r,
R +∞ 00
0
and that −∞ f (s)ds = lim r→+∞ b∗ (r) − b∗ (−r) ≤ 2L ∗ . So we have | f (r)| ≤ L ∗ ∀r ∈ R. Using
R Tn 00 X x 0
s
ds. Indeed
Itô formula and the boundness of f we will obtain the L p -boundness of 1h 0 f
h
we have
1
2
Tn
Z
b
∗
(X sy ) −
b
∗
(X sx )ds
0
‚
2
≤h
f
X Txn
h
Œ
−f
x‹
h
≤
1
2
+h
Tn
Z
b
∗
(X sx
+ h) − b
∗
(X sx )ds
1
=
2
0
Z
Tn
f
0
0
X sx
h
b(X sx )ds
+h
Z
f
00
X sx
Tn
0
h
0
f
0
Tn
Z
X sx
h
dWs
ds
Stochastic flows of diffeomorphisms
217
∗
∗
≤ L h X Tn − x + L h
Z
Tn
|b(X sx )|ds
Z
+h
f
∗
≤ L h(2n + n) + h
Z
Tn
f
0
X sx
E
0

p
X sx

0


b∗ (X sy ) − b∗ (X sx )ds  ≤ hp Cn,p 1 + E 
Z
Tn
h
dWs
dWs
h
0
Tn
0
0
0
Thus we have:
Z
Tn
f
0
0
X sx
2
h
p
2

00

ds  ≤ hp Cn,p
Therefore, thanks to Kolmogorov’s lemma we have proved that
t∧Tn
Z
sgn(X sx − a)b(X sx )ds
0
admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α,
for α < 21 .
R t∧Tn
To complete the proof we have to show that 0
sgn(X sx − a)dWs admits a modification jointly
continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 21 . We have:

p
Z
t∧Tn
sgn(X sx − a) − sgn(X sy − b)dW s 
E sup
t≥0

0

≤ Cp E 

0

+C p E 


≤ Cp E 
Z
0
Tn
Tn
Z

≤ Cp E 

Tn
Z
0
Z
0
Tn
!p
2

| sgn(X sx − a) − sgn(X sy − b)|2 ds 
!p
2

| sgn(X sx − a) − sgn(X sx − b)|2 ds 
!p
2

| sgn(X sx − b) − sgn(X sy − b)|2 ds 

!p
Z
2


1a≤X sx <b ds  + C p E 
0
p
Tn
!p
2

1 b−eK T ( y−x)<X sx ≤b ds 
p
≤ C p,n |a − b| 2 + C p,n |x − y| 2
R t∧Tn
This inequalities and Kolmogorov’s lemma prove that 0
sgn(X sx − a)dWs admits a modification
jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 21 . The proof is
complete.
From now on we will consider only the continuous version of L ta (X x ).
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Electronic Communications in Probability
Corollary 3.3. Let T ≥ 0 and x ≤ y. Then the process
a
sup sup L t+s
(X u ) − L ta (X u )
s → sup
(2)
t∈[0,T ] u∈[x, y] a∈R
is continuous.
Proof. Note that ∀s ≤ t, and ∀u ∈ R it holds |X su − u| ≤ L t + sups≤t |Ws |.
Thus a 6∈ [u − L t − sups≤t |Ws |, u + L t + sups≤t |Ws |] implies L ta (X u ) = 0. Denote by As the random
compact set [x − L(T +s)−sup r≤T +s |Wr |, y + L(T +s)+sup r≤T +s |Wr |]. Note that a.s. s < r implies
As ⊂ A r . Thus ∀s ≤ r it holds:
a
sup sup L t+s
(X u ) − L ta (X u ) = sup
sup
t∈[0,T ] u∈[x, y] a∈R
a
sup sup L t+s
(X u ) − L ta (X u )
t∈[0,T ] u∈[x, y] a∈A r
Thanks to this equality and to the compactness of [0, T ]×[x, y]×A r , the continuity of the process
(2) is proved on the interval [0, r]. Because of the arbitrariness of r the claim is proved.
4
Existence of the stochastic flow of diffeomorphisms
We will now prove the non-coalescence property of the solutions of equation (1). This result has
been already proved in [6] under more general hypothesis. However, for the sake of completeness
we will give a proof based on the continuity of L ta (X x ). The following lemma, which appears in
[8], and in [7] with a complete proof, will be useful.
Lemma 4.1. Let X be a continuous semimartingale, and denote by ⟨X ⟩ t , its quadratic variation. Let
f : R+ × R × Ω → R be a bounded measurable function. Then a.s., ∀t ≥ 0
Z
t
f (s, X s , ·)d⟨X ⟩ t =
Z
0
Z
da
R
t
f (s, a, ·)d Lsa (X )
0
Proposition 4.2. ∀x ∈ R, ∀h > 0, and T ≥ 0, a.s. X Tx+h − X Tx > 0.
a
Proof. Fix x ∈ R, h > 0 and T ≥ 0. From corollary 3.3, it follows that the process s → sup t∈[0,T ] supa∈R L t+s
(X x )−
a
x
L t (X ), is continuous and vanishing in 0. Therefore ∀" ∈ (0, 1) a.s. exists s",x (ω) > 0 such that
"
a
∀a ∈ R and t ∈ [0, T ]. So, a.s. it holds
s ∈ [0, s",x (ω)] implies L t+s
(X x ) − L ta (X x ) < 2L ∗ (1+")
∀t ∈ [0, T ] :
inf
r∈[t,t+s",x (ω)]
=
(X rx+h
Z
−
X rx ) − (X tx+h
−
X tx )
t+s",x (ω)
Z
b∗ (X ux ) − b∗ (X ux + (X tx+h − X tx ))du
≥
t
”
€
€
ŠŠ— h
a
b∗ (a) − b∗ a + X tx+h − X tx
× L t+s
",x (ω)
i
(X x ) − L ta (X x ) da
R
≥ −kL ·t+s
",x (ω)
a
≥ − sup L t+s
a∈R
€ €
ŠŠ
(X x ) − L ·t (X x )k∞ × kb∗ (·) − b∗ · + X tx+h − X tx k1
€
Š
"
x
a
x
(X x+h − X tx )
(X
)
−
L
(X
)
× 2L ∗ X tx+h − X tx ≥ −
t
",x (ω)
1+" t
Stochastic flows of diffeomorphisms
219
This inequality implies:
(X rx+h − X rx ) ≥
inf
(X tx+h − X tx )
(3)
1+"
r∈[t,t+s",x (ω)]
In the same way we obtain:
sup
r∈[t,t+s",x (ω)]
(X rx+h − X rx ) ≤ (1 + ")(X tx+h − X tx ) ≤
T
In particular a.s. N",T,x (ω) := d s
",x (ω)
(X tx+h − X tx )
1−"
(4)
e < ∞, and thus a.s. we have
X Tx+h − X Tx ≥ h
N",T,x (ω)
1
1+"
>0
Remark 2. Using corollary 3.3, with the same argument of the preceding proof, it is possible to
prove that given an interval [x, y], ∀" > 0 a.s. exists s",x, y (ω) > 0 such that s ∈ [0, s",x, y (ω)]
"
a
implies L t+s
(X u ) − L ta (X u ) < 2L ∗ (1+")
∀a ∈ R, t ∈ [0, T ], and u ∈ [x, y]. This fact will be used in
the next theorem, which is crucial to prove the existence of a flow of class C 1,α .
Theorem 4.3. Let x, y, t ∈ R such that x < y, and t ≥ 0. Then, a.s.
Œ
‚Z
‚Z
Œ
Œ ‚ y
X t − X tx
≤ sup exp
inf exp
L ta (X u )Db(da)
L ta (X u )D b(d a) ≤
u∈[x, y]
y−x
u∈[x, y]
R
R
Proof. Step 1. Fix x < y and t ≥ 0. Fix " > 0, and let s",x, y (ω) be defined as in remark 2.
Moreover define N",t,x, y (ω) := d s t (ω) e. Define , ∀ i ∈ N, t i (ω) = (i × s",x, y (ω)) ∧ t. Obviously we
",x, y
have, a.s., that i ≥ N",t,x, y (ω) implies t i (ω) = t. Define g : Ω × R+ → R+ as:
g(h) = sup
sup sup |Lsr+a (X z ) − Lsa (X z )|
sup
r∈[0,h] z∈[x, y] s∈[0,t] a∈R
Note that, with the same reasoning used in corollary 3.3 and remark 2, it is possible to show that
g is a.s. continuous, increasing and vanishing in 0. Let z, w ∈ [x, y] such that z < w. Then it
holds:
‚ w
ΠZ t
∞ Z t i+1
X
X t − X tz
b(X sw ) − b(X sz )
b(X sw ) − b(X sz )
",z,w
=
ds
=
ds := I0
ln
w − Xz
w − Xz
w−z
X
X
s
s
s
s
0
i=0 t
i
Observe that in the last summation a.s. only a finite number of terms are different from 0. Define:
",z,w
I1
:=
∞
X
i=0
",z,w
I2
:=
∞
X
i=0
Z
t i+1
ti
1
X twi − X tzi
Z
t i+1
b(X sw ) − b(X sz )
X twi − X tzi
ti
– ‚
b
‚
X sz
+
ds
X twi − X tzi
1+"
ŒŒ
™
−
b(X sz )
ds
220
Electronic Communications in Probability
",z,w
I3
:=
∞
X
Z
t i+1
b
X twi − X tzi
ti
i=0
‚
–
1
∗
‚
+
X sz
X twi − X tzi
‚
ŒŒ
−b
1−"
∗
‚
+
X sz
X twi − X tzi
1+"
ŒŒ™
ds
Note that the following estimate holds:
",z,w
I2
",z,w
− I3
",z,w
≤ I1
",z,w
≤ I2
",z,w
+ I3
(5)
Finally define the random set:
1 w
1 w
X t i − X tzi , −
X t i − X tzi
Ai,",z,w := −
1−"
1+"
and


A
ρi,",z,w
(a) := 

1 − "2

× 1Ai,",z,w  (a)
2" X twi − X tzi
A
The following properties are immediate: ρi,",z,w
≥ 0,
1
[− 1−"
(w
R
R
A
ρi,",z,w
(a)da = 1, and has support con-
A
A
tained in
− z)e , 0]. We will use the following notation: ρ̌i,",z,w
(a) = ρi,",z,w
(−a).
Similarly we define:
1 w
z
Bi,",z,w := −
X ti − X ti , 0
1+"
Kt
and


1+"


B
× 1Bi,",z,w  (a)
ρi,",z,w
(a) :=  X twi − X tz1
B
A
ρi,",z,w
satisfies properties similar to those of ρi,",z,w
. In particular its support is contained in
1
[− 1+"
(w − z)e K t , 0].
Step 2. Thanks to estimates (3) and (4) we have:
",z,w
|I0
",z,w
− I1
|=
∞
X
Z
b(X sw ) − b(X sz )
X sw
ti
i=0
≤"
t i+1
∞
X
Z
i=0
t i+1
−
X sz
b(X sw ) − b(X sz )
X twi − X tzi
ti
−
b(X sw ) − b(X sz )
X twi − X tzi
ds
ds
Using the decomposition b∗ = 2b1 − b, and the relation, which holds for α ≤ β, |b(β) − b(α)| ≤
b∗ (β) − b∗ (α), we obtain:
!
!
Z t i+1 1 w
Z t i+1
∞
∞
X
X
b(X sw ) − b(X sz )
b (X s ) − b1 (X sz )
",z,w
",z,w
ds − "
ds
|I0
− I1 | ≤ 2"
X twi − X tzi
X twi − X tzi
t
t
i=0
i=0
i
i
≤
2"
1−"
",z,w
K t − "I1
(6)
Stochastic flows of diffeomorphisms
221
Step 3. From the occupation time formula we have:
",z,w
I3
∞
X
=
Z
i=0
‚
1
R
(X twi − X tzi )
b
∗
‚
a+
X twi − X tzi
‚
ŒŒ
−b
1+"
∗
‚
a+
X twi − X tzi
ŒŒ
1−"
×(L tai+1 (X z ) − L tai (X z ))da
∞
X
=
Z 2"
R
1 − "2
i=0
=
≤
2"
1−"
€
Š
A
D b∗ ∗ ρi,",z,w
(a) × (L tai+1 (X z ) − L tai (X z ))da
X
∞ Z z
z
·
·
A
(X
))(a)
Db∗ (da)
(X
)
−
L
∗
(L
ρ̌
ti
t i+1
i,",z,w
2
i=0
R
X
∞ Z h
i
2"
A
·
z
·
z
a
z
a
z
ρ̌
∗
L
(X
)
−
L
(X
)
(a)
−
L
(X
)
−
L
(X
)
i,",z,w
t
t
t
t
i+1
i
i+1
i
1 − "2
i=0 R
×D b (d a) +
≤
Z
2"
1 − "2
Z
2"
∗
1 − "2
L ta (X z )D b∗ (d a)
+ 4L
∗
R
L ta (X z )Db∗ (da)
R
2"
1
Kt
g
(w − z)e
N",t,x, y
1−"
1 − "2
(7)
Step 4. It holds:
",z,w
I2
=
∞
X
Z
i=0
– ‚
1
R
(X twi − X tzi )
=
∞
X
Z
i=0
=
b
a+
X twi − X tzi
1+"
ŒŒ
™
h
i
− b(a) × L tai+1 (X z ) − L tai (X z ) da
h
i
€
Š
B
D b ∗ ρi,",z,w
(a) × L tai+1 (X z ) − L tai (X z ) da
R
∞ Z h
X
i=0
‚
i
B
L ·t i+1 (X z ) − L ·t i (X z ) ∗ ρ̌i,",z,w
(a)Db(da)
R
From this equality it follows:
",z,w
|I2
Z
−
L ta (X z )Db(da)|
R
=
∞ Z
X
i=0
nh
i
h
io
B
L ·t i+1 (X z ) − L ·t i (X z ) ∗ ρ̌i,",z,w
(a) − L tai+1 (X z ) − L tai (X z ) Db(da)
R
≤ 4L ∗ g
1
1+"
(w − z)e K t
N",t,x, y
(8)
222
Electronic Communications in Probability
Step 5. From (5), and from (6), (7) and (8), follows that, assuming " ∈ (0, 21 ), there exists a
constant M dependent only on x, y, t, and on the constants K, L and L ∗ , such that
Z
",z,w
|I0
L ta (X z )D b(da)| ≤ M "(1 + sup sup L ta (X u )) + M g(M (w − z))N",t,x, y
−
(9)
u∈[x, y] a∈R
R
We call ∆ a finite partition of [x, y] if, for some n ∈ N, ∆ = {x = z0 < z1 < · · · < zi < zi+1 < · · · <
zn = y}. Moreover we define |∆| := maxzi ∈∆\ y (zi+1 − zi ). We denote by Λ the set of finite partition
of [x, y]. Obviously it holds:
‚ y
Œ
‚ zi+1
‚ zi+1
z Œ
z Œ
X t − X tx
Xt − Xti
Xt − Xti
≤
≤ inf max
(10)
sup min
∆∈Λ zi ∈∆\ y
zi+1 − zi
y−x
zi+1 − zi
∆∈Λ zi ∈∆\ y
Using (9), we have, ∀" ∈ (0, 12 ) :
z
‚
z
X t i+1 − X t i
inf max
zi+1 − zi
∆∈Λ zi ∈∆\ y
Œ
‚Z
≤ inf max exp
∆∈Λ zi ∈∆\ y
Œ
L ta (X zi )Db(da)
R
Œ
‚
× exp
M "(1 + sup sup
u∈[x, y] a∈R
‚Z
Œ
L ta (X u )Db(da)
≤ sup exp
u∈[x, y]
L ta (X u )) +
M g(M |∆|)N",t,x, y
‚
× exp
Œ
M "(1 + sup sup
u∈[x, y] a∈R
R
L ta (X u ))
With the same reasoning we obtain:
‚ zi+1
‚Z
Œ
z Œ
Xt − Xti
z
a
sup inf
≥ inf exp
L t (X )Db(da)
z∈[x, y]
zi+1 − zi
∆∈Λ zi ∈∆\ y
R
‚
Œ
× exp −M "(1 + sup sup
u∈[x, y] a∈R
L ta (X u ))
Thanks to the arbitrariness of ", and thanks to relation (10) the proof is complete.
Using the preceding theorem we can now prove the existence of the stochastic flow of class C 1,α .
All results we have proved refers to solutions starting from t = 0. This choice was made to simplify
notations. In the next theorem we will treat the general case. So, we will consider solutions of the
following equation:
s,x
s,x
dX t = b(X t )d t + dWt
(11)
s,x
X (s) = x
0,x
Obviously for X t
theorem 4.3 holds.
Theorem 4.4. Assume conditions 1 and 2 of section 2, and let T > 0. Then there exists a map
(s, t, x, ω) → φs,t (x)(ω) defined for 0 ≤ s ≤ t ≤ T , x ∈ R, ω ∈ Ω with values in R, such that
s,x
s,x
1. given any 0 ≤ s ≤ T , x ∈ R the process X s,x = (X t s ≤ t ≤ T ) defined as X t
continuous Fs,t −measurable solution of equation (11).
= φs,t (x) is a
Stochastic flows of diffeomorphisms
223
−1
−1
2. a.s. φs,t is a diffeomorphism ∀0 ≤ s ≤ t ≤ T and φs,t , φs,t
, Dφs,t , and Dφs,t
are continuous
in (s, t, x), and of class C α in x, for α < 12 .
3. a.s. φs,t (x) = φu,t (φs,u (x)) ∀0 ≤ s ≤ u ≤ t ≤ T and x ∈ R, and φs,s (x) = x.
Moreover an explicit expression for Dφs,t (x) is given by:
Dφs,t (x) = exp
‚Z
−1
L ta (X φ (0,s,x) ) −
−1
Lsa (X φ (0,s,x) )
Œ
Db(da)
R
Proof. Step 1. We first give the proof under the assumption b1 ∈ W 1,∞ . It will take the first three
steps. Let D be the set of dyadic numbers, and let D T := D ∩ [0, T ]. Define ∀x ∈ D and ∀t ∈ D T ,
0,x
Xet (ω) = X tx (ω). Note the two following facts:
1. Being D a countable set, there exists a negligible set A0 such that, ∀ω 6∈ A0 and ∀x ∈ D X ·x
is a continuous solution of equation (1). In particular since it holds
X tx = x +
Z
t
b(X sx )ds + Wt
0
∀ω 6∈ A0 and x ∈ D, the family of processes {X .x } x∈D is uniformly equicontinuous.
2. Thanks to the countability of the set D × D × D T , there exists a negligible set A1 ⊃ A0 such
that ∀ω 6∈ A1 ∀x, y ∈ D such that x < y, and ∀t ∈ D T it holds
!
‚Z
Œ
‚Z
Œ
0, y
0,x
Xet − Xet
a
u
a
u
≤ sup exp
L t (X )Db(da)
inf exp
L t (X )D b(d a) ≤
u∈[x, y]
y−x
u∈[x, y]
R
R
These facts imply that, ∀ω 6∈ A1 is well-defined the continuous extension of Xe·0,· (ω) on R × [0, T ].
Denote by φ0,t (x)(ω) this extension. Note that, ∀ω 6∈ A1 , the family {φ0,· (x)(ω)} x∈R is uniformly
equicontinuous; more precisely, ∀ω 6∈ A1 , ∀0 ≤ s ≤ t ≤ T
|φ0,t (x) − φ0,s (x)| ≤ (t − s)L + sup |Wr+(t−s) − Wr |
(12)
r∈[0,T ]
In particular we have that a.s. |φ0,t (x) − x| is bounded in x. This fact, together with continuity
in x, implies that, ∀ω 6∈ A1 , φ0,t (·) is surjective. Moreover it’s immediate to verify that, ∀ω 6∈ A1 ,
and ∀x, y ∈ R, such that x < y, it holds:
‚Z
Œ ‚
Œ
‚Z
Œ
φ0,t ( y) − φ0,t (x)
a
u
a
u
≤ sup exp
L t (X )Db(da)
L t (X )D b(d a) ≤
inf exp
u∈[x, y]
y−x
u∈[x, y]
R
R
This fact, together with the continuity of L ta (X x ) with respect to (a, t, x) implies that ∀ω 6∈ A1 ,
€R
Š
∀0 ≤ t ≤ T, φ0,t (x)(ω) is differentiable in x, and its derivative is exp R L ta (X x )Db(da) . So
∀t ∈ [0, T ] a.s. φ0,t is a surjective and differentiable function whose derivative is strictly positive
€R
Š
everywhere, and therefore is a diffeomorphism. Moreover exp R L ta (X x )Db(da) is of class C α
with respect to x for α < 12 .
Step 2. We now prove that ∀x ∈ R a.s. φ0,t (x) = X tx ∀0 ≤ t ≤ T. Fix x ∈ R. Because of the a.s.
continuity of both φ0,t (x) and X tx with respect to t, and because of the countability of D T , it is
224
Electronic Communications in Probability
sufficient to prove that ∀t ∈ D T , a.s. φ0,t (x) = X tx . So fix t ∈ D T , and let {x n }n∈N be a sequence of
0,x
x
dyadic numbers converging to x. By construction we have that ∀ω 6∈ A1 φ0,t (x n ) = Xet n = X t n and
xn
φ0,t (x) = limn→∞ φ0,t (x n ) = limn→∞ X t . Moreover thanks to theorem 4.3, there is a negligible
set Anx such that ∀ω 6∈ Anx it holds
‚Z
inf exp
u∈[x,x n ]
Œ
L ta (X u )Db(da)
‚
≤
R
Denote by A x the negligible set
S∞
i
i=0 A x .
x
X t n − X tx
Œ
xn − x
‚Z
≤ sup exp
u∈[x,x n ]
Œ
L ta (X u )Db(da)
R
Therefore ∀ω 6∈ A x ∪ A1 we have the equality
x
X tx = lim X t n = φ0,t (x)
n→∞
This implies that φ0,t (x) is a continuous F0,t −measurable solution of equation (1).
−1
Step 3. We will denote by φ0,t
(ω) the inverse function of φ0,t (ω). Define ∀0 ≤ s ≤ t ≤ T
−1
φs,t = φ0,t ◦ φ0,s . It’s immediate to verify that property 3 if satisfied. It’s also easy to show that φs,t
−1
−1
is a diffeomorphism, and its inverse function is φs,t
= φ0,s ◦ φ0,t
. Moreover, let {(sn , t n , x n )}n∈N be
a sequence such that 0 ≤ sn ≤ t n ≤ T, x n ∈ R and (sn , t n , x n ) → (s, t, x). Obviously we have
−1
−1
φsn ,t n (x n ) = (φ0,t n ◦ φ0,t
) ◦ φs,t ◦ (φ0,s ◦ φ0,s
(x n ))
n
−1
−1
Thanks to (12), φ0,t n ◦ φ0,t
and φ0,s ◦ φ0,s
converge to the identity; this observation prove
n
−1
that φs,t (x) is jointly continuous in (s, t, x). This implies the continuity of φ0,t
with respect to
−1
(t, x), and thus the continuity of φs,t in (s, t, x). Having proved that the derivative of φ0,t (x) is
€R
Š
exp R L ta (X x )D b(d a) , we have that
Dφs,t (x) = exp
‚Z
−1
L ta (X φ (0,s,x) ) −
−1
Lsa (X φ (0,s,x) )
Œ
Db(da)
R
−1
Dφs,t
(x)
‚ Z
Œ
a
φ −1 (0,s,x)
a
φ −1 (0,s,x)
= exp −
L t (X
) − Ls (X
) Db(da)
R
which are jointly continuous in (s, t, x), and of class C α for α < 21 . The property (2) is proved. We
have already proved property (1) if s = 0. To prove property (1) in the general case, fix x ∈ R and
s ≥ 0. Thus applying the Itô formula, we obtain that φs,t (x) is a solution of equation (11). This
fact, and the pathwise uniqueness imply property (1).
Step 4. We give now the proof under the condition b2 ∈ W 1,∞ .
ft := WT −t − WT . Note that W
f is a Brownian motion and the completed σ-algebra generated
Let W
fu − W
fr for s ≤ r ≤ u ≤ t is Fes,t := F T −t,T −s . Note that −b satisfies the conditions used in the
by W
es,t (x), continuous Fes,t −measurable solution of the equation
first three steps. So there exists φ
¨
s,x
s,x
ft
dX t = −b(X t )d t + d W
(13)
s,x
X (s) = x
satisfying property (2) and (3).
e−1
Define, for 0 ≤ s ≤ t ≤ T ψs,t (x)(ω) := φ
T −t,T −s (x)(ω). It’s immediate to verify that ψ satisfies
property (2) and (3). Moreover it’s easy to verify that, for any x ∈ R and 0 ≤ s ≤ T, ψs,· (x)(ω) is
a continuous Fs,t −measurable solution of the equation solution of problem (11).
Stochastic flows of diffeomorphisms
Remark 3. A classic approach to the problem of the existence of a C 1,α stochastic flow is the use
of Itô formula to remove the drift (see [9] and later works on this approach, or a variant in [3]).
Nevertheless, under the assumptions of this paper, there are some difficulties to use this approach.
Indeed, suppose it is possible to prove the existence of a solution of the following parabolic equation
¨
∂ t u(t, x) + 21 ∂ x x u(t, x) + b(x)∂ x u(t, x) = 0
(14)
u(T, x) = x
and that the solution is sufficiently smooth, so that we can apply Itô formula:
1
du(t, X t ) = ∂ t u(t, X t )d t + ∂ x u(t, X t )b(X t )d t + ∂ x u(t, X t )dWt + ∂ x x u(t, X t )d t = ∂ x u(t, X t )dWt
2
Suppose moreover that u(t, ·) is a diffeomorphism. Through the change of variable Yt = u(t, X t )
the original problem has been reduced to the problem of the existence of the flow of the following
stochastic equation
d Yt = σ(t, Yt )dWt
where σ(t, y) = ∂ x u(t, u−1
t ( y)). To solve this problem using well-known theorems, we should prove
σ ∈ C 1,α for some α > 0. But σ ∈ C 1,α implies ∂ x x u(t, ·) ∈ C α . So the discontinuity of the term
b(x)∂ x u(t, x) would be balanced by the term ∂ t u(t, x). However, examples such that ∂ t u(t, x) is
continuous, and b(x) is discontinuous can be shown. Consider, for example, b(x) = sgn(x). It is
t,x
t,x
t,x
possible to prove that u(t, x) = E[X T ] and ∂ t u(t, x) = P(X T < 0) − P(X T > 0). This term, as
α
shown in [4], is continuous in x. Thus we have obtained ∂ x x u(t, ·) 6∈ C . So, even if it is possible
to prove the existence and smoothness of the solution of equation (14), it is not possible to prove
that σ ∈ C 1,α , and it is not possible to prove the existence of the stochastic flow through well-known
theorems.
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