getdoce4f9. 168KB Jun 04 2011 12:05:06 AM

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

SubmittedJuly 2,2009, accepted in final formMay 24,2010 AMS 2000 Subject classification: 60H10

Keywords: Stochastic flows, Local time

Abstract

The existence of a stochastic flow of classC1,α_{, forα <}1

2, 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 reg-ularity 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 classC1,α. Because of the greater regularity of stochastic flows compared to deterministic flows, there are cases in which a PDE ad-mits 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 Xx

t =b(Xtx)d t+dWt

Xx_{(0) =x} (1)

A complete proof of existence of the stochastic flows of classC1,αis known only whenbis Hölder continuous and bounded, see[3]. In the 1-dimensional case, an important example that deals with discontinuousbhas been studied in[5]. Moreover there are preliminary results in[6]. The class of bounded variation (BV) fields bemerges 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,

whenb∈BVand the negative part of the distributional divergence ofbis 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 classC1,α, for

α < 1₂, in dimension one, whenb∈BV and the positive or the negative part of the distributional derivative of bis bounded. This result, although restricted to the 1-dimensional case, is stronger than the deterministic one both because we accept a bound on b′from any side, and because we construct a flow of classC1,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 b′. 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_)andb=b1_−b2_{, withb}1_andb2_{increasing and bounded functions}

2. b1∈W1,∞orb2∈W1,∞

We will first suppose b1 _∈W1,∞_{. 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 C1,α, for

α < 1

2. 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∈W1,∞with the hypothesisb2∈W1,∞. 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

– sup 0≤s≤t

(X_sx_∧τ−Xsy∧τ)2

™

≤n(x−y)2

whereτ is a stopping time depending on x and y. This doesn’t appear sufficient to apply Kol-mogorov’s lemma.

Note that, as a standard consequence of the pathwise uniqueness we have that ∀h > 0, a.s.,

Xx+h

t ≥Xtx. Moreover, assumingb1∈W

1,∞_{, we have}

(X_tx+h−X_tx)−(X_sx+h−X_sx) =

Zt

s

b(X_rx+h)−b(X_rx)d r≤ Z t

s

kD b1k∞(Xrx+h−X x r)d r

From this inequality, using Gronwall’s lemma we obtain:

(X_tx+h−X_tx)≤e(t−s)kD b1k∞_(Xx+h

s −X x s)

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 classC1,αforα <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 byFs,t the completedσ

-algebra generated byWu−Wr fors≤ r≤u≤ t. We will use the following notation: L=kbk∞,

K=kD b1k_∞,b∗=b1+b2,L∗=kb∗k_∞. We will denote byXx

t the unique solution of the stochastic differential equation (1).

3

Local-time continuity with respect to the initial data

Definition 3.1. Let Xx

t be the unique solution of equation (1), and let a ∈R. We will denote by La_t(Xx)its local time at a, i.e. the continuous and increasing process such that

|X_tx−a|=|x−a|+

Z t

0

sgn(X_sx−a)d X_sx+La_t(Xx)

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: LetX_t =X₀+A_t+M_t be a continuous semimartingale, where

M_t is a continuous local martingale, vanishing in 0 andA_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:

E

 

Z t

0

1{a<Xs≤b}d〈M〉s

p

≤C_p,K|a−b|p

Theorem 3.2. There exists a modification of La

t(Xx)which is jointly continuous in(a,t,x),and it is Hölder continuous in(a,x),of orderα,forα <1

2.

Proof. Define an increasing sequence of stopping times as follows:

T_n:=inft≥0 : |W_t| ∨t L≥n

We haveT_n↑ ∞a.s. Denote byX_Tnthe unique stopped solution of equation (1). It is sufficient to prove the theorem forLa

t∧Tn(X

x_{). Note that∀t≥0|X}

t∧Tn−x| ≤2n, andXTnsatisfies the hypothesis

of remark 1. By definition of La_t(Xx)it follows:

La_t∧T

n(X

x_{) =|X}x

t∧Tn−a| − |x−a| −

Zt∧Tn

0

sgn(X_sx−a)b(X_sx)ds− Z t∧Tn

0

sgn(X_sx−a)dWs

Thanks to the inequality |X_tx−Xty| ≤ eK 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 orderα, forα < 1

2. The second term is obviously continuous in(a,t,x)and Hölder continuous in(a,x), of orderα, forα < 1

of orderα, forα < 1

2. We will apply Kolmogorov’s lemma to the spaceC([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>aandy>x. In the other cases}

the following estimates are similar. It holds:

E

Itô formula and the boundness of f′ we will obtain theLp_{-boundness of} 1

≤L∗hX_Tn−x+L∗h

Thus we have:

E

Therefore, thanks to Kolmogorov’s lemma we have proved that Z t∧Tn

s −a)dWsadmits a modification jointly

continuous in(a,t,x), and Hölder continuous in(a,x), of orderα, forα <1₂. We have:

This inequalities and Kolmogorov’s lemma prove thatRt∧Tn

Corollary 3.3. Let T≥0and x≤y. Then the process s→ sup

t∈[0,T] sup

u∈[x,y] sup

a∈R

La_t+s(Xu)−La_t(Xu) (2)

is continuous.

Proof. Note that∀s≤t, and∀u∈Rit holds|X_su−u| ≤L t+sup_s≤t|Ws|.

Thusa6∈[u−L t−sup_s≤t|Ws|,u+L t+sups≤t|Ws|]impliesLat(X

u_{) =0. Denote byA}

sthe random

compact set[x−L(T+s)−sup_r≤T+s|W_r|,y+L(T+s)+sup_r≤T+s|W_r|]. Note that a.s.s<rimplies

A_s⊂A_r. Thus∀s≤rit holds: sup

t∈[0,T] sup

u∈[x,y] sup

a∈R

La_t+s(Xu)−La_t(Xu) = sup

t∈[0,T] sup

u∈[x,y] sup

a∈Ar

La_t+s(Xu)−La_t(Xu)

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 ofrthe 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 La

t(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

Zt

0

f(s,Xs,·)d〈X〉t =

Z

R d a

Z t

0

f(s,a,·)d L_sa(X)

Proposition 4.2. ∀x∈R_{,∀h>0,and T≥0,a.s. X}x+h

T −XTx>0.

Proof. Fixx∈R_{,h>0 andT}≥0. From corollary 3.3, it follows that the processs→sup_t∈[0,T]supa∈RLa_t+s(Xx)− La_t(Xx), is continuous and vanishing in 0. Therefore∀ǫ∈(0, 1)a.s. existssǫ,x(ω)>0 such that

s ∈[0,sǫ,x(ω)]implies Lat+s(X x_)−La

t(X

x_{) <} ǫ

2L∗_(1+ǫ) ∀a∈ R and t ∈[0,T]. So, a.s. it holds

∀t∈[0,T]:

inf

r∈[t,t+sǫ,x(ω)]

(X_rx+h−X_rx)−(X_tx+h−X_tx)≥

Z t+sǫ,x(ω)

t

b∗(X_ux)−b∗(X_ux+ (X_tx+h−X_tx))du

=

Z

R

”

b∗(a)−b∗€a+€X_tx+h−X_txŠŠ—× h

La_t+s

ǫ,x(ω)(X

x_)−La t(X

x₎i_{d a}

≥ −kL·_t+s

ǫ,x(ω)(X

x_)−L·

t(X x_)k

∞× kb∗(·)−b∗ €

·+€X_tx+h−X_txŠŠk₁

≥ −

sup

a∈R La_t+s

ǫ,x(ω)(X

x_)−La t(X

x₎

×2L∗€X_tx+h−X_txŠ≥ − ǫ 1+ǫ(X

x+h t −X

This inequality implies:

inf

r∈[t,t+sǫ,x(ω)]

(X_rx+h−X_rx)≥(X

x+h t −Xtx)

1+ǫ (3)

In the same way we obtain:

sup

r∈[t,t+sǫ,x(ω)]

(X_rx+h−X_rx)≤(1+ǫ)(X_tx+h−X_tx)≤(X

x+h t −X

x t)

1−ǫ (4)

In particular a.s.Nǫ,T,x(ω):=⌈_s T

ǫ,x(ω)

⌉<∞, and thus a.s. we have

X_Tx+h−X_Tx≥h

1 1+ǫ

Nǫ,T,x(ω)

>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],∀ǫ >0 a.s. exists sǫ,x,y(ω)>0 such thats ∈[0,sǫ,x,y(ω)]

impliesLa_t+s(Xu)−La_t(Xu)< ǫ

2L∗_(1+ǫ) ∀a∈R,t∈[0,T], andu∈[x,y]. This fact will be used in

the next theorem, which is crucial to prove the existence of a flow of classC1,α.

Theorem 4.3. Let x,y,t∈R_{such that x< y,and t}≥0.Then, a.s.

inf

u∈[x,y]exp ‚Z

R

La_t(Xu)D b(d a)

Œ ≤

‚

X_ty−Xx t y−x

Œ ≤ sup

u∈[x,y] exp

‚Z

R

La_t(Xu)D b(d a)

Œ

Proof. Step 1. Fix x < y and t ≥ 0. Fix ǫ > 0, and lets_ǫ,x,y(ω) be defined as in remark 2. Moreover defineN_ǫ,t,x,y(ω):=⌈ t

sǫ,x,y(ω)⌉. Define ,∀i∈ N_,t

i(ω) = (i×sǫ,x,y(ω))∧t. Obviously we

have, a.s., thati≥N_ǫ,t,x,y(ω)impliest_i(ω) =t. Define g:Ω×R+_→R+_as:

g(h) = sup

r∈[0,h] sup

z∈[x,y] sup

s∈[0,t] sup

a∈R

|L_sr+a(Xz)−L_sa(Xz)|

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 thatz <w. Then it holds:

ln ‚

Xw t −Xzt w−z

Œ

=

Z t

0

b(Xw

s)−b(Xsz) Xw

s −Xsz

ds=

∞ X

i=0 Z ti+1

ti

b(Xw

s )−b(Xsz) Xw

s −Xsz

ds:=I₀ǫ,z,w

Observe that in the last summation a.s. only a finite number of terms are different from 0. Define:

I₁ǫ,z,w:=

∞ X

i=0 Z ti+1

ti

b(X_sw)−b(X_sz)

Xw ti−X

z ti

ds

I₂ǫ,z,w:=

∞ X

i=0 Zti+1

ti

1

Xw ti−X

z ti

–

b

‚

X_sz+

‚

Xw_t

i−X

z ti

1+ǫ

ŒŒ

−b(X_sz)

™

I₃ǫ,z,w:=

Note that the following estimate holds:

I₂ǫ,z,w−I₃ǫ,z,w≤I₁ǫ,z,w≤I₂ǫ,z,w+I₃ǫ,z,w (5) Finally define the random set:

Ai,ǫ,z,w:=

Similarly we define:

Bi,ǫ,z,w:=

Step 2.Thanks to estimates (3) and (4) we have:

Step 3.From the occupation time formula we have:

Step 4.It holds:

I₂ǫ,z,w=

From this equality it follows:

Step 5. From (5), and from (6), (7) and (8), follows that, assuming ǫ ∈(0,1

2), there exists a constantM dependent only onx,y,t, and on the constantsK, LandL∗, such that

|I₀ǫ,z,w− Z

R

La_t(Xz)D b(d a)| ≤Mǫ(1+ sup

u∈[x,y] sup

a∈R

La_t(Xu)) +M g(M(w−z))Nǫ,t,x,y (9)

We call∆a finite partition of[x,y]if, for somen∈N_{,∆ ={x=z0<z1<· · ·<zi<zi+1<· · ·<}

zn=y}. Moreover we define|∆|:=maxzi∈∆\y(zi+1−zi). We denote byΛthe set of finite partition

of[x,y]. Obviously it holds:

sup ∆∈Λ

min

zi∈∆\y

‚

Xzi+1 t −X

zi

t z_i+1−z_i

Œ ≤

‚

X_ty−X_tx y−x

Œ ≤ inf

∆∈Λ zmaxi∈∆\y

‚

Xzi+1 t −X

zi

t z_i+1−z_i

Œ

(10)

Using (9), we have,∀ǫ∈(0,1₂):

inf ∆∈Λzmaxi∈∆\y

‚

Xzi+1 t −X

zi

t zi+1−zi

Œ ≤ inf

∆∈Λzmaxi∈∆\y

exp ‚Z

R

La_t(Xzi_{)D b(d a)}

Œ

×exp ‚

Mǫ(1+ sup

u∈[x,y] sup

a∈R

La_t(Xu)) +M g(M|∆|)N_ǫ,t,x,y

Œ

≤ sup

u∈[x,y] exp

‚Z

R

La_t(Xu)D b(d a)

Œ ×exp

‚

Mǫ(1+ sup

u∈[x,y] sup

a∈R

La_t(Xu))

Œ

With the same reasoning we obtain:

sup ∆∈Λ

inf

zi∈∆\y

‚

Xzi+1 t −X

zi

t zi+1−zi

Œ ≥ inf

z∈[x,y]exp ‚Z

R

La_t(Xz)D b(d a)

Œ

×exp ‚

−Mǫ(1+ sup

u∈[x,y] sup

a∈R

La_t(Xu))

Œ

Thanks to the arbitrariness ofǫ, and thanks to relation (10) the proof is complete.

Using the preceding theorem we can now prove the existence of the stochastic flow of classC1,α_.

All results we have proved refers to solutions starting fromt=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:

d X_ts,x=b(X_ts,x)d t+dW_t

Xs,x(s) =x (11)

Obviously forX0,t xtheorem 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)→φ_s,t(x)(ω)defined for0≤s≤t≤T , x∈R_{,ω∈Ωwith values in}R_{, such that}

1. given any0≤s≤T , x ∈R_{the process X}s,x_{= (X}s,x

t s≤t≤T)defined as X s,x

2. a.s. φs,t is a diffeomorphism∀0≤s≤t≤T andφs,t,φ−1s,t, Dφs,t, and Dφ−1s,t are continuous in(s,t,x),and of class Cαin x,forα <1

2.

3. a.s.φ_s,t(x) =φ_u,t(φ_s,u(x))∀0≤s≤u≤t≤T and x∈R_,andφ

s,s(x) =x. Moreover an explicit expression for Dφ_s,t(x)is given by:

Dφs,t(x) =exp

‚Z

R

La_t(Xφ−1(0,s,x))−L_sa(Xφ−1(0,s,x))D b(d a)

Œ

Proof. Step 1.We first give the proof under the assumptionb1∈W1,∞. It will take the first three steps. LetDbe the set of dyadic numbers, and letDT:=D∩[0,T]. Define∀x∈Dand∀t∈DT,

e

X_t0,x(ω) =X_tx(ω). Note the two following facts:

1. BeingDa countable set, there exists a negligible setA0such that,∀ω6∈A0and∀x∈D X·x is a continuous solution of equation (1). In particular since it holds

X_tx=x+

Z t

0

b(X_sx)ds+W_t

∀ω6∈A0andx∈D, the family of processes{X.x}x∈Dis uniformly equicontinuous.

2. Thanks to the countability of the setD×D×DT_{, there exists a negligible setA}

1⊃A0such that∀ω6∈A₁∀x,y∈Dsuch that x< y, and∀t∈DT _{it holds}

inf

u∈[x,y]exp ‚Z

R

La_t(Xu)D b(d a)

Œ ≤ Xe

0,y t −Xe

0,x t y−x

!

≤ sup

u∈[x,y] exp

‚Z

R

La_t(Xu)D b(d a)

Œ

These facts imply that,∀ω6∈A1is well-defined the continuous extension ofXe·0,·(ω)onR×[0,T]. Denote byφ_0,t(x)(ω)this extension. Note that,∀ω6∈A₁, the family{φ_0,·(x)(ω)}_x∈Ris uniformly

equicontinuous; more precisely,∀ω6∈A₁,∀0≤s≤t≤T

|φ0,t(x)−φ0,s(x)| ≤(t−s)L+ sup r∈[0,T]

|Wr+(t−s)−Wr| (12)

In particular we have that a.s. |φ0,t(x)−x|is bounded in x. This fact, together with continuity

inx, implies that,∀ω6∈A1,φ0,t(·)is surjective. Moreover it’s immediate to verify that,∀ω6∈A1, and∀x,y∈R_{, such that x<y, it holds:}

inf

u∈[x,y]exp ‚Z

R

La_t(Xu)D b(d a)

Œ ≤

‚

φ0,t(y)−φ0,t(x) y−x

Œ ≤ sup

u∈[x,y] exp

‚Z

R

La_t(Xu)D b(d a)

Œ

This fact, together with the continuity of La_t(Xx)with respect to(a,t,x)implies that∀ω6∈A1, ∀0≤ t ≤ T,φ0,t(x)(ω)is differentiable in x, and its derivative is exp

€R

RL a t(X

x_{)D b(d a)}Š_{. So}

∀t∈[0,T]a.s.φ_0,t is a surjective and differentiable function whose derivative is strictly positive everywhere, and therefore is a diffeomorphism. Moreover exp€R_RLa

t(Xx)D b(d a)

Š

is of classCα

with respect toxforα <1

2.

Step 2. We now prove that∀x∈R_{a.s. φ0,t(x) =X}x

t ∀0≤t≤T. Fix x∈R. Because of the a.s.

sufficient to prove that∀t∈DT, a.s.φ0,t(x) =Xtx. So fixt∈D

T_{, and let{x}

n}n∈Nbe a sequence of

dyadic numbers converging tox. By construction we have that∀ω6∈A₁φ_0,t(x_n) =Xe0,xn

t =X xn

t and φ_0,t(x) =lim_n→∞φ0,t(xn) =limn→∞X

xn

t . Moreover thanks to theorem 4.3, there is a negligible

setAn

x such that∀ω6∈Anxit holds

inf

u∈[x,xn]

exp ‚Z

R

La_t(Xu)D b(d a)

Œ ≤

‚

Xxn

t −Xtx xn−x

Œ

≤ sup

u∈[x,xn]

exp ‚Z

R

La_t(Xu)D b(d a)

Œ

Denote byA_x the negligible setS∞_i=0Ai

x. Therefore∀ω6∈Ax∪A1we have the equality

X_tx= lim

n→∞X

xn

t =φ0,t(x)

This implies thatφ_0,t(x)is a continuousF_0,t−measurable solution of equation (1).

Step 3. We will denote by φ_0,−1_t(ω) the inverse function of φ_0,t(ω). Define ∀0 ≤ s ≤ t ≤ T φ_s,t=φ0,t◦φ0,−1s. It’s immediate to verify that property 3 if satisfied. It’s also easy to show thatφs,t

is a diffeomorphism, and its inverse function isφ−1_s,t =φ0,s◦φ−10,t. Moreover, let{(sn,tn,xn)}n∈Nbe

a sequence such that 0≤sn≤tn≤T,xn∈Rand(sn,tn,xn)→(s,t,x). Obviously we have φsn,tn(xn) = (φ0,tn◦φ

−1

0,t)◦φs,t◦(φ0,s◦φ−10,sn(xn))

Thanks to (12), φ0,tn ◦φ

−1

0,t and φ0,s◦φ0,−1sn converge to the identity; this observation prove

that φs,t(x)is jointly continuous in (s,t,x). This implies the continuity of φ0,−1t with respect to

(t,x), and thus the continuity ofφ−1

s,t in(s,t,x). Having proved that the derivative ofφ0,t(x)is

exp€R_RL_ta(Xx)D b(d a)Š, we have that

Dφs,t(x) =exp

‚Z

R

La_t(Xφ−1(0,s,x))−L_sa(Xφ−1(0,s,x))D b(d a)

Œ

Dφ_s−1_,t(x) =exp ‚

− Z

R

La t(X

φ−1_(0,s,x)

)−La s(X

φ−1_(0,s,x)

)

D b(d a)

Œ

which are jointly continuous in(s,t,x), and of classCα_{forα <}1

2. The property (2) is proved. We have already proved property (1) ifs=0. To prove property (1) in the general case, fixx∈R_and

s≥0. Thus applying the Itô formula, we obtain thatφ_s,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∈W1,∞.

LetWft :=WT−t−WT. Note thatWfis a Brownian motion and the completedσ-algebra generated

byWfu−Wfr fors≤r≤u≤tisFes,t:=FT−t,T−s. Note that−bsatisfies the conditions used in the

first three steps. So there existsφes,t(x), continuousFes,t−measurable solution of the equation

¨

d X_ts,x=−b(X_ts,x)d t+dfW_t

Xs,x_{(s) =x} (13)

satisfying property (2) and (3).

Define, for 0≤s≤t ≤T ψs,t(x)(ω):=φe−1T−t,T−s(x)(ω). It’s immediate to verify thatψsatisfies

property (2) and (3). Moreover it’s easy to verify that, for anyx∈R_{and 0≤s≤T,ψs,·(x)(ω)is}

Remark 3. A classic approach to the problem of the existence of a C1,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

¨

∂_tu(t,x) +1

2∂x xu(t,x) +b(x)∂xu(t,x) =0

u(T,x) =x (14)

and that the solution is sufficiently smooth, so that we can apply Itô formula:

du(t,Xt) =∂tu(t,Xt)d t+∂xu(t,Xt)b(Xt)d t+∂xu(t,Xt)dWt+

1

2∂x xu(t,Xt)d t=∂xu(t,Xt)dWt

Suppose moreover that u(t,·)is a diffeomorphism. Through the change of variable Yt = u(t,Xt) 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

whereσ(t,y) =∂_xu(t,u−1_t (y)).To solve this problem using well-known theorems, we should prove σ∈ C1,α for someα > 0.But σ∈C1,αimplies∂x xu(t,·)∈Cα.So the discontinuity of the term b(x)∂xu(t,x) would be balanced by the term ∂tu(t,x). However, examples such that ∂tu(t,x)is continuous, and b(x)is discontinuous can be shown. Consider, for example, b(x) =sgn(x). It is possible to prove that u(t,x) =E[X_Tt,x]and∂tu(t,x) =P(XTt,x <0)−P(X

t,x

T >0).This term, as shown in[4], is continuous in x.Thus we have obtained∂x xu(t,·)6∈Cα.So, even if it is possible to prove the existence and smoothness of the solution of equation(14),it is not possible to prove thatσ∈C1,α,and it is not possible to prove the existence of the stochastic flow through well-known theorems.

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