ELECTRONIC
COMMUNICATIONS in PROBABILITY
ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN
MOTION AND WITH MONOTONE DRIFT
BRAHIM BOUFOUSSI1
Cadi Ayyad University FSSM, Department of Mathematics, P.B.O 2390 Marrakesh, Morocco. email: [email protected]
YOUSSEF OUKNINE2
Cadi Ayyad University FSSM, Department of Mathematics P.B.O 2390 Marrakesh, Morocco. email: [email protected]
Submitted 3 March 2003, accepted in final form 31 July 2003 AMS 2000 Subject classification: 60H10, 60G18
Keywords: Fractional Brownian motion. Stochastic integrals. Girsanov transform.
Abstract Let {BH
t , t ∈ [0, T]} be a fractional Brownian motion with Hurst parameter H > 12. We
prove the existence of a weak solution for a stochastic differential equation of the form Xt= x+BH
t +
Rt
0(b1(s, Xs) +b2(s, Xs))ds, whereb1(s, x) is a H¨older continuous function of order
strictly larger than 1− 1
2H inxand thanH−
1
2 in time andb2is a real bounded nondecreasing
and left (or right) continuous function.
1
Introduction
LetBH={BH
t , t∈[0, T]}be a fractional Brownian motion with Hurst parameterH ∈(0,1). That is,BH is a centered Gaussian process with covariance
RH(t, s) =E(BH t BsH) =
1 2
n
|t|2H+|s|2H− |t−s|2Ho.
If H = 1
2 the process B
H is a standard Brownian motion. Consider the following stochastic differential equation
Xt=x+BtH+
Z t
0
(b1(s, Xs) +b2(s, Xs))ds, (1.1)
whereb1, b2 : [0, T]×R−→Rare Borel functions. The purpose of this paper is to prove, by
approximation arguments, the existence of a weak solution to this equation if H > 12, under the following weak regularity assumptions on the coefficients:
1RESEARCH SUPPORTED BY PSR PROGRAM 2001, CADI AYYAD UNIVERSITY. 2RESEARCH SUPPORTED BY PSR PROGRAM 2001, CADI AYYAD UNIVERSITY.
(H1) b1is H¨older continuous of order 1> α >1−21H inxand of orderγ > H−12 in time:
|b1(t, x)−b1(s, y)| ≤C(|x−y|α+|t−s|γ). (1.2)
(H2) sup
s∈[0,T]
sup x∈R
|b2(s, x)|≤M <∞.
(H3) ∀s∈[0, T], b2(s, .) is a nondecreasing and left (or right) continuous function.
The same approximation arguments can be used to consider the case where b2 satisfies the
following assumptions: (H′
2) sup
s∈[0,T]
sup x∈R
|b2(s, x)|≤M(1 +|x|)
(H′
3) for alls∈[0, T], b2(s, .) is a nonincreasing and continuous function
If b2 ≡ 0 and H = 12 (the process BH is a standard Brownian motion), the existence of a
strong solution is well-known by the results of Zvonkin [18], Veretennikov [16] and Bahlali [2]. See also the work by Nakao [11] and its generalization by Ouknine [14]. In the case of Equation (1.1) driven by the fractional Brownian motion withb2≡0, the weak existence and
uniqueness are established in [13] using a suitable version of Girsanov theorem; the existence of a strong solution could be deduced from an extension of Yamada-Watanabe’s theorem or by a direct arguments.
In the general caseH >1/2,to establish existence and uniqueness result, a H¨older type space-time condition is imposed on the drift. Recently, Mishura and Nualart [9] gave an existence and uniqueness result for one discontinuous function namely thesgnfunction. Their approach relies on the Novikov criterion and it is valid for 1+4√5 > H >1/2.
Our aim is to establish existence and uniqueness result for general monotone function including sgnfunction andH >1/2.
The paper is organized as follows. In Section 2 we give some preliminaries on fractional calculus and fractional Brownian motion. In Section 3 we formulate a Girsanov theorem and show the existence of a weak solution to Equation (1.1). As a consequence we deduce the uniqueness in law and the pathwise uniqueness. Finally Section 4 discusses the existence of a strong solution.
2
Preliminaries
2.1
Fractional calculus
An exhaustive survey on classical fractional calculus can be found in [15]. We recall some basic definitions and results.
Forf ∈L1([a, b]) andα >0 theleft fractional Riemann-Liouville integral off of order αon
(a, b) is given at almost allxby
Iaα+f(x) = 1 Γ(α)
Z x
a
(x−y)α−1f(y)dy,
where Γ denotes the Euler function.
This integral extends the usual n-order iterated integrals off for α=n ∈N. We have the
Iaα+(I β a+f) =I
α+β a+ f.
The fractional derivative can be introduced as inverse operation. We assume 0< α <1 and p > 1. We denote by Iα
a+(Lp) the image ofLp([a, b]) by the operator Iaα+ . Iff ∈ Iaα+(Lp), the functionφsuch thatf =Iα
a+φis unique inLp and it agrees with theleft-sided Riemann-Liouville derivative of f of order αdefined by
Daα+f(x) = 1 Γ(1−α)
d dx
Z x
a
f(y) (x−y)αdy. The derivative of f has the following Weil representation:
Dα
a+f(x) = 1 Γ(1−α)
µ
f(x) (x−a)α +α
Z x
a
f(x)−f(y) (x−y)α+1dy
¶
1(a,b)(x), (2.1)
where the convergence of the integrals at the singularity x=y holds inLp-sense. When αp > 1 any function in Iα
a+(L
p) is ³α−1
p
´
- H¨older continuous. On the other hand, any H¨older continuous function of orderβ > α has fractional derivative of order α. That is, Cβ([a, b])⊂Iα
a+(Lp) for all p >1. Recall that by construction forf ∈Iα
a+(Lp),
Iaα+(Dαa+f) =f and for general f ∈L1([a, b]) we have
Dα
a+(Iaα+f) =f.
Iff ∈Iaα++β (L1),α≥0, β≥0, α+β≤1 we have the second composition formula Daα+(Daβ+f) =Dαa++βf.
2.2
Fractional Brownian motion
LetBH ={BH
t , t∈[0, T]}be a fractional Brownian motion with Hurst parameter 0< H <1 defined on the probability space (Ω,F, P). For each t ∈[0, T] we denote byFBH
t the σ-field generated by the random variables BH
s ,s∈[0, t] and the sets of probability zero.
We denote byEthe set of step functions on [0, T]. LetHbe the Hilbert space defined as the
closure ofEwith respect to the scalar product
1[0,t],1[0,s]®H=RH(t, s).
The mapping 1[0,t] −→ BHt can be extended to an isometry between H and the Gaussian spaceH1(BH) associated withBH. We will denote this isometry byϕ−→BH(ϕ).
The covariance kernelRH(t, s) can be written as
RH(t, s) =
Z t∧s
0
KH(t, r)KH(s, r)dr,
where KH is a square integrable kernel given by (see [3]): KH(t, s) = Γ(H+1
2)
−1(t−s)H−1
2F(H−1 2,
1
2 −H, H+ 1 2,1−
F(a, b, c, z) being the Gauss hypergeometric function. Consider the linear operatorK∗
H from
EtoL2([0, T]) defined by
(KH∗ϕ)(s) =KH(T, s)ϕ(s) +
Z T
s
(ϕ(r)−ϕ(s))∂KH
∂r (r, s)dr. For any pair of step functionsϕand ψinEwe have (see [1])
hK∗
Hϕ, KH∗ψiL2([0,T])=hϕ, ψiH.
As a consequence, the operator K∗
H provides an isometry between the Hilbert spacesH and L2([0, T]). Hence, the processW ={Wt, t∈[0, T]} defined by
Wt=BH((KH∗)−
1
(1[0,t])) (2.2)
is a Wiener process, and the processBH has an integral representation of the form
BHt =
Z t
0
KH(t, s)dWs, (2.3)
because¡K∗
H1[0,t]
¢
(s) =KH(t, s)1[0,t](s).
On the other hand, the operatorKHonL2([0, T]) associated with the kernelKHis an
isomor-phism from L2([0, T]) onto IH+1/2
0+ (L2([0, T])) and it can be expressed in terms of fractional integrals as follows (see [3]):
(KHh)(s) = I02+Hs 1 2−HI
1 2−H
0+ s H−1
2h, ifH ≤1/2, (2.4)
(KHh)(s) = I01+sH− 1 2IH−
1 2
0+ s 1
2−Hh, ifH ≥1/2, (2.5) whereh∈L2([0, T]).
We will make use of the following definition ofFt-fractional Brownian motion.
2.1 Definition. Let {Ft, t∈[0, T]} be a right-continuous increasing family of σ-fields on
(Ω,F, P) such that F0 contains the sets of probability zero. A fractional Brownian motion
BH ={BH
t , t∈[0, T]}is called anFt-fractional Brownian motion if the processW defined in (2.2) is anFt-Wiener process.
3
Existence of strong solution for SDE with monotone
drift.
In this section we are interested by the special caseb1≡0. We will prove by approximation
arguments that there is a strong solution of equation (1.1). We will discuss two cases: 1)b2(s, .) satisfies (H2) and (H3).
2)b2(s, .) satisfies (H′2) and (H′3).
1– The first case:
To treat the first situation, let us suppose that b2(s, .) is nondecreasing and left continuous
3.1 Lemma. Letb: [0, T]×R→R, a bounded measurable function such that for anys∈[0, T],
b(s, .)is a nondecreasing and left continuous function. Then there exists a family of measurable functions
Proof. First assume that b(s, .) is left continuous and let us choose for anyn≥1 bn(s, x) =n
Z x
x−1
n
b(s, y)dy .
Sinceb(s, .) is nondecreasing thenbn(s, .) is also a nondecreasing function for any fixedn≥1. Letx, y∈R, we clearly have for anyn≥1,
Now let x0 ∈ R and take an increasing sequence of real numbers xn converging to x0. We
want to show that for any s ∈ [0, T], lim
n→∞bn(s, xn) = b(s, x0). It is enough to prove that
there exists a subsequencebϕ(n)(s, xϕ(n)) which converges to b(s, x0). To do this, remark first
that since b(s, .) is left continuous we have lim
n→∞bn(s, x0) =b(s, x0). Now let us consider any
strictly increasing sequencex′
n converging tox0such thatx0−x′n=o(1n). We clearly get by
and for any fixed n≥1 the functionbn(s, .) is nondecreasing, we have
We deduce by (3.2) and the left continuity ofb(s, .), lim
n→∞bϕ(n)(s, xϕ(n)) =b(s, x0).
Which ends the proof.
Let (BH)t
≥1be a fractional Brownian motion with Hurst parameterH ∈(12,1). We consider
the following SDE
Xt=x+BHt +
Z t
0
b2(s, Xs)ds , 0≤t≤T . (3.4)
3.2 Theorem. Suppose thatb2 satisfies the assumptions(H2)and(H3). Then there exists a
strong solution to the equation (3.4).
Proof. Assume that b2 : [0, T]×R −→ R is a measurable and bounded function which is
nondecreasing and left continuous with respect to the space variablex. Forn≥1, letbn be as in lemma 3.1 and consider the following SDE
Xtn =x+BtH+
Z t
0
bn(s, Xsn)ds , 0≤t≤T . (3.5)
By standard Picard’s iteration argument, one may show that for anyn≥1, the equation (3.5) has a strong solution which we denote byXn.
Letn > m, we denote by ∆t=Xn
t −Xtm. Using the monotony argument onbn, we have ∆t≥
Z t
0
bm(s, Xn
s)−bm(s, Xsm)ds , ≥
Z t
0
(bm(s, Xsn)−bm(s, Xsm))I{∆s≤0}ds ,
≥2mM
Z t
0
∆sI{∆s≤0}ds≥ −2mM
Z t
0
∆−s ds .
(3.6)
We then get
∆−t ≤2mM
Z t
0
∆−s ds . (3.7)
By Gronwall’s lemma, we have for almost allw and for anyt∈[0, T], the sequence (Xn t(w)) is a nondecreasing function of nwhich is bounded sincebn is. Therefore it has a limit when n→ ∞and we set
lim n→∞X
n
t(ω) =Xt(ω),
which entails in particular that X is FBtH− adapted. Applying the convergence result in
Lemma 3.1 and the boundedness ofbn we get by Lebesgue’s dominated convergence theorem,
Xt=x+BHt +
Z t
0
3.1 Remark. To show that Equation (3.4) has a weak solution, a continuity condition is imposed on the drift in [13]. Here, the function b2 may have a countable set of discontinuity points.
The solution constructed in Theorem 3.2 is the minimal one.
3.2 Remark. Let b2 : [0, T]×R −→ R be a bounded measurable function, which is
nonde-creasing and right continuous. In this case we consider a denonde-creasing sequence of Lipschitz continuous functions which approximate the drift. One may take
bn(s, x) =n
Z x+1
n
x
b2(s, y)dy .
For any fixed (s, x)∈[0, T]×R, the sequence (bn(s, x))
n≥1is nonincreasing and for any fixed
n≥1 ands∈[0, T] the function bn(s, .) is nondecreasing. The same arguments as in Lemma 3.1 can be used to prove that for any sequence (xn)n≥1 decreasing tox, we have
lim
n→∞bn(s, xn) =b2(s, x).
This allows us to construct the maximal solution to the equation (3.4). 2– The second case:
In this case we use the following lemma:
3.3 Lemma. Letb(., .) : [0, T]×R−→Rbe a continuous function with linear growth, that is
there exists a constant M <∞ such that ∀(s, x)∈[0, T]×R,|b(s, x)|≤M(1+|x|). Then
the sequence of functions
bn(s, x) = sup
y∈Q(b(s, y) −n|x−y|), is well defined for n≥M and it satisfies
• For any sequencexnconverging tox∈R, we have
lim
n→∞bn(s, xn) =b(s, x),
• n7→bn(s, x) is nonincreasing, for all x∈R, s∈[0, T]
• |bn(s, x)−bn(s, y)| ≤n|x−y| for alln≥M, s∈[0, T], x, y∈R
• |bn(s, x)| ≤M(1+|x|), for all(s, x)∈[0, T]×R, n≥M .
For the proof of this lemma we refer for example to [8].
3.4 Theorem. Assume thatb2 satisfies conditionsH2′ andH3′. Then there exists a unique
strong solution to the equation (3.4).
Proof. For anyn≥1, letbn be as in Lemma 3.3. Sincebn is Lipschitz and linear growth, the result in [13] assures the existence of a strong solutionXn to the equation
Xtn=x+BtH+
Z t
0
bn(s, Xsn)ds .
Since (bn)n≥1is nonincreasing, comparison theorem entails that (Xn)n≥1 is a.s nonincreasing.
a.s toX, which is clearly a strong solution to the SDE (3.4). Moreover, ifX1andX2 are two
solutions of (3.4), using the fact that b2(s, .) is nonincreasing, we get by applying Tanaka’s
formula to the continuous semi-martingaleX1−X2,
(Xt1−Xt2)+=
Then we have the pathwise uniqueness of the solution.
4
Existence of a weak solution
4.1
Girsanov transform
As in the previous section, let BH be a fractional Brownian motion with Hurst parameter 0< H <1 and denote bynFBH
t , t∈[0, T]
o
its natural filtration.
Given an adapted process with integrable trajectories u= {ut, t ∈ [0, T]} and consider the transformation As a consequence we deduce the following version of the Girsanov theorem for the fractional Brownian motion, which has been obtained in [3, Theorem 4.9]:
4.1 Theorem. Consider the shifted process (4.1) defined by a processu={ut, t∈[0, T]} with integrable trajectories. Assume that:
i) R0·usds∈I0H++1/2(L
Proof. By the standard Girsanov theorem applied to the adapted and square integrable pro-cessKH−1¡R0·usds¢we obtain that the processWfdefined in (4.2) is anFBH
t - Brownian motion under the probability Pe . Hence, the result follows.
From (2.5) the inverse operator KH−1 is given by
KH−1h=sH−12DH− 1 2
0+ s 1
2−Hh′, ifH >1/2 (4.3) for allh∈IH+
1 2
0+ (L2([0, T])). Then ifH >
1
2 we needu∈I
H−1/2
0+ (L2([0, T])), and a sufficient condition for i) is the fact that the trajectories of uare H¨older continuous of orderH−12+ε for some ε >0.
4.2
Existence of a weak solution
Consider the stochastic differential equation:
Xt=x+BtH+
Z t
0
(b1(s, Xs) +b2(s, Xs))ds, 0≤t≤T, (4.4)
where b1 andb2 are Borel functions on [0, T]×Rsatisfying the conditionsH1 forb1 andH2
and H3 (resp. H′2 andH′3) forb2. By aweak solution to equation (4.4) we mean a couple of
adapted continuous processes¡BH, X¢on a filtered probability space (Ω,F, P,{Ft, t∈[0, T]}), such that:
i) BH is anF
t -fractional Brownian motion in the sense of Definition 2.1. ii) X andBH satisfy (4.4).
4.2 Theorem. Suppose thatb1andb2are Borel functions on[0, T]×Rsatisfying the conditions
H1forb1,H2 andH3 (resp. H2′ andH3′) forb2. Then Equation (4.4) has a weak solution.
Proof. LetX2be the strong solution of (3.4) and setBeH
t =BtH−
Rt
0b1(s, X 2
s)ds.We claim that the processus=−b1(s, Xs2) satisfies conditions i) and ii) of Theorem 4.1. If this claim is true, under the probability measureP ,e BeH is anFBH
t -fractional Brownian motion, and
³ e
BH, X2´
is a weak solution of (4.4) on the filtered probability space ³Ω,F,P ,e
n
FtBH, t∈[0, T]
o´
. Set
vs=−KH−1
µZ ·
0
b1(r, Xr2)dr
¶
(s).
We will show that the processvsatisfies conditions i) and ii) of Theorem 4.1. Along the proof cH will denote a generic constant depending only onH. LetH > 12, by (4.3), the processv is clearly adapted and we have
vs = −sH−1 2DH−
1 2
0+ s 1 2−Hb
where
As consequence, taking into account thatα <1, we have for anyλ >1
E
In order to estimate the termβ(s), we apply the H¨older continuity condition (1.2) and we get
|β(s)| ≤ cHsH−12
By Fernique’s Theorem, taking into account thatα <1, for anyλ >1 we have
E
4.3
Uniqueness in law and pathwise uniqueness
In this subsection we will prove uniqueness in law of weak solution under the conditionH1for
b1,H2′ andH3′ forb2. The main result is
4.3 Theorem. Suppose that b1 andb2 are Borel functions on [0, T]×R. satisfying the
con-ditions H1 for b1,H2′ and H3′ for b2. Then we have the uniqueness in distribution for the
solution of Equation (4.4).
Proof. It is clear thatX2is pathwise unique, hence the uniqueness in law holds whenb1≡0.
Let ¡X, BH¢ be a solution of the stochastic differential equation (4.4) defined in the filtered probability space (Ω,F, P,{Ft, t∈[0, T]}). Define
us=
µ
KH−1
Z ·
0
b1(r, Xr))dr
¶
(s).
LetPe defined by
dPe dP = exp
Ã
−
Z T
0
usdWs−1 2
Z T
0
u2sds
!
. (4.6)
We claim that the process ussatisfies conditions i) and ii) of Theorem 4.1. In fact, us is an adapted process and taking into account that Xt has the same regularity properties as the fBm we deduce thatR0Tu2
sds <∞almost surely. Finally, we can apply again Novikov theorem in order to show thatE³dPe
dP
´
= 1, because by Gronwall’s lemma
kXk∞≤¡|x|+°°BH°°
∞+C1T
¢
eC2T,
and
|Xt−Xs| ≤ |BtH−BHs |+C3|t−s|(1 +kXk∞)
for some constants Ci,i= 1,2,3.
By the classical Girsanov theorem the process
f
Wt=Wt+
Z t
0
urdr
is anFt-Brownian motion under the probability Pe. In terms of the processWtf we can write
Xt=x+
Z t
0
KH(t, s)dWsf +
Z t
0
b2(s, Xs)ds ,
Set
e
BsH=
Z t
0
KH(t, s)dWsf . Then X satisfies the following SDE,
Xt=x+BeHt +
Z t
0
As a consequence, the processesX andX2 have the same distribution under the probability
In conclusion we have proved the uniqueness in law, which is equivalent to pathwise uniqueness (see [13] Theorem 5)
4.1 Remark. In the caseH <1/2, a deep study is made between stochastic differential equation with continuous coefficient and unit drift and anticipating ones (cf [4]).
AcknowledgementThe work was partially carried out during a stay of Youssef Ouknine at the IPRA (PAU, France) (Institut pluridisciplinaire de recherches appliqu´ees). He would like to thank the IPRA for hospitality and support.
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