Approximation of solutions of SDEs driven by a fractional ... - arXiv

arXiv:1701.01244v1 [math.PR] 5 Jan 2017

Modern Stochastics: Theory and Applications 3 (2016) 303–313 DOI: 10.15559/16-VMSTA69

Approximation of solutions of SDEs driven by a fractional Brownian motion, under pathwise uniqueness Oussama El Barrimia,1,∗, Youssef Oukninea,b a

Cadi Ayyad University, Faculty of Sciences Semlalia, Av. My Abdellah, 2390, Marrakesh, Morocco b Hassan II Academy of Sciences and Technology Rabat, Morocco [email protected] (O. El Barrimi), [email protected] (Y. Ouknine) Received: 29 July 2016, Revised: 12 December 2016, Accepted: 13 December 2016, Published online: 20 December 2016

Abstract Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod’s selection theorem. Keywords Fractional Brownian motion, Stochastic differential equations 2010 MSC 60G15, 60G22

1

Introduction

Consider a fractional Brownian motion (fBm), a self-similar Gaussian process with stationary increments. It was introduced by Kolmogorov [5] and studied by Mandelbrot and Van Ness [6]. The fBm with Hurst parameter H ∈ (0, 1) is a centered Gaussian process with covariance function 1 RH (t, s) = E BtH BsH = t2H + s2H − |t − s|2H . 2 ∗ Corresponding

author. author is supported by the CNRST “Centre National pour la Recherche Scientifique et Technique”, grant No. I 003/034, Rabat, Morocco. 1 This

© 2016 The Author(s). Published by VTeX. Open access article under the CC BY license.

www.i-journals.org/vmsta

304

O. El Barrimi, Y. Ouknine

If H = 1/2, then the process B 1/2 is a standard Brownian motion. When H 6= 12 , B H is neither a semimartingale nor a Markov process, so that many of the techniques employed in stochastic analysis are not available for an fBm. The self-similarity and stationarity of increments make the fBm an appropriate model for many applications in diverse fields from biology to finance. We refer to [7] for details on these notions. Consider the following stochastic differential equation (SDE) ( dXt = b(t, Xt ) dt + dBtH , (1) X 0 = x ∈ Rd , where b : [0, T ] × Rd → Rd is a measurable function, and B H is a d-dimensional fBm with Hurst parameter H < 1/2 whose components are one-dimensional independent fBms defined on a probability space (Ω, F , {Ft }t∈[0,T ] , P ), where the filtration {Ft }t∈[0,T ] is generated by BtH , t ∈ [0, T ], augmented by the P -null sets. It has been proved in [2] that if b satisfies the assumption ∞ (2) [0, T ]; L1 Rd ∩ L∞ Rd , b ∈ L1,∞ ∞ := L

1 , then Eq. (1) has a unique strong solution, which will be assumed for H < 2(3d−1) throughout this paper. Notice that if the drift coefficient is Lipschitz continuous, then Eq. (1) has a unique strong solution, which is continuous with respect to the initial condition. Moreover, the solution can be constructed using various numerical schemes. Our purpose in this paper is to establish some stability results under the pathwise uniqueness of solutions and under weak regularity conditions on the drift coefficient b. We mention that a considerable result in this direction has been established in [1] when an fBm is replaced by a standard Brownian motion. The paper is organized as follows. In Section 2, we introduce some properties, notation, definitions, and preliminary results. Section 3 is devoted to the study of the variation of solution with respect to the initial data. In the last section, we drop the continuity assumption on the drift and try to obtain the same result as in Section 3.

2

Preliminaries

In this section, we give some properties of an fBm, definitions, and some tools used in the proofs. For any H < 1/2, let us define the square-integrable kernel # " 1 Z H− 2 3 1 1 1 −H t t H− H− − H− s2 (u − s) 2 u 2 du , t > s, KH (t, s) = cH s 2 s 2H ]1/2 , t > s. where cH = [ (1−2H)β(1−2H,H+ 1 2 )) Note that H− 21 3 1 ∂KH t (t − s)H− 2 . (t, s) = cH H − ∂t 2 s

Approximation of solutions of SDEs driven by fBm, under pathwise uniqueness

305

Let B H = {BtH , t ∈ [0, T ]} be an fBm defined on (Ω, F , {Ft }t∈[0,T ] , P ). We denote by ζ the set of step functions on [0, T ]. Let H be the Hilbert space defined as the closure of ζ with respect to the scalar product h1[0,t] , 1[0,s] iH = RH (t, s). The mapping 1[0,t] → BtH can be extended to an isometry between H and the Gaussian subspace of L2 (Ω) associated with B H , and such an isometry is denoted by ϕ → B H (ϕ). ∗ Now we introduce the linear operator KH from ζ to L2 ([0, T ]) defined by Z b ∂KH ∗ (t, s) dt. ϕ(t) − ϕ(s) KH ϕ (s) = KH (b, s)ϕ(s) + ∂t s

∗ The operator KH is an isometry between ζ and L2 ([0, T ]), which can be extended to the Hilbert space H. Define the process W = {Wt , t ∈ [0, T ]} by ∗ −1 Wt = B H KH 1[0,t] .

Then W is a Brownian motion; moreover, B H has the integral representation Z t BtH = KH (t, s) dW (s). 0

H+ 1

We need also to define an isomorphism KH from L2 ([0, T ]) onto I0+ 2 (L2 ) associated with the kernel KH (t, s) in terms of the fractional integrals as follows: 1

1

−H H− 1 2

2 −H I 2 (KH ϕ)(s) = I02H + s 0+

s

ϕ,

ϕ ∈ L2 [0, T ] .

Note that, for ϕ ∈ L2 ([0, T ]), I0α+ is the left fractional Riemann-Liouville integral operator of order α defined by Z x 1 (x − y)α−1 ϕ(y) dy, I0α+ ϕ(x) = Γ (α) 0 where Γ is the gamma function (see [3] for details). The inverse of KH is given by 1 1 1 −H −1 KH ϕ (s) = s 2 −H D02+ sH− 2 D02H + ϕ(s),

H+ 21

where for ϕ ∈ I0+ order α defined by

H+ 12

ϕ ∈ I0+

L2 ,

(L2 ), D0α+ is the left-sided Riemann Liouville derivative of

D0α+ ϕ(x)

d 1 = Γ (1 − α) dx

Z

0

x

ϕ(y) dy. (x − y)α

If ϕ is absolutely continuous (see [8]), then 1 1 −H 1 −1 KH ϕ (s) = sH− 2 I02+ s 2 −H ϕ′ (s).

(3)

306


Definition 2.1. On a given probability space (Ω, F , P ), a process X is called a strong solution to (1) if (1) X is {Ft }t∈[0,T ] adapted, where {Ft }t∈[0,T ] is the filtration generated by BtH , t ∈ [0, T ]; (2) X satisfies (1). Definition 2.2. A sextuple (Ω, F , {Ft }t∈[0,T ] , P, X, B H ) is called a weak solution to (1) if (1) (Ω, F , P ) is a probability space equipped with the filtration {Ft }t∈[0,T ] that satisfies the usual conditions; (2) X is an {Ft }t∈[0,T ] -adapted process, and B H is an {Ft }t∈[0,T ] -fBm; (3) X and B H satisfy (1). Definition 2.3 (Pathwise uniqueness). We say that pathwise uniqueness holds for e B H ) are two weak solutions of Eq. (1) defined Eq. (1) if whenever (X, B H ) and (X, e are indistinon the same probability space (Ω, F , (Ft )t∈[0,T ] , P ), then X and X guishable. The main tool used in the proofs is Skorokhod’s selection theorem given by the following lemma.

Lemma 2.4. ([4], p. 9) Let (S, ρ) be a complete separable metric space, and let P , Pn , n = 1, 2, . . ., be probability measures on (S, B(S)) such that Pn converges e Fe, Pe ), we can construct weakly to P as n → ∞. Then, on a probability space (Ω, S-valued random variables X, Xn , n = 1, 2, . . ., such that: (i) Pn = Pe Xn , n = 1, 2, . . ., and P = Pe X , where Pe Xn and PeX are respectively the laws of Xn and X; (ii) Xn converges to X Pe -a.s.

We will also make use of the following result, which gives a criterion for the tightness of sequences of laws associated with continuous processes. Lemma 2.5. ([4], p. 18) Let {Xtn , t ∈ [0, T ]}, n = 1, 2, . . ., be a sequence of ddimensional continuous processes satisfying the following two conditions: (i) There exist positive constants M and γ such that E[|X n (0)|γ ] ≤ M for every n = 1, 2, . . .; (ii) there exist positive constants α, β, Mk , k = 1, 2, . . ., such that, for every n ≥ 1 and all t, s ∈ [0, k], k = 1, 2, . . ., α E Xtn − Xsn ≤ Mk |t − s|1+β .

e Fe, Pe ), and d-dimenThen, there exist a subsequence (nk ), a probability space (Ω, nk e e e such that sional continuous processes X, X , k = 1, 2, . . ., defined on Ω e nk and X nk coincide; (1) The laws of X e nk converges to X et uniformly on every finite time interval Pe-a.s. (2) X t


3

307

Variation of solutions with respect to initial conditions

The purpose of this section is to ensure the continuous dependence of the solution with respect to the initial condition when the drift b is continuous and bounded. Note that, in the case of ordinary differential equation, the continuity of the coefficient is sufficient to ensure this dependence. Next, we give a theorem that will be essential in establishing the desired result. Theorem 3.1. Let b be a continuous bounded function. Then, under the pathwise uniqueness for SDE (1), we have h 2 i lim E sup Xt (x) − Xt (x0 ) = 0. x→x0

0≤t≤T

Before we proceed to the proof of Theorem 3.1, we state the following technical lemma.

Lemma 3.2. Let X n be the solution of (1) corresponding to the initial condition 1 , there exists a positive constant Cp such that, for all xn . Then, for every p > 2H s, t ∈ [0, T ], 2p E Xtn − Xsn ≤ Cp |t − s|2pH . Proof. Fix s < t in [0, T ]. We have

2p " Z # t n 2p 2p Xt − Xsn ≤ Cp b u, Xun du + BtH − BsH . s

Due to the stationarity of the increments and the scaling property of an fBm and the boundedness of b, we get that 2p E Xtn − Xsn ≤ Cp |t − s|2p + |t − s|2pH ≤ Cp |t − s|2pH ,

which finishes the proof. Let us now turn to the proof of Theorem 3.1. Proof. Suppose that the result of the theorem is false. Then there exist a constant δ > 0 and a sequence xn converging to x0 such that h 2 i inf E sup Xt (xn ) − Xt (x0 ) ≥ δ. n

0≤t≤T

Let X n (respectively, X) be the solution of (1) corresponding to the initial condition xn (respectively, x0 ). According to Lemma 3.2, the sequence (X n , X, B H ) satisfies conditions (i) and (ii) of Lemma 2.5. Then, by Skorokhod’s selection theorem there e Fe, Pe), and stochastic proexist a subsequence {nk , k ≥ 1}, a probability space (Ω, H H,k f f e Ye , B e ), (X k , Y k , B e ), k ≥ 1, defined on (Ω, e Fe, Pe) such that: cesses (X, fk , Yfk , B e H,k ) and (X nk , X, B H ) coincide; (α) for each k ≥ 1, the laws of (X

308


e k , Ye k , B e H,k ) converges to (X, e Ye , B e H ) uniformly on every finite time in(β) (X terval Pe -a.s.

Thanks to property (α), we have, for k ≥ 1 and t > 0, 2 Z t ek H,k k e = 0. e ds − B b s, X E Xt − xk − t s 0

e k satisfies the following SDE: In other words, X t Z t etk = xk + esk ds + B etH,k . b s, X X 0

Similarly,

Yetk = x0 +

Using (β), we deduce that

lim

k→∞

and lim

k→∞

Z

t

0

Z

t

0

Z

t 0

e H,k . b s, Yesk ds + B t

e k ds = b s, X s

b s, Yesk ds =

Z

t

0

Z

0

t

es ) ds b(s, X b(s, Yes ) ds

in probability and uniformly in t ∈ [0, T ]. e and Ye satisfy the same SDE on (Ω, e F, e Pe) with the same Thus, the processes X H e driving noise Bt and the initial condition x0 . Then, by pathwise uniqueness, we et = Yet for all t ∈ [0, T ], Pe-a.s. conclude that X On the other hand, by uniform integrability we have that h 2 i δ ≤ lim inf E max Xt (xn ) − Xt (x0 ) n 0≤t≤T h k i e e − Ye k 2 = lim inf E max X t t k 0≤t≤T i h e max |X et − Yet |2 , ≤ E 0≤t≤T

which is a contradiction. Then the desired result follows.

4

The case of discontinuous drift coefficient

In this section, we drop the continuity assumption on the drift coefficient and only assume that b is bounded. The goal of this section is to generate the same result as in Theorem 3.1 without the continuity assumption. Next, in order to use the fractional Girsanov theorem given in [8, Thm. 2], we should first check that the conditions imposed in the latter are satisfied in our context. This will be done in the following lemma.


309

Lemma 4.1. Suppose that XR is a solution of SDE (1), and let b be a bounded function. · −1 Then the process v = KH ( 0 b(r, Xr ) dr) enjoys the following properties: (1) vs ∈ L2 ([0, T ]), P -a.s.; RT (2) E[exp{ 12 0 |vs |2 ds}] < ∞.

Proof. (1) In light of (3), we can write 1 1 −H 1 |vs | = sH− 2 I02+ s 2 −H b(s, Xs ) Z s 1 1 1 H− 21 = s (s − r)− 2 −H r 2 −H b(r, Xr ) dr 1 Γ ( 2 − H) 0 Z s 1 1 1 H− 12 ≤ kbk∞ 1 (s − r)− 2 −H r 2 −H dr s Γ ( 2 − H) 0

Γ ( 23 − H) 1 −H s2 Γ (2 − 2H) Γ ( 23 − H) 1 −H , T2 ≤ kbk∞ Γ (2 − 2H) = kbk∞

where k · k∞ denotes the norm in L∞ ([0, T ]; L∞ (Rd )). As a result, we get that Z

T

|vs |2 ds < ∞,

P -a.s.

0

(2) The second item is obtained easily by the following estimate: " ( Z )# 1 T 1 2 E exp |vs | ds ≤ exp CH T 2(1−H) kbk2∞ , 2 0 2 where CH =

Γ ( 32 −H)2 Γ (2−2H)2 ,

which finishes the proof.

Next, we will establish the following Krylov-type inequality that will play an essential role in the sequel. Lemma 4.2. Suppose that X is a solution of SDE (1). Then, there exists β > 1 + dH such that, for any measurable nonnegative function g : [0, T ] × Rd 7→ Rd+ , we have E

Z

0

T

g(t, Xt ) dt ≤ M

Z

0

T

Z

β

g (t, x) dx dt Rd

!1/β

where M is a constant depending only on T , d, β, and H. Proof. Let W be a d-dimensional Brownian motion such that Z t H Bt = KH (t, s) dWs . 0

,

(4)

310


For the process v introduced in Lemma 4.1, let us define Pb by ( Z ) Z T 1 T 2 dPb = exp − vt dWt − vt dt := ZT−1 . dP 2 0 0 Then, in light of Lemma 4.1 together with the fractional Girsanov theorem [8, Thm. 2], we can conclude that Pb is a probability measure under which the process X − x is an fBm. Now, applying Hölder’s inequality, we have ( Z ) Z T T b ZT E g(t, Xt ) dt = E g(t, Xt ) dt 0

0

( Z α 1/α b b Z E ≤C E T

T

ρ

g (t, Xt ) dt

0

)1/ρ

,

(5)

where 1/α + 1/ρ = 1, and C is a positive constant depending only on T , α, and ρ. b α ] satisfies the following property: From [2, Lemma 4.3] we can see that E[Z T α b Z ≤ CH,d,T kbk∞ < ∞, (6) E T where CH,d,T is a continuous increasing function depending only on H, d, and T . On the other hand, applying again Hölder’s inequality with 1/γ + 1/γ ′ = 1 and γ > dH + 1, we obtain Z T Z TZ −d/2 2 2H b E g ρ (t, Xt ) dt = g ρ (t, y) 2πt2H exp−ky−xk /2t dy dt 0 0 Rd !1/γ ′ Z TZ ′ ′ 2 2H −dγ /2 ≤ 2πt2H exp−γ ky−xk /2t dy dt Rd

0

×

Z

T

Z

ργ

g (t, y) dy dt

Rd

0

!1/γ

.

(7)

A direct calculation gives Z −d/2 (1−γ ′ ) dH −dγ ′ /2 ′ 2 2H ′ t . exp−γ ky−xk /2t dy = (2π)d/2−dγ /2 γ ′ 2πt2H Rd

Plugging this into (7), we get b E

Z

T

Z

ρ

g (t, Xt ) dt ≤

0

T

d/2−dγ ′ /2

(2π)

0

×

Z

0

T

Z

ργ

g (t, y) dy dt

Rd

≤ (2π)d/2−dγ

γ

′ −d/2 (1−γ ′ ) dH

′

/2

γ

t

dt

!1/γ

′ ′ −d/2 1/γ

Z

0

T

t(1−γ

′

!1/γ ′

) dH

dt

!1/γ ′


×

Z

T

Z

g ργ (t, y) dy dt

Rd

0

′

≤ C γ , T, d, H

Z

0

T

Z

!1/γ

ργ

g (t, y) dy dt Rd

311

!1/γ

.

Finally, combining this with (5) and (6), we get estimate (4) with β = ργ. The proof is now complete. Now we are able to state the main result of this section. Theorem 4.3. If the pathwise uniqueness holds for Eq. (1), then without the continuity assumption on the drift coefficient, the conclusion of Theorem 3.1 remains valid. Proof. The proof is similar to that of Theorem 3.1. The only difficulty is to show that Z t Z t es ) ds e k ds = b(s, X lim b s, X s k→∞

0

0

in probability. In other words, for ǫ > 0, we will show that " Z # t k e e lim sup P b s, Xs − b(s, Xs ) ds > ǫ = 0. 0 k→∞

(8)

Let us first define

bδ (t, x) = δ −d φ(x/δ) ∗ b(t, x),

where ∗ denotes the convolution on RRd , and φ is an infinitely differentiable function with support in the unit ball such that φ(x) dx = 1. Applying Chebyshev’s inequality, we obtain " Z # t k es − b(s, X es ) ds > ǫ P b s, X 0 "Z # t 2 1 k e e ≤ 2E b s, Xs − b(s, Xs ) ds ǫ 0 # ( "Z t 2 4 k δ k e ds e − b s, X b s, X ≤ 2 E s s ǫ 0 "Z # t 2 es ) ds bδ s, X esk − bδ (s, X +E 0

+E

"Z

0

=

t

b (s, X es ) − b(s, X es ) 2 ds δ

#)

4 (J1 + J2 + J3 ). ǫ2

esk to X es uniformly on From the continuity of bδ in x and from the convergence of X e every finite time interval P a.s. it follows that J2 converges to 0 as k → ∞ for every δ > 0.

312


On the other hand, let θ : Rd → R+ be a smooth truncation function such that θ(z) = 1 in the unit ball and θ(z) = 0 for |z| > 1. By applying Lemma 4.2 we obtain Z t e k 2 ds e k − b s, X e k /R bδ s, X J1 = E θ X s s s 0 Z t e k 2 ds e k − b s, X e k /R bδ s, X 1−θ X +E s s s 0 Z t

δ

e k /R ds, 1−θ X (9) ≤ N b − b β,R + 2CE s 0

where N does not depend on δ and k, and k · kβ,R denotes the norm in Lβ ([0, T ] × B(0, R)). The last expression in the right-hand side of the last inequality satisfies the following estimate: Z t h i k esk /R ds ≤ sup P sup X es > R . 1−θ X (10) E k≥1

0

s≤t

e k |p ] < ∞ for all p > 1, and thus But we know that supk≥1 E[sups≤t |X s h i k es > R = 0. lim sup P sup X R→∞ k≥1

(11)

s≤t

Substituting estimate (10) into (9), letting δ → 0, and using (11), we deduce that the convergence of the term J1 follows. e it suffices to use the same arguments Finally, since estimate (10) also holds for X, as before to obtain the convergence of the term J3 , which completes the proof. Acknowledgements We thank the reviewer for his thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the paper. References [1] Bahlali, K., Mezerdi, B., Ouknine, Y.: Pathwise uniqueness and approximation of solutions of stochastic differential equations. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds.) Séminaire de Probabilités XXXII. Springer, Berlin (1998). MR1655150. doi:10.1007/ BFb0101757 [2] Banos, D., Nilssen, T., Proske, F.: Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE with singular drift. arXiv:1509.01154 (2015) [3] Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1998). MR1677455. doi:10.1023/A:1008634027843 [4] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. NorthHolland, Amsterdam (1981). MR1011252


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