ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT

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1 Elect. Comm. in Probab ELECTRONIC COMMUNICATIONS in PROBABILITY ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WIT MONOTONE DRIFT BRAIM BOUFOUSSI 1 Cadi Ayyad University FSSM, Department of Mathematics, P.B.O 239 Marrakesh, Morocco. boufoussi@ucam.ac.ma YOUSSEF OUKNINE 2 Cadi Ayyad University FSSM, Department of Mathematics P.B.O 239 Marrakesh, Morocco. ouknine@ucam.ac.ma Submitted 3 March 23, accepted in final form 31 July 23 AMS 2 Subject classification: 61, 6G18 Keywords: Fractional Brownian motion. Stochastic integrals. Girsanov transform. Abstract Let {Bt, t [, T ]} be a fractional Brownian motion with urst parameter > 1 2. We prove the existence of a weak solution for a stochastic differential equation of the form X t = x Bt b 1s, X s b 2 s, X s ds, where b 1 s, x is a ölder continuous function of order strictly larger than in x and than 1 2 in time and b 2 is a real bounded nondecreasing and left or right continuous function. 1 Introduction Let B = {B t, t [, T ]} be a fractional Brownian motion with urst parameter, 1. That is, B is a centered Gaussian process with covariance R t, s = EB t B s = 1 2 { t 2 s 2 t s 2}. If = 1 2 the process B is a standard Brownian motion. Consider the following stochastic differential equation X t = x B t b 1 s, X s b 2 s, X s ds, 1.1 where b 1, b 2 : [, T ] R R are Borel functions. The purpose of this paper is to prove, by approximation arguments, the existence of a weak solution to this equation if > 1 2, under the following weak regularity assumptions on the coefficients: 1 RESEARC SUPPORTED BY PSR PROGRAM 21, CADI AYYAD UNIVERSITY. 2 RESEARC SUPPORTED BY PSR PROGRAM 21, CADI AYYAD UNIVERSITY. 122

2 On a SDE driven by a fractional Brownian motion b 1 is ölder continuous of order 1 > α > in x and of order γ > sup s [,T ] x R sup b 2 s, x M <. in time: b 1 t, x b 1 s, y C x y α t s γ s [, T ], b 2 s,. is a nondecreasing and left or right continuous function. The same approximation arguments can be used to consider the case where b 2 satisfies the following assumptions: 2 sup s [,T ] x R sup b 2 s, x M1 x 3 for all s [, T ], b 2 s,. is a nonincreasing and continuous function If b 2 and = 1 2 the process B 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 with b 2, 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 case > 1/2, to establish existence and uniqueness result, a ölder type spacetime condition is imposed on the drift. Recently, Mishura and Nualart [9] gave an existence and uniqueness result for one discontinuous function namely the sgn function. Their approach relies on the Novikov criterion and it is valid for > > 1/2. Our aim is to establish existence and uniqueness result for general monotone function including sgn function and > 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. For f L 1 [a, b] and α > the left fractional Riemann-Liouville integral of f of order α on a, b is given at almost all x by I α a fx = 1 Γα x a x y α 1 fydy, where Γ denotes the Euler function. This integral extends the usual n-order iterated integrals of f for α = n N. We have the first composition formula

3 124 Electronic Communications in Probability Ia α Iβ a f = I αβ a f. The fractional derivative can be introduced as inverse operation. We assume < α < 1 and p > 1. We denote by Ia α L p the image of L p [a, b] by the operator I α a. If f I α a L p, the function φ such that f = Ia α φ is unique in L p and it agrees with the left-sided Riemann- Liouville derivative of f of order α defined by Da α fx = 1 d Γ1 α dx x a a fy x y α dy. The derivative of f has the following Weil representation: Da α fx = 1 fx x Γ1 α x a α α fx fy dy 1 x y α1 a,b x, 2.1 where the convergence of the integrals at the singularity x = y holds in L p -sense. When αp > 1 any function in Ia α L p is α 1 p - ölder continuous. On the other hand, any ölder continuous function of order β > α has fractional derivative of order α. That is, C β [a, b] Ia α L p for all p > 1. Recall that by construction for f Ia α L p, and for general f L 1 [a, b] we have I α a Dα a f = f Da α Iα a f = f. If f I αβ a L 1, α, β, α β 1 we have the second composition formula D α a Dβ a f = D αβ a f. 2.2 Fractional Brownian motion Let B = {Bt, t [, T ]} be a fractional Brownian motion with urst parameter < < 1 defined on the probability space Ω, F, P. For each t [, T ] we denote by Ft B the σ-field generated by the random variables Bs, s [, t] and the sets of probability zero. We denote by E the set of step functions on [, T ]. Let be the ilbert space defined as the closure of E with respect to the scalar product 1[,t], 1 [,s] = R t, s. The mapping 1 [,t] Bt can be extended to an isometry between and the Gaussian space 1 B associated with B. We will denote this isometry by ϕ B ϕ. The covariance kernel R t, s can be written as R t, s = s K t, rk s, rdr, where K is a square integrable kernel given by see [3]: K t, s = Γ t s 1 2 F 1 2, 1 2, 1 2, 1 t s,

4 On a SDE driven by a fractional Brownian motion 125 F a, b, c, z being the Gauss hypergeometric function. Consider the linear operator K from E to L 2 [, T ] defined by T Kϕs = K T, sϕs s For any pair of step functions ϕ and ψ in E we have see [1] K ϕ, K ψ L2 [,T ] = ϕ, ψ. ϕr ϕs K r, sdr. r As a consequence, the operator K provides an isometry between the ilbert spaces and L 2 [, T ]. ence, the process W = {W t, t [, T ]} defined by W t = B K 1 1 [,t] 2.2 is a Wiener process, and the process B has an integral representation of the form B t = K t, sdw s, 2.3 because K 1 [,t] s = K t, s 1 [,t] s. On the other hand, the operator K on L 2 [, T ] associated with the kernel K is an isomorphism from L 2 [, T ] onto I 1/2 L 2 [, T ] and it can be expressed in terms of fractional integrals as follows see [3]: K hs = I 2 s 1 2 I 1 2 s 1 2 h, if 1/2, 2.4 K hs = I 1 s 1 2 I 1 2 s 1 2 h, if 1/2, 2.5 where h L 2 [, T ]. We will make use of the following definition of F t -fractional Brownian motion. 2.1 Definition. Let {F t, t [, T ]} be a right-continuous increasing family of σ-fields on Ω, F, P such that F contains the sets of probability zero. A fractional Brownian motion B = {B t, t [, T ]} is called an F t -fractional Brownian motion if the process W defined in 2.2 is an F t -Wiener process. 3 Existence of strong solution for SDE with monotone drift. In this section we are interested by the special case b 1. We will prove by approximation arguments that there is a strong solution of equation 1.1. We will discuss two cases: 1 b 2 s,. satisfies 2 and 3. 2 b 2 s,. satisfies 2 and 3. 1 The first case: To treat the first situation, let us suppose that b 2 s,. is nondecreasing and left continuous function. We will use the following approximation lemma:

5 126 Electronic Communications in Probability 3.1 Lemma. Let b : [, T ] R R, a bounded measurable function such that for any s [, T ], bs,. is a nondecreasing and left continuous function. Then there exists a family of measurable functions {b n s, x ; n 1, s [, T ], x R} such that For any sequence x n increasing to x R, we have lim b n s, x n = b s, x, ds a.e. x b n s, x is nondecreasing, for all n 1, s [, T ] n b n s, x is nondecreasing, for all x R, s [, T ] b n s, x b n s, y 2n M x y for all n 1, s [, T ] sup sup sup n 1 s [,T ] x R b n s, x M. Proof. First assume that bs,. is left continuous and let us choose for any n 1 b n s, x = n x x 1 n bs, y dy. Since bs,. is nondecreasing then b n s,. is also a nondecreasing function for any fixed n 1. Let x, y R, we clearly have for any n 1, b n s, x b n s, y 2 n M x y. 3.1 Obviously, we get that b n is uniformly bounded by the constant M. Let n < m, s [, T ] and x R, we have b m s, x b n s, x = m n m n = m n x x 1 m x x 1 m x x 1 m x 1 m bs, y dy n bs, y dy x 1 n bs, y dy m n m bs, x 1 m, bs, y bs, x 1m dy. Now let x R and take an increasing sequence of real numbers x n converging to x. We want to show that for any s [, T ], lim b ns, x n = bs, x. It is enough to prove that there exists a subsequence b ϕn s, x ϕn which converges to bs, x. To do this, remark first that since bs,. is left continuous we have lim b ns, x = bs, x. Now let us consider any strictly increasing sequence x n converging to x such that x x n = o 1 n. We clearly get by 3.1 s [, T ], lim b ns, x n = bs, x. 3.2 We may choose a sequence ϕn n such that x n x ϕn. Since b n s, x n 1 is increasing and for any fixed n 1 the function b n s,. is nondecreasing, we have b n s, x n b ϕn s, x ϕn bs, x ϕn. 3.3

6 On a SDE driven by a fractional Brownian motion 127 We deduce by 3.2 and the left continuity of bs,., Which ends the proof. lim b ϕns, x ϕn = bs, x. Let B t 1 be a fractional Brownian motion with urst parameter 1 2, 1. We consider the following SDE X t = x B t b 2 s, X s ds, t T Theorem. Suppose that b 2 satisfies the assumptions 2 and 3. Then there exists a strong solution to the equation 3.4. Proof. Assume that b 2 : [, T ] R R is a measurable and bounded function which is nondecreasing and left continuous with respect to the space variable x. For n 1, let b n be as in lemma 3.1 and consider the following SDE X n t = x B t b n s, X n s ds, t T. 3.5 By standard Picard s iteration argument, one may show that for any n 1, the equation 3.5 has a strong solution which we denote by X n. Let n > m, we denote by t = X n t X m t. Using the monotony argument on b n, we have t 2 mm b m s, X n s b m s, X m s ds, b m s, X n s b m s, X m s I { s } ds, s I { s } ds 2 mm s ds. 3.6 We then get t 2 mm s ds. 3.7 By Gronwall s lemma, we have for almost all w and for any t [, T ], the sequence X n t w is a nondecreasing function of n which is bounded since b n is. Therefore it has a limit when n and we set lim Xn t ω = X t ω, which entails in particular that X is F B t adapted. Applying the convergence result in Lemma 3.1 and the boundedness of b n we get by Lebesgue s dominated convergence theorem, X t = x B t b 2 s, X s ds.

7 128 Electronic Communications in Probability 3.1 Remark. To show that Equation 3.4 has a weak solution, a continuity condition is imposed on the drift in [13]. ere, the function b 2 may have a countable set of discontinuity points. The solution constructed in Theorem 3.2 is the minimal one. 3.2 Remark. Let b 2 : [, T ] R R be a bounded measurable function, which is nondecreasing and right continuous. In this case we consider a decreasing sequence of Lipschitz continuous functions which approximate the drift. One may take b n s, x = n x 1 n x b 2 s, y dy. For any fixed s, x [, T ] R, the sequence b n s, x n 1 is nonincreasing and for any fixed n 1 and s [, T ] the function b n s,. is nondecreasing. The same arguments as in Lemma 3.1 can be used to prove that for any sequence x n n 1 decreasing to x, we have lim b ns, x n = b 2 s, x. This allows us to construct the maximal solution to the equation The second case: In this case we use the following lemma: 3.3 Lemma. Let b.,. : [, T ] R R be a continuous function with linear growth, that is there exists a constant M < such that s, x [, T ] R, bs, x M 1 x. Then the sequence of functions b n s, x = sup bs, y n x y, y Q is well defined for n M and it satisfies For any sequence x n converging to x R, we have lim b n s, x n = b s, x, n b n s, x is nonincreasing, for all x R, s [, T ] b n s, x b n s, y n x y for all n M, s [, T ], x, y R b n s, x M1 x, for all s, x [, T ] R, n M. For the proof of this lemma we refer for example to [8]. 3.4 Theorem. Assume that b 2 satisfies conditions 2 and 3. Then there exists a unique strong solution to the equation 3.4. Proof. For any n 1, let b n be as in Lemma 3.3. Since b n is Lipschitz and linear growth, the result in [13] assures the existence of a strong solution X n to the equation X n t = x B t b n s, X n s ds. Since b n n 1 is nonincreasing, comparison theorem entails that X n n 1 is a.s nonincreasing. By the linear growth condition on b n and Gronwall s lemma we may deduce that X n converges

8 On a SDE driven by a fractional Brownian motion 129 a.s to X, which is clearly a strong solution to the SDE 3.4. Moreover, if X 1 and X 2 are two solutions of 3.4, using the fact that b 2 s,. is nonincreasing, we get by applying Tanaka s formula to the continuous semi-martingale X 1 X 2, X 1 t X 2 t = Then we have the pathwise uniqueness of the solution. signx 1 s X 2 s b 2 s, X 1 s b 2 s, X 2 s ds. 4 Existence of a weak solution 4.1 Girsanov transform As in the previous section, { let B be a } fractional Brownian motion with urst parameter < < 1 and denote by Ft B, t [, T ] its natural filtration. Given an adapted process with integrable trajectories u = {u t, t [, T ]} and consider the transformation We can write B t = B t = B t = B t u s ds = K t, sd W s, u s ds. 4.1 K t, sdw s where W t = W t u s ds r dr. 4.2 Notice that u sds belongs a.s to L 2 [, T ] if and only if u sds I 1/2 L 2 [, T ]. 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 process u = {u t, t [, T ]} with integrable trajectories. Assume that: i u sds I 1/2 L 2 [, T ], almost surely. ii Eξ T = 1, where ξ T = exp T u s ds sdw s 1 T 2 u s ds u s ds 2 sds. Then the shifted process B is an F B t - fractional Brownian motion with urst parameter under the new probability P defined by d P dp = ξ T.

9 13 Electronic Communications in Probability Proof. By the standard Girsanov theorem applied to the adapted and square integrable process u sds we obtain that the process W defined in 4.2 is an Ft B - Brownian motion under the probability P. ence, the result follows. From 2.5 the inverse operator is given by h = s 1 2 D 1 2 s 1 2 h, if > 1/2 4.3 for all h I 1 2 L 2 [, T ]. Then if > 1 2 we need u I 1/2 L 2 [, T ], and a sufficient condition for i is the fact that the trajectories of u are ölder continuous of order 1 2 ε for some ε >. 4.2 Existence of a weak solution Consider the stochastic differential equation: X t = x B t b 1 s, X s b 2 s, X s ds, t T, 4.4 where b 1 and b 2 are Borel functions on [, T ] R satisfying the conditions 1 for b 1 and 2 and 3 resp. 2 and 3 for b 2. By a weak solution to equation 4.4 we mean a couple of adapted continuous processes B, X on a filtered probability space Ω, F, P, {F t, t [, T ]}, such that: i B is an F t -fractional Brownian motion in the sense of Definition 2.1. ii X and B satisfy Theorem. Suppose that b 1 and b 2 are Borel functions on [, T ] R satisfying the conditions 1 for b 1, 2 and 3 resp. 2 and 3 for b 2. Then Equation 4.4 has a weak solution. Proof. Let X 2 be the strong solution of 3.4 and set B t = Bt b 1s, Xs 2 ds. We claim that the process u s = b 1 s, Xs 2 satisfies conditions i and ii of Theorem 4.1. If this claim is true, B, X 2 under the probability measure P, B is an F B t is a weak solution of 4.4 on the filtered probability space Set v s = b 1 r, Xr 2 dr s. -fractional Brownian motion, and Ω, F, P {, Ft B, t [, T ] We will show that the process v satisfies conditions i and ii of Theorem 4.1. Along the proof c will denote a generic constant depending only on. Let > 1 2, by 4.3, the process v is clearly adapted and we have v s = s 1 2 D 1 2 s 1 2 b 1 s, X 2 s := c αs βs, }.

10 On a SDE driven by a fractional Brownian motion 131 where and Using the estimate and the equality we obtain αs = b 1 s, X 2 s s s 1 2 b1 s, X 2 s 1 2 s 1 2 βs = 1 2 s 1 2 s s s s 1 2 r 1 2 s r 1 2 dr b 1 s, X 2 s b 1 r, X 2 s s r 1 2 r 1 2 dr. b 1 r, X 2 s b 1 r, X 2 r s r 1 2 r 1 2 dr. b 1 s, Xs 2 b, x C s γ Xs 2 α s r 1 2 s 1 2 s r 1 2 dr = c s 1 2, αs c s 2 [ b 1 1, x C s γ ] X 2 s α Cs γ 1 2 c s 2 C 1 X 2 α Csγ b 1, x. As consequence, taking into account that α < 1, we have for any λ > 1 E exp λ T αs 2 ds <. 4.5 In order to estimate the term βs, we apply the ölder continuity condition 1.2 and we get s X βs c s s Xr 2 α r s γ r 1 s r 1 2 s r dr s c s B 1 s Br α 2 s r α 1 2 r s γ r 1 s r 1 2 s r dr c s 1 2 α ε G α, where we have fixed ε < 1 α 1 2 and we denote Bs B r G = sup s<r 1 s r ε. By Fernique s Theorem, taking into account that α < 1, for any λ > 1 we have E exp λ T βs 2 ds <, and we deduce condition ii of Theorem 4.1 by means of Novikov criterion.

11 132 Electronic Communications in Probability 4.3 Uniqueness in law and pathwise uniqueness In this subsection we will prove uniqueness in law of weak solution under the condition 1 for b 1, 2 and 3 for b 2. The main result is 4.3 Theorem. Suppose that b 1 and b 2 are Borel functions on [, T ] R. satisfying the conditions 1 for b 1, 2 and 3 for b 2. Then we have the uniqueness in distribution for the solution of Equation 4.4. Proof. It is clear that X 2 is pathwise unique, hence the uniqueness in law holds when b 1. Let X, B be a solution of the stochastic differential equation 4.4 defined in the filtered probability space Ω, F, P, {F t, t [, T ]}. Define u s = b 1 r, X r dr s. Let P defined by d P T dp = exp u s dw s 1 T u 2 2 sds. 4.6 We claim that the process u s satisfies conditions i and ii of Theorem 4.1. In fact, u s is an adapted process and taking into account that X t has the same regularity properties as the fbm we deduce that T u2 sds < almost surely. Finally, we can apply again Novikov theorem in order to show that E d P dp = 1, because by Gronwall s lemma X x B C 1 T e C2T, and X t X s B t B s C 3 t s 1 X for some constants C i, i = 1, 2, 3. By the classical Girsanov theorem the process W t = W t is an F t -Brownian motion under the probability P. In terms of the process W t we can write Set X t = x Then X satisfies the following SDE, K t, s d W s B s = X t = x B t u r dr K t, sd W s. b 2 s, X s ds b 2 s, X s ds,

12 On a SDE driven by a fractional Brownian motion 133 As a consequence, the processes X and X 2 have the same distribution under the probability P. In fact, if Ψ is a bounded measurable functional on C[, T ], we have E P ΨX = Ψξ dp ξd P Ω d P T = E P ΨX exp b 1 r, X r dr sdw s 1 T 2 b 1 r, X r dr sds 2 T = E P ΨX exp b 1 r, X r dr sd W s 1 T 2 b 1 r, X r dr sds 2 T = E P ΨX 2 exp b 1 r, Xr 2 dr sdw s 1 2 T = E P ΨX 2. b 1 r, X 2 r dr 2 sds In conclusion we have proved the uniqueness in law, which is equivalent to pathwise uniqueness see [13] Theorem Remark. In the case < 1/2, a deep study is made between stochastic differential equation with continuous coefficient and unit drift and anticipating ones cf [4]. Acknowledgement The work was partially carried out during a stay of Youssef Ouknine at the IPRA PAU, France Institut pluridisciplinaire de recherches appliquées. e would like to thank the IPRA for hospitality and support. References [1] E. Alòs, O. Mazet and D. Nualart: Stochastic calculus with respect to Gaussian processes, Annals of Probability [2] K. Bahlali: Flows of homeomorphisms of stochastic differential equations with mesurable drift, Stoch. Stoch. Reports, Vol [3] L. Decreusefond and A. S. Üstunel: Stochastic Analysis of the fractional Brownian Motion. Potential Analysis , [4] L. Denis, M. Erraoui and Y. Ouknine: Existence and uniqueness of one dimensional SDE driven by fractional noise. preprint 22. [5] X. M. Fernique: Regularité des trajectoires de fonctions aléatoires gaussiennes. In: École d Eté de Saint-Flour IV 1974, Lecture Notes in Mathematics 48, 2-95.

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