Backward Stochastic Differential Equations With Young Drift
Probability, Uncertainty and Quantitative Risk (2017) 2:5DOI 10.1186/s41546-017-0016-5Probability, Uncertaintyand Quantitative RiskRESEARCHOpen AccessBackward stochastic differential equations withYoung driftJoscha Diehl · Jianfeng ZhangReceived: 28 October 2016 / Accepted: 26 March 2017 / The Author(s). 2017 Open Access This article is distributed under the terms of the Creative CommonsAttribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit tothe original author(s) and the source, provide a link to the Creative Commons license, and indicate ifchanges were made.Abstract We show the well-posedness of backward stochastic differential equationscontaining an additional drift driven by a path of finite q-variation with q [1, 2).In contrast to previous work, we apply a direct fixpoint argument and do notrely on any type of flow decomposition. The resulting object is an effective toolto study semilinear rough partial differential equations via a Feynman–Kac typerepresentation.Keywords Rough paths theory · Young integration · BSDE · rough PDEIntroductionStochastic differential equations (SDEs) driven by Brownian motion W and an additional deterministic path η of low regularity (so called “mixed SDEs”) have beenwell-studied. In (Guerra and Nualart 2008), the well-posedness of such SDEs isestablished if η has finite q-variation with q [1, 2). 1 The integral with respectto the latter is handled via fractional calculus. Independently, in (Diehl 2012) thesame problem is studied using Young integration for the integral with respect to η.Interestingly, both approaches need to establish (unique) existence of solutions viathe Yamada–Watanabe theorem. A direct proof using a contraction argument is notobvious to implement.1 See Section “Appendix - Young integration” for background on the variation norm and Young integration.J. Diehl ( )Max–Planck Institute for Mathematics in the Sciences, Leipzig, Germanye-mail: diehl@mis.mpg.deJ. ZhangDepartment of Mathematics, University of Southern California, Los Angeles, California, USAe-mail: jianfenz@usc.edu
Page 2 of 17J. Diehl, J. ZhangFor paths of q-variation with q (2, 3), integration has to be dealt with viathe theory of rough paths. Motivated by a problem in stochastic filtering, (Dan andet al 2013) gives a formal meaning to the mixed SDE by using a flow decomposition which separates the stochastic integration from the deterministic rough pathintegration. It is not shown that the resulting object actually satisfies any integralequation.In (Diehl et al. 2015), well-posedness of the corresponding mixed SDE is established by first constructing a joint rough path “above” W and η. The deterministictheory of rough paths then allows mixed SDEs to be solved. The main difficulty inthat work is the proof of exponential integrability of the resulting process, which isneeded for applications. In (Diehl et al. 2014), these results have been used to studylinear “rough” partial differential equations via Feyman-Kac formulae.Backward stochastic differential equations (BSDEs) were introduced by Bismutin 1973. In (Bismut 1973), he applied linear BSDEs to stochastic optimal control.In 1990, Pardoux and Peng (Pardoux and Peng 1990) then considered non-linearequations. A solution to a BSDE with driver f and random variable ξ L 2 (FT ) isan adapted pair of processes (Y, Z ) in suitable spaces, satisfying T TYt ξ f (r, Yr , Z r )ds Z r d Wr , t T.ttUnder appropriate conditions on f and ξ , they showed the existence of a uniquesolution to such an equation. One important use for BSDEs is their application tosemilinear partial differential equations. This “nonlinear Feynman–Kac” formula is,for example, studied in (Pardoux and Peng 1992).In this work, we are interested in showing well-posedness of the followingequation T T TYt ξ f (r, Yr , Z r )dr g(Yr )dηr Zr d W r .(1)tttHere W is a multidimensional Brownian motion, η is a multidimensional (deterministic) path of finite q-variation, q [1, 2), and ξ is a bounded random variable,measurable at time T.Such equations have previously been studied in (Diehl and Friz 2012). In that workη is even allowed to be a rough path, i.e., every q 1 is feasible. The drawback ofthat approach is that no intrinsic meaning is given to the equation. Indeed a solutionto (1) is only defined as the limit of smooth approximations, which is shown to existusing a flow decomposition. In the current work we solve (1) directly via a fixpointargument. The resulting object solves the integral equation, where the integral withrespect to η is a pathwise Young integral. In Section “Main result”, we state and provethis main result.It is well-known that BSDEs provide a stochastic representation for solutions tosemi-linear parabolic partial differential equations (PDEs), in what is sometimescalled the “nonlinear Feynman–Kac formula” (Pardoux and Peng 1992). In Section“Application to rough PDEs”, we extend this representation to rough PDEs; therebygiving a novel and short proof for their well-posedness in the Young regime.In Section “Appendix - Young integration”, we recall the notions of p-variationand Young integration.
Probability, Uncertainty and Quantitative Risk (2017) 2:5Page 3 of 17Main resultLet ( , F , Ft , P) be a stochastic basis, where Ft is the usual filtration of a standard d-dimensional Brownian motion W. Denote by Et [·] : E[· Ft ] the conditionalexpectation at time t.We shall need the following spaces.Definition 1 For p 2, define B p to be the space of adapted process Y : [0, T ] R with2 1/2 Y p,2 : ess sup Et Y 2p var;[t,T ] ess sup YT .ωt,ωDenote by BMO the space of all progressively measurable Z : [0, T ] Rdwith T Z BMO : ess supEt Z r 2 dr .t,ωtTheorem 2 Let T 0, ξ L (FT ), q [1, 2) and η C 0,q var ([0, T ], Re ).Assume f : [0, T ] R Rd R, satisfies for some C f 0, P a.s.,sup f (t, 0, 0) C ft [0,T ] f (t, y, z) f (t, y , z ) C f y y z z .Let g1 , . . . , ge Cb2 (R). Let p 2 such that 1/ p 1/q 1.(i) There exists a unique Y B p , Z BMO such that T T Yt ξ f (r, Yr , Z r )dr g(Yr )dηr ttTZ r d Wr ,(2)twhere the dη integral is a well-defined (pathwise) Young integral.(ii) If, for i 1, 2, T T Tf i s, Ysi , Z si ds g(Ys )dηs Z si d Ws ,Yti ξi tttand ξ1 ξ2 , f 1 f 2 , then Y 1 Y 2 .(iii) The solution mappingL (FT ) C q var ([0, T ], Re ) B p BMO(ξ, η) (Y, Z ),is locally uniformly continuous.2 The space C 0,p var ([t, T ], R) and the norm · p var;[t,T ] are reviewed in Section “Appendix - Youngintegration”.
Page 4 of 17J. Diehl, J. Zhang(iv) Fixing f, g there exists for every M 0 a C(M) 0 such that for ξ, ξ FT with ξ , ξ , η q var;[0,T ] M we have for the correspondingsolutions (Y, Z ), (Y , Z ) 2 1/2Y0 Y0 C(M)E ξ ξ .Remark 1 The refined continuity statement in (iv) will be important for ourapplication to rough PDEs in Section “Application to rough PDEs”.Remark 2 Note that the coefficient g preceding the Young path is not allowed todepend on Z. This stems from the fact that we want this integral to be a well-definedYoung integral, and Z, in general, does not possess enough regularity for this (apriori, it is only known to be predictable and square integrable).In special cases, it turns out that the classical BSDE (without the Young integral) issolved with a Z that is a quite regular path in time, and one could hope for somethingsimilar for the “rough” BSDE.Since we do not want to impose such regularity constraints, which would eitherinvolve a Markovian setting with smooth coefficients or a study of Malliavindifferentiability, we do not pursue this direction.Remark 3 The use of the space of essentially bounded processes Y and BMOprocesses Z (Definition 1) is essential for our proof.In classical BSDE theory, these spaces usually only appear when studyingequations with a driver f that is quadratic in z and with a bounded terminal condition.The f we consider is Lipschitz, so our need for these spaces stems from theinterplay with the Young integral.Indeed, the map Y g(Y ) is only locally Lipschitz in p-variation norm(Lemma 2), which, in general, presents a problem when trying to close estimatesinvolving the expectation of the processes under consideration. Here the fact that wehave a bound on the essential supremum of Y comes to the rescue, as it allows us topull a term out of the expectation, see (7). This explains the norm for Y.In order to bound the p-variation norm of the stochastic integral, we apply theconditional version of the Burkholder–Davis–Gundy inequality for p-variation, see(5). This explains the use of the BMO norm for Z.Note that this in stark contrast to the theory of SDEs, where one, in general, doesnot have a handle on the essential supremum of solutions. Hence, as explained in theintroduction, for SDEs with a Young drift, a fixpoint procedure has not yet been established. On the other hand, it is not clear how to treat the BSDEs under considerationhere using classical L 2 -type theory with possibly unbounded terminal condition.Interestingly, the flow decomposition used in (Diehl and Friz 2012) leads to atransformed BSDE that is quadratic in Z. Hence, also there the terminal conditionneeds to be bounded.Proof For R 0 defineB(R) : (Y, Z ) : Y p,2 R, Z BMO R .
Probability, Uncertainty and Quantitative Risk (2017) 2:5Page 5 of 17For Y B p , Z BMO define (Y, Z ) : (Ỹ , Z̃ ), where T T Ỹt ξ f (r, Yr , Z r )dr g(Yr )dηr ttTZ̃ r d Wr .tThis is well-defined, as is standard in the BSDE literature (see, for example,(Pardoux and Peng 1990)), by setting T TỸt : Et ξ f (r, Yr , Z r )dr g(Yr )dηr ,ttand letting Z̃ be the integrand in the Itô representation of the martingale t tỸt f (r, Yr , Z r )dr g(Yr )dηr .00In what follows, A B means there exists a constant C 0 that is independentof η, ξ such that A C B. The constant is bounded for g C 2 , C f bounded.bUnique existence on small intervalWe first show that for a T 0 small enough, leaves a ball invariant, i.e., for a Tsmall enough and R large enough (B(R)) B(R).Let (Ỹ , Z̃ ) (Y, Z ), then · f (r, Yr , Z r )dr p var;[t,T ]t · f (r, Yr , Z r )dr t1 var;[t,T ] T f (r, Yr , Z r ) drt T t f (r, 0, 0) dr T Y ;[t,T ] T T Y p var;[t,T ] T YT TT Z r dr.tUsing the Young estimate (Theorem 5 in the Appendix), we estimate · · g(Yr )dηr g(Yr )dηr p var;[t,T ]t tq var;[t,T ] Z r drt(3)(4) 1 Y p var;[t,T ] η q var;[t,T ] .The Burkholder–Davis–Gundy inequality for p-variation (Friz and Victoir 2010,Theorem 14.12) gives 2 T · 2 Et Z̃ r d Wr Et Z̃ r dr .(5)tp var;[t,T ]t
Page 6 of 17J. Diehl, J. ZhangNow the dη integral satisfies the usual product rule, so with Itô’s formula we get Ỹt2 ξ 2 2 TTf (r, Yr , Z r )Ỹr dr 2t Tg(Yr )Ỹr dηr t T2Ỹr Z̃ s d Wr t Z̃ r 2 dr.tBy Lemma 2 (refer again to the Appendix) g(Y )Ỹ p var;[t,T ] g Ỹ p var;[t,T ] g(Y ) p var;[t,T ] Ỹ ;[t,T ] g Ỹ p var;[t,T ] Dg Y p var;[t,T ] Ỹ p var;[t,T ] YT Ỹ p var;[t,T ] Ỹ 2p var;[t,T ] R 2 .Taking the conditional expectation, we get Et Ỹt2 T Et 2 Z̃ s ds Et ξ Et T( f (r, 0, 0) Yr Z r ) Ỹr dr 1 Et Ỹ p var Ỹ 2p var R 2 .(6)2tt η q varNow EttT ( f (r, 0, 0) Yr Z r ) Ỹr dr T Et f (r, 0, 0) Yr Z r Ỹr dr2222t T f (r, 0, 0) 2 dr T Et Y 2 Et Z r 2 dr T Et Ỹ 2 tt 222 1 T Et Y p var YT R T Et Ỹ p var ỸT 2 1 T R 2 T Et ξ 2 R T Et Ỹ 2p var T Et ξ 2 . EtTWe trivially estimate 2 · f (r, Yr , Z r )dr tp var;[t,T ] 2 · Et g(Yr )dηr tp var;[t,T ] 2· , Et Z̃ r d Wr Et Ỹ 2p var;[t,T ] Ettp var;[t,T ]
Probability, Uncertainty and Quantitative Risk (2017) 2:5Page 7 of 17which we can bound, using (3), (4), and (5), by a constant times T 222222T T Et Y p var;[t,T ] T Et ξ (1 T ) E Z̃ r drt 1 Et Y 2p var;[t,T ] η 2q var;[t,T ] T 22 2222 T T R T Et ξ (1 T ) E Z̃ r drt 1 R η 2q var;[t,T ] .2Combining with (6), we get Et Ỹt2 Et 1 T T η q varT Z̃ s 2 dst Et [ξ 2 ] 1 T R 2 R T Et Ỹ 2p var 1 Et Ỹ p var Ỹ 2p var R R 22 1 T T 2 Et [ξ 2 ] 1 T R 2 R T T 2 T 2 R 2 T 2 Et ξ 2 (1 T ) ET Z̃ r 2 drt 1 R 2 η 2q var;[t,T ] η q var 1/221/2 1 T T R T Et ξ T E T 2 T 2 R 2 T 2 Et ξ 2 T EtUsing a T 1/2 Z̃ r dr2tT (1 R) η q var;[t,T ] Z̃ r 2 dr 1 R 2 η 2q var;[t,T ] R R 21 a 2and picking T 0 such that T T 2 1/2, we get T Et Z̃ s 2 ds c 1 F(T ) R R 2 ,twith F(T ) 0, as T 0 (here we use that η q var;[0,T ] 0 for T 0, see(Friz and Victoir 2010, Theorem 5.31)).Then 1/2 1/2Et Ỹ 2p var;[t,T ] T T R T Et ξ 2 T 1/2 1 F(T ) R R 2 (1 R) η q var;[t,T ] ,which can be made smaller than R/2 by first picking R large and then T small. Soindeed the ball stays invariant.We now show that for a T small enough, is a contraction on B(R). So let(Y, Z ), (Y , Z ) B(R) be given. Note that, since YT YT , we have for everyt [0, T ] Yt Yt (YT Yt ) (YT Yt ) Y Y p var;[t,T ] .
Page 8 of 17J. Diehl, J. ZhangHence, Yt Yt Et Yt Yt Et Y Y p var;[t,T ] 1/2 Et Y Y 2p var;[t,T ].So thatess sup Y (ω) Y (ω) Y Y p,2 .ωLet (Ỹ , Z̃ ) (Y, Z ), (Ỹ , Z̃ ) (Y , Z ). Using the Young estimate(Theorem 5) and Lemma 1 (in the Appendix below), we have for some constant c,that can change from line to line, T Ỹ Ỹ p var;[t,T ] cT Y Y p var;[t,T ] c Z r Z r drtwhere M Hence, Z̃ d W, M Y Y p var;[t,T ] η q var c 1 Y p var;[t,T ] Y Y η q var M M p var;[t,T ] ,Z̃ d W . 1/2 1/2 Et Ỹ Ỹ 2p var;[t,T ] cEt Y Y p var;[t,T ]T η q var 1/2 T Z r Z r 2 dr cT 1/2 Ett 1/2 c Et Y 2p var;[t,T ]sup Y (ω) Y (ω) η q var 1/2 cEt M M 2p var;[t,T ] T cT 1/2 Ett Z r Z r 2 drω 1/2 c T η q var Y Y p,2 c (1 R) η q var Y Y p,2 Et Tt( Z̃ s Z̃ s )2 dsSo, for a T small enough, 1 Ỹ Ỹ p,2 Y Y p,2 Z Z BMO Z̃ Z̃ BMO .4On the other hand, T2 Ỹt Ỹt 2( f (Ys , Z s ) f (Ys , Z s ))(Ys Ys ) dst 2Tt 2tT (g(Ys ) g(Ys ))(Ỹs Ỹs ) dηs (Ỹs Ys )(Z s Z s ) d Bs Tt Z̃ s Z̃ s 2 ds. 1/2.(7)
Probability, Uncertainty and Quantitative Risk (2017) 2:5Page 9 of 17Note that by Lemma 2 and then Lemma 1 g(Y ) g(Y ) Y Y p var;[t,T ] g(Y ) g(Y ) Y Y p var;[t,T ] g(Y ) g(Y ) p var;[t,T ] Y Y Y Y 2p var;[t,T ] (1 Y p var;[t,T ] ) Y Y 2p var;[t,T ] (1 R) Y Y 2p var;[t,T ] .Hence, the Young integral is bounded by a constant times η q var;[t,T ] (1 R) Y Y 2p var;[t,T ] .We estimate the Lebesgue integral as T T ( f(Ys , Z s ) f (Ys , Z s ))(Ys Ys ) ds Ys Ys Z s Z s Ys Ys dstt T1 T 1 Y Y λ Z s Z s 2ds.λtSo, after taking the conditional expectation, T 1/2 1/21 1/2 Et Z̃ s Z̃ s 2 ds T 1/2 1 Et Y Y 2p var;[t,T ]λt T 1/2 λEt Z r Z r 2 drt1/2 η q var (1 1/2R)1/2 Et Ỹ Ỹ 2p var;[t,T ].That is 1 1/2 Z̃ Z̃ BMO T 1/2 1 Y Y p,2 λ Z Z BMOλ η q (1 R)1/2 Y Y p,2Picking a small λ and then a small T, we get Z̃ Z̃ BMO 11 Y Y p,2 Z Z BMO44Define the modified norm Y, Z : Y p,2 2 Z BMO .Then, Ỹ Ỹ , Z̃ Z̃ 11 Y Y p,2 Z Z BMO Z̃ Z̃ BMO Y Y p,242137 Z Z BMO Y Y p,2 Z Z BMO2447 Y Y , Z Z .8
Page 10 of 17J. Diehl, J. ZhangWe therefore have a contraction and thereby existence of a unique solution onsmall enough time intervals.Continuity on small time intervalThis follows from virtually the same argument as the contraction mappingargument.Comparison on small time intervalLet C B 0 be given, and pick T T (C B ) so small that the BSDE is wellposed for any f, g with g C 2 , C f C B and any η C q var , ξ FT withb η q var;[0,T ] , ξ C B .Let ξ1 , ξ2 FT be given with ξ1 C B and η C q var with η q var;[0,T ] CB.Let ηn be a sequence of smooth paths approximating η in q-variation norm, withn η q var;[0,T ] C B for all n 1.Let Y1n (resp. Y2n ) be the classical BSDE solution with driving path ηn and data(ξ1 , f 1 , g) (resp. (ξ2 , f 2 , g)). Then, by a standard comparison theorem (for example,see (El Karoui et al. 1997)),Y1n Y2n .By continuity we know that Y1n Y1 p,2 Y2n Y2 p,2 0.In particular, almost surely, Y1n Y1 Y2n Y2 0.Hence Y1 Y2 .Unique existence on arbitrary time intervalWe show existence for arbitrary T 0. Denoteξ : ess supξ, ξ : ess infξ, f : ess sup f, f : ess inf f.ωωBy assumption T ξ ξ ωω f 2 f 2 (t, 0, 0)dt .0Consider the following Young ODEs: T Yt ξ f (s, Y s , 0) Yt ξ tTg(Y s )dηs ;tT f (s, Y s , 0) ttTg(Y s )dηs .Note that (Y , 0) and (Y , 0) solve the following BSDEs respectively: T T TYt ξ f (s, Y s , Z s ) g(Y s )dηs Z s d Ws ;t Yt ξ ttT f (s, Y s , Z s ) ttT g(Y s )dηs tTZ s d Ws .
Probability, Uncertainty and Quantitative Risk (2017) 2:5Page 11 of 17Choose δ such that the BSDE (2) is well-posed on a time interval of length δwhenever the terminal condition is bounded by Y Y .Let π : 0 t0 · · · tn T be a partition such that ti 1 ti δ for alli. First, by the preceding arguments, BSDE (2) on [tn 1 , tn ] with terminal conditionξ is well-posed and we denote the solution by (Y n , Z n ). By comparison we haveY tn 1 Ytnn 1 Y tn 1 . Hence, we can again start the BSDE from Ytn 1 at time tn 1and solve back to time tn 2 .Repeating the arguments backwards in time we obtain the existence of a (unique)solution on [0, T ].ContinuityUsing the previous step, we can use the continuity result on small intervals to getthe continuity of the solution map on arbitrary intervals, that is, point (iii) is proven.We finish by showing the continuity statement (iv). Since the dη-term is moredifficult than the dt-term, we will assume f 0 for ease of presentation.First note that since ξ , ξ M, the local uniform continuity of thesolution map in Theorem 2 we get Y n p,2 C0 (M).Let 1αr : 0 y g θ Yr (1 θ)Yr dθ.Note that α p var;[t,T ] C1 (M) Y p var;[t,T ] Y p var;[t,T ]So that Et α p var;[t,T ] C2 (M),for some constant C2 (M). LetY : Y Y . Then (almost surely) Y ;[t,T ] Y ;[t,T ] Y ;[t,T ] Y p,2 Y p,2 2C0 .Nowd Yt αt Yt dηt Z t d Wt .By Itô’s formula, together with the classical product rule for the dη-term, we get t t αr dηrαr dηrd expYt expZ t d Wt ,00so that if the latter is an honest martingale we get T αr dηr Y0 E expYT E exp 20Let us calculate the conditional moments of t : T 1/2αr dηrE[( YT )2 ]1/2 .0 Ttαr dηr .
Page 12 of 17FirstJ. Diehl, J. Zhang Et q var;[t,T ] cY oung η q var;[t,T ] Et α p var;[t,T cY oung η q;[t,T ] C2 .Further, by the product rule, T( t )m 1 (m 1)t rm αr dηr ,so that Et ( )m 1 q var;[t,T ] cY oung (m 1) η q var;[t,T ] Et m p var;[t,T ] α ;[t,T ] cY oung (m 1) η q var;[t,T ] m ;[t,T ] α p var;[t,T ] Et m p var;[t,T ] g sup Es m p var;[s,T ] Et [ α p var;[t,T ] ]s [t,T ] cY oung (m 1) η q var;[t,T ] Et m p var;[t,T ] g sup Es m p var;[s,T ] C2s [
to study semilinear rough partial differential equations via a Feynman–Kac type representation. Keywords Rough paths theory ·Young integration ·BSDE ·rough PDE Introduction Stochastic differential equations (SDEs) driven by Brownian motion W andanaddi-tional deterministic path η of low regularity (so called “mixed SDEs”) have been .
Introduction 1.1 Introduction to Backward Stochastic Differential Equa-tions What is Backward Stochastic Differential Equations? The most classical form of backward stochastic differential equation (BSDE) is Y t Z T t f(s;Y s;Z s)ds Z T t Z sdW s (1.1.1) where F FW, the terminal condition is a Rd-valued FW T-measurable random variable .
Lecture 21: Stochastic Differential Equations In this lecture, we study stochastic di erential equations. See Chapter 9 of [3] for a thorough treatment of the materials in this section. 1. Stochastic differential equations We would like to solve di erential equations of the form dX (t;X(t))dtX (t; (t))dB(t)
STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL EQUATIONS 5 In discrete stochastic processes, there are many random times similar to (2.3). They are non-anticipating, i.e., at any time n, we can determine whether the cri-terion for such a random time is met or not solely by the “history” up to time n.
processes 1.Basics of stochastic processes 2.Markov processes and generators 3.Martingale problems 4.Exisence of solutions and forward equations 5.Stochastic integrals for Poisson random measures 6.Weak and strong solutions of stochastic equations 7.Stochastic equations for Markov processes in Rd 8.Convergenc
(iii) introductory differential equations. Familiarity with the following topics is especially desirable: From basic differential equations: separable differential equations and separa-tion of variables; and solving linear, constant-coefficient differential equations using characteristic equations.
Basic 3 Forward outside and inside edges on a circle, forward crossovers, backward one-foot glide, backward snowplow stop, backward half swizzle pumps on a circle, moving forward to backward and backward to forward two-foot turn, beginning two-foot spin Basic 4 Basic forward outside and forward inside consecutive edges, backward edges on a circle,
1 Introduction Let (f , -, P; {Yt}t o) be a filtered probability space satisfying the usual conditions. Assume that a standard d-dimensional Brownian motion { W } _ o is defined on this space. Consider the following forward-backward stochastic differential equations: t t
GHAMI Asia HARRIS GHAVIMI HARTLEYClarita GIL Maria GIRMA Turufat GOMES Marcio GOMEZ Luis GOMEZ Jessica GOMEZ Marie GOTTARDI Giannino GORDON Natasha GREAVES Cynthia GREENWOOD Peter GRIFFIN Daniel HABIB Assema Kedir HABIB Fatuma Kedir HABIB Jemal Kedir HABIB Merema Kedir HABIB Mehammed Kedir HABIB Mojda HABIB Shemsu Kedir HADDADI Rkia HADGAY Ismal HAKIM Hamid HAKIM Mohamed HAMDAN Rkia HAMDAN .
