Predicting Football Match Outcomes with Machine Learning Approaches

MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.2023.k.211 Existence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with Stochastic Linear Growth Coefficients Mostapha Abdelouahab Saouli  Laboratory of Applied Mathematics, Department of Mathematics, Kasdi Merbah University, B. P. 511, Ouargla, 30000, Algeria [email protected], [email protected]  Abstract We deal with fractional mean field backward stochastic differential equations with hurst parameter H∈(1 2,1) when the coefficient fsatisfy a stochastic Lipschitz conditions, we prove the existence and uniqueness of solution and provide a comparison theorem. Via an approximation and comparison theorem, we show the existence of a minimal solution when the drift satisfies a stochastic growth condition. Keywords: Fractional Brownian Motion, Backward Stochastic Differential Equations, Stochastic linear Growth, Stochastic Lipschitz-continuous, Itˆo’s Fractional Formula, Comparison Theorem. Received: 01 October 2023 Accepted: 30 October 2023 Online: 07 November 2023 Published: 20 December 2023 1 Introduction Fractional Brownian motion (fBm) with Hurst parameter H∈(0,1) is a zero mean Gaussian process BH={BH t, t ≥0}with the covariance function EBH sBH t=1 2t2H+s2H− |t−s|2H. For H=1 2, the process BHis a classical Brownian motion. This process is a self-similar, i.e. BHat has the same law as BH at for any a > 0.In the case H > 1 2, the process BHexhibits long range dependence. These properties make this process a useful driving noise in models arising in finance, physics, telecommunication networks and other fields. However, since BHwith H > 1 2is not a semimartingale, we cannot use the classical theory of stochastic calculus to define the fractional stochastic integral. Essentially, two different types of integrals with respect to fBm have been defined and developed. The first one is the pathwise Riemann Stieltjes integral which exists if the integrand has a continuous paths of order α > 1−H(see Young, [15]). This integral has the properties of Stratonovich integral, which leads to difficulties in the applications. The second one, introduced in [5] is the divergence operator (Skorokhod integral), defined as the adjoint of the derivative operator in the framework of the Malliavin calculus. Since this stochastic integral satisfies the zero mean property and it can be expressed as the limit of Riemann sums defined using Wick products, it was later developed by many authors. Backward stochastic differential equations (BSDEs in short) driven by Brownian motion were introduced by Bismut [2] for the linear case. In 1990, the nonlinear backward stochastic differential equations were introduced by Pardoux and Peng [12]. Since then, these pioneer works are extensively used in many fields like mathematical finance [6], stochastic optimal control and stochastic games [8]. At the same time, for better applications, BSDE itself has been developed into many different branches. For example, Buckdahn et al.[3,4] introduced the so-called mean-field BSDEs (MF-BSDEs), owing to the fact that mathematical mean-field approaches have important applications in many domains, such as economics, physics and game theory. BSDE driven by fractional Brownian motion were introduced by Bender [1] for the linear case. The nonlinear BSDEs with respect to fBm were first studied by Hu [7], Hu and Peng [9], they obtained the existence and uniqueness of the solution but with some restrictive assumption. Then Maticiuc and Nie [11] improved their result and omitted this assumption. They also developed a theory of backward stochastic variational inequalities, i.e. they proved an existence and uniqueness of the solution of reflected BSDEs driven by fBm. In this paper we study the nonlinear dynamics systems governed by the mean field BSDEs driven by fBm with Hurst parameter H > 1 2. First, we establish existence and uniqueness of solutions of such an equation under stochastic Lipschitzian condition and establish a comparison theorem. Also by the method developed in [10], we prove that kind of equation has a minimal solution under continuous and stochastic linear growth conditions. The organization of our paper is as follows: The existence and uniqueness result for the solution of fractional mean-field backward SDE under stochastic Lipschitz condition and comparison theorem are given in section 2. Finally, section 3 is devoted to the existence of minimal solution for fractional mean-field BSDE under continuous and stochastic linear growth. 211 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX 2 Fractional Mean-Field Backward SDE In this section, we recall the existence and uniqueness result and the comparison theorem for fractional mean-field BSDEs under stochastic Lipschitz conditions. 2.1 Definitions and Notations For a fixed real t∈[0, T] and suppose that BH={BH t, t ≥0}is a one-dimensional fBm defined on (Ω,F,P) with Hurst parameter 1 2< H < 1 and F= (Ft)0≤t≤Tis the natural filtration generated by BH.Assume that •η0is a given constant. •b,σ: [0, T]→Rare continuous deterministic functions, σis differentiable and such that σ(t)= 0 ∀t∈[0, T], note that, since ||σ||2 t=H(2H−1) Rt 0Rt 0|u−v|2H−2σ(u)σ(v)dudv, we have d dt ||σ||2 t=σ(t) ˆσ(t)≥0, where ˆσ(t) = Rt 0ϕ(t−v)σ(v)dv. Let ¯ Ω,¯ F,¯ P= (Ω ×Ω,Ft⊗Ft,P⊗P) be the (non-completed) product of (Ω,F,P) with itself. We denote the filtration of this product space by ¯ F=¯ Ft=Ft⊗Ft,0≤t≤T. A random variable ξ∈L0(Ω,F,P;Rn) originally defined on Ω is extended canonically to ¯ Ω: ´ ξ(´ω, ω) = ξ(´ω),(´ω, ω)∈¯ Ω=Ω×Ω. For every θ∈L1¯ Ω,¯ F,¯ P, the variable θ(·, ω) : Ω → Rbelongs to L1¯ Ω,¯ F,¯ P,P(dω)−a.s, . We denote its expectation by E′(θ(·, ω)) = RΩθ(´ω, ω)P(d´ω). Notice that    E′(θ) = E′(θ(·, ω)) ∈L1(Ω,F,P), and E(θ) = R¯ Ωθd¯ P=RΩE′(θ(·, ω)) P(dω) = E(E′(θ)) . Let ηtbe a solution of the following SDE with respect to fractional Brownian motion dηt=b(t)dt +σ(t)dBH t, η0=x0.(1) Let f: Ω×[0, T]×R57−→ Rbe measurable functions. Now we consider the following mean field backward SDE with respect to fBm: (dYt=−E′f(t, ηt, Yt, Zt,´ Yt,´ Zt)dt +ZtdBH t, YT=ξ(2) Before giving the definition of solutions of BSDE (2), we introduce for a fixed δ > 0 the following sets: •L2(δ, FT,R) is the space of R-valued and FTmeasurable random variables such that EeδA(T)|ξ|2<∞. •C1,2 pol ([0, T]×R) is the space of all C1,2-functions over [0, T]×R, which together with their derivatives are of polynomial growth. •V[0,T ]=nY=ψ(·, η) ; ψ∈ C1,2 pol ([0, T]×R)o, dψ dt is bounded, t∈[0, T]. •˜ VH,a [0,T ]and ˜ VH [0,T ]denote the completion of V[0,T ] under the following norm, respectively: ||Y||a v △ = EZT 0 a2(t)t2H−1eδA(t)|Yt|2dt! 1 2 , ||Z||v △ = EZT 0 t2H−1eδA(t)|Zt|2dt! 1 2 , where δ > 0 is a constant. It is easy to see that ˜ VH [0,T ]⊂˜ VH,a [0,T ]⊂L2(0, T, R). Definition. A solution of equation (2) is a pair of processes (Y, Z) which belongs to the space ˜ V 1 2,a [0,T ]ט VH [0,T ] and satisfies (2) . The setting of our problem is to find a pair of processes (Y·, Z·)∈˜ V 1 2,a [0,T ]ט VH [0,T ]satisfying the BSDE (2). In the following, we will prove the existence and uniqueness of Eq. (2) . 2.2 Fractional Mean-Field BSDE with Stochastic Lipschitz Coefficients. Assume the coefficient f: Ω ×[0, T]×R5→R and the terminal value ξ: Ω →Rsatisfy the following assumptions, for δ > 0: We say that the coefficient fsatisfies assumptions (H1) if the following holds: (H1.1) There exist two non-negative processes {α(t)}0≤t≤Tand {β(t)}0≤t≤Tsuch that: 1. For any 0 ≤t≤T,α(t) and β(t) are Ftmeasurable. 2. For all 0 ≤t≤T x ∈R,(y, ´y)∈R2,˜y, ˜y′∈R2, (z, ´z)∈R2and ˜z, ˜z′∈R2,we have   f(t, ω, x, y, z, ˜y, ˜z)−f(t, ω, x, ´y, ´z, ˜y ′ ,˜z ′) ≤α(t)|y−´y|+˜y−˜y ′+β(t)|z−´z|+˜z−˜z ′. (H1.2) For all 0 ≤t≤T,a2(t) = α(t) + β2(t)>0, and A(t) = Rt 0a2(s)ds < ∞. (H1.3) The integrability condition holds: E ZT 0 eδA(t)|f(t, ω, x, 0,0,0,0)|2 a2(t)dt!<∞. (H1.4) ξ∈L2(δ, FT,R). To solve equation (2), we investigate first the case, where the generator does not depend on the unknown processes Yand Z. Namely, we consider the stochastic equation Yt=ξ+ZT t E′(f(s, ηs)) ds −ZT t ZsdBH s,0≤t≤T, (3) where f(t, ω, ηt)∈L2(δ, Ft,R) satisfies the following integrability condition: (H1.3)′ERT 0eδA(t)|f(t,ω,ηt)|2 a2(t)dt<∞. 212 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Saouli,gE xistence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with... Proposition. For any δ > 0, there exists a unique solution (Y, Z)∈˜ V 1 2,a [0,T ]ט VH [0,T ]to equation (3). Moreover, there exists a positive constant Cδ,M depending on δand Msuch that for any 0 ≤t≤T, EeδA(t)|Yt|2+EZT t eδA(s)a2(s)|Ys|2 +EZT t eδA(s)s2H−1|Zs|2ds ≤Cδ,M θ(ξ, t, T),(4) where θ(ξ, t, T) = EeδA(T)|ξ|2+2 δRT teδA(s)|f(s,ηs)|2 a2(s)ds. Proof. Note that by [9], the stochastic equation (3) admits a unique solution (Y, Z)∈˜ V 1 2,a [0,T ]ט VH [0,T ]. It remains to show that Y∈˜ V 1 2,a [0,T ]. Itˆo’s formula applied to equation (3) yields for 0≤t≤Tand δ > 0, eδA(t)|Yt|2+δZT t a2(s)eδA(s)|Ys|2ds =eδA(T)|ξ|2+ 2 ZT t eδA(s)YsE′(f(s, ηs)) ds −2ZT t eδA(s)YsZsdBH s−2ZT t eδA(s)DH sYsZsds. Taking expectation, we get E eδA(t)|Yt|2+δZT t a2(s)eδA(s)|Ys|2ds! =E eδA(T)|ξ|2+ 2 ZT t eδA(s)Ysf(s, ηs)ds! −2E ZT t eδA(s)DH sYsZsds!.(5) Then using the inequality 2ab ≤a2ϵ+b2 ϵ, we have 2Ysf(s, ηs)≤δ 2a2(s)|Ys|2+2 δ |f(s, ηs)|2 a2(s). It is known (see for example Hu and Peng, [9], Maticiuc and Nie, [11]) that DH tYt= (ˆσ(t)/σ (t)) Zt. Moreover by Remark 6 in Maticiuc and Nie [11], there exists M > 0 such that for all t∈[0, T], t2H−1/M ≤ˆσ(t)/σ (t)≤Mt2H−1. EeδA(t)|Yt|2+δ 2EZT t a2(s)eδA(s)|Ys|2ds +2 MEZT t eδA(s)s2H−1|Zs|2ds ≤θ(ξ, t, T).(6) Choosing δ, M > 2, we deduce from (6), EeδA(t)|Yt|2+EZT t a2(s)eδA(s)|Ys|2ds +EZT t eδA(s)s2H−1|Zs|2ds ≤Cδ,M θ(ξ, t, T). This implies in particular that (Y, Z)∈˜ V 1 2,a [0,T ]ט VH [0,T ] and (4) follows. Theorem. Assume that assumptions (H1) hold. Then, for δsufficiently large, the fractional MF-BSDE (2) has a unique solution (Y, Z)∈˜ V 1 2,a [0,T ]ט VH [0,T ]. Proof. Existence part: We consider the sequence (Yn, Zn)n≥0defined by (−dY n+1 t=E′f(t, ηt, Y n t, Zn t,´ Yn t,´ Zn t)dt −Zn+1 tdBH t Yn+1 T=ξ, 0≤t≤T. (7) Since for a fixed n∈N, the coefficient fof the fractional MF-BSDE (3.7) does not depend on the solution Yn+1, Zn+1it follows from the previous proposition that the sequence (Yn, Zn)n≥0is well defined in ˜ V 1 2,a [0,T ]ט VH [0,T ]. Our strategy consists of proving that (Yn, Zn)n≥0is a Cauchy sequence. To this, let n≥1 and define for a process π∈ {Y, Z}, ∆πn+1 =πn+1 −πnand ∆fn+1(s, ηs) = E′f(s, ηs, Y n+1 s, Zn+1 s,´ Yn+1 s,´ Zn+1 s) −E′f(s, ηs, Y n s, Zn s,´ Yn s,´ Zn s). It is readily seen that the pair Yn+1 t, Zn+1 tt∈[0,T ] solves the following fractional MF-BSDE: ∆Yn+1 t=ZT t ∆fn(s, ηs)ds−ZT t ∆Zn+1 sdBH s,0≤t≤T. (8) Itˆo’s formula, applied to eδA(t)∆Yn+1 t 2, yields for 0≤t≤Tand δ > 0, EeδA(t)∆Yn+1 t 2+δEZT t a2(s)eδA(s)∆Yn+1 s 2ds +2EZT t eδA(s)DH s∆Yn+1 s∆Zn+1 sds = 2EZT t eδA(s)∆Yn+1 s∆fn(s, ηs)ds. It is known (see for example Hu and Peng, [9], Maticiuc and Nie, [11]) that DH t∆Yn+1 t= (ˆσ(t)/σ (t)) ∆Zn+1 t. Moreover by Remark 6 in Maticiuc and Nie [11], there exists M > 0 such that for all t∈[0, T], t2H−1/M ≤ˆσ(t)/σ (t)≤Mt2H−1. EeδA(t)∆Yn+1 t 2+δEZT t a2(s)eδA(s)∆Yn+1 s 2ds +2 MEZT t s2H−1eδA(s)|∆Zn s|2ds (9) = 2EZT t eδA(s)∆Yn+1 s∆fn(s, ηs)ds. 213 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX By assumption (H1.1) (2), we deduce that 2EZT t eδA(s)∆Yn+1 s∆fn(s, ηs)ds ≤2EZT t eδA(s)∆Yn+1 s|∆fn(s, ηs)|ds, ≤2EZT t eδA(s)α(s)∆Yn+1 s|∆Yn s|ds +2EZT t eδA(s)α(s)∆Yn+1 sE(|∆Yn s|)ds (10) +2EZT t eδA(s)β(s)∆Yn+1 s|∆Zn s|ds +2EZT t eδA(s)β(s)∆Yn+1 sE(|∆Zn s|)ds. Therefore by choosing δ≥1, using H¨older’s inequality and Jensen’s inequality we get EeδA(t)∆Yn+1 t 2+δEZT t a2(s)eδA(s)∆Yn+1 s 2ds +2 MEZT t s2H−1eδA(s)|∆Zn s|2ds ≤4EZT t a2(s){eδA(s)E∆Yn+1 s 21 2 ×eδA(s)E(|∆Yn s|)21 2}ds (11) +4EZT tβ2(s)eδA(s)E∆Yn+1 s 21 2 ×eδA(s)E(|∆Zn s|)21 2ds. Denote x(s) = eδA(s)E∆Yn+1 s 2 1 2,Then from (3.10) x(t)2≤4ZT t a2(s)x(s)eδA(s)E(|∆Yn s|)2 1 2ds +4 ZT t β(s)x(s)eδA(s)E(|∆Zn s|)2 1 2ds. Using Lemma 20 in Maticiuc and Nie [11] above inequality, it follows that x(t)≤4ZT t a2(s)eδA(s)E(|∆Yn s|)2 1 2ds +4 ZT t β(s)eδA(s)E(|∆Zn s|)2 1 2ds. Therefore for t∈[tk, T] x(t)2≤32 ZT t a2(s)eδA(s)E(|∆Yn s|)2 1 2ds!2 +32 ZT t β(s)eδA(s)E(|∆Zn s|)2 1 2ds!2 . Now we compute ZT tk x(s)2ds ≤32 (T−tk)ZT tk a2(s)eδA(s)E(|∆Yn s|)21 2ds2 (12) +32 (T−tk)ZT tk β(s)eδA(s)E(|∆Zn s|)21 2ds2 , = : Γ. For the term Γ in (11) ZT tk a2(s)eδA(s)E(|∆Yn s|)21 2ds2 ≤ZT tk a2(s)ds ·ZT tk a2(s)eδA(s)E(|∆Yn s|)2ds,(13) and ZT tk β(s)eδA(s)E(|∆Zn s|)21 2ds2 =ZT tk β(s) √s2H−1 √s2H−1eδA(s)E(|∆Zn s|)21 2ds2 , ≤ZT tk a2(s) s2H−1ds ·ZT tk s2H−1eδA(s)E(|∆Zn s|)2ds. (14) Combining (12) and (13), it follows that ZT tk x(s)2ds ≤FZT tk a2(s)eδA(s)E(|∆Yn s|)2ds (15) +GZT tk s2H−1eδA(s)E(|∆Zn s|)2ds, where F= 32 (T−tk)RT tka2(s)ds < ∞ and G= 32 (T−tk)RT tk a2(s) s2H−1ds < ∞.And similarly ZT tk 1 s2H−1x(s)2ds ≤˜ FZT tk a2(s)eδA(s)E(|∆Yn s|)2ds (16) +˜ GZT tk s2H−1eδA(s)E(|∆Zn s|)2ds, where ˜ F= 32 T2−2H−t2−2H k 2−2HRT tka2(s)ds < ∞and ˜ G= 32 T2−2H−t2−2H k 2−2HRT tk a2(s) s2H−1ds < ∞. Now from (9) and (10) 214 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Saouli,gE xistence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with... EeδA(t)∆Yn+1 t 2 +EZT tδ2−2s2H−1−2 δs2H−1a2(s) ×eδA(s)∆Yn+1 s 2ds (17) +2 ME ZT t s2H−1eδA(s)∆Zn+1 s 2ds! ≤2δEZT t eδA(s) ×a2(s)|∆Yn s|2ds +s2H−1|∆Zn s|2ds. Choosing δ > 0 such that δ−2 δ−2 δs2H−1>1 for any t≤s≤Tand using the inequality (14), and note that M > 2, we have EZT tk eδA(s)a2(s)∆Yn+1 s 2+s2H−1∆Zn+1 s 2ds ≤δMF EZT tk a2(s)eδA(s)E(|∆Yn s|)2ds +δ(MG + 2) EZT tk s2H−1eδA(s)E(|∆Zn s|)2ds. Now choosing δ > 0 and taking klarge enough such that δMF ≤1 4and δ(MG + 2) ≤1 4,we deduce EZT tk eδA(s)a2(s)∆Yn+1 s 2ds +EZT tk eδA(s)s2H−1∆Zn+1 s 2ds ≤1 2EZT tk eδA(s)a2(s)E(|∆Yn s|)2ds +1 2EZT tk eδA(s)s2H−1E(|∆Zn s|)2ds. As a consequence, we deduce that (Yn, Zn)n≥0is a Cauchy sequence in ˜ V 1 2,a [0,T ]ט VH [0,T ]. Then there exists a pair (Y, Z)∈˜ V 1 2,a [0,T ]ט VH [0,T ]being a limit of (Yn, Zn)n≥1, i.e. EZT 0 a2(s)eδA(s)|Yn s−Ys|2ds +EZT 0 s2H−1eδA(s)|Zn s−Zs|2ds →0,as n→ ∞. It remains to show that the pair (Y, Z) satisfies equation (2) on the interval [0, T]. We have for any t∈[tk, T], Yn+1 t−ξ−ZT t f(s, ηs, Y n s, Zn s,´ Yn s,´ Zn s)ds → n→∞ Yt−ξ−ZT t f(s, ηs, Ys, Zs,´ Ys,´ Zs)ds, in L2(Ω,F,P). And Zn t1[t,T ]→Zt1[t,T ]in L2(Ω,F,H). Arguing as in the proof of Theorem 23 in Maticiuc and Nie [11] we show that (Y, Z) satisfies (2) on [tk, T]. The next step is to solve the equation on [tk−1, tk]. With the same arguments, repeating the above technique we obtain a uniqueness of the solution of MF-BSDE with respect to fBm on the whole interval [0, T]. Uniqueness part: Let Y1 ·, Z1 ·and Y2 ·, Z2 ·two solutions of fractional MF-BSDEs (2), then by Itˆo’s formula applied to eδA(t)Y1 t−Y2 t 2it follows that, ∀t∈[0, T], EeδA(t)Y1 t−Y2 t 2+δEZT t a2(s)eδA(s)Y1 s−Y2 s 2ds = 2EZT t eδA(s)Y1 s−Y2 s ×f(s, ηs, Y 1 s, Z1 s,´ Y1 s,´ Z1 s)−f(s, ηs, Y 2 s, Z2 s,´ Y2 s,´ Z2 s)ds −2EZT t eδA(s)DH sY1 s−Y2 sZ1 s−Z2 sds, then, we can write EeδA(t)Y1 t−Y2 t 2 +EZT tδ−4−2δM s2H−1a2(s)eδA(s)Y1 s−Y2 s 2ds +2δ−2 δM EZT t eδA(s)s2H−1Z1 s−Z2 s 2ds ≤0, which can be chosen δ, M such that δ−4−2δM s2H−1>0 for any t≤s≤Tand 2δ−2 δM >0.Thus, we deduce that EeδA(t)Y1 t−Y2 t 2 +EZT t eδA(s)a2(s)Y1 s−Y2 s 2+s2H−1Z1 s−Z2 s 2ds ≤0. This implies Y1 t=Y2 tand Z1 s=Z2 s. The result follows. 2.3 Comparison Theorem In this subsection we study a comparison theorem for the fractional MF-BSDEs of the following form: (−dY i t=E′fi(t, ηt, Y i t, Zi t,´ Yi t,´ Zi t)dt −Zi tdBH t Yi T=ξi,0≤t≤T. (18) where for any i∈ {1,2},fi: Ω ×[0, T]×R5→R. We assume in addition that (H1.5)   ξ1≤ξ2, f1(s, η, y, z, ´y, ´z)≤f2(s, η, y, z, ´y, ´z), ∀(s, η, y, z, ´y, ´z)∈[0, T]×R5. We have the following theorem: Theorem. Suppose that ξ1, f1and ξ2, f2satisfy (H1.1)−(H1.5). If Yi s, Zi s,i= 1,2are solutions to Eq. (18), then we have ∀t∈[0, T], Y 1≤Y2,P−a.s. 215 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Proof. Let us define ∆Yt=Y2 t−Y1 t, ∆Zt=Z2 t−Z1 t, ∆´ Yt=´ Y2 t−´ Y1 t, ∆ ´ Zt=´ Z2 t−´ Z1 t,∆ξ=ξ2−ξ1and ∆ft, ηt,∆Yt,∆Zt,∆´ Yt,∆´ Zt =E ′f2t, ηt, Y 2 t, Z2 t,´ Y2 t,´ Z2 t −E ′f1(t, ηt, Y 1 t, Z1 t,´ Y1 t,´ Z1 t). It follows that (∆Yt,∆Zt)t∈[0,T ]satisfies the fractional MF-BSDE for any 0 ≤t≤T ∆Yt = ∆ξ+ZT t ∆fs, ηs,∆Ys,∆Zs,∆´ Ys,∆´ Zsds −ZT t ∆ZsdBH s. Applying Itˆo’s formula to eδA(t)∆Y− t 2, we obtain EeδA(t)∆Y− t 2+δEZT t a2(s)eδA(s)∆Y− s 2ds +2 MEZT t 1{∆Ys<0}eδA(s)s2H−1|∆Zs|2ds =EeδA(T)∆ξ−−2EZT t 1{∆Ys<0}eδA(s)∆Y− s× ∆fs, ηs,∆Ys,∆Zs,∆´ Ys,∆´ Zsds. Since E′f2t, ηt, Y 2 t, Z2 t,´ Y2 t,´ Z2 t≥E′f1(t, ηt, Y 2 t, Z2 t,´ Y2 t,´ Z2 t) and ∆ξ=ξ1−ξ2≥0, we have EeδA(t)∆Y− t 2+δEZT t a2(s)eδA(s)∆Y− s 2ds +2 MEZT t 1{∆Ys<0}eδA(s)s2H−1|∆Zs|2ds ≤2EZT t 1{∆Ys<0}eδA(s)∆Y− s ×(E ′f1s, ηs, Y 2 s, Z2 s,´ Y2 s,´ Z2 s −E ′f1(s, ηs, Y 1 s, Z1 s,´ Y1 s,´ Z1 s))ds. From (H1.1) ,(H1.2) and Young’s inequality, we have ∆Y− sE ′f1s, ηs, Y 2 s, Z2 s,´ Y2 s,´ Z2 s −f1(s, ηs, Y 1 s, Z1 s,´ Y1 s,´ Z1 s) ≤1 24 + 4M s2H−1a2(s)∆Y− s 2+s2H−1 2M|∆Zs|2. Finally, it follows that EeδA(t)∆Y− t 2+RT tδ−4−4M s2H−1a2(s)eδA(s)∆Y− s 2ds +1 MERT t1{∆Ys<0}eδA(s)s2H−1|∆Zs|2ds≤0 Therefore, choosing δ > 0 and M > 0, such that δ−4−4M s2H−1≥0,we derive that ∆Y− t= 0 P−a.s. for all t∈[0, T], which implies that ∆Yt=Y2 t−Y1 t≥0 P−a.s. for all t∈[0, T]. 3 Fractional MF-BSDE with Continuous and Stochastic Linear Growth Coefficients. The objective of this section is to prove an existence theorem for MF-BSDEs (2) with Hurst parameter H > 1 2when the coefficient fis continuous with stochastic linear growth. More precisely, the coefficient f: Ω ×[0, T]×R4→Ris measurable and the terminal value ξ: Ω →Ris FT−measurable satisfying the following assumptions for δ > 0. (A1) The following hold: i) For fixed ωand t,f(t, ω, x, ·,·,·) is continuous. ii) For all (t, ω, x, y, z, ´y)∈[0, T]×Ω×R4 |f(t, ω, x, y, z, ´y)| ≤ φ(t)+r(t) (|x|+|y|+|´y|)+θ(t) (|z|). where φ, r and θare three nonnegative processes such that for a.e. t ∈[0, T ], φ(t), r (t) and θ(t) Ft−measurable. iii) For all (t, ω, x, y, z, ´y)∈[0, T]×Ω×R4, f(t, ω, x, y, z, ´y) is Ft−measurable. (A2) For any t∈[0, T], a2(t) = r(t) + θ2(t)>0, and A(t) = Zt 0 a2(s)ds < ∞. (A3) One has EeδA(T)|ξ|2+EZT 0 eδA(t)|φ(t)|2 a2(t)+a2(t)|ηt|2dt < ∞. To reach our objective, we first give the following useful approximation lemma, which generalizes the corresponding result of Lepeltier and San Martin [10]. Lemma. Let f: Ω ×[0, T]×R4→Rbe a measurable function such that: For a.s. every (t, ω)∈[0, T ]×Ω, f (t, ω, x, y, z, ´y) is a continuous. For every (t, ω, x, y, z, ´y)∈[0, T]×Ω×R4 |f(t, ω, x, y, z, ´y)| ≤φ(t) + r(t) (|x|+|y|+|´y|) + θ(t)|z|. where φ, r and θare three nonnegative processes such that for a.e. t ∈[0, T], φ(t), r (t) and θ(t) Ft−measurable. Then exists the sequence of fonction fn fn(t, ω, x, y, z, ´y) = inf (˜y,˜z,˜y′,˜z′)∈Q (ft, ω, x, ˜y, ˜z, ˜y′ +nr(t)|y−˜y|+´y−˜y′+θ(t)|z−˜z|), are well defined for n≥1 and satisfy the following conditions (i) For all n≥1,(t, ω, x, y, z, ´y)∈[0, T]×Ω×R5, |fn(t, ω, x, y, z, ´y)| ≤φ(t) + r(t) (|x|+|y|+|´y|) + θ(t)|z|. 216 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Saouli,gE xistence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with... (ii) For any (t, ω, x, y, z, ´y), fn(t, ω, x, y, z, ´y) is nondecreasing in n. (iii) For all (t, ω, x, y, z, ´y)∈[0, T]×Ω×R4, if (t, ω, x, yn, zn,´yn)→(t, ω, x, y, z, ´y),then fn(t, ω, x, yn, zn,´yn)→f(t, ω, x, y, z, ´y). (iv) For any n≥1,(t, ω)∈[0, T ]×Ω,for all (t, ω, x, y, z, ´y)∈[0, T]×Ω×R4and t, ω, x, ˜y, ˜z, ˜y′∈[0, T]×Ω×R4, we have fn(t, ω, x, y, z, ´y)−fnt, ω, x, ˜y, ˜z, ˜y′ ≤nr(t)|y−˜y|+´y−˜y′+θ(t)|z−˜z|. 3.1 Existence Result Now, by the approximation method of the function f(previous lemma) and the comparison theorem, we establish the following existence theorem. Theorem. Assume (A1)-(A3). Then, for δsufficiently large, the MF-BSDE (2) has a minimal solution Y −, Z −∈˜ V 1 2,a [0,T ]ט VH [0,T ]. Proof. We only prove that fractional MF-BSDE (2) has a minimal solution. Since (A1) holds, it follows from previous lemma that there exists a sequence of stochastic Lipschitz-continuous functions fnassociated with f, which is non-decreasing in n. Since g(t) = φ(t) + r(t) (|x|+|y|+|´y|) + θ(t)|z| is stochastic Lipschitz, the existence and uniqueness result in the previous section implies that there exists a unique solution (U, V )∈˜ V 1 2,a [0,T ]ט VH [0,T ]for fractional MF-BSDEs with data(ξ, g). Now, for any n≥1, let anand Anbe two random processes with positive values defined by a2 n(t) = nr (t) + n2θ2(t)>0, and An(t) = Zt 0 a2 n(s)ds < ∞. Then, in view of (A1) −(A2), an(t) and An(t) are Ft−measurable, for a.e. t∈[0, T] such that for any n≥1, 0 < a < anand A < An< n2A. Thus, it is clair to deduce that for any n≥1, ˜ V 1 2,an [0,T ]⊂˜ V 1 2,a [0,T ].Moreover, from (A3), (ξ, fn) satisfies the following conditions for any n≥1: EeδAn(T)|ξ|2≤Eeδn2A(T)|ξ|2 <∞, and EZT 0 eδAn(t)|fn(t, ω, x, 0,0,0)|2 a2 n(t)dt ≤ZT 0 eδn2A(t)|φ(t)|2 a2(t)dt < ∞. Therefore, we get again from the previous section that for every n≥1 there exists a unique solution (Yn, Zn)∈˜ V 1 2,an [0,T ]ט VH [0,T ]for fractional MF-BSDE with data: (−dY n t=E′fn(t, ηt, Y n t, Zn t,´ Yn t)dt −Zn tdBH t, Yn T=ξ, 0≤t≤T. (19) Consequently, for any n≥1, (Yn, Zn)∈˜ V 1 2,a [0,T ]ט VH [0,T ]. On the other hand, since for fixed (t, ω, x, y, z, ´y) and all n≥1, fn(t, ω, x, y, z, ´y)≤fn+1 (t, ω, x, y, z, ´y) ≤φ(t) + r(t) (|x|+|y|+|´y|) + θ(t)|z|, it follows from the comparison theorem that for every n≥1, Yn≤Yn+1 ≤U, dP⊗dt −a.s. (20) The idea of the proof is to establish that the limit of the sequence (Yn, Zn) is a solution of the fractional MF-BSDE (2). To this end, we will sketch the proof in four steps. Step 1: A priori estimates. There exists a constant C > 0 independent of n such that EeδA(t)|Yn t|2+EZT t eδA(s)a2(s)|Yn s|2ds +EZT t eδA(s)s2H−1|Zn s|2ds ≤C, (21) where Cis a positive constant which may be different from line to line. Indeed, for any δ > 0, Itˆo’s formula applied to eδA(t)|Yn t|2provides eδA(t)|Yn t|2 =eδA(T)|ξ|2+ 2 ZT t eδA(s)Yn s ×E′fn(s, ηs, Y n s, Zn s,´ Yn s)ds −2ZT t eδA(s)Yn sZn sdBH s−2ZT t eδA(s)DH sYn sZn sds −δZT t eδA(s)|Yn s|2ds. Taking expectation, we get EeδA(t)|Yn t|2+δEZT t a2(s)eδA(s)|Yn s|2ds +2EZT t eδA(s)DH sYn sZn sds =EeδA(T)|ξ|2+ 2EZT t eδA(s)Yn s ×E′fn(s, ηs, Y n s, Zn s,´ Yn s)ds. It is known (see example Hu and Peng, [9], Maticiuc and Nie, [11]) that DH tYn t= (ˆσ(t)/σ (t)) Zn t. Moreover by Remark 6 in Maticiuc and Nie [11], there exists M > 0 such that for all t∈[0, T], t2H−1/M ≤ ˆσ(t)/σ (t)≤Mt2H−1. 217 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Then, we have EeδA(t)|Yn t|2+δEZT t a2(s)eδA(s)|Yn s|2ds +2 MEZT t s2H−1eδA(s)|Zn s|2ds ≤EeδA(T)|ξ|2+ 2EZT t eδA(s)Yn s ×E′fn(s, ηs, Y n s, Zn s,´ Yn s)ds, (22) assumption (A2) and property (iv) from previous lemma together with Young’s inequality imply, 2Yn sE′fn(s, ηs, Y n s, Zn s,´ Yn s) ≤5 + M s2H−1a2(s)|Yn s|2 +s2H−1 M|Zn s|2+|φ(s)|2 a2(s). Therefore, for sufficiently large δ > 0, choosing δsuch that δ−5−M s2H−1>1, we obtain for M > 1 EZT t a2(s)eδA(s)|Yn s|2ds +EZT t s2H−1eδA(s)|Zn s|2ds ≤CE eδA(T)|ξ|2+ZT t eδA(s)|φ(s)|2 a2(s)ds! <∞. Finally, we get EeδA(t)|Yn t|2 +EZT 0 a2(s)eδA(s)|Yn s|2ds +EZT 0 s2H−1eδA(s)|Zn s|2ds ≤CE eδA(T)|ξ|2+ZT 0 eδA(s)|φ(s)|2 a2(s)ds! <∞. Step 2: Convergence result. From (20) and (21), there exists a process Ynsuch that Yn t↗Yta.s. for all t∈[0, T]. Therefore, it follows from Fatou’s lemma together with the dominated convergence theorem that (ERT 0eδA(s)|Ys|2ds ≤C, and limn→∞ RT 0a2(s)eδA(s)|Yn s−Ys|2ds = 0.(23) Next, for all n≥1, by Itˆo’s formula applied to eδA(t)Yn+1 t−Yn t 2, we get eδA(t)Yn+1 t−Yn t 2 = 2 ZT t eδA(s)Yn+1 s−Yn s ×(E′fn+1 s, ηs, Y n+1 s, Zn+1 s,´ Yn+1 s −E′fns, ηs, Y n s, Zn s,´ Yn s)ds −2ZT t eδA(s)Yn+1 s−Yn sZn+1 s−Zn sdBH s −2ZT t eδA(s)DH sYn+1 s−Yn sZn+1 s−Zn sds −δZT t eδA(s)Yn+1 s−Yn s 2ds. Letting t= 0, it follows from the uniform linear growth condition on the sequence fn, (property (ii) in previous lemma), Cauchy-Schawrtz inequality and assumption (A2) that EYn+1 0−Yn 0 2 +δEZT 0 eδA(s)a2(s)Yn+1 s−Yn s 2ds +2 MEZT 0 eδA(s)s2H−1Zn+1 s−Zn s 2ds ≤CEZT 0 eδA(s)(|φ(s)|2 a2(s) +a2(s)Yn+1 s 2+|Yn s|2+|ηs|2 +Zn+1 s 2+|Zn s|2ds)1 2(24) × EZT 0 eδA(s)Yn+1 s−Yn s 2ds! 1 2 . Therefore, from (20) and assumption (A3), we provide the existence of a constant ˜ C > 0 independent of n such that EZT 0 eδA(s)s2H−1Zn+1 s−Zn s 2ds ≤˜ C EZT 0 eδA(s)Yn+1 s−Yn s 2ds! 1 2 . Consequently, it follows from (23) that (Zn)n≥1is a Cauchy sequence in ˜ VH [0,T ]. Then there exists an Ft−jointly measurable process Z∈˜ VH [0,T ]such that lim n→∞ EZT 0 eδA(s)s2H−1|Zn s−Zs|2ds = 0. Step 3: (Y, Z) verifies MF-BSDE driven by fBm (2). Since (Yn, Zn)→(Y, Z) in ˜ V 1 2,a [0,T ]ט VH [0,T ], along a subsequence which we still denote (Yn, Zn), we get (Yn, Zn)→(Y, Z)dt ⊗dPa.e., 218 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Saouli,gE xistence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with... and there exists χ∈˜ VH [0,T ]such that for all n≥1, |Zn|< χ dt ⊗dPa.e. Therefore, by the previous lemmma, we have fnt, ηt, Y n t, Zn t,´ Yn t →ft, ηt, Yt, Zt,´ Ytdt ⊗dPa.e., Moreover, from condition (ii) in the previous lemma and (20), we have fnt, ηt, Y n t, Zn t,´ Yn t ≤Σ (t)<∞dt ⊗dPa.e., where Σ (t) = φ(t) + r(t)Y1 t+ ´ Y1 t+|Ut| +θ(t)|χt|. Then it follows from the dominated convergence theorem that EZT t fns, ηs, Y n s, Zn s,´ Yn sds → n→∞ EZT t fs, ηs, Ys, Zs,´ Ysds. Finally, passing to the limit on both sides of fractional MF-BSDE (19), we get that (Y, Z) is a solution of MFBSDE (2) . Step 4: Minimal solution. Let ˜ Y , ˜ Z∈˜ V 1 2,a [0,T ]ט VH [0,T ] be any solution of fractional MF-BSDE (2) and let us consider for any n≥1 the fractional MF-BSDE (19) with its unique solution (Yn, Zn), which converges to (Y, Z). Since fn≤ffor all n≥1, we get by virtue of the comparison theorem that Yn≤˜ Yfor all n≥1. Therefore, Y≤˜ Y. That proves that (Y, Z) is the minimal solution for fractional MF-BSDE (2). 4 Conclusion In the first part of this work, we have studied mean field backward stochastic differential equations driven by fBm with Hurst parameter H > 1 2under stochastic Lipschitz condition. In the second part of the paper we establish the existence of minimal solution to the mean field backward stochastic differential equations driven by fBm with Hurst parameter H > 1 2under continuous and stochastic linear growth condition. Motivated by the works of [3,4,9,10,14,13], we have proved an existence result to this kind of equations, in which the coefficient fis assumed to be continuous and stochastic linear growth condition, more precisely, we have treated the stochastic Lipschitz case. So our method in continuous and stochastic linear growth condition case is similar techniques developed in [10] with some suitable changes due to the difference between the processes and the spaces. 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