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# An explicit Euler method for McKean-Vlasov SDEs driven by fractional Brownian motion

In this paper, we establish the theory of chaos propagation and propose an Euler-Maruyama scheme for McKean-Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst exponent H ∈ (0,1). Meanwhile, upper bounds for errors in the Euler method is obtained. A numerical example is demonstrated to verify the theoretical results.

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05/26/2022

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## 1 Introduction

The pioneering work of McKean–Vlasov stochastic differential equations (SDEs) has been done by McKean in [21, 19, 20] connected with a mathematical foundation of the Boltzmann equation. Due to their widespread applications in many fields, McKean–Vlasov SDEs have been researched by many scholars. In [17], existence and uniqueness are proved for distribution-dependent SDEs with non-degenerate noise under integrability conditions on distribution-dependent coefficients. Some theories about McKean–Vlasov SDEs were investigated, including ergodicity [11], Harnack inequality [27], and the Bismut formula [12, 25]. And the integration by parts formulae on Wiener space for solutions of the SDEs with general McKean–Vlasov interaction was derived in [8]. In [6]

, Buckdahn et al. characterized the function on the coefficients of the stochastic differential equation under appropriate regularity conditions as the unique classical solution of a nonlocal partial differential equation of mean-field type. A complete probabilistic analysis of a large class of stochastic differential games with mean field interactions was provided in

[7].

It is well known that the explicit solutions to McKean–Vlasov SDEs are difficult to be shown. Hence, the numerical methods for McKean–Vlasov SDEs driven by standard Brownian motion are studied by many scholars [1, 2, 4, 5, 9]. Moreover, it should be noted that SDEs driven by fractional Brownian motion (fBm) have wider applications [3, 23, 24]. On the other hand, the numerical methods for SDEs driven by fBm have attracted increasing interest; see [15, 16, 18, 28, 30] for example. Galeati et al. [14] examined the distribution-dependent stochastic differential equations with erratic, potentially distributional drift, driven by an additive fBm of Hurst parameter , and they established strong well-posedness under a variety of assumptions on the drift. To our knowledge, the numerical method for McKean–Vlasov SDEs driven by fBm has not been discussed yet. As we know, propagation of chaos plays a key role to approximate the Mckean-Vlasov SDEs. This paper aims at establishing the theory about propagation of chaos and the strong convergence rate in sense of EM method for McKean–Vlasov SDEs driven by fBm under the globally Lipschitz condition.

In this paper, we consider the following -dimensional McKean–Vlasov SDEs driven by fBm of the form

 dXt=b(Xt,L(Xt))dt+σ(L(Xt))dBHt, (1.1)

where the coefficients . Here, the initial value with , and is a -dimensional fBm with Hurst parameter . As we know, the covariance of is

 RH(t,s)=E(BHtBHs)=12(t2H+s2H−|t−s|2H),∀t,s∈[0,T].

The fBm corresponds to a standard Brownian motion when . Further, the fBm is not a semi-martingale or a Markov process unless . Therefore, when working with the fBm , many of the powerful features are unavailable. We rewrite (1.1) to the system of noninteracting particles

 (1.2)

where denotes the law of the process at time . Compared with the standard SDEs, McKean–Vlasov SDEs provide an additional complexity, that is, it is required to approximate the law at each time step. Although there are other technologies, the most common one is the so-called interacting particle system

 Xi,Nt=Xi0+∫t0b(Xi,Ns,μX,Ns)ds+∫t0σ(μX,Ns)dBH,is, (1.3)

where the empirical measures is defined by

 μX,Nt(⋅):=1NN∑j=1δXj,Nt(⋅),

and denotes the Dirac measure at point .

The structure of this work is as follows. The mathematical preliminaries on the McKean–Vlasov SDEs driven by fBm are presented in Section 2. Section 3 gives the main theorem and its proof. Numerical simulations are provided in Section 4.

## 2 Mathematical Preliminaries

Throughout the article, we will always work on a finite time interval

and consider an underlying probability space

. More precisely, is the Banach space of continuous functions vanishing at 0 equipped with the supremum norm, is the Borel -algebra and is the unique probability measure on such that the canonical process is a -dimensional fBm with Hurst parameter . For any , let

 ∥V(x)∥Lp:=(E|V(x)|p)1/p.

We use and for the Euclidean norm and inner product, respectively, and let and . The notation for and

means the tensor product of

and . We will denote the set of all probability measures on by

 Pp(Rd):={μ∈P(Rd):∫Rd|x|pμ(dx)<∞}

For and any , the -Wasserstein distance is defined by,

 Wθ(μ,ν):=(infπ∈Π(μ,ν)∫Rd×Rd|x−y|θπ(dx,dy))1/θ

where is the set of couplings of and , and is a Polish space under the -Wasserstein metric.

Let with . For and , the left-sided fractional Riemann-Liouville integral of of order on is defined as

 Iαa+g(x)=1Γ(α)∫xa(x−y)α−1g(y)dy,

and the right-sided fractional Riemann-Liouville integral of of order on is defined as

 Iαb−g(x)=(−1)−αΓ(α)∫bx(y−x)α−1g(y)dy,

where and denotes the Gamma function. For further details about fractional integral and derivative, we refer the reader to [3, 26].

###### Assumption 2.1

There exists a positive constant such that

 |b(x,μ)−b(y,ν)|≤L(|x−y|+Wθ(μ,ν)), (2.1)

for all and .

 |σ(μ)−σ(ν)|≤LWθ(μ,ν), (2.2)

for all . And for initial experience distribution ,

 |b(0,δ0)|∨|σ(δ0))|≤L. (2.3)

Furthermore, from Assumption 2.1, there exists a positive constant such that

 |b(x,μ)|≤L(1+|x|+Wθ(μ,δ0)), (2.4)

and

 |σ(μ)|≤L(1+Wθ(μ,δ0)). (2.5)

## 3 Main Result

###### Lemma 3.1

[12] When and hold, the solution of in (1.1) exists and is unique. Moreover, when and is independent of distribution, the solution of in (1.1) exists and is unique.

###### Lemma 3.2

For two empirical measures and . Then,

 E(Wθθ(μNt,νNt))≤E(1NN∑j=1∣∣Z1,j,Nt−Z2,j,Nt∣∣θ).

This lemma follows from constructing a simple transport plan .

### 3.1 Case H>1/2

###### Theorem 3.3

Let Assumption 2.1 holds and , then

 supt∈[0,T]E(|Xit|q)+supt∈[0,T]E(|Xi,Nt|q)≤Cq,H,T,L(1+E|Xi0|q),

where is a positive constant dependent on .

Proof. From (1.3) and elementary inequality, we get

 E(|Xi,Nt|q)≤3q−1E(|Xi0|q)+3q−1E(∣∣∣∫t0b(Xi,Ns,μX,Ns)ds∣∣∣q)+3q−1E(∣∣∣∫t0σ(μX,Ns)dBH,is∣∣∣q). (3.1)

For the second part at the right side of (3.1), using the Hölder inequality, we obtain

 E(∣∣∣∫t0b(Xi,Ns,μX,Ns)ds∣∣∣q)≤Tq−1E(∫t0∣∣b(Xi,Ns,μX,Ns)∣∣qds).

Applying (2.4), we see that

 E(∣∣∣∫t0b(Xi,Ns,μX,Ns)ds∣∣∣q)≤Tq−1E(∫t0Lq(1+|Xi,Ns|+Wθ(μX,Ns,δ0))qds)≤CT,q,L+CT,q,L∫t0E(|Xi,Ns|q)ds+CT,q,L∫t0E(Wqθ(μX,Ns,δ0))ds. (3.2)

For the last part of the right end of (3.1), apply Theorem 1.1 in [22], we get

 E(∣∣∣∫t0σ(μX,Ns)dBH,is∣∣∣q)≤Cq,H(∫t0∣∣σ(μX,Ns)∣∣1/H ds)qH. (3.3)

Using the Hölder inequality,

 E(∣∣∣∫t0σ(μX,Ns)dBH,is∣∣∣q)≤Cq,H,T∫t0∣∣σ(μX,Ns)∣∣q ds.

By (2.5), we have

 E(∣∣∣∫t0σ(μX,Ns)dBH,is∣∣∣q)≤Cq,H,T∫t0LqE(1+Wθ(μX,Ns,δ0))q ds≤Cq,H,T,L+Cq,H,T,L∫t0E(Wqθ(μX,Ns,δ0)) ds. (3.4)

Through a similar proof in [10, Lemma 2.3], for one can observe that

 Wθθ(μX,Ns,δ0)=1NN∑j=1∣∣Xj,Ns∣∣θ. (3.5)

Combine (3.2) and (3.4) into (3.1) and (3.5), we get

 E(|Xi,Nt|q)≤CqE(|Xi0|q)+CT,q,L+CT,q,L∫t0E(|Xi,Ns|q) ds+CT,q,L∫t0E(Wqθ(μX,Ns,δ0)) ds+Cq,H,T,L+Cq,H,T,L∫t0E(Wqθ(μX,Ns,δ0)) ds≤Cq,H,T,L(1+E|Xi0|q)+CT,q,L∫t0E(|Xi,Ns|q) ds+CT,q,LE⎛⎝(1NN∑j=1∣∣Xj,Ns∣∣θ)q/θ⎞⎠.

For the last term, we apply the Minkowski inequality and since all are identically distributed, we have

 E⎛⎝(1NN∑j=1∣∣Xj,Ns∣∣θ)q/θ⎞⎠≤(1NN∑j=1∥∥|Xj,Ns|θ∥∥Lq/θ)q/θ=(1NN∑j=1(E|Xj,Ns|q)θ/q)q/θ=E(|Xi,Ns|q).

Thus, we get

 E(|Xi,Nt|q)≤Cq,H,T,L(1+E|Xi0|q)+Cq,H,T,L∫t0E(|Xi,Ns|q) ds.

Then applying the Grönwall inequality, we obtain

 E(|Xi,Nt|q)≤Cq,H,T,L(1+E|Xi0|q).

Similarly, we can show

 E(|Xit|q)≤Cq,H,T,L(1+E|Xi0|q).

The proof is complete.

###### Theorem 3.4 (Propagation of Chaos)

Let Assumption 2.1 be satisfied. If for some , then it holds that

where the constant depends on , , , and but does not depend on N.

Proof. It follows from (1.2) and (1.3) that

 Xit−Xi,Nt=∫t0(b(Xis,LXis)−b(Xi,Ns,μX,Ns)) ds+∫t0(σ(LXis)−σ(μX,Ns)) dBH,is.

Using the elementary inequality, we can show that

 E(∣∣Xit−Xi,Nt∣∣p)≤2p−1E(∣∣∣∫t0b(Xis,LXis)−b(Xi,Ns,μX,Ns) ds∣∣∣p)+2p−1E(∣∣∣∫t0σ(LXis)−σ(μX,Ns) dBH,is∣∣∣p). (3.6)

For the first part on the right of (3.6), use the Hölder inequality and Assumption 2.1, we get

 (3.7)

For the second part on the right of (3.6), through the same technique as (3.3), we get

 E(∣∣∣∫t0σ(LXis)−σ(μX,Ns) dBH,is∣∣∣p)≤Cp,T,HE((∫t0|σ(LXis)−σ(μX,Ns)|1/H ds)pH).

Apply Assumption 2.1, we obtain

 E(∣∣∣∫t0σ(LXis)−σ(μX,Ns) dBH,is∣∣∣p)≤Cp,T,H,L∫t0E(Wpθ(LXis,μX,Ns)) ds. (3.8)

Combining this with (3.7), then

For the part of Wasserstein distance, we note and we obtain

By Lemma 3.2, we see

 E(Wpθ(LXis,μX,Ns))≤Cp,θE⎛⎝(1NN∑j=1∣∣Xjt−Xj,Nt∣∣θ)p/θ⎞⎠+Cp,θE(Wpθ(LXis,μXs)).

Thought the fact that for , we have

where we use the Minkowski inequality in the last inequality. Then through simple sorting, we have

What’s particularly interesting is that

is controlled by the Wasserstein distance estimate in

[13, Theorem 1]. Therefore,

where

 Mq(LXis)=∫Rd∣∣Xis∣∣qLXis( dXis).

By Theorem 3.3, we note that . Thus,

Then, applying the Grönwall inequality completes the proof.

#### 3.1.1 EM Method for Interacting Particle System

Now, we define a uniform mesh with , where for . The numerical solutions are then generated by the EM method

 Yi,Ntk+1=Yi,Ntk+b(Yi,Ntk,μY,Ntk)Δ+σ(μY,Ntk)ΔBH,itk, (3.9)

where the empirical measures We show two versions of extension of the numerical solution at the discrete time points to . The first is the piecewise constant extension given by

 ¯Yi,Nt=Yi,Ntk,tk≤t

and the second is the continuous extension of the EM method defined by

 Yi,Nt=¯Yi,Ntk+∫ttkb(¯Yi,Ns,¯μY,Ns) ds+∫ttkσ(¯μY,Ns) dBH,is. (3.11)

Here, From (3.11), for all , we have

 Yi,Nt=Xi0+∫t0b(¯Yi,Ns,¯μY,Ns) ds+∫t0σ(¯μY,Ns) dBH,is. (3.12)
###### Theorem 3.5

Let Assumption 2.1 holds. For some , then

 supt∈[0,T]E(|Yi,Nt|q)≤Cq,H,T,L(1+E|Xi0|q),

where is a positive constant dependent on but independent of .

Proof. Similar to the proof of Theorem 3.3, we can show

 E(|Yi,Nt|q)≤3q−1E(|Xi0|q)+3q−1E(∣∣∣∫t0b(¯Yi,Ns,¯μY,Ns)ds∣∣∣q)+3q−1E(∣∣∣∫t0σ(¯μY,Ns)dBH,is∣∣∣q)≤Cq,H,T,L(1+E|Xi0|q)+Cq,H,T,L∫t0E(|¯Yi,Ns|q) ds≤Cq,H,T,L(1+E|Xi0|q)+Cq,H,T,L∫t0sup0≤τ≤sE(|Yi,Ns|q) ds.

Therefore, for , we have

 sup0≤τ≤tE(|Yi,Nτ|q)≤Cq,H,T,L(1+E|Xi0|q)+Cq,H,T,L∫t0sup0≤τ≤sE(|Yi,Ns|q) ds.

By the Grönwall inequality, we see

 sup0≤τ≤tE(|Yi,Nτ|q)≤Cq,H,T,L(1+E|Xi0|q),∀t∈[0,T].

Therefore, the assertion holds.

###### Lemma 3.6

Assume (2.4) and (2.5) hold, for a constant , then

 E(∣∣Yi,Nt−¯Yi,Nt∣∣p)≤Cκ,H,p,LΔpH, (3.13)

where is a positive constant dependent on but independent of .

Proof. From (3.11), we separate the left hand side of (3.13) into two parts

 E(∣∣Yi,Nt−¯Yi,Nt∣∣p)=E(∣∣∣∫ttkb(¯Yi,Ns,¯μY,Ns) ds+∫ttkσ(¯μY,Ns) dBH,is∣∣∣p)≤2p−1E(∣∣∣∫ttkb(¯Yi,Ns,¯μY,Ns) ds∣∣∣p)+2p−1E(∣∣∣∫ttkσ(¯μY,Ns) dBH,is∣∣∣p). (3.14)

Let us first consider the first part on the right of (3.14), by using the Hölder inequality and (2.4), we obtain

 E(∣∣∣∫ttkb(¯Yi,Ns,¯μY,Ns) ds∣∣∣p)≤Δp−1E(∫ttk∣∣b(¯Yi,Ns,¯μY,Ns)∣∣p ds)≤Δp−1E(∫ttkLp(1+∣∣¯Yi,Ns∣∣+Wθ(¯μY,Ns,δ0))p ds)≤LpΔp+Cp,LΔp−1∫ttkE(∣∣¯Yi,Ns∣∣p) ds+Cp,LΔp−1∫ttkE(Wpθ(¯μY,Ns,δ0)) ds. (3.15)

For the second part on the right of (3.14), by choosing , we note the fact that . Then

 E(∣∣∣∫ttkσ(¯μY,Ns) dBH,is∣∣∣p)=C−pκE(∣∣∣∫ttk(∫ts(t−r)−κ(r−s)κ−1 dr)σ(¯μY,Ns) dBH,is∣∣∣p).

Thanks to Stochastic Fubini Theorem for the Wiener Integrals with regard to fBm [23, Theorem 1.13.1], we obtain

 ∫ttk(∫ts(t−r)−κ(r−s)κ−1 dr)σ(¯μY,Ns) dBH,is=∫ttk(t−r)−κ(∫r0(r−s)κ−1σ(¯μY,Ns) dBH,is) dr.

Therefore, by the Hölder inequality, we get

 E(∣∣∣∫ttkσ(¯μY,Ns) dBH,is∣∣∣p)≤C−pκE⎛⎝(∫ttk(t−r)−pκ/(p−1) dr)p−1∫ttk∣∣∣∫r0(r−s)κ−1σ(¯μY,Ns) dBH,is∣∣∣p dr⎞⎠≤C−pκ(p−1)(p−1−pκ)p−1Δp−1−pκE(∫ttk∣∣∣∫r0(r−s)κ−1σ(¯μY,Ns) dBH,is∣∣∣p dr).

Applying the Theorem 1.1 in [22], we see

Thus, we obtain

By [12, Lemma 3.2] with , and , we have

where we use the Hölder inequality in the last inequality. Then by (2.5), we know that

 (3.16)

Combining (3.15) and (3.16) into (3.14) yields

 (3.17)

By Lemma 3.2 and the Minkowski inequality , we note that

 (3.18)

Substituting (3.18) into (3.17), we get

Applying the Theorem 3.5 to above inequality gives that

 E(∣∣Yi,Nt−¯Yi,Nt∣∣p)≤Cκ,H,p,LΔpH.

Thus, we obtain the desired result.

###### Theorem 3.7

Let Assumption 2.1 holds, for , then

 E(∣∣Xi,Nt−Yi,Nt∣∣p)≤C