On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient

We develop an explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations using the notion of derivative with respect to measure introduced by Lions and discussed in \cite{cardaliaguet2013}. The drift coefficient is allowed to grow super-linearly in the space variable. Further, both drift and diffusion coefficients are assumed to be only once differentiable in variables corresponding to space and measure. The rate of strong convergence is shown to be equal to $1.0$ without using It\^o's formula for functions depending on measure. The challenges arising due to the dependence of coefficients on measure are tackled and our findings are consistent with the analogous results for stochastic differential equations.


Introduction
Let (Ω, F , {F t } {t≥0} , P ) be a filtered probability space satisfying the usual conditions. Assume that {W t } {t≥0} is an m-dimensional Brownian motion. Consider the following d-dimensional McKean-Vlasov stochastic differential equation (MV-SDE), almost surely for any t ∈ [0, T ] where µ X t denotes the law of the random variable X t . When the law µ X t is known, MV-SDEs reduce to SDEs with added dependency on time variable. MV-SDEs are widely used in physics, biology and neural activities, see for example, [2,4,5,10,11]. The main purpose of this article is to study Milstein-type numerical approximation of MV-SDEs (1) in strong sense when the drift coefficient is allowed to grow super-linearly in space variable. There is a significant interest in the strong approximation of SDEs due to its importance in Multilevel Monte Carlo path simulations for SDEs, see [12]. It is well known that the classical Euler scheme for SDE with super-linear coefficients diverges in finite time, see for example [14] and hence such divergence can obviously be observed in the case of MV-SDEs. The numerical approximation of SDEs in strong sense are well understood in the literature for global and non-global Lipschitz coefficients, see for example [9,13,15,16,17,18,21,22,24,26] and references therein. Recently, authors in [19] developed an explicit tamed Euler scheme and an implicit Euler scheme for simulating MV-SDEs when the drift coefficient satisfies non-global Lipschitz condition (and hence can grow super-linearly) and the diffusion coefficient satisfies global Lipschitz condition in space variable. We develop an explicit Milstein-type scheme for MV-SDEs (1) and study its strong convergence. The drift coefficient is assumed to satisfy polynomial Lipschitz condition i.e., it is allowed to grow super-linearly and the diffusion coefficient satisfies global Lipschitz condition in the space variable. For the variable corresponding to measure, coefficients satisfy global Lipschitz condition and are bounded in Wasserstein metric. Moreover, derivatives in space variable of drift and diffusion coefficients are respectively assumed to be polynomial Lipschitz and Lipschitz. Also, the derivative of drift and diffusion coefficients with respect to measures satisfy Lipschitz condition and are bounded in Wasserstein metric. Further, the rate of strong convergence in L 2 -norm is shown to be equal to 1 without using Itô's formula for functions depending on measure, see [8], which is consistent with the analogous result available in literature for SDEs. Novel techniques have been developed to tackle challenges arising due to the presence of the law µ t in the coefficients. The following meanfield stochastic Ginzburg Landau equation fits well in our framework, almost surely for any t ∈ [0, T ]. This equation has been investigated in [19] and its variant (without mean-field term) in [23]. We also remark that the technique developed in this article can be used to investigate the higher order numerical approximations of MV-SDEs. To the best of author's knowledge, this is the first paper dealing with Milstein-type scheme for MV-SDEs using the notion of derivatives with respect to measures introduced by Lions in his lectures at the Collège de France and reproduced in [6].
1.1. Notations. We now introduce the notations used in this article. The notation ·, · stands for the inner product in R d . We use the same notation | · | for both Euclidean and Hilbert-Schmidt norms and its meaning should be clear from the context. Also, σx denotes the usual matrix multiplication of σ ∈ R d×m and x ∈ R d . With a slight abuse of notation, b (l) and σ (l) are used to denote l-th element of b ∈ R d and l-th column vector of σ ∈ R d×m respectively which is clear from the context and should not cause any confusion in reader's mind. Further, σ (k,l) stands for (k, l)-th element of σ ∈ R d×m . For a function f : R d → R, ∂ x f stands for gradient of f . ⌊·⌋ stands for the floor function. The symbol δ x (·) denotes the Dirac measure at point x ∈ R d . Moreover, P 2 (R d ) denotes the space of probability measures µ on the measurable space Then, P 2 (R d ) is a Polish space under the L 2 -Wasserstein metric given by where Π(µ 1 , µ 2 ) is the set of all couplings of µ 1 , µ 2 ∈ P 2 (R d ). Throughout this article, K stands for a generic constant which may vary from place to place.
1.2. Differentiability of functions of measures. There are many different notions for differentiating functions of measures, see for example [1,25]. In this article, we use the notion of differentiability introduced by Lions in his lectures at the Collège de France which has been reproduced in [6]. We give a brief description of the concept of measure derivative for functions defined on the Wasserstein space there exists an atomless, Polish probability space (Ω,F,P ) and a random variable . By Theorem 6.5 (structure of the gradient) in [6], if f is of class C 1 , then there exists a function ∂ µ f (ν 0 ) :

Assumptions and Main results
Let (Ω, {F t } {t≥0} , F , P ) be a filtered probability space satisfying the usual conditions, i.e., the probability space (Ω, F , P ) is complete, F 0 contains all P -null sets of F and filtration is right continuous. Let {W t } {t≥0} be an m-dimensional Brownian motion adapted to the filtration {F t } {t≥0} . Assume that b : R d × P 2 (R d ) → R d and σ : R d × P 2 (R d ) → R d×m are measurable functions. We consider the following McKean-Vlasov Stochastic Differential Equation (MV-SDE) defined on (Ω, {F t } {t≥0} , F , P ), almost surely for any t ∈ [0, T ] where µ X s denotes the law of X s , i.e. µ X s := P •X −1 s for every s ∈ [0, T ] and X 0 stands for an R d -valued and F 0 -measurable random variable.
We make the following assumptions on the coefficients and the initial value.
2.1. Propagation of Chaos and Interacting Particle System. For a fixed N ∈ N, let {W i } i∈{1,...,N } be N independent Brownian motions that are also independent of W . Consider N-dimensional system of interacting particles given by, almost surely for any t ∈ [0, T ] and i ∈ {1, . . . , N}, where For the propagation of chaos result, consider the system of non-interacting particles given by, almost surely for any t ∈ [0, T ] and i ∈ {1, . . . , N}, where µ X i s = µ X s for every i ∈ {1, . . . , N} because X i 's are independent.
The proof of the following proposition can be found in [19,20].
Proposition 1. Let Assumptions 1, 2 and 3 be satisfied. Then, there exists a unique solution to MV-SDE (2) and the following holds, where the constant K > 0 does not depend on N. For any x ∈ R d and µ ∈ P 2 (R d ), define b n x, µ := b x, µ 1 + n −1 |x| ρ+2 (5) for every n ∈ N. We propose the following explicit Milstein-type scheme for MV-SDE (2), where Λ By combining Propositions 1 and 2, we obtain the following theorem.
for any n, N ∈ N where constant K > 0 does not depend on n, N ∈ N.
We conclude this section by listing following remarks which are consequences of the assumptions mentioned above. Remark 1. From Assumptions 2 and 4, for any x, y ∈ R d , µ ∈ P 2 (R d ) and k ∈ {1, . . . , d}.
Remark 2. From Remark 1 and equation (5), for all x ∈ R d , µ ∈ P 2 (R d ) and n ∈ N where the constant K > 0 does not depend on n ∈ N.

Moment Bounds
Before establishing the moment bound of the Milstein-type scheme (6) in Lemma 4, we first establish following lemmas and corollaries. Proof. By Cauchy-Schwarz inequality and Burkholder-Gundy-Davis inequality, and then the application of Remark 3 completes the proof.
Lemma 2. Let Assumptions 3 and 5 be satisfied. Then, where constant K > 0 does not depend on n and N.
Proof. On using Cauchy-Schwarz inequality and Burkholder-Gundy-Davis inequality, and then the proof is completed by using Remark 3.
As a consequence of Remark 3, Lemma 1 and Lemma 2, one obtains the following corollary.

Corollary 1. Let Assumptions 3 and 5 be satisfied. Then,
for any s ∈ [0, T ], i ∈ {1, · · · , N} and n, N ∈ N where the constant K > 0 does not depend on n and N.
Lemma 3. Let Assumptions 2 to 5 be satisfied. Then, , · · · , N} and n, N ∈ N where the constant K > 0 does not depend on n and N.
Proof. From equation (6), one can get the following estimate, and then the application of Hölder's inequality and Burkholder-Gundy-Davis inequality gives, which on using Remark 2 and Corollary 1 completes the proof.
for any n, N ∈ N where the constant K > 0 does not depend on n and N.
Proof. By the application of Itô's formula, almost surely, which on the application of Burkholder-Gundy-Davis inequality and Cauchy-Schwarz inequality yields the following estimate, for any i ∈ {1, . . . , N}, n, N ∈ N, and u ∈ [0, T ]. Also, one uses Remark 1, Cauchy-Schwarz inequality and Young's inequality to obtain the following estimate, (1 + |X i,N,n r | 2 ) p/2 ds for any n, N ∈ N and u ∈ [0, T ]. Finally, the proof is completed by using the Gronwall's inequality.

Rate of Convergence
In this section, we shall prove Proposition 2. For this, we require some lemmas and corollaries which are shown below. Notice that as a consequence of Lemmas 1, 2 and 4, we obtain the following corollaries.
Corollary 2. Let Assumptions 1 to 5 be satisfied. Then, for any s ∈ [0, T ], i ∈ {1, · · · , N} and n, N ∈ N where the constant K > 0 does not depend on n and N.
Corollary 6. Let Assumptions 1 to 5 be satisfied. Then, for any s ∈ [0, T ], i ∈ {1, · · · , N} and n, N ∈ N where the constant K > 0 does not depend on n and N.
The following lemma is very useful in this article.
i.e., φ ′ (t 0 ) exists. Notice that the second term on the right hand side of the above expression is Gateaux derivative of F atZ + t 0 (Z −Z) in the direction of Z −Z.
Also, φ is continuous on [0.1]. Hence, by mean value theorem, there exists a completes the proof.
As a special case of the above lemma, we obtain the following corollary.
By Lemma 5, which completes the proof.
Lipschitz conditions i.e., there exists a constant L > 0 such that, , for all x, y,x,ȳ ∈ R d and µ,μ ∈ P 2 (R d ). Then, Proof. By Corollary 7, Cauchy-Schwarz inequality, assumptions on f and Young's inequality, one obtains, The proof is completed by the following estimate on Wasserstein metric, polynomial Lipschitz condition i.e., there exists a constant L > 0 such that, , for all x, y,x,ȳ ∈ R d and µ,μ ∈ P 2 (R d ). Then, Proof. The proof follows by adapting the arguments of Lemma 6. Proof. From equation (7), almost surely for any s ∈ [0, T ] and n, N ∈ N. Further, from equation (6), almost surely for any s ∈ [0, T ] and n, N ∈ N. On substituting values from equations (9) and (10) in equation (8) and then on using Lemma 6 and Cauchy-Schwarz inequality, one gets, for any s ∈ [0, T ] and n, N ∈ N. By using Young's inequality, T 1 is estimated as, for any s ∈ [0, T ] and n, N ∈ N. Further, using equation (6), b n X i,N,n κn(r) , µ X,N,n κn(r) dr Notice that the second term on the right hand side of the above expression is zero. Thus, from Young's inequality, Remark 2 and equations (3) and (6), one obtains n (X i,N,n κn(r) , µ X,N,n κn(r) ) dr for any s ∈ [0, T ] and n, N ∈ N. Using Cauchy-Schwarz inequality, Young's inequality, Assumption 2 and Remark 1, T 21 can be estimated by, for any s ∈ [0, T ] and n, N ∈ N.
Using Cauchy-Schwarz inequality, Assumption 2 and Remark 1, T 22 can be estimated as, for any s ∈ [0, T ] and n, N ∈ N.