Weak solutions of mean-field stochastic differential equations

Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China. Email: [email protected] Based on joint works with Hui Min (Beijing University of Technology) Workshop “PDE and Probability Methods for Interactions”, Inria, Sophia Antipolis, France, March 30-31, 2017.

1 / 44

Contents

1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

2 / 44

1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

3 / 44

Objective of the talk

Let 𝑇 be a fixed time horizon, 𝑏, 𝜎 measurable mappings defined over appropriate spaces. We are interested in a weak solution of Mean-Field (McKean-Vlasov) SDE : For 𝑡 ∈ [0, 𝑇 ], 𝜉 ∈ 𝐿2 (Ω, ℱ0 , 𝑃 ; R𝑑 ), ∫︁ 𝑋𝑡 = 𝜉 +

𝑡

∫︁

𝑡

𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )𝑑𝑠 + 0

𝜎(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )𝑑𝐵𝑠 ,

(1.1)

0

where 𝑄 is a probability measure with respect to which 𝐵 is a B.M. Remark: 𝑄𝑋·∧𝑠 is the law of 𝑋·∧𝑠 w.r.t. 𝑄.

3 / 44

Brief state of art 1) Such Mean-Field SDEs have been intensively studied: ∙ For a longer time as limit equ. for systems with a large number of particles (propagation of chaos)(Bossy, M´el´eard, Sznitman, Talay,...); ∙ Mean-Field Games, since 2006-2007 (Lasry, Lions,...); 2) Mean-Field SDEs/FBSDEs and associated nonlocal PDEs: ∙ Preliminary works in 2009 (AP, SPA); ∙ Classical solution of non-linear PDE related with the mean-field SDE: Buckdahn, Peng, Li, Rainer (2014); Chassagneux, Crisan, Delarue (2014); ∙ For the case with jumps: Li, Hao (2016); Li (2016); ∙ Weak solution: Oelschl¨ager(1984), Funaki (1984), G¨artner (1988), Lacker (2015), Carmona, Lacker (2015), Li, Hui (2016, 2017)...... 4 / 44

Objective of the talk

Our objectives: To prove the existence and the uniqueness in law of the weak solution of mean-field SDE (1.1): * when the coefficient 𝑏 is bounded, measurable and with a modulus of continuity w.r.t the measure, while 𝜎 is independent of the measure and Lipschitz. * when the coefficients (𝑏, 𝜎) are bounded and continuous.

5 / 44

Preliminaries

We consider + (Ω, ℱ, 𝑃 ) - complete probability space; + 𝑊 B.M. over (Ω, ℱ, 𝑃 ) (for simplicity: all processes 1-dimensional); + F-filtration generated by 𝑊 , and augmented by ℱ0 . 𝑝-Wasserstein metric on ∫︁ 𝒫𝑝 (R) := {𝜇 | 𝜇 probab. on (R, ℬ(R)) with

|𝑥|𝑝 𝜇(𝑥) < +∞};

R

𝑊𝑝 (𝜇, 𝜈):= inf

{︀(︀

∫︁ R×R

)︀ 1 }︀ |𝑥|𝑝 𝜌(𝑑𝑥𝑑𝑦) 𝑝 , 𝜌(· × R) = 𝜇, 𝜌(R × ·) = 𝜈 . (1.2)

6 / 44

Preliminaries Generalization of the def. of a weak sol. of a classical SDE (see, e.g., Karatzas and Shreve, 1988) to (1.1):

Definition 1.1 ̃︀ 𝑄, 𝐵, 𝑋) is a weak solution of SDE (1.1), if ̃︀ ℱ, ̃︀ F, A six-tuple (Ω, ̃︀ = {ℱ̃︀𝑡 }0≤𝑡≤𝑇 is a ̃︀ ℱ, ̃︀ 𝑄) is a complete probability space, and F (i) (Ω, ̃︀ ℱ, ̃︀ 𝑄) satisfying the usual conditions. filtration on (Ω, ̃︀ (ii) 𝑋 = {𝑋𝑡 }0≤𝑡≤𝑇 is a continuous, F-adapted R-valued process; ̃︀ 𝑄)-BM. 𝐵 = {𝐵𝑡 }0≤𝑡≤𝑇 is an (F, ∫︀ 𝑇 (iii) 𝑄{ 0 (|𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )| + |𝜎(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )|2 )𝑑𝑠 < +∞} = 1, and equation (1.1) is satisfied, 𝑄-a.s.

7 / 44

Preliminaries

Definition 1.2 We say that uniqueness in law holds for the mean-field SDE (1.1), if for any two weak solutions (Ω𝑖 , ℱ 𝑖 , F𝑖 , 𝑄𝑖 , 𝐵 𝑖 , 𝑋 𝑖 ), 𝑖 = 1, 2, we have 𝑄1𝑋 1 = 𝑄2𝑋 2 , i.e., the two processes 𝑋 1 and 𝑋 2 have the same law.

8 / 44

1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

9 / 44

Case 1: Existence of a weak solution Let 𝑏, 𝜎 satisfy the following assumption (H1): (i) 𝑏 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R) × 𝒫1 (R) → R is bounded and measurable; (ii) 𝜎 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R) → R is bounded, measurable, and s.t., for all (𝑡, 𝜙) ∈ [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R), 1/𝜎(𝑡, 𝜙) is bounded in (𝑡, 𝜙); (iii) (Modulus of continuity) ∃𝜌 : R+ → R+ increasing, continuous, with 𝜌(0+) = 0 s.t., for all 𝑡 ∈ [0, 𝑇 ], 𝜙 ∈ 𝐶([0, 𝑇 ]; R), 𝜇, 𝜈 ∈ 𝒫1 (R), |𝑏(𝑡, 𝜙·∧𝑡 , 𝜇) − 𝑏(𝑡, 𝜙·∧𝑡 , 𝜈)| ≤ 𝜌(𝑊1 (𝜇, 𝜈)); (iv) ∃𝐿 ≥ 0 s.t., for all 𝑡 ∈ [0, 𝑇 ], 𝜙, 𝜓 ∈ 𝐶([0, 𝑇 ]; R), |𝜎(𝑡, 𝜙·∧𝑡 ) − 𝜎(𝑡, 𝜓·∧𝑡 )| ≤ 𝐿 sup |𝜙𝑠 − 𝜓𝑠 |. 0≤𝑠≤𝑡

9 / 44

Case 1: Existence of a weak solution We want to study weak solutions of the following mean-field SDE: ∫︁ 𝑡 ∫︁ 𝑡 𝑋𝑡 = 𝜉 + 𝜎(𝑠, 𝑋·∧𝑠 )𝑑𝐵𝑠 + 𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋𝑠 )𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], 0

(2.1)

0

where (𝐵𝑡 )𝑡∈[0,𝑇 ] is a BM under the probability measure 𝑄. Now we can give the main statement of this section.

Theorem 2.1 Under assumption (H1) mean-field SDE (2.1) has a weak solution ̃︀ 𝑄, 𝐵, 𝑋). ̃︀ ℱ, ̃︀ F, (Ω, Proof: Girsanov’s Theorem. Schauder’s Fixed Point Theorem.

10 / 44

Case 1: Existence of a weak solution Let us give two examples. Example 1. Take diffusion coefficient 𝜎 ≡ 𝐼𝑑 and drift coefficient ∫︀ ̂︀𝑏(𝑠, 𝜙·∧𝑠 , 𝜇𝑠 ) := 𝑏(𝑠, 𝜙·∧𝑠 , 𝜓𝑑𝜇𝑠 ), 𝜙 ∈ 𝐶([0, 𝑇 ]), 𝜇 ∈ 𝒫1 (R), 𝑠 ∈ [0, 𝑇 ]; the function 𝜓 ∈ 𝐶([0, 𝑇 ]; R) is arbitrarily given but fixed, and Lipschitz. Then our mean-field SDE (2.1) can be written as follows: ∫︁

𝑡

𝑏(𝑠, 𝑋·∧𝑠 , 𝐸𝑄 [𝜓(𝑋𝑠 )])𝑑𝑠, 𝑡 ∈ [0, 𝑇 ].

𝑋𝑡 = 𝐵𝑡 +

(2.2)

0

Here 𝑏 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]) × R → R is bounded, meas., Lips. in 𝑦. Then, the coefficients ̂︀𝑏 and 𝜎 satisfy (H1), and from Theorem 2.1, we obtain ̃︀ 𝑄, 𝐵, 𝑋). ̃︀ ℱ, ̃︀ F, that the mean-field SDE (2.2) has a weak solution (Ω, 11 / 44

Case 1: Existence of a weak solution

Example 2. Take diffusion coefficient 𝜎 ≡ 𝐼𝑑 and drift coefficient ∫︀ ̂︀𝑏(𝑠, 𝜙·∧𝑠 , 𝜇𝑠 ) := 𝑏(𝑠, 𝜙·∧𝑠 , 𝑦)𝜇𝑠 (𝑑𝑦), 𝜙 ∈ 𝐶([0, 𝑇 ]), 𝜇𝑠 ∈ 𝒫1 (R), 𝑠 ∈ [0, 𝑇 ], i.e., we consider the following mean-field SDE: ∫︁ 𝑡 ∫︁ 𝑏(𝑠, 𝑋·∧𝑠 , 𝑦)𝑄𝑋𝑠 (𝑑𝑦)𝑑𝑠, 𝑡 ∈ [0, 𝑇 ].

𝑋𝑡 = 𝐵𝑡 + 0

(2.3)

R

Here the coefficient 𝑏 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]) × R → R is bounded, meas. and Lips. in 𝑦. Then, the coefficients ̂︀𝑏 and 𝜎 satisfy (H1), and from Theorem ̃︀ 𝑄, 𝐵, 𝑋). ̃︀ ℱ, ̃︀ F, 2.1 the mean-field SDE (2.3) has a weak solution (Ω,

12 / 44

Case 1: Uniqueness in law of weak solutions Let the functions 𝑏 and 𝜎 satisfy the following assumption (H2): (i) 𝑏 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R) × 𝒫1 (R) → R is bounded and measurable; (ii) 𝜎 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R) → R is bounded and measurable, and |1/𝜎(𝑡, 𝜙)| ≤ 𝐶, (𝑡, 𝜙) ∈ [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R), for some 𝐶 ∈ R+ ; (iii) (Modulus of continuity) There exists a continuous and increasing function 𝜌 : R+ → R+ with

∫︁

𝑑𝑢 = +∞, 0+ 𝜌(𝑢) such that, for all 𝑡 ∈ [0, 𝑇 ], 𝜙 ∈ 𝐶([0, 𝑇 ]; R), 𝜇, 𝜈 ∈ 𝒫1 (R), 𝜌(𝑟) > 0, for all 𝑟 > 0, and

|𝑏(𝑡, 𝜙·∧𝑡 , 𝜇) − 𝑏(𝑡, 𝜙·∧𝑡 , 𝜈)|2 ≤ 𝜌(𝑊1 (𝜇, 𝜈)2 ); (iv) ∃𝐿 ≥ 0 such that, for all 𝑡 ∈ [0, 𝑇 ], 𝜙, 𝜓 ∈ 𝐶([0, 𝑇 ]; R), |𝜎(𝑡, 𝜙·∧𝑡 ) − 𝜎(𝑡, 𝜓·∧𝑡 )| ≤ 𝐿 sup |𝜙𝑠 − 𝜓𝑠 |. 0≤𝑠≤𝑡 13 / 44

Case 1: Uniqueness in law of weak solutions Obviously, under assumption (H2) the coefficients 𝑏 and 𝜎 also satisfy (H1). Thus, due to Theorem 2.1, the following mean-field SDE ∫︁ 𝑋𝑡 = 𝜉 +

𝑡

∫︁

0

𝑡

𝜎(𝑠, 𝑋·∧𝑠 )𝑑𝐵𝑠 , 𝑡 ∈ [0, 𝑇 ],

𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋𝑠 )𝑑𝑠 +

(2.1)

0

has a weak solution.

Theorem 2.2 Suppose that assumption (H2) holds, and let (Ω𝑖 , ℱ 𝑖 , F𝑖 , 𝑄𝑖 , 𝐵 𝑖 , 𝑋 𝑖 ), 𝑖 = 1, 2, be two weak solutions of mean-field SDE (2.1). Then (𝐵 1 , 𝑋 1 ) and (𝐵 2 , 𝑋 2 ) have the same law under their respective probability measures, i.e., 𝑄1(𝐵 1 ,𝑋 1 ) = 𝑄2(𝐵 2 ,𝑋 2 ) . 14 / 44

Case 1: Uniqueness in law of weak solutions Sketch of the proof: For 𝜙 ∈ 𝐶([0, 𝑇 ]; R), 𝜇 ∈ 𝒫1 (R), we define ̃︀𝑏(𝑠, 𝜙·∧𝑠 , 𝜇) = 𝜎 −1 (𝑠, 𝜙·∧𝑠 )𝑏(𝑠, 𝜙·∧𝑠 , 𝜇), and we introduce ⎧ ∫︁ 𝑡 ⎪ 𝑖 𝑖 ⎪ ̃︀𝑏(𝑠, 𝑋 𝑖 , 𝑄𝑖 𝑖 )𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], ⎪ ⎨ 𝑊𝑡 = 𝐵𝑡 + ·∧𝑠 𝑋𝑠 0 ∫︁ ∫︁ 𝑇 ⎪ 1 𝑇 ̃︀ ⎪ 𝑖 𝑖 𝑖 𝑖 𝑖 ̃︀ ⎪ 𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋 𝑖 )𝑑𝐵𝑠 − |𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑖𝑋 𝑖 )|2 𝑑𝑠}, ⎩ 𝐿𝑇 = exp{− 𝑠 𝑠 2 0 0 (2.4) 𝑖 = 1, 2. Then from the Girsanov Theorem we know that (𝑊𝑡𝑖 )𝑡∈[0,𝑇 ] is an ̃︀ 𝑖 = 𝐿𝑖 𝑄𝑖 , 𝑖 = 1, 2, respectively. F𝑖 -B.M. under the probability measure 𝑄 𝑇

From (H2), for each 𝑖, we have a unique strong solution 𝑋 𝑖 of the SDE ∫︁ 𝑡 𝑖 𝜎(𝑠, 𝑋·∧𝑠 )𝑑𝑊𝑠𝑖 , 𝑡 ∈ [0, 𝑇 ]. (2.5) 𝑋𝑡𝑖 = 𝑋0𝑖 + 0 15 / 44

Case 1: Uniqueness in law of weak solutions It is by now standard that ∃ a meas. and non-anticipating function Φ : [0, 𝑇 ] × R × 𝐶([0, 𝑇 ]; R) → R not depending on 𝑖 = 1, 2, s.t. ̃︀ 𝑖 -a.s. (and, 𝑄𝑖 -a.s.), 𝑖 = 1, 2. (2.6) 𝑋𝑡𝑖 = Φ𝑡 (𝑋0𝑖 , 𝑊 𝑖 ), 𝑡 ∈ [0, 𝑇 ], 𝑄 ∫︀ 𝑡 Then from (2.4) that 𝑊𝑡𝑖 = 𝐵𝑡𝑖 + 0 ̃︀𝑏(𝑠, Φ·∧𝑠 (𝑋0𝑖 , 𝑊 𝑖 ), 𝑄𝑖𝑋 𝑖 )𝑑𝑠, 𝑖 = 1, 2. 𝑠 Hence, putting 𝑓 (𝑠, 𝜙·∧𝑠 ) = ̃︀𝑏(𝑠, 𝜙·∧𝑠 , 𝑄1 ), (𝑠, 𝜙) ∈ [0, 𝑇 ] × 𝐶([0, 𝑇 ]; 𝑋𝑠

R), from (2.4) and (2.6) we have ∫︁ 𝑡 ⎧ 1 1 ⎪ ⎪ 𝑊 = 𝐵 + 𝑓 (𝑠, Φ·∧𝑠 (𝑋01 , 𝑊 1 ))𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], ⎨ 𝑡 𝑡 0 ∫︁ 𝑡 ⎪ ⎪ ̃︀𝑡2 + ⎩ 𝑊𝑡2 = 𝐵 𝑓 (𝑠, Φ·∧𝑠 (𝑋02 , 𝑊 2 ))𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], 0

where, 𝑡 ∈ [0, 𝑇 ], ∫︁ 𝑡 (︁ )︁ ̃︀𝑏(𝑠, Φ·∧𝑠 (𝑋 2 , 𝑊 2 ), 𝑄2 2 ) − ̃︀𝑏(𝑠, Φ·∧𝑠 (𝑋 2 , 𝑊 2 ), 𝑄1 1 ) 𝑑𝑠. ̃︀ 2 = 𝐵 2 + 𝐵 𝑡 𝑡 0 0 𝑋𝑠 𝑋𝑠 0

16 / 44

Case 1: Uniqueness in law of weak solutions ¯ : [0, 𝑇 ] × R × 𝐶([0, 𝑇 ]; R) → R meas. s.t., for both 𝐵 1 , 𝐵 ̃︀ 2 , Hence, ∃Φ ¯ 𝑡 (𝑋 1 , 𝑊 1 ) and 𝐵 ̃︀ 2 = Φ ¯ 𝑡 (𝑋 2 , 𝑊 2 ), 𝑡 ∈ [0, 𝑇 ]. 𝐵𝑡1 = Φ 0 𝑡 0

(2.8)

Now we define ⎧ ̂︀ 2𝑡 𝑑𝐵𝑡2 , 𝑡 ̂︀ 2𝑡 = −(̃︀𝑏(𝑠, Φ·∧𝑠 (𝑋02 , 𝑊 2 ), 𝑄2 2 ) − ̃︀𝑏(𝑠, Φ·∧𝑠 (𝑋02 , 𝑊 2 ), 𝑄1 1 ))𝐿 ⎨ 𝑑𝐿 𝑋𝑠 𝑋𝑠 ⎩ 𝐿 ̂︀ 20 = 1. (2.9) ̃︀ 2 is an Brownian motion From the Girsanov Theorem we know that 𝐵 ̂︀ 2 = 𝐿 ̂︀ 2 𝑄2 . Moreover, putting under the probability measure 𝑄 𝑇

17 / 44

Case 1: Uniqueness in law of weak solutions

⎧ ∫︁ 𝑇 ∫︁ 𝑇 ⎪ ̃︀ 2 = exp{− 𝑓 (𝑠, Φ·∧𝑠 (𝑋 2 , 𝑊 2 ))𝑑𝑊 2 + 1 |𝑓 (𝑠, Φ·∧𝑠 (𝑋 2 , 𝑊 2 ))|2 𝑑𝑠}, ⎨𝐿 𝑇 0 𝑠 0 20 0 ⎪ ⎩ ¯2 ̃︀ 2 𝑄 ̂︀ 2 , 𝑄 =𝐿 𝑇

(2.10) we have that (𝑊𝑡1 )𝑡∈[0,𝑇 ]

(𝑊𝑡2 )𝑡∈[0,𝑇 ]

is a B.M. under both ̃︀ 1 . is a B.M. under 𝑄

̃︀ 2 𝑄

and

¯ 2, 𝑄

while

On the other hand, since 𝑓 is bounded and meas., we can prove that ̃︀ : R × 𝐶([0, 𝑇 ]; R) → R, s.t. ∃ a meas. function Φ ̃︀ 0𝑖 , 𝑊 𝑖 ) = Φ(𝑋

∫︁

𝑇

𝑓 (𝑠, Φ·∧𝑠 (𝑋0𝑖 , 𝑊 𝑖 ))𝑑𝑊𝑠𝑖 , 𝑄𝑖 -a.s., 𝑖 = 1, 2.

0

18 / 44

Case 1: Uniqueness in law of weak solutions

Therefore, recalling the definition of 𝐿1𝑇 and (2.10), we have ⎧ ∫︁ 𝑇 ∫︁ ⎪ 1 𝑇 1 1 1 1 ⎪ ⎪ |𝑓 (𝑠, Φ·∧𝑠 (𝑋01 , 𝑊 1 ))|2 𝑑𝑠}, ⎨ 𝐿𝑇 = exp{− 𝑓 (𝑠, Φ·∧𝑠 (𝑋0 , 𝑊 ))𝑑𝑊𝑠 + 2 0 0 ∫︁ 𝑇 ∫︁ 𝑇 ⎪ 1 ⎪ ⎪ ̃︀ 2 = exp{− 𝑓 (𝑠, Φ·∧𝑠 (𝑋 2 , 𝑊 2 ))𝑑𝑊 2 + |𝑓 (𝑠, Φ·∧𝑠 (𝑋02 , 𝑊 2 ))|2 𝑑𝑠}, ⎩𝐿 𝑇 0 𝑠 2 0 0 (2.11) ̂︀ and we see that ∃ a meas. function Φ : R × 𝐶([0, 𝑇 ]; R) → R, s.t. ̂︀ 01 , 𝑊 1 ), 𝑄1 -a.s., and 𝐿 ̃︀ 2𝑇 = Φ(𝑋 ̂︀ 02 , 𝑊 2 ), 𝑄2 -a.s. (and, 𝑄 ¯ 2 -a.s.). 𝐿1𝑇 = Φ(𝑋 (2.12)

19 / 44

Case 1: Uniqueness in law of weak solutions Consequently, as 𝑋0𝑖 is ℱ0𝑖 -measurable, 𝑖 = 1, 2 and 𝑄1𝑋 1 = 𝑄2𝑋 2 , 0

0

from (2.8), (2.10), (2.11) and (2.12) we have that, for all bounded measurable function 𝐹 : 𝐶([0, 𝑇 ]; R𝑑 )2 → R, 1 ¯ 1 , 𝑊 1 ), 𝑊 1 )] 𝐸𝑄1 [𝐹 (𝐵 1 , 𝑊 1 )] = 𝐸𝑄̃︀1 [ 𝐹 (Φ(𝑋 0 1 ̂︀ Φ(𝑋 , 𝑊 1) 0

= 𝐸𝑄¯ 2 [

1 ̂︀ 2 , 𝑊 2 ) Φ(𝑋 0

̃︀ 2 , 𝑊 2 )]. ¯ 02 , 𝑊 2 ), 𝑊 2 )] = 𝐸 ̂︀2 [𝐹 (𝐵 𝐹 (Φ(𝑋 𝑄

That is, ̂︀ 2 𝑄1(𝐵 1 ,𝑊 1 ) = 𝑄 ̃︀ 2 ,𝑊 2 ) . (𝐵

(2.13)

Taking into account (2.6), we have ̂︀ 2 𝑄1(𝐵 1 ,𝑊 1 ,𝑋 1 ) = 𝑄 ̃︀ 2 ,𝑊 2 ,𝑋 2 ) , (𝐵

(2.14)

̂︀ 2 2 . and, in particular, 𝑄1𝑋 1 = 𝑄 𝑋 20 / 44

Case 1: Uniqueness in law of weak solutions On the other hand, we can prove ∫︀ ̂︀ 2 2 , 𝑄2 2 )2 ≤ 𝐶 𝑠 𝜌(𝑊1 (𝑄1 1 , 𝑄2 2 )2 )𝑑𝑟; ∙𝑊1 (𝑄1𝑋 1 , 𝑄2𝑋 2 )2 = 𝑊1 (𝑄 0 𝑋 𝑋 𝑋 𝑋 𝑠

𝑠

𝑟

𝑠

𝑠

𝑟

∙ The continuity of 𝑠 → 𝑊1 (𝑄1𝑋 1 , 𝑄2𝑋 2 ). 𝑠

𝑠

Putting 𝑢(𝑠) := 𝑊1 (𝑄1𝑋 1 , 𝑄2𝑋 2 ), 𝑠 ∈ [0, 𝑇 ], then we have from above, 𝑠 ∫︀ 𝑠𝑠 𝑢(𝑠)2 ≤ 𝐶 0 𝜌(𝑢(𝑟)2 )𝑑𝑟, 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇 . ∫︀ 𝑑𝑢 From (H2)-(iii), 0+ 𝜌(𝑢) = +∞, it follows from Bihari’s inequality that 𝑢(𝑠) = 0, for any 𝑠 ∈ [0, 𝑇 ], that is, 𝑄1𝑋 1 = 𝑄2𝑋 2 , 𝑠 ∈ [0, 𝑇 ]. Thus, from 𝑠 𝑠 ̃︀ 2 = 𝐵 2 , 𝐿 ̂︀ 2 = 1, and, consequently, (2.7) and (2.9) it follows that 𝐵 𝑇 2 2 2 2 ̂︀ ̂︀ 𝑄 = 𝑄 . Then, 𝑄 = 𝑄 2 2 2 , and from (2.14) ̃︀ 2 ,𝑊 2 ,𝑋 2 ) (𝐵

(𝐵 ,𝑊 ,𝑋 )

𝑄1(𝐵 1 ,𝑊 1 ,𝑋 1 ) = 𝑄2(𝐵 2 ,𝑊 2 ,𝑋 2 ) . This implies, in particular, 𝑄1(𝐵 1 ,𝑋 1 ) = 𝑄2(𝐵 2 ,𝑋 2 ) .

(2.15)  21 / 44

1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

22 / 44

Case 2: Preliminaries Definition 3.1 (see, e.g., Karatzas, Shreve, 1988) A probability 𝑃̂︀ on (𝐶([0, 𝑇 ]; R), ℬ(𝐶([0, 𝑇 ]; R))) is a solution to the local martingale problem associated with 𝒜′ , if for every 𝑓 ∈ 𝐶 1,2 ([0, 𝑇 ]×R; R), 𝑀𝑡𝑓

∫︁ := 𝑓 (𝑡, 𝑦(𝑡)) − 𝑓 (0, 𝑦(0)) −

𝑡

(𝜕𝑠 + 𝒜′ )𝑓 (𝑠, 𝑦(𝑠))𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], (3.1)

0

is a continuous local martingale w.r.t (F𝑦 , 𝑃̂︀), where 𝑦 = (𝑦(𝑡))𝑡∈[0,𝑇 ] is the coordinate process on 𝐶([0, 𝑇 ]; R), the considered filtration F𝑦 = (ℱ𝑡𝑦 )𝑡∈[0,𝑇 ] is that generated by 𝑦 = (𝑦(𝑡))𝑡∈[0,𝑇 ] and augmented by all 𝑃̂︀-null sets, and 𝒜′ is defined by, 𝑦 ∈ 𝐶([0, 𝑇 ]; R), 1 𝒜′ 𝑓 (𝑠, 𝑦) = 𝑏(𝑠, 𝑦)𝜕𝑥 𝑓 (𝑠, 𝑦(𝑠)) + 𝜎 2 (𝑠, 𝑦)𝜕𝑥2 𝑓 (𝑠, 𝑦(𝑠)). 2

(3.2) 22 / 44

Case 2: Preliminaries Let us first recall a well-known result concerning the equivalence between the weak solution of a functional SDE and the solution to the corresponding local martingale problem (see, e.g., Karatzas, Shreve, 1988).

Lemma 3.1 ̃︀ 𝑃̃︀, 𝑊 ̃︀ ℱ, ̃︀ F, ̃︁ , 𝑋) to the following The existence of a weak solution (Ω, functional SDE with given initial distribution 𝜇 on ℬ(R): ∫︁ 𝑋𝑡 = 𝜉 +

𝑡

∫︁

0

𝑡

̃︁𝑠 , 𝑡 ∈ [0, 𝑇 ], 𝜎(𝑠, 𝑋·∧𝑠 )𝑑𝑊

𝑏(𝑠, 𝑋·∧𝑠 )𝑑𝑠 + 0

is equivalent to the existence of a solution 𝑃̂︀ to the local martingale problem (3.1) associated with 𝒜′ defined by (3.2), with 𝑃̂︀𝑦(0) = 𝜇. The both solutions are related by 𝑃̂︀ = 𝑃̃︀ ∘ 𝑋 −1 , i.e., the probability measure 𝑃̂︀ is the law of the weak solution 𝑋 on (𝐶([0, 𝑇 ]; R), ℬ(𝐶([0, 𝑇 ]; R))).

23 / 44

Case 2: Preliminaries Recall the definition of the derivative of 𝑓 : 𝒫2 (R) → R w.r.t probability measure 𝜇 ∈ 𝒫2 (R) (in the sense of P.L.Lions)(P.L.Lions’ lectures at Coll`ege de France, also see the notes of Cardaliaguet).

Definition 3.2 (i) 𝑓̃︀ : 𝐿2 (Ω, ℱ, 𝑃 ; R) → R is Fr´echet differentiable at 𝜉 ∈ 𝐿2 (Ω, ℱ, 𝑃 ), if ∃ a linear continuous mapping 𝐷𝑓̃︀(𝜉)(·) ∈ 𝐿(𝐿2 (Ω, ℱ, 𝑃 ; R); R), s.t. 𝑓̃︀(𝜉 + 𝜂)−𝑓̃︀(𝜉) = 𝐷𝑓̃︀(𝜉)(𝜂) + 𝑜(|𝜂|𝐿2 ), with |𝜂|𝐿2 → 0 for 𝜂 ∈ 𝐿2 (Ω, ℱ, 𝑃 ). (ii) 𝑓 : 𝒫2 (R) → R is differentiable at 𝜇 ∈ 𝒫2 (R), if for 𝑓̃︀(𝜉) := 𝑓 (𝑃𝜉 ), 𝜉 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R), there is some 𝜁 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R) with 𝑃𝜁 = 𝜇 such that 𝑓̃︀ : 𝐿2 (Ω, ℱ, 𝑃 ; R) → R is Fr´echet differentiable in 𝜁. 24 / 44

Case 2: Preliminaries From Riesz’ Representation Theorem there exists a 𝑃 -a.s. unique variable 𝜗 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R) such that 𝐷𝑓̃︀(𝜁)(𝜂) = (𝜗, 𝜂)𝐿2 = 𝐸[𝜗𝜂], for all 𝜂 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R). P.L. Lions proved that there is a Borel function ℎ : R → R such that 𝜗 = ℎ(𝜁), 𝑃 -a.e., and function ℎ depends on 𝜁 only through its law 𝑃𝜁 . Therefore, 𝑓 (𝑃𝜉 ) − 𝑓 (𝑃𝜁 ) = 𝐸[ℎ(𝜁) · (𝜉 − 𝜁)] + 𝑜(|𝜉 − 𝜁|𝐿2 ), 𝜉 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R).

Definition 3.3 We call 𝜕𝜇 𝑓 (𝑃𝜁 , 𝑦) := ℎ(𝑦), 𝑦 ∈ R, the derivative of function 𝑓 : 𝒫2 (R) → R at 𝑃𝜁 , 𝜁 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R). Remark: 𝜕𝜇 𝑓 (𝑃𝜁 , 𝑦) is only 𝑃𝜁 (𝑑𝑦)-a.e. uniquely determined. 25 / 44

Case 2: Preliminaries Definition 3.4 We say that 𝑓 ∈ 𝐶 1 (𝒫2 (R)), if for all 𝜉 ∈ 𝐿2 (Ω, ℱ, 𝑃 ; R) there exists a 𝑃𝜉 -modification of 𝜕𝜇 𝑓 (𝑃𝜉 , .), also denoted by 𝜕𝜇 𝑓 (𝑃𝜉 , .), such that 𝜕𝜇 𝑓 : 𝒫2 (R) × R → R is continuous w.r.t the product topology generated by the 2-Wasserstein metric over 𝒫2 (R) and the Euclidean norm over R, and we identify this modified function 𝜕𝜇 𝑓 as the derivative of 𝑓 . The function 𝑓 is said to belong to 𝐶𝑏1,1 (𝒫2 (R)), if 𝑓 ∈ 𝐶 1 (𝒫2 (R)) is s.t. 𝜕𝜇 𝑓 : 𝒫2 (R) × R → R is bounded and Lipschitz continuous, i.e., there exists some constant 𝐶 ≥ 0 such that (i) |𝜕𝜇 𝑓 (𝜇, 𝑥)| ≤ 𝐶, 𝜇 ∈ 𝒫2 (R), 𝑥 ∈ R; (ii) |𝜕𝜇 𝑓 (𝜇, 𝑥)−𝜕𝜇 𝑓 (𝜇′ , 𝑥′ )|≤𝐶(𝑊2 (𝜇, 𝜇′ )+|𝑥−𝑥′ |), 𝜇, 𝜇′ ∈ 𝒫2 (R), 𝑥, 𝑥′ ∈ R. 26 / 44

Case 2: Preliminaries

Definition 3.5 We say that 𝑓 ∈ 𝐶 2 (𝒫2 (R)), if 𝑓 ∈ 𝐶 1 (𝒫2 (R)) and 𝜕𝜇 𝑓 (𝜇, .) : R → R is differentiable, and its derivative 𝜕𝑦 𝜕𝜇 𝑓 : 𝒫2 (R)×R→R ⊗ R is continuous, for every 𝜇 ∈ 𝒫2 (R). Moreover, 𝑓 ∈ 𝐶𝑏2,1 (𝒫2 (R)), if 𝑓 ∈ 𝐶 2 (𝒫2 (R))

⋂︀

𝐶𝑏1,1 (𝒫2 (R)) and its

derivative 𝜕𝑦 𝜕𝜇 𝑓 : 𝒫2 (R) × R → R ⊗ R is bounded and Lipschitzcontinuous. Remark: 𝐶𝑏2,1 (R × 𝒫2 (R)), 𝐶𝑏1,2,1 ([0, 𝑇 ]×R×𝒫2 (R); R) are similarly defined.

27 / 44

Case 2: Preliminaries Now we can give our It^ o’s formula.

Theorem 3.1 Let 𝜎 = (𝜎𝑠 ), 𝛾 = (𝛾𝑠 ), 𝑏 = (𝑏𝑠 ), 𝛽 = (𝛽𝑠 ) R-valued adapted stochastic processes, such that ∫︀ 𝑇 3 (i) There exists a constant 𝑞 > 6 s.t. 𝐸[( 0 (|𝜎𝑠 |𝑞 + |𝑏𝑠 |𝑞 )𝑑𝑠) 𝑞 ] < +∞; ∫︀ 𝑇 (ii) 0 (|𝛾𝑠 |2 + |𝛽𝑠 |)𝑑𝑠 < +∞, P-a.s. Let 𝐹 ∈ 𝐶𝑏1,2,1 ([0, 𝑇 ] × R × 𝒫2 (R)). Then, for the It^ o processes 𝑡

∫︁ 𝑋𝑡 = 𝑋0 +

𝑡

∫︁

𝑏𝑠 𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], 𝑋0 ∈ 𝐿2 (Ω, ℱ0 , 𝑃 ),

𝜎𝑠 𝑑𝑊𝑠 + 0

∫︁ 𝑌𝑡 = 𝑌0 +

0

𝑡

∫︁ 𝛾𝑠 𝑑𝑊𝑠 +

0

𝑡

𝛽𝑠 𝑑𝑠, 𝑡 ∈ [0, 𝑇 ], 𝑌0 ∈ 𝐿2 (Ω, ℱ0 , 𝑃 ),

0 28 / 44

Case 2: Preliminaries Theorem 3.1 (continued) we have 𝐹 (𝑡, 𝑌𝑡 , 𝑃𝑋𝑡 ) − 𝐹 (0, 𝑌0 , 𝑃𝑋0 ) ∫︁ 𝑡 (︂ 1 = 𝜕𝑟 𝐹 (𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 ) + 𝜕𝑦 𝐹 (𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 )𝛽𝑟 + 𝜕𝑦2 𝐹 (𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 )𝛾𝑟2 2 0 )︂ 1 2 ¯ ¯ ¯ ¯ + 𝐸[(𝜕𝜇 𝐹 )(𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 , 𝑋𝑟 )𝑏𝑟 + 𝜕𝑧 (𝜕𝜇 𝐹 )(𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 , 𝑋𝑟 )¯ 𝜎𝑟 ] 𝑑𝑟 2 ∫︁ 𝑡 + 𝜕𝑦 𝐹 (𝑟, 𝑌𝑟 , 𝑃𝑋𝑟 )𝛾𝑟 𝑑𝑊𝑟 , 𝑡 ∈ [0, 𝑇 ]. 0

¯ ¯𝑏, 𝜎 Here (𝑋, ¯ ) denotes an independent copy of (𝑋, 𝑏, 𝜎), defined on a P.S. ¯ ℱ, ¯ 𝑃¯ ). The expectation 𝐸[·] ¯ on (Ω, ¯ ℱ, ¯ 𝑃¯ ) concerns only r.v. endowed (Ω, with the superscript¯. 29 / 44

Case 2: Preliminaries (H3) The coefficients (𝜎, 𝑏) ∈ 𝐶𝑏1,2,1 ([0, 𝑇 ] × R × 𝒫2 (R); R × R).

Theorem 3.2 (Buckdahn, Li, Peng and Rainer, 2014) Let Φ ∈ 𝐶𝑏2,1 (R × 𝒫2 (R)), then under assumption (H3) the following PDE: ⎧ 1 ⎪ 0 = 𝜕𝑡 𝑉 (𝑡, 𝑥, 𝜇) + 𝜕𝑥 𝑉 (𝑡, 𝑥, 𝜇)𝑏(𝑥, 𝜇) + 𝜕𝑥2 𝑉 (𝑡, 𝑥, 𝜇)𝜎 2 (𝑥, 𝜇) ⎪ ⎪ ⎪ 2 ⎪ ∫︁ ⎪ ⎪ ⎪ ⎪ + (𝜕𝜇 𝑉 )(𝑡, 𝑥, 𝜇, 𝑦)𝑏(𝑦, 𝜇)𝜇(𝑑𝑦) ⎪ ⎪ ⎪ R∫︁ ⎨ 1 𝜕𝑦 (𝜕𝜇 𝑉 )(𝑡, 𝑥, 𝜇, 𝑦)𝜎 2 (𝑦, 𝜇)𝜇(𝑑𝑦), + ⎪ ⎪ 2 ⎪ R ⎪ ⎪ ⎪ ⎪ ⎪ (𝑡, 𝑥, 𝜇) ∈ [0, 𝑇 ) × R × 𝒫2 (R); ⎪ ⎪ ⎪ ⎪ ⎩ 𝑉 (𝑇, 𝑥, 𝜇) = Φ(𝑥, 𝜇), (𝑥, 𝜇) ∈ R × 𝒫2 (R). has a unique classical solution 𝑉 (𝑡, 𝑥, 𝜇) ∈ 𝐶𝑏1,2,1 ([0, 𝑇 ] × R × 𝒫2 (R); R). 30 / 44

Case 2: Existence of a weak solution

Let 𝑏 and 𝜎 satisfy the following assumption: (H4) 𝑏, 𝜎 : [0, 𝑇 ] × R × 𝒫2 (R) → R are continuous and bounded. We want to study weak solution of the following mean-field SDE: ∫︁ 𝑡 ∫︁ 𝑡 𝜎(𝑠, 𝑋𝑠 , 𝑄𝑋𝑠 )𝑑𝐵𝑠 , 𝑡 ∈ [0, 𝑇 ], 𝑋𝑡 = 𝜉 + 𝑏(𝑠, 𝑋𝑠 , 𝑄𝑋𝑠 )𝑑𝑠 0

(3.3)

0

where 𝜉 ∈ 𝐿2 (Ω, ℱ0 , 𝑃 ; R) obeys a given distribution law 𝑄𝜉 = 𝜈 ∈ 𝒫2 (R) and (𝐵𝑡 )𝑡∈[0,𝑇 ] is a B.M. under the probability measure 𝑄.

31 / 44

Case 2: Existence of a weak solution Extension of the corresponding local martingale problem:

Definition 3.6 A probability measure 𝑃̂︀ on (𝐶([0, 𝑇 ]; R), ℬ(𝐶([0, 𝑇 ]; R))) is a solution to the local martingale problem (resp., martingale problem) associated with ̃︀ if for every 𝑓 ∈ 𝐶 1,2 ([0, 𝑇 ] × R; R) (resp., 𝑓 ∈ 𝐶 1,2 ([0, 𝑇 ] × R; R)), the 𝒜, 𝑏

process ∫︁

𝑓

𝐶 (𝑡, 𝑦, 𝜇) := 𝑓 (𝑡, 𝑦(𝑡)) − 𝑓 (0, 𝑦(0)) −

𝑡 (︀

)︀ ̃︀ (𝑠, 𝑦(𝑠), 𝜇(𝑠))𝑑𝑠, (𝜕𝑠 + 𝒜)𝑓

0

(3.4) is a continuous local

(F𝑦 , 𝑃̂︀)-martingale

(resp., continuous

(F𝑦 , 𝑃̂︀)-

martingale), 32 / 44

Case 2: Existence of a weak solution

Definition 3.6 (continued) where 𝜇(𝑡) = 𝑃̂︀𝑦(𝑡) is the law of the coordinate process 𝑦 = (𝑦(𝑡))𝑡∈[0,𝑇 ] on 𝐶([0, 𝑇 ]; R) at time 𝑡, the filtration F𝑦 is that generated by 𝑦 and completed, and 𝒜̃︀ is defined by ̃︀ )(𝑠, 𝑦, 𝜈) := 𝜕𝑦 𝑓 (𝑠, 𝑦)𝑏(𝑠, 𝑦, 𝜈) + 1 𝜕𝑦2 𝑓 (𝑠, 𝑦)𝜎 2 (𝑠, 𝑦, 𝜈), (𝒜𝑓 2

(3.5)

̃︀ )(𝑠, 𝑦(𝑠), 𝜇(𝑠)) abbreviates (𝑠, 𝑦, 𝜈) ∈ [0, 𝑇 ] × R × 𝒫2 (R). Here ((𝜕𝑠 + 𝐴)𝑓 ̃︀ )(𝑠, 𝑦(𝑠), 𝜇(𝑠)) := (𝜕𝑠 𝑓 )(𝑠, 𝑦(𝑠)) + (𝐴𝑓 ̃︀ )(𝑠, 𝑦(𝑠), 𝜇(𝑠)). ((𝜕𝑠 + 𝐴)𝑓

33 / 44

Case 2: Existence of a weak solution

Proposition 3.1 ̃︀ 𝑄, 𝐵, 𝑋) to equation (3.3) with ̃︀ ℱ, ̃︀ F, The existence of a weak solution (Ω, initial distribution 𝜈 on ℬ(R) is equivalent to the existence of a solution 𝑃̂︀ to the local martingale problem (3.4) associated with 𝒜̃︀ defined by (3.5), with 𝑃̂︀𝑦(0) = 𝜈.

34 / 44

Case 2: Existence of a weak solution

Lemma 3.2 Let the probability measure 𝑃̂︀ on (𝐶([0, 𝑇 ]; R), ℬ(𝐶([0, 𝑇 ]; R))) be a ̃︀ Then, for the solution to the local martingale problem associated with 𝒜. second order differential operator (︀

∫︁ ̃︀ 𝒜𝑓 )(𝑠, 𝑦, 𝜈) := (𝒜𝑓 )(𝑠, 𝑦, 𝜈) + (𝜕𝜇 𝑓 )(𝑠, 𝑦, 𝜈, 𝑧)𝑏(𝑠, 𝑧, 𝜈)𝜈(𝑑𝑧) R ∫︁ 1 + 𝜕𝑧 (𝜕𝜇 𝑓 )(𝑠, 𝑦, 𝜈, 𝑧)𝜎 2 (𝑠, 𝑧, 𝜈)𝜈(𝑑𝑧), 2 R (3.6)

35 / 44

Case 2: Existence of a weak solution Lemma 3.2 (continued) applying to functions 𝑓 ∈ 𝐶 1,2 ([0, 𝑇 ] × R × 𝒫2 (R); R) we have that, for every such 𝑓 ∈ 𝐶 1,2 ([0, 𝑇 ] × R × 𝒫2 (R); R), the process 𝐶 𝑓 (𝑡, 𝑦, 𝜇) :=𝑓 (𝑡, 𝑦(𝑡), 𝜇(𝑡)) − 𝑓 (0, 𝑦(0), 𝜇(0)) ∫︁ 𝑡 − (𝜕𝑠 + 𝒜)𝑓 (𝑠, 𝑦(𝑠), 𝜇(𝑠))𝑑𝑠, 𝑡 ∈ [0, 𝑇 ],

(3.7)

0

is a continuous local (F𝑦 , 𝑃̂︀)-martingale, where 𝜇(𝑡) = 𝑃̂︀𝑦(𝑡) is the law of the coordinate process 𝑦 = (𝑦(𝑡))𝑡∈[0,𝑇 ] on 𝐶([0, 𝑇 ]; R) at time 𝑡, the filtration F𝑦 is that generated by 𝑦 and completed. Moreover, if 𝑓 ∈𝐶 1,2,1 ([0, 𝑇 ]×R×𝒫2 (R); R), this process 𝐶 𝑓 is an (F𝑦 , 𝑃̂︀)-martingale. 𝑏

36 / 44

Case 2: Existence of a weak solution Now we can give the main statement of this section.

Theorem 3.3 Under assumption (H4) mean-field SDE (3.3) has a weak solution ̃︀ 𝑄, ̃︀ ℱ, ̃︀ F, ̃︀ 𝐵, 𝑋). (Ω, Remark 2. If 𝑏, 𝜎 : [0, 𝑇 ] × 𝐶([0, 𝑇 ]; R) × 𝒫2 (𝐶([0, 𝑇 ]; R)) → R are bounded and continuous, then the following mean-field SDE ∫︁ 𝑡 ∫︁ 𝑡 𝑋𝑡 = 𝜉 + 𝑏(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )𝑑𝑠 + 𝜎(𝑠, 𝑋·∧𝑠 , 𝑄𝑋·∧𝑠 )𝑑𝐵𝑠 , 𝑡 ∈ [0, 𝑇 ], 0

0

(1.1) 𝐿2 (Ω, ℱ

where 𝜉 ∈ 0 , 𝑃 ) obeys a given distribution law 𝑄𝜉 = 𝜈, has a weak ̃︀ 𝑄, ̃︀ ℱ, ̃︀ F, ̃︀ 𝑋, 𝐵). solution (Ω, 37 / 44

Case 2: Uniqueness in law of weak solutions

Now we want to study the uniqueness in law for the weak solution of the mean-field SDE (3.3).

Definition 3.7 We call 𝒞 ⊂ 𝑏ℬ(R) = {𝜑 | 𝜑 : R → R bounded Borel-measurable function} a determining class on R, if for any two finite measures 𝜈1 and 𝜈2 on ∫︀ ∫︀ ℬ(R), R𝑑 𝜑(𝑥)𝜈1 (𝑑𝑥) = R𝑑 𝜑(𝑥)𝜈2 (𝑑𝑥) for all 𝜑 ∈ 𝒞 implies 𝜈1 = 𝜈2 . Remark: The class 𝐶0∞ (R) is a determining class on R.

38 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 For given 𝑓 ∈ 𝐶0∞ (R), we consider the Cauchy problem 𝜕 𝑣(𝑡, 𝑥, 𝜈) = 𝒜𝑣(𝑡, 𝑥, 𝜈), (𝑡, 𝑥, 𝜈) ∈ [0, 𝑇 ] × R × 𝒫2 (R), 𝜕𝑡 𝑣(0, 𝑥, 𝜈) = 𝑓 (𝑥), 𝑥 ∈ R,

(3.8)

where ∫︁ ̃︀ 𝒜𝑣(𝑡, 𝑥, 𝜈) = (𝒜𝑣)(𝑡, 𝑥, 𝜈) +

(𝜕𝜇 𝑣)(𝑡, 𝑥, 𝜈, 𝑢)𝑏(𝑡, 𝑢, 𝜈)𝜈(𝑑𝑢) ∫︁ 1 + 𝜕𝑧 (𝜕𝜇 𝑣)(𝑡, 𝑥, 𝜈, 𝑢)𝜎 2 (𝑡, 𝑢, 𝜈)𝜈(𝑑𝑢), 2 R 1 ̃︀ (𝒜𝑣)(𝑡, 𝑥, 𝜈) = 𝜕𝑦 𝑣(𝑡, 𝑥, 𝜈)𝑏(𝑡, 𝑥, 𝜈) + 𝜕𝑦2 𝑣(𝑡, 𝑥, 𝜈)𝜎 2 (𝑡, 𝑥, 𝜈), 2 R

(𝑡, 𝑥, 𝜈) ∈ [0, ∞) × R × 𝒫2 (R). 39 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 (continued) We suppose that, for all 𝑓 ∈ 𝐶0∞ (R), (3.8) has a solution ⋂︀ 𝑣𝑓 ∈ 𝐶𝑏 ([0, ∞) × R × 𝒫2 (R)) 𝐶𝑏1,2,1 ((0, ∞) × R × 𝒫2 (R)). Then, the local martingale problem associated with 𝒜̃︀ (Recall Definition 3.6) and with the initial condition 𝛿𝑥 has at most one solution. Remark: Theorem 3.4 generalizes a well-known classical uniqueness for weak solutions to the case of mean-field SDE.

Corollary 3.1 Under the assumption of Theorem 3.4, we have for the mean-field SDE (3.3) the uniqueness in law, that is, for any weak solutions, 𝑖 = 1, 2 (Ω𝑖 , ℱ 𝑖 , F𝑖 , 𝑄𝑖 , 𝐵 𝑖 , 𝑋 𝑖 ), of SDE (3.3), we have 𝑄1𝑋 1 = 𝑄2𝑋 2 . 40 / 44

Uniqueness in law of weak solutions Sketch of proof of Theorem 3.4: Let 𝑇 > 0, denote by 𝑦=(𝑦(𝑡))𝑡∈[0,𝑇 ] the coordinate process on 𝐶([0, 𝑇 ]; R). Let 𝑃 1 and 𝑃 2 be two arbitrary solutions of the local martingale problem associated with 𝒜̃︀ and initial 𝑙 condition 𝑥 ∈ R: 𝑃𝑦(0) = 𝛿𝑥 , 𝑙 = 1, 2.

Consequently, due to Lemma 3.2, for any 𝑔 ∈ 𝐶𝑏1,2,1 ([0, 𝑇 ] × R × 𝒫2 (R)), 𝐶

𝑔

(𝑡, 𝑦, 𝑃𝑦𝑙 )

:=

𝑙 𝑔(𝑡, 𝑦(𝑡), 𝑃𝑦(𝑡) )−𝑔(0, 𝑥, 𝛿𝑥 )−

∫︁ 0

𝑡 𝑙 (𝜕𝑠 +𝒜)𝑔(𝑠, 𝑦(𝑠), 𝑃𝑦(𝑠) )𝑑𝑠,

(3.9) 𝑃 𝑙 -martingale,

𝐶0∞ (R),

𝑙 = 1, 2, 𝑡 ∈ [0, 𝑇 ]. For given 𝑓 ∈ let ⋂︀ 1,2,1 𝑣𝑓 ∈ 𝐶𝑏 ([0, 𝑇 ] × R × 𝒫2 (R)) 𝐶𝑏 ((0, 𝑇 ) × R × 𝒫2 (R)) be a solution of

is a

the Cauchy problem (3.8). 41 / 44

Uniqueness in law of weak solutions Then putting 𝑔(𝑡, 𝑧, 𝜈) := 𝑣𝑓 (𝑇 −𝑡, 𝑧, 𝜈), 𝑡 ∈ [0, 𝑇 ], 𝑧 ∈ R, 𝜈 ∈ 𝒫2 (R), defines a function 𝑔 of class ⋂︀ 𝐶𝑏 ([0, 𝑇 ] × R × 𝒫2 (R)) 𝐶𝑏1,2,1 ((0, 𝑇 ) × R × 𝒫2 (R)) which satisfies 𝜕𝑠 𝑔(𝑠, 𝑧, 𝜈)+𝒜𝑔(𝑠, 𝑧, 𝜈) = 0, 𝑔(𝑇, 𝑧, 𝜈) = 𝑓 (𝑧), (𝑠, 𝑧, 𝜈)∈[0, 𝑇 ]×R×𝒫2 (R). From (3.9) we see that {𝐶 𝑔 (𝑠, 𝑦, 𝑃𝑦𝑙 ), 𝑠 ∈ [0, 𝑇 ]} is an (F𝑦 , 𝑃 𝑙 )∫︀ martingale. Hence, for 𝐸 𝑙 [·] = Ω𝑙 (·)𝑑𝑃 𝑙 , 𝑙 𝐸 𝑙 [𝑓 (𝑦(𝑇 ))] = 𝐸 𝑙 [𝑔(𝑇, 𝑦(𝑇 ), 𝑃𝑦(𝑇 ) )] = 𝑔(0, 𝑥, 𝛿𝑥 ), 𝑥 ∈ R, 𝑙 = 1, 2,

that is 𝐸 1 [𝑓 (𝑦(𝑇 ))] = 𝐸 2 [𝑓 (𝑦(𝑇 ))], for all 𝑓 ∈ 𝐶0∞ (R). Combining this 1 = 𝑃 2 , for every 𝑡 ≥ 0. with the arbitrariness of 𝑇 ≥ 0, we have that 𝑃𝑦(𝑡) 𝑦(𝑡) 42 / 44

Uniqueness in law of weak solutions Consequently, 𝑃 1 , 𝑃 2 are solutions of the same classical martingale problem, associated with 𝒜̃︀ = 𝒜̃︀𝑙 , 𝑙 = 1, 2, 𝒜̃︀𝑙 𝜑(𝑡, 𝑧) = 𝜕𝑦 𝜑(𝑡, 𝑧)̃︀𝑏𝑙 (𝑡, 𝑧) + 𝜕𝑦2 𝜑(𝑡, 𝑧)(̃︀ 𝜎 𝑙 (𝑡, 𝑧))2 , 𝜑 ∈ 𝐶 1,2 ([0, 𝑇 ] × R; R), with the coefficients 𝜎 ̃︀1 = 𝜎 ̃︀2 , ̃︀𝑏1 = ̃︀𝑏2 (without mean field term), 𝑙 𝑙 𝜎 ̃︀𝑙 (𝑡, 𝑧) = 𝜎(𝑡, 𝑧, 𝑃𝑦(𝑡) ), ̃︀𝑏𝑙 (𝑡, 𝑧) = 𝑏(𝑡, 𝑧, 𝑃𝑦(𝑡) ), (𝑡, 𝑧) ∈ [0, 𝑇 ] × R, 1 = 𝑃 2 , 𝑡 ∈ [0, 𝑇 ]. and we have seen that 𝑃𝑦(𝑡) 𝑦(𝑡)

.............. 𝑃 1 = 𝑃 2 , i.e., the local martingale problem has at most one solution.  43 / 44

Thank you very much!

谢谢!

44 / 44