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 2642...

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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 β€œPDE and Probability Methods for Interactions”, Inria, Sophia Antipolis, France, March 30-31, 2017.

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Contents

1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

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Objective of the talk

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Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

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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 𝑑 ∈ [0, 𝑇 ], πœ‰ ∈ 𝐿2 (Ω, β„±0 , 𝑃 ; R𝑑 ), ∫︁ 𝑋𝑑 = πœ‰ +

𝑑

∫︁

𝑑

𝑏(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )𝑑𝑠 + 0

𝜎(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )𝑑𝐡𝑠 ,

(1.1)

0

where 𝑄 is a probability measure with respect to which 𝐡 is a B.M. Remark: π‘„π‘‹Β·βˆ§π‘  is the law of π‘‹Β·βˆ§π‘  w.r.t. 𝑄.

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

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Preliminaries

We consider + (Ω, β„±, 𝑃 ) - complete probability space; + π‘Š B.M. over (Ω, β„±, 𝑃 ) (for simplicity: all processes 1-dimensional); + F-filtration generated by π‘Š , and augmented by β„±0 . 𝑝-Wasserstein metric on ∫︁ 𝒫𝑝 (R) := {πœ‡ | πœ‡ probab. on (R, ℬ(R)) with

|π‘₯|𝑝 πœ‡(π‘₯) < +∞};

R

π‘Šπ‘ (πœ‡, 𝜈):= inf

{οΈ€(οΈ€

∫︁ RΓ—R

)οΈ€ 1 }οΈ€ |π‘₯|𝑝 𝜌(𝑑π‘₯𝑑𝑦) 𝑝 , 𝜌(Β· Γ— R) = πœ‡, 𝜌(R Γ— Β·) = 𝜈 . (1.2)

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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 ΜƒοΈ€ 𝑄, 𝐡, 𝑋) is a weak solution of SDE (1.1), if ΜƒοΈ€ β„±, ΜƒοΈ€ F, A six-tuple (Ω, ΜƒοΈ€ = {ℱ̃︀𝑑 }0≀𝑑≀𝑇 is a ΜƒοΈ€ β„±, ΜƒοΈ€ 𝑄) is a complete probability space, and F (i) (Ω, ΜƒοΈ€ β„±, ΜƒοΈ€ 𝑄) satisfying the usual conditions. filtration on (Ω, ΜƒοΈ€ (ii) 𝑋 = {𝑋𝑑 }0≀𝑑≀𝑇 is a continuous, F-adapted R-valued process; ΜƒοΈ€ 𝑄)-BM. 𝐡 = {𝐡𝑑 }0≀𝑑≀𝑇 is an (F, βˆ«οΈ€ 𝑇 (iii) 𝑄{ 0 (|𝑏(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )| + |𝜎(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )|2 )𝑑𝑠 < +∞} = 1, and equation (1.1) is satisfied, 𝑄-a.s.

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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 , β„± 𝑖 , F𝑖 , 𝑄𝑖 , 𝐡 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.

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1

Objective of the talk

2

Case 1: The drift coefficient is bounded and measurable.

3

Case 2: The coefficients are bounded, continuous.

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Case 1: Existence of a weak solution Let 𝑏, 𝜎 satisfy the following assumption (H1): (i) 𝑏 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R) Γ— 𝒫1 (R) β†’ R is bounded and measurable; (ii) 𝜎 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R) β†’ R is bounded, measurable, and s.t., for all (𝑑, πœ™) ∈ [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R), 1/𝜎(𝑑, πœ™) is bounded in (𝑑, πœ™); (iii) (Modulus of continuity) βˆƒπœŒ : R+ β†’ R+ increasing, continuous, with 𝜌(0+) = 0 s.t., for all 𝑑 ∈ [0, 𝑇 ], πœ™ ∈ 𝐢([0, 𝑇 ]; R), πœ‡, 𝜈 ∈ 𝒫1 (R), |𝑏(𝑑, πœ™Β·βˆ§π‘‘ , πœ‡) βˆ’ 𝑏(𝑑, πœ™Β·βˆ§π‘‘ , 𝜈)| ≀ 𝜌(π‘Š1 (πœ‡, 𝜈)); (iv) βˆƒπΏ β‰₯ 0 s.t., for all 𝑑 ∈ [0, 𝑇 ], πœ™, πœ“ ∈ 𝐢([0, 𝑇 ]; R), |𝜎(𝑑, πœ™Β·βˆ§π‘‘ ) βˆ’ 𝜎(𝑑, πœ“Β·βˆ§π‘‘ )| ≀ 𝐿 sup |πœ™π‘  βˆ’ πœ“π‘  |. 0≀𝑠≀𝑑

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Case 1: Existence of a weak solution We want to study weak solutions of the following mean-field SDE: ∫︁ 𝑑 ∫︁ 𝑑 𝑋𝑑 = πœ‰ + 𝜎(𝑠, π‘‹Β·βˆ§π‘  )𝑑𝐡𝑠 + 𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝑄𝑋𝑠 )𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], 0

(2.1)

0

where (𝐡𝑑 )π‘‘βˆˆ[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 ΜƒοΈ€ 𝑄, 𝐡, 𝑋). ΜƒοΈ€ β„±, ΜƒοΈ€ F, (Ω, Proof: Girsanov’s Theorem. Schauder’s Fixed Point Theorem.

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Case 1: Existence of a weak solution Let us give two examples. Example 1. Take diffusion coefficient 𝜎 ≑ 𝐼𝑑 and drift coefficient βˆ«οΈ€ ̂︀𝑏(𝑠, πœ™Β·βˆ§π‘  , πœ‡π‘  ) := 𝑏(𝑠, πœ™Β·βˆ§π‘  , πœ“π‘‘πœ‡π‘  ), πœ™ ∈ 𝐢([0, 𝑇 ]), πœ‡ ∈ 𝒫1 (R), 𝑠 ∈ [0, 𝑇 ]; the function πœ“ ∈ 𝐢([0, 𝑇 ]; R) is arbitrarily given but fixed, and Lipschitz. Then our mean-field SDE (2.1) can be written as follows: ∫︁

𝑑

𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝐸𝑄 [πœ“(𝑋𝑠 )])𝑑𝑠, 𝑑 ∈ [0, 𝑇 ].

𝑋𝑑 = 𝐡𝑑 +

(2.2)

0

Here 𝑏 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]) Γ— R β†’ R is bounded, meas., Lips. in 𝑦. Then, the coefficients ̂︀𝑏 and 𝜎 satisfy (H1), and from Theorem 2.1, we obtain ΜƒοΈ€ 𝑄, 𝐡, 𝑋). ΜƒοΈ€ β„±, ΜƒοΈ€ 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 βˆ«οΈ€ ̂︀𝑏(𝑠, πœ™Β·βˆ§π‘  , πœ‡π‘  ) := 𝑏(𝑠, πœ™Β·βˆ§π‘  , 𝑦)πœ‡π‘  (𝑑𝑦), πœ™ ∈ 𝐢([0, 𝑇 ]), πœ‡π‘  ∈ 𝒫1 (R), 𝑠 ∈ [0, 𝑇 ], i.e., we consider the following mean-field SDE: ∫︁ 𝑑 ∫︁ 𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝑦)𝑄𝑋𝑠 (𝑑𝑦)𝑑𝑠, 𝑑 ∈ [0, 𝑇 ].

𝑋𝑑 = 𝐡𝑑 + 0

(2.3)

R

Here the coefficient 𝑏 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]) Γ— R β†’ R is bounded, meas. and Lips. in 𝑦. Then, the coefficients ̂︀𝑏 and 𝜎 satisfy (H1), and from Theorem ΜƒοΈ€ 𝑄, 𝐡, 𝑋). ΜƒοΈ€ β„±, ΜƒοΈ€ F, 2.1 the mean-field SDE (2.3) has a weak solution (Ω,

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Case 1: Uniqueness in law of weak solutions Let the functions 𝑏 and 𝜎 satisfy the following assumption (H2): (i) 𝑏 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R) Γ— 𝒫1 (R) β†’ R is bounded and measurable; (ii) 𝜎 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R) β†’ R is bounded and measurable, and |1/𝜎(𝑑, πœ™)| ≀ 𝐢, (𝑑, πœ™) ∈ [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R), for some 𝐢 ∈ R+ ; (iii) (Modulus of continuity) There exists a continuous and increasing function 𝜌 : R+ β†’ R+ with

∫︁

𝑑𝑒 = +∞, 0+ 𝜌(𝑒) such that, for all 𝑑 ∈ [0, 𝑇 ], πœ™ ∈ 𝐢([0, 𝑇 ]; R), πœ‡, 𝜈 ∈ 𝒫1 (R), 𝜌(π‘Ÿ) > 0, for all π‘Ÿ > 0, and

|𝑏(𝑑, πœ™Β·βˆ§π‘‘ , πœ‡) βˆ’ 𝑏(𝑑, πœ™Β·βˆ§π‘‘ , 𝜈)|2 ≀ 𝜌(π‘Š1 (πœ‡, 𝜈)2 ); (iv) βˆƒπΏ β‰₯ 0 such that, for all 𝑑 ∈ [0, 𝑇 ], πœ™, πœ“ ∈ 𝐢([0, 𝑇 ]; R), |𝜎(𝑑, πœ™Β·βˆ§π‘‘ ) βˆ’ 𝜎(𝑑, πœ“Β·βˆ§π‘‘ )| ≀ 𝐿 sup |πœ™π‘  βˆ’ πœ“π‘  |. 0≀𝑠≀𝑑 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 ∫︁ 𝑋𝑑 = πœ‰ +

𝑑

∫︁

0

𝑑

𝜎(𝑠, π‘‹Β·βˆ§π‘  )𝑑𝐡𝑠 , 𝑑 ∈ [0, 𝑇 ],

𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝑄𝑋𝑠 )𝑑𝑠 +

(2.1)

0

has a weak solution.

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

Case 1: Uniqueness in law of weak solutions Sketch of the proof: For πœ™ ∈ 𝐢([0, 𝑇 ]; R), πœ‡ ∈ 𝒫1 (R), we define ̃︀𝑏(𝑠, πœ™Β·βˆ§π‘  , πœ‡) = 𝜎 βˆ’1 (𝑠, πœ™Β·βˆ§π‘  )𝑏(𝑠, πœ™Β·βˆ§π‘  , πœ‡), and we introduce ⎧ ∫︁ 𝑑 βŽͺ 𝑖 𝑖 βŽͺ ̃︀𝑏(𝑠, 𝑋 𝑖 , 𝑄𝑖 𝑖 )𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], βŽͺ ⎨ π‘Šπ‘‘ = 𝐡𝑑 + Β·βˆ§π‘  𝑋𝑠 0 ∫︁ ∫︁ 𝑇 βŽͺ 1 𝑇 ΜƒοΈ€ βŽͺ 𝑖 𝑖 𝑖 𝑖 𝑖 ΜƒοΈ€ βŽͺ 𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝑄𝑋 𝑖 )𝑑𝐡𝑠 βˆ’ |𝑏(𝑠, π‘‹Β·βˆ§π‘  , 𝑄𝑖𝑋 𝑖 )|2 𝑑𝑠}, ⎩ 𝐿𝑇 = exp{βˆ’ 𝑠 𝑠 2 0 0 (2.4) 𝑖 = 1, 2. Then from the Girsanov Theorem we know that (π‘Šπ‘‘π‘– )π‘‘βˆˆ[0,𝑇 ] is an ΜƒοΈ€ 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 ∫︁ 𝑑 𝑖 𝜎(𝑠, π‘‹Β·βˆ§π‘  )π‘‘π‘Šπ‘ π‘– , 𝑑 ∈ [0, 𝑇 ]. (2.5) 𝑋𝑑𝑖 = 𝑋0𝑖 + 0 15 / 44

Case 1: Uniqueness in law of weak solutions It is by now standard that βˆƒ a meas. and non-anticipating function Ξ¦ : [0, 𝑇 ] Γ— R Γ— 𝐢([0, 𝑇 ]; R) β†’ R not depending on 𝑖 = 1, 2, s.t. ΜƒοΈ€ 𝑖 -a.s. (and, 𝑄𝑖 -a.s.), 𝑖 = 1, 2. (2.6) 𝑋𝑑𝑖 = Φ𝑑 (𝑋0𝑖 , π‘Š 𝑖 ), 𝑑 ∈ [0, 𝑇 ], 𝑄 βˆ«οΈ€ 𝑑 Then from (2.4) that π‘Šπ‘‘π‘– = 𝐡𝑑𝑖 + 0 ̃︀𝑏(𝑠, Ξ¦Β·βˆ§π‘  (𝑋0𝑖 , π‘Š 𝑖 ), 𝑄𝑖𝑋 𝑖 )𝑑𝑠, 𝑖 = 1, 2. 𝑠 Hence, putting 𝑓 (𝑠, πœ™Β·βˆ§π‘  ) = ̃︀𝑏(𝑠, πœ™Β·βˆ§π‘  , 𝑄1 ), (𝑠, πœ™) ∈ [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; 𝑋𝑠

R), from (2.4) and (2.6) we have ∫︁ 𝑑 ⎧ 1 1 βŽͺ βŽͺ π‘Š = 𝐡 + 𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋01 , π‘Š 1 ))𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], ⎨ 𝑑 𝑑 0 ∫︁ 𝑑 βŽͺ βŽͺ ̃︀𝑑2 + ⎩ π‘Šπ‘‘2 = 𝐡 𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋02 , π‘Š 2 ))𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], 0

where, 𝑑 ∈ [0, 𝑇 ], ∫︁ 𝑑 (︁ )︁ ̃︀𝑏(𝑠, Ξ¦Β·βˆ§π‘  (𝑋 2 , π‘Š 2 ), 𝑄2 2 ) βˆ’ ̃︀𝑏(𝑠, Ξ¦Β·βˆ§π‘  (𝑋 2 , π‘Š 2 ), 𝑄1 1 ) 𝑑𝑠. ΜƒοΈ€ 2 = 𝐡 2 + 𝐡 𝑑 𝑑 0 0 𝑋𝑠 𝑋𝑠 0

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Case 1: Uniqueness in law of weak solutions Β― : [0, 𝑇 ] Γ— R Γ— 𝐢([0, 𝑇 ]; R) β†’ R meas. s.t., for both 𝐡 1 , 𝐡 ΜƒοΈ€ 2 , Hence, βˆƒΞ¦ Β― 𝑑 (𝑋 1 , π‘Š 1 ) and 𝐡 ΜƒοΈ€ 2 = Ξ¦ Β― 𝑑 (𝑋 2 , π‘Š 2 ), 𝑑 ∈ [0, 𝑇 ]. 𝐡𝑑1 = Ξ¦ 0 𝑑 0

(2.8)

Now we define ⎧ Μ‚οΈ€ 2𝑑 𝑑𝐡𝑑2 , 𝑑 Μ‚οΈ€ 2𝑑 = βˆ’(̃︀𝑏(𝑠, Ξ¦Β·βˆ§π‘  (𝑋02 , π‘Š 2 ), 𝑄2 2 ) βˆ’ ̃︀𝑏(𝑠, Ξ¦Β·βˆ§π‘  (𝑋02 , π‘Š 2 ), 𝑄1 1 ))𝐿 ⎨ 𝑑𝐿 𝑋𝑠 𝑋𝑠 ⎩ 𝐿 Μ‚οΈ€ 20 = 1. (2.9) ΜƒοΈ€ 2 is an Brownian motion From the Girsanov Theorem we know that 𝐡 Μ‚οΈ€ 2 = 𝐿 Μ‚οΈ€ 2 𝑄2 . Moreover, putting under the probability measure 𝑄 𝑇

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Case 1: Uniqueness in law of weak solutions

⎧ ∫︁ 𝑇 ∫︁ 𝑇 βŽͺ ΜƒοΈ€ 2 = exp{βˆ’ 𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋 2 , π‘Š 2 ))π‘‘π‘Š 2 + 1 |𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋 2 , π‘Š 2 ))|2 𝑑𝑠}, ⎨𝐿 𝑇 0 𝑠 0 20 0 βŽͺ ⎩ Β―2 ΜƒοΈ€ 2 𝑄 Μ‚οΈ€ 2 , 𝑄 =𝐿 𝑇

(2.10) we have that (π‘Šπ‘‘1 )π‘‘βˆˆ[0,𝑇 ]

(π‘Šπ‘‘2 )π‘‘βˆˆ[0,𝑇 ]

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

ΜƒοΈ€ 2 𝑄

and

Β― 2, 𝑄

while

On the other hand, since 𝑓 is bounded and meas., we can prove that ΜƒοΈ€ : R Γ— 𝐢([0, 𝑇 ]; R) β†’ R, s.t. βˆƒ a meas. function Ξ¦ ΜƒοΈ€ 0𝑖 , π‘Š 𝑖 ) = Ξ¦(𝑋

∫︁

𝑇

𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋0𝑖 , π‘Š 𝑖 ))π‘‘π‘Šπ‘ π‘– , 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 ∫︁ βŽͺ 1 𝑇 1 1 1 1 βŽͺ βŽͺ |𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋01 , π‘Š 1 ))|2 𝑑𝑠}, ⎨ 𝐿𝑇 = exp{βˆ’ 𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋0 , π‘Š ))π‘‘π‘Šπ‘  + 2 0 0 ∫︁ 𝑇 ∫︁ 𝑇 βŽͺ 1 βŽͺ βŽͺ ΜƒοΈ€ 2 = exp{βˆ’ 𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋 2 , π‘Š 2 ))π‘‘π‘Š 2 + |𝑓 (𝑠, Ξ¦Β·βˆ§π‘  (𝑋02 , π‘Š 2 ))|2 𝑑𝑠}, ⎩𝐿 𝑇 0 𝑠 2 0 0 (2.11) Μ‚οΈ€ and we see that βˆƒ a meas. function Ξ¦ : R Γ— 𝐢([0, 𝑇 ]; R) β†’ R, s.t. Μ‚οΈ€ 01 , π‘Š 1 ), 𝑄1 -a.s., and 𝐿 ΜƒοΈ€ 2𝑇 = Ξ¦(𝑋 Μ‚οΈ€ 02 , π‘Š 2 ), 𝑄2 -a.s. (and, 𝑄 Β― 2 -a.s.). 𝐿1𝑇 = Ξ¦(𝑋 (2.12)

19 / 44

Case 1: Uniqueness in law of weak solutions Consequently, as 𝑋0𝑖 is β„±0𝑖 -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 : 𝐢([0, 𝑇 ]; R𝑑 )2 β†’ R, 1 Β― 1 , π‘Š 1 ), π‘Š 1 )] 𝐸𝑄1 [𝐹 (𝐡 1 , π‘Š 1 )] = 𝐸𝑄̃︀1 [ 𝐹 (Ξ¦(𝑋 0 1 Μ‚οΈ€ Ξ¦(𝑋 , π‘Š 1) 0

= 𝐸𝑄¯ 2 [

1 Μ‚οΈ€ 2 , π‘Š 2 ) Ξ¦(𝑋 0

ΜƒοΈ€ 2 , π‘Š 2 )]. Β― 02 , π‘Š 2 ), π‘Š 2 )] = 𝐸 Μ‚οΈ€2 [𝐹 (𝐡 𝐹 (Ξ¦(𝑋 𝑄

That is, Μ‚οΈ€ 2 𝑄1(𝐡 1 ,π‘Š 1 ) = 𝑄 ΜƒοΈ€ 2 ,π‘Š 2 ) . (𝐡

(2.13)

Taking into account (2.6), we have Μ‚οΈ€ 2 𝑄1(𝐡 1 ,π‘Š 1 ,𝑋 1 ) = 𝑄 ΜƒοΈ€ 2 ,π‘Š 2 ,𝑋 2 ) , (𝐡

(2.14)

Μ‚οΈ€ 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 βˆ«οΈ€ Μ‚οΈ€ 2 2 , 𝑄2 2 )2 ≀ 𝐢 𝑠 𝜌(π‘Š1 (𝑄1 1 , 𝑄2 2 )2 )π‘‘π‘Ÿ; βˆ™π‘Š1 (𝑄1𝑋 1 , 𝑄2𝑋 2 )2 = π‘Š1 (𝑄 0 𝑋 𝑋 𝑋 𝑋 𝑠

𝑠

π‘Ÿ

𝑠

𝑠

π‘Ÿ

βˆ™ The continuity of 𝑠 β†’ π‘Š1 (𝑄1𝑋 1 , 𝑄2𝑋 2 ). 𝑠

𝑠

Putting 𝑒(𝑠) := π‘Š1 (𝑄1𝑋 1 , 𝑄2𝑋 2 ), 𝑠 ∈ [0, 𝑇 ], then we have from above, 𝑠 βˆ«οΈ€ 𝑠𝑠 𝑒(𝑠)2 ≀ 𝐢 0 𝜌(𝑒(π‘Ÿ)2 )π‘‘π‘Ÿ, 0 ≀ 𝑠 ≀ 𝑑 ≀ 𝑇 . βˆ«οΈ€ 𝑑𝑒 From (H2)-(iii), 0+ 𝜌(𝑒) = +∞, it follows from Bihari’s inequality that 𝑒(𝑠) = 0, for any 𝑠 ∈ [0, 𝑇 ], that is, 𝑄1𝑋 1 = 𝑄2𝑋 2 , 𝑠 ∈ [0, 𝑇 ]. Thus, from 𝑠 𝑠 ΜƒοΈ€ 2 = 𝐡 2 , 𝐿 Μ‚οΈ€ 2 = 1, and, consequently, (2.7) and (2.9) it follows that 𝐡 𝑇 2 2 2 2 Μ‚οΈ€ Μ‚οΈ€ 𝑄 = 𝑄 . Then, 𝑄 = 𝑄 2 2 2 , and from (2.14) ΜƒοΈ€ 2 ,π‘Š 2 ,𝑋 2 ) (𝐡

(𝐡 ,π‘Š ,𝑋 )

𝑄1(𝐡 1 ,π‘Š 1 ,𝑋 1 ) = 𝑄2(𝐡 2 ,π‘Š 2 ,𝑋 2 ) . This implies, in particular, 𝑄1(𝐡 1 ,𝑋 1 ) = 𝑄2(𝐡 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 (𝐢([0, 𝑇 ]; R), ℬ(𝐢([0, 𝑇 ]; R))) is a solution to the local martingale problem associated with π’œβ€² , if for every 𝑓 ∈ 𝐢 1,2 ([0, 𝑇 ]Γ—R; R), 𝑀𝑑𝑓

∫︁ := 𝑓 (𝑑, 𝑦(𝑑)) βˆ’ 𝑓 (0, 𝑦(0)) βˆ’

𝑑

(πœ•π‘  + π’œβ€² )𝑓 (𝑠, 𝑦(𝑠))𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], (3.1)

0

is a continuous local martingale w.r.t (F𝑦 , 𝑃̂︀), where 𝑦 = (𝑦(𝑑))π‘‘βˆˆ[0,𝑇 ] is the coordinate process on 𝐢([0, 𝑇 ]; R), the considered filtration F𝑦 = (ℱ𝑑𝑦 )π‘‘βˆˆ[0,𝑇 ] is that generated by 𝑦 = (𝑦(𝑑))π‘‘βˆˆ[0,𝑇 ] and augmented by all 𝑃̂︀-null sets, and π’œβ€² is defined by, 𝑦 ∈ 𝐢([0, 𝑇 ]; R), 1 π’œβ€² 𝑓 (𝑠, 𝑦) = 𝑏(𝑠, 𝑦)πœ•π‘₯ 𝑓 (𝑠, 𝑦(𝑠)) + 𝜎 2 (𝑠, 𝑦)πœ•π‘₯2 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 ΜƒοΈ€ 𝑃̃︀, π‘Š ΜƒοΈ€ β„±, ΜƒοΈ€ F, ̃︁ , 𝑋) to the following The existence of a weak solution (Ω, functional SDE with given initial distribution πœ‡ on ℬ(R): ∫︁ 𝑋𝑑 = πœ‰ +

𝑑

∫︁

0

𝑑

̃︁𝑠 , 𝑑 ∈ [0, 𝑇 ], 𝜎(𝑠, π‘‹Β·βˆ§π‘  )π‘‘π‘Š

𝑏(𝑠, π‘‹Β·βˆ§π‘  )𝑑𝑠 + 0

is equivalent to the existence of a solution 𝑃̂︀ to the local martingale problem (3.1) associated with π’œβ€² defined by (3.2), with 𝑃̂︀𝑦(0) = πœ‡. The both solutions are related by 𝑃̂︀ = 𝑃̃︀ ∘ 𝑋 βˆ’1 , i.e., the probability measure 𝑃̂︀ is the law of the weak solution 𝑋 on (𝐢([0, 𝑇 ]; R), ℬ(𝐢([0, 𝑇 ]; R))).

23 / 44

Case 2: Preliminaries Recall the definition of the derivative of 𝑓 : 𝒫2 (R) β†’ R w.r.t probability measure πœ‡ ∈ 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, β„±, 𝑃 ; R) β†’ R is FrΒ΄echet differentiable at πœ‰ ∈ 𝐿2 (Ω, β„±, 𝑃 ), if βˆƒ a linear continuous mapping 𝐷𝑓̃︀(πœ‰)(Β·) ∈ 𝐿(𝐿2 (Ω, β„±, 𝑃 ; R); R), s.t. 𝑓̃︀(πœ‰ + πœ‚)βˆ’π‘“ΜƒοΈ€(πœ‰) = 𝐷𝑓̃︀(πœ‰)(πœ‚) + π‘œ(|πœ‚|𝐿2 ), with |πœ‚|𝐿2 β†’ 0 for πœ‚ ∈ 𝐿2 (Ω, β„±, 𝑃 ). (ii) 𝑓 : 𝒫2 (R) β†’ R is differentiable at πœ‡ ∈ 𝒫2 (R), if for 𝑓̃︀(πœ‰) := 𝑓 (π‘ƒπœ‰ ), πœ‰ ∈ 𝐿2 (Ω, β„±, 𝑃 ; R), there is some 𝜁 ∈ 𝐿2 (Ω, β„±, 𝑃 ; R) with π‘ƒπœ = πœ‡ such that 𝑓̃︀ : 𝐿2 (Ω, β„±, 𝑃 ; R) β†’ R is FrΒ΄echet differentiable in 𝜁. 24 / 44

Case 2: Preliminaries From Riesz’ Representation Theorem there exists a 𝑃 -a.s. unique variable πœ— ∈ 𝐿2 (Ω, β„±, 𝑃 ; R) such that 𝐷𝑓̃︀(𝜁)(πœ‚) = (πœ—, πœ‚)𝐿2 = 𝐸[πœ—πœ‚], for all πœ‚ ∈ 𝐿2 (Ω, β„±, 𝑃 ; R). P.L. Lions proved that there is a Borel function β„Ž : R β†’ R such that πœ— = β„Ž(𝜁), 𝑃 -a.e., and function β„Ž depends on 𝜁 only through its law π‘ƒπœ . Therefore, 𝑓 (π‘ƒπœ‰ ) βˆ’ 𝑓 (π‘ƒπœ ) = 𝐸[β„Ž(𝜁) Β· (πœ‰ βˆ’ 𝜁)] + π‘œ(|πœ‰ βˆ’ 𝜁|𝐿2 ), πœ‰ ∈ 𝐿2 (Ω, β„±, 𝑃 ; R).

Definition 3.3 We call πœ•πœ‡ 𝑓 (π‘ƒπœ , 𝑦) := β„Ž(𝑦), 𝑦 ∈ R, the derivative of function 𝑓 : 𝒫2 (R) β†’ R at π‘ƒπœ , 𝜁 ∈ 𝐿2 (Ω, β„±, 𝑃 ; R). Remark: πœ•πœ‡ 𝑓 (π‘ƒπœ , 𝑦) is only π‘ƒπœ (𝑑𝑦)-a.e. uniquely determined. 25 / 44

Case 2: Preliminaries Definition 3.4 We say that 𝑓 ∈ 𝐢 1 (𝒫2 (R)), if for all πœ‰ ∈ 𝐿2 (Ω, β„±, 𝑃 ; R) there exists a π‘ƒπœ‰ -modification of πœ•πœ‡ 𝑓 (π‘ƒπœ‰ , .), also denoted by πœ•πœ‡ 𝑓 (π‘ƒπœ‰ , .), such that πœ•πœ‡ 𝑓 : 𝒫2 (R) Γ— R β†’ 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 πœ•πœ‡ 𝑓 as the derivative of 𝑓 . The function 𝑓 is said to belong to 𝐢𝑏1,1 (𝒫2 (R)), if 𝑓 ∈ 𝐢 1 (𝒫2 (R)) is s.t. πœ•πœ‡ 𝑓 : 𝒫2 (R) Γ— R β†’ R is bounded and Lipschitz continuous, i.e., there exists some constant 𝐢 β‰₯ 0 such that (i) |πœ•πœ‡ 𝑓 (πœ‡, π‘₯)| ≀ 𝐢, πœ‡ ∈ 𝒫2 (R), π‘₯ ∈ R; (ii) |πœ•πœ‡ 𝑓 (πœ‡, π‘₯)βˆ’πœ•πœ‡ 𝑓 (πœ‡β€² , π‘₯β€² )|≀𝐢(π‘Š2 (πœ‡, πœ‡β€² )+|π‘₯βˆ’π‘₯β€² |), πœ‡, πœ‡β€² ∈ 𝒫2 (R), π‘₯, π‘₯β€² ∈ R. 26 / 44

Case 2: Preliminaries

Definition 3.5 We say that 𝑓 ∈ 𝐢 2 (𝒫2 (R)), if 𝑓 ∈ 𝐢 1 (𝒫2 (R)) and πœ•πœ‡ 𝑓 (πœ‡, .) : R β†’ R is differentiable, and its derivative πœ•π‘¦ πœ•πœ‡ 𝑓 : 𝒫2 (R)Γ—Rβ†’R βŠ— R is continuous, for every πœ‡ ∈ 𝒫2 (R). Moreover, 𝑓 ∈ 𝐢𝑏2,1 (𝒫2 (R)), if 𝑓 ∈ 𝐢 2 (𝒫2 (R))

β‹‚οΈ€

𝐢𝑏1,1 (𝒫2 (R)) and its

derivative πœ•π‘¦ πœ•πœ‡ 𝑓 : 𝒫2 (R) Γ— R β†’ R βŠ— R is bounded and Lipschitzcontinuous. Remark: 𝐢𝑏2,1 (R Γ— 𝒫2 (R)), 𝐢𝑏1,2,1 ([0, 𝑇 ]Γ—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 = (πœŽπ‘  ), 𝛾 = (𝛾𝑠 ), 𝑏 = (𝑏𝑠 ), 𝛽 = (𝛽𝑠 ) R-valued adapted stochastic processes, such that βˆ«οΈ€ 𝑇 3 (i) There exists a constant π‘ž > 6 s.t. 𝐸[( 0 (|πœŽπ‘  |π‘ž + |𝑏𝑠 |π‘ž )𝑑𝑠) π‘ž ] < +∞; βˆ«οΈ€ 𝑇 (ii) 0 (|𝛾𝑠 |2 + |𝛽𝑠 |)𝑑𝑠 < +∞, P-a.s. Let 𝐹 ∈ 𝐢𝑏1,2,1 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R)). Then, for the It^ o processes 𝑑

∫︁ 𝑋𝑑 = 𝑋0 +

𝑑

∫︁

𝑏𝑠 𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], 𝑋0 ∈ 𝐿2 (Ω, β„±0 , 𝑃 ),

πœŽπ‘  π‘‘π‘Šπ‘  + 0

∫︁ π‘Œπ‘‘ = π‘Œ0 +

0

𝑑

∫︁ 𝛾𝑠 π‘‘π‘Šπ‘  +

0

𝑑

𝛽𝑠 𝑑𝑠, 𝑑 ∈ [0, 𝑇 ], π‘Œ0 ∈ 𝐿2 (Ω, β„±0 , 𝑃 ),

0 28 / 44

Case 2: Preliminaries Theorem 3.1 (continued) we have 𝐹 (𝑑, π‘Œπ‘‘ , 𝑃𝑋𝑑 ) βˆ’ 𝐹 (0, π‘Œ0 , 𝑃𝑋0 ) ∫︁ 𝑑 (οΈ‚ 1 = πœ•π‘Ÿ 𝐹 (π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ ) + πœ•π‘¦ 𝐹 (π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ )π›½π‘Ÿ + πœ•π‘¦2 𝐹 (π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ )π›Ύπ‘Ÿ2 2 0 )οΈ‚ 1 2 Β― Β― Β― Β― + 𝐸[(πœ•πœ‡ 𝐹 )(π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ , π‘‹π‘Ÿ )π‘π‘Ÿ + πœ•π‘§ (πœ•πœ‡ 𝐹 )(π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ , π‘‹π‘Ÿ )Β― πœŽπ‘Ÿ ] π‘‘π‘Ÿ 2 ∫︁ 𝑑 + πœ•π‘¦ 𝐹 (π‘Ÿ, π‘Œπ‘Ÿ , π‘ƒπ‘‹π‘Ÿ )π›Ύπ‘Ÿ π‘‘π‘Šπ‘Ÿ , 𝑑 ∈ [0, 𝑇 ]. 0

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

Case 2: Preliminaries (H3) The coefficients (𝜎, 𝑏) ∈ 𝐢𝑏1,2,1 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R); R Γ— R).

Theorem 3.2 (Buckdahn, Li, Peng and Rainer, 2014) Let Ξ¦ ∈ 𝐢𝑏2,1 (R Γ— 𝒫2 (R)), then under assumption (H3) the following PDE: ⎧ 1 βŽͺ 0 = πœ•π‘‘ 𝑉 (𝑑, π‘₯, πœ‡) + πœ•π‘₯ 𝑉 (𝑑, π‘₯, πœ‡)𝑏(π‘₯, πœ‡) + πœ•π‘₯2 𝑉 (𝑑, π‘₯, πœ‡)𝜎 2 (π‘₯, πœ‡) βŽͺ βŽͺ βŽͺ 2 βŽͺ ∫︁ βŽͺ βŽͺ βŽͺ βŽͺ + (πœ•πœ‡ 𝑉 )(𝑑, π‘₯, πœ‡, 𝑦)𝑏(𝑦, πœ‡)πœ‡(𝑑𝑦) βŽͺ βŽͺ βŽͺ R∫︁ ⎨ 1 πœ•π‘¦ (πœ•πœ‡ 𝑉 )(𝑑, π‘₯, πœ‡, 𝑦)𝜎 2 (𝑦, πœ‡)πœ‡(𝑑𝑦), + βŽͺ βŽͺ 2 βŽͺ R βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ (𝑑, π‘₯, πœ‡) ∈ [0, 𝑇 ) Γ— R Γ— 𝒫2 (R); βŽͺ βŽͺ βŽͺ βŽͺ ⎩ 𝑉 (𝑇, π‘₯, πœ‡) = Ξ¦(π‘₯, πœ‡), (π‘₯, πœ‡) ∈ R Γ— 𝒫2 (R). has a unique classical solution 𝑉 (𝑑, π‘₯, πœ‡) ∈ 𝐢𝑏1,2,1 ([0, 𝑇 ] Γ— R Γ— 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 ] Γ— R Γ— 𝒫2 (R) β†’ R are continuous and bounded. We want to study weak solution of the following mean-field SDE: ∫︁ 𝑑 ∫︁ 𝑑 𝜎(𝑠, 𝑋𝑠 , 𝑄𝑋𝑠 )𝑑𝐡𝑠 , 𝑑 ∈ [0, 𝑇 ], 𝑋𝑑 = πœ‰ + 𝑏(𝑠, 𝑋𝑠 , 𝑄𝑋𝑠 )𝑑𝑠 0

(3.3)

0

where πœ‰ ∈ 𝐿2 (Ω, β„±0 , 𝑃 ; R) obeys a given distribution law π‘„πœ‰ = 𝜈 ∈ 𝒫2 (R) and (𝐡𝑑 )π‘‘βˆˆ[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 (𝐢([0, 𝑇 ]; R), ℬ(𝐢([0, 𝑇 ]; R))) is a solution to the local martingale problem (resp., martingale problem) associated with ΜƒοΈ€ if for every 𝑓 ∈ 𝐢 1,2 ([0, 𝑇 ] Γ— R; R) (resp., 𝑓 ∈ 𝐢 1,2 ([0, 𝑇 ] Γ— R; R)), the π’œ, 𝑏

process ∫︁

𝑓

𝐢 (𝑑, 𝑦, πœ‡) := 𝑓 (𝑑, 𝑦(𝑑)) βˆ’ 𝑓 (0, 𝑦(0)) βˆ’

𝑑 (οΈ€

)οΈ€ ΜƒοΈ€ (𝑠, 𝑦(𝑠), πœ‡(𝑠))𝑑𝑠, (πœ•π‘  + π’œ)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 πœ‡(𝑑) = 𝑃̂︀𝑦(𝑑) is the law of the coordinate process 𝑦 = (𝑦(𝑑))π‘‘βˆˆ[0,𝑇 ] on 𝐢([0, 𝑇 ]; R) at time 𝑑, the filtration F𝑦 is that generated by 𝑦 and completed, and π’œΜƒοΈ€ is defined by ΜƒοΈ€ )(𝑠, 𝑦, 𝜈) := πœ•π‘¦ 𝑓 (𝑠, 𝑦)𝑏(𝑠, 𝑦, 𝜈) + 1 πœ•π‘¦2 𝑓 (𝑠, 𝑦)𝜎 2 (𝑠, 𝑦, 𝜈), (π’œπ‘“ 2

(3.5)

ΜƒοΈ€ )(𝑠, 𝑦(𝑠), πœ‡(𝑠)) abbreviates (𝑠, 𝑦, 𝜈) ∈ [0, 𝑇 ] Γ— R Γ— 𝒫2 (R). Here ((πœ•π‘  + 𝐴)𝑓 ΜƒοΈ€ )(𝑠, 𝑦(𝑠), πœ‡(𝑠)) := (πœ•π‘  𝑓 )(𝑠, 𝑦(𝑠)) + (𝐴𝑓 ΜƒοΈ€ )(𝑠, 𝑦(𝑠), πœ‡(𝑠)). ((πœ•π‘  + 𝐴)𝑓

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Case 2: Existence of a weak solution

Proposition 3.1 ΜƒοΈ€ 𝑄, 𝐡, 𝑋) to equation (3.3) with ΜƒοΈ€ β„±, ΜƒοΈ€ 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 π’œΜƒοΈ€ defined by (3.5), with 𝑃̂︀𝑦(0) = 𝜈.

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Case 2: Existence of a weak solution

Lemma 3.2 Let the probability measure 𝑃̂︀ on (𝐢([0, 𝑇 ]; R), ℬ(𝐢([0, 𝑇 ]; R))) be a ΜƒοΈ€ Then, for the solution to the local martingale problem associated with π’œ. second order differential operator (οΈ€

∫︁ ΜƒοΈ€ π’œπ‘“ )(𝑠, 𝑦, 𝜈) := (π’œπ‘“ )(𝑠, 𝑦, 𝜈) + (πœ•πœ‡ 𝑓 )(𝑠, 𝑦, 𝜈, 𝑧)𝑏(𝑠, 𝑧, 𝜈)𝜈(𝑑𝑧) R ∫︁ 1 + πœ•π‘§ (πœ•πœ‡ 𝑓 )(𝑠, 𝑦, 𝜈, 𝑧)𝜎 2 (𝑠, 𝑧, 𝜈)𝜈(𝑑𝑧), 2 R (3.6)

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Case 2: Existence of a weak solution Lemma 3.2 (continued) applying to functions 𝑓 ∈ 𝐢 1,2 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R); R) we have that, for every such 𝑓 ∈ 𝐢 1,2 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R); R), the process 𝐢 𝑓 (𝑑, 𝑦, πœ‡) :=𝑓 (𝑑, 𝑦(𝑑), πœ‡(𝑑)) βˆ’ 𝑓 (0, 𝑦(0), πœ‡(0)) ∫︁ 𝑑 βˆ’ (πœ•π‘  + π’œ)𝑓 (𝑠, 𝑦(𝑠), πœ‡(𝑠))𝑑𝑠, 𝑑 ∈ [0, 𝑇 ],

(3.7)

0

is a continuous local (F𝑦 , 𝑃̂︀)-martingale, where πœ‡(𝑑) = 𝑃̂︀𝑦(𝑑) is the law of the coordinate process 𝑦 = (𝑦(𝑑))π‘‘βˆˆ[0,𝑇 ] on 𝐢([0, 𝑇 ]; R) at time 𝑑, the filtration F𝑦 is that generated by 𝑦 and completed. Moreover, if 𝑓 ∈𝐢 1,2,1 ([0, 𝑇 ]Γ—R×𝒫2 (R); R), this process 𝐢 𝑓 is an (F𝑦 , 𝑃̂︀)-martingale. 𝑏

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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 ΜƒοΈ€ 𝑄, ΜƒοΈ€ β„±, ΜƒοΈ€ F, ΜƒοΈ€ 𝐡, 𝑋). (Ω, Remark 2. If 𝑏, 𝜎 : [0, 𝑇 ] Γ— 𝐢([0, 𝑇 ]; R) Γ— 𝒫2 (𝐢([0, 𝑇 ]; R)) β†’ R are bounded and continuous, then the following mean-field SDE ∫︁ 𝑑 ∫︁ 𝑑 𝑋𝑑 = πœ‰ + 𝑏(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )𝑑𝑠 + 𝜎(𝑠, π‘‹Β·βˆ§π‘  , π‘„π‘‹Β·βˆ§π‘  )𝑑𝐡𝑠 , 𝑑 ∈ [0, 𝑇 ], 0

0

(1.1) 𝐿2 (Ω, β„±

where πœ‰ ∈ 0 , 𝑃 ) obeys a given distribution law π‘„πœ‰ = 𝜈, has a weak ΜƒοΈ€ 𝑄, ΜƒοΈ€ β„±, ΜƒοΈ€ F, ΜƒοΈ€ 𝑋, 𝐡). 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 π’ž βŠ‚ 𝑏ℬ(R) = {πœ‘ | πœ‘ : R β†’ R bounded Borel-measurable function} a determining class on R, if for any two finite measures 𝜈1 and 𝜈2 on βˆ«οΈ€ βˆ«οΈ€ ℬ(R), R𝑑 πœ‘(π‘₯)𝜈1 (𝑑π‘₯) = R𝑑 πœ‘(π‘₯)𝜈2 (𝑑π‘₯) for all πœ‘ ∈ π’ž implies 𝜈1 = 𝜈2 . Remark: The class 𝐢0∞ (R) is a determining class on R.

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Case 2: Uniqueness in law of weak solutions Theorem 3.4 For given 𝑓 ∈ 𝐢0∞ (R), we consider the Cauchy problem πœ• 𝑣(𝑑, π‘₯, 𝜈) = π’œπ‘£(𝑑, π‘₯, 𝜈), (𝑑, π‘₯, 𝜈) ∈ [0, 𝑇 ] Γ— R Γ— 𝒫2 (R), πœ•π‘‘ 𝑣(0, π‘₯, 𝜈) = 𝑓 (π‘₯), π‘₯ ∈ R,

(3.8)

where ∫︁ ΜƒοΈ€ π’œπ‘£(𝑑, π‘₯, 𝜈) = (π’œπ‘£)(𝑑, π‘₯, 𝜈) +

(πœ•πœ‡ 𝑣)(𝑑, π‘₯, 𝜈, 𝑒)𝑏(𝑑, 𝑒, 𝜈)𝜈(𝑑𝑒) ∫︁ 1 + πœ•π‘§ (πœ•πœ‡ 𝑣)(𝑑, π‘₯, 𝜈, 𝑒)𝜎 2 (𝑑, 𝑒, 𝜈)𝜈(𝑑𝑒), 2 R 1 ΜƒοΈ€ (π’œπ‘£)(𝑑, π‘₯, 𝜈) = πœ•π‘¦ 𝑣(𝑑, π‘₯, 𝜈)𝑏(𝑑, π‘₯, 𝜈) + πœ•π‘¦2 𝑣(𝑑, π‘₯, 𝜈)𝜎 2 (𝑑, π‘₯, 𝜈), 2 R

(𝑑, π‘₯, 𝜈) ∈ [0, ∞) Γ— R Γ— 𝒫2 (R). 39 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 (continued) We suppose that, for all 𝑓 ∈ 𝐢0∞ (R), (3.8) has a solution β‹‚οΈ€ 𝑣𝑓 ∈ 𝐢𝑏 ([0, ∞) Γ— R Γ— 𝒫2 (R)) 𝐢𝑏1,2,1 ((0, ∞) Γ— R Γ— 𝒫2 (R)). Then, the local martingale problem associated with π’œΜƒοΈ€ (Recall Definition 3.6) and with the initial condition 𝛿π‘₯ 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 , β„± 𝑖 , F𝑖 , 𝑄𝑖 , 𝐡 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(𝑑))π‘‘βˆˆ[0,𝑇 ] the coordinate process on 𝐢([0, 𝑇 ]; R). Let 𝑃 1 and 𝑃 2 be two arbitrary solutions of the local martingale problem associated with π’œΜƒοΈ€ and initial 𝑙 condition π‘₯ ∈ R: 𝑃𝑦(0) = 𝛿π‘₯ , 𝑙 = 1, 2.

Consequently, due to Lemma 3.2, for any 𝑔 ∈ 𝐢𝑏1,2,1 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R)), 𝐢

𝑔

(𝑑, 𝑦, 𝑃𝑦𝑙 )

:=

𝑙 𝑔(𝑑, 𝑦(𝑑), 𝑃𝑦(𝑑) )βˆ’π‘”(0, π‘₯, 𝛿π‘₯ )βˆ’

∫︁ 0

𝑑 𝑙 (πœ•π‘  +π’œ)𝑔(𝑠, 𝑦(𝑠), 𝑃𝑦(𝑠) )𝑑𝑠,

(3.9) 𝑃 𝑙 -martingale,

𝐢0∞ (R),

𝑙 = 1, 2, 𝑑 ∈ [0, 𝑇 ]. For given 𝑓 ∈ let β‹‚οΈ€ 1,2,1 𝑣𝑓 ∈ 𝐢𝑏 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R)) 𝐢𝑏 ((0, 𝑇 ) Γ— R Γ— 𝒫2 (R)) be a solution of

is a

the Cauchy problem (3.8). 41 / 44

Uniqueness in law of weak solutions Then putting 𝑔(𝑑, 𝑧, 𝜈) := 𝑣𝑓 (𝑇 βˆ’π‘‘, 𝑧, 𝜈), 𝑑 ∈ [0, 𝑇 ], 𝑧 ∈ R, 𝜈 ∈ 𝒫2 (R), defines a function 𝑔 of class β‹‚οΈ€ 𝐢𝑏 ([0, 𝑇 ] Γ— R Γ— 𝒫2 (R)) 𝐢𝑏1,2,1 ((0, 𝑇 ) Γ— R Γ— 𝒫2 (R)) which satisfies πœ•π‘  𝑔(𝑠, 𝑧, 𝜈)+π’œπ‘”(𝑠, 𝑧, 𝜈) = 0, 𝑔(𝑇, 𝑧, 𝜈) = 𝑓 (𝑧), (𝑠, 𝑧, 𝜈)∈[0, 𝑇 ]Γ—R×𝒫2 (R). From (3.9) we see that {𝐢 𝑔 (𝑠, 𝑦, 𝑃𝑦𝑙 ), 𝑠 ∈ [0, 𝑇 ]} is an (F𝑦 , 𝑃 𝑙 )βˆ«οΈ€ martingale. Hence, for 𝐸 𝑙 [Β·] = Ω𝑙 (Β·)𝑑𝑃 𝑙 , 𝑙 𝐸 𝑙 [𝑓 (𝑦(𝑇 ))] = 𝐸 𝑙 [𝑔(𝑇, 𝑦(𝑇 ), 𝑃𝑦(𝑇 ) )] = 𝑔(0, π‘₯, 𝛿π‘₯ ), π‘₯ ∈ R, 𝑙 = 1, 2,

that is 𝐸 1 [𝑓 (𝑦(𝑇 ))] = 𝐸 2 [𝑓 (𝑦(𝑇 ))], for all 𝑓 ∈ 𝐢0∞ (R). Combining this 1 = 𝑃 2 , for every 𝑑 β‰₯ 0. with the arbitrariness of 𝑇 β‰₯ 0, we have that 𝑃𝑦(𝑑) 𝑦(𝑑) 42 / 44

Uniqueness in law of weak solutions Consequently, 𝑃 1 , 𝑃 2 are solutions of the same classical martingale problem, associated with π’œΜƒοΈ€ = π’œΜƒοΈ€π‘™ , 𝑙 = 1, 2, π’œΜƒοΈ€π‘™ πœ‘(𝑑, 𝑧) = πœ•π‘¦ πœ‘(𝑑, 𝑧)̃︀𝑏𝑙 (𝑑, 𝑧) + πœ•π‘¦2 πœ‘(𝑑, 𝑧)(ΜƒοΈ€ 𝜎 𝑙 (𝑑, 𝑧))2 , πœ‘ ∈ 𝐢 1,2 ([0, 𝑇 ] Γ— R; R), with the coefficients 𝜎 ΜƒοΈ€1 = 𝜎 ΜƒοΈ€2 , ̃︀𝑏1 = ̃︀𝑏2 (without mean field term), 𝑙 𝑙 𝜎 ̃︀𝑙 (𝑑, 𝑧) = 𝜎(𝑑, 𝑧, 𝑃𝑦(𝑑) ), ̃︀𝑏𝑙 (𝑑, 𝑧) = 𝑏(𝑑, 𝑧, 𝑃𝑦(𝑑) ), (𝑑, 𝑧) ∈ [0, 𝑇 ] Γ— R, 1 = 𝑃 2 , 𝑑 ∈ [0, 𝑇 ]. and we have seen that 𝑃𝑦(𝑑) 𝑦(𝑑)

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

Thank you very much!

θ°’θ°’!

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