# 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...

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PDE and Probability Methods for Interactionsβ, 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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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

Let π be a fixed time horizon, π, π measurable mappings defined over appropriate spaces. We are interested in a weak solution of Mean-Field (McKean-Vlasov) SDE : For π‘ β [0, π ], π β πΏ2 (β¦, β±0 , π ; Rπ ), β«οΈ ππ‘ = π +

π‘

β«οΈ

π‘

π(π , πΒ·β§π  , ππΒ·β§π  )ππ  + 0

π(π , πΒ·β§π  , ππΒ·β§π  )ππ΅π  ,

(1.1)

0

where π is a probability measure with respect to which π΅ is a B.M. Remark: ππΒ·β§π  is the law of πΒ·β§π  w.r.t. π.

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Brief state of art 1) Such Mean-Field SDEs have been intensively studied: β For a longer time as limit equ. for systems with a large number of particles (propagation of chaos)(Bossy, MΒ΄elΒ΄eard, Sznitman, Talay,...); β Mean-Field Games, since 2006-2007 (Lasry, Lions,...); 2) Mean-Field SDEs/FBSDEs and associated nonlocal PDEs: β Preliminary works in 2009 (AP, SPA); β Classical solution of non-linear PDE related with the mean-field SDE: Buckdahn, Peng, Li, Rainer (2014); Chassagneux, Crisan, Delarue (2014); β For the case with jumps: Li, Hao (2016); Li (2016); β 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 π is bounded, measurable and with a modulus of continuity w.r.t the measure, while π is independent of the measure and Lipschitz. * when the coefficients (π, π) are bounded and continuous.

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Preliminaries

We consider + (β¦, β±, π ) - complete probability space; + π B.M. over (β¦, β±, π ) (for simplicity: all processes 1-dimensional); + F-filtration generated by π , and augmented by β±0 . π-Wasserstein metric on β«οΈ π«π (R) := {π | π probab. on (R, β¬(R)) with

|π₯|π π(π₯) < +β};

R

ππ (π, π):= inf

{οΈ(οΈ

β«οΈ RΓR

)οΈ 1 }οΈ |π₯|π π(ππ₯ππ¦) π , π(Β· Γ R) = π, π(R Γ Β·) = π . (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 ΜοΈ π, π΅, π) is a weak solution of SDE (1.1), if ΜοΈ β±, ΜοΈ F, A six-tuple (β¦, ΜοΈ = {β±ΜοΈπ‘ }0β€π‘β€π is a ΜοΈ β±, ΜοΈ π) is a complete probability space, and F (i) (β¦, ΜοΈ β±, ΜοΈ π) satisfying the usual conditions. filtration on (β¦, ΜοΈ (ii) π = {ππ‘ }0β€π‘β€π is a continuous, F-adapted R-valued process; ΜοΈ π)-BM. π΅ = {π΅π‘ }0β€π‘β€π is an (F, β«οΈ π (iii) π{ 0 (|π(π , πΒ·β§π  , ππΒ·β§π  )| + |π(π , πΒ·β§π  , ππΒ·β§π  )|2 )ππ  < +β} = 1, and equation (1.1) is satisfied, π-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 (β¦π , β± π , Fπ , ππ , π΅ π , π π ), π = 1, 2, we have π1π 1 = π2π 2 , i.e., the two processes π 1 and π 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 π, π satisfy the following assumption (H1): (i) π : [0, π ] Γ πΆ([0, π ]; R) Γ π«1 (R) β R is bounded and measurable; (ii) π : [0, π ] Γ πΆ([0, π ]; R) β R is bounded, measurable, and s.t., for all (π‘, π) β [0, π ] Γ πΆ([0, π ]; R), 1/π(π‘, π) is bounded in (π‘, π); (iii) (Modulus of continuity) βπ : R+ β R+ increasing, continuous, with π(0+) = 0 s.t., for all π‘ β [0, π ], π β πΆ([0, π ]; R), π, π β π«1 (R), |π(π‘, πΒ·β§π‘ , π) β π(π‘, πΒ·β§π‘ , π)| β€ π(π1 (π, π)); (iv) βπΏ β₯ 0 s.t., for all π‘ β [0, π ], π, π β πΆ([0, π ]; R), |π(π‘, πΒ·β§π‘ ) β π(π‘, πΒ·β§π‘ )| β€ πΏ sup |ππ  β ππ  |. 0β€π β€π‘

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Case 1: Existence of a weak solution We want to study weak solutions of the following mean-field SDE: β«οΈ π‘ β«οΈ π‘ ππ‘ = π + π(π , πΒ·β§π  )ππ΅π  + π(π , πΒ·β§π  , πππ  )ππ , π‘ β [0, π ], 0

(2.1)

0

where (π΅π‘ )π‘β[0,π ] is a BM under the probability measure π. 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 ΜοΈ π, π΅, π). ΜοΈ β±, ΜοΈ F, (β¦, Proof: Girsanovβs Theorem. Schauderβ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 π β‘ πΌπ and drift coefficient β«οΈ ΜοΈπ(π , πΒ·β§π  , ππ  ) := π(π , πΒ·β§π  , ππππ  ), π β πΆ([0, π ]), π β π«1 (R), π  β [0, π ]; the function π β πΆ([0, π ]; R) is arbitrarily given but fixed, and Lipschitz. Then our mean-field SDE (2.1) can be written as follows: β«οΈ

π‘

π(π , πΒ·β§π  , πΈπ [π(ππ  )])ππ , π‘ β [0, π ].

ππ‘ = π΅π‘ +

(2.2)

0

Here π : [0, π ] Γ πΆ([0, π ]) Γ R β R is bounded, meas., Lips. in π¦. Then, the coefficients ΜοΈπ and π satisfy (H1), and from Theorem 2.1, we obtain ΜοΈ π, π΅, π). ΜοΈ β±, ΜοΈ F, that the mean-field SDE (2.2) has a weak solution (β¦, 11 / 44

Case 1: Existence of a weak solution

Example 2. Take diffusion coefficient π β‘ πΌπ and drift coefficient β«οΈ ΜοΈπ(π , πΒ·β§π  , ππ  ) := π(π , πΒ·β§π  , π¦)ππ  (ππ¦), π β πΆ([0, π ]), ππ  β π«1 (R), π  β [0, π ], i.e., we consider the following mean-field SDE: β«οΈ π‘ β«οΈ π(π , πΒ·β§π  , π¦)πππ  (ππ¦)ππ , π‘ β [0, π ].

ππ‘ = π΅π‘ + 0

(2.3)

R

Here the coefficient π : [0, π ] Γ πΆ([0, π ]) Γ R β R is bounded, meas. and Lips. in π¦. Then, the coefficients ΜοΈπ and π satisfy (H1), and from Theorem ΜοΈ π, π΅, π). ΜοΈ β±, ΜοΈ F, 2.1 the mean-field SDE (2.3) has a weak solution (β¦,

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Case 1: Uniqueness in law of weak solutions Let the functions π and π satisfy the following assumption (H2): (i) π : [0, π ] Γ πΆ([0, π ]; R) Γ π«1 (R) β R is bounded and measurable; (ii) π : [0, π ] Γ πΆ([0, π ]; R) β R is bounded and measurable, and |1/π(π‘, π)| β€ πΆ, (π‘, π) β [0, π ] Γ πΆ([0, π ]; R), for some πΆ β R+ ; (iii) (Modulus of continuity) There exists a continuous and increasing function π : R+ β R+ with

β«οΈ

ππ’ = +β, 0+ π(π’) such that, for all π‘ β [0, π ], π β πΆ([0, π ]; R), π, π β π«1 (R), π(π) > 0, for all π > 0, and

|π(π‘, πΒ·β§π‘ , π) β π(π‘, πΒ·β§π‘ , π)|2 β€ π(π1 (π, π)2 ); (iv) βπΏ β₯ 0 such that, for all π‘ β [0, π ], π, π β πΆ([0, π ]; R), |π(π‘, πΒ·β§π‘ ) β π(π‘, πΒ·β§π‘ )| β€ πΏ sup |ππ  β ππ  |. 0β€π β€π‘ 13 / 44

Case 1: Uniqueness in law of weak solutions Obviously, under assumption (H2) the coefficients π and π also satisfy (H1). Thus, due to Theorem 2.1, the following mean-field SDE β«οΈ ππ‘ = π +

π‘

β«οΈ

0

π‘

π(π , πΒ·β§π  )ππ΅π  , π‘ β [0, π ],

π(π , πΒ·β§π  , πππ  )ππ  +

(2.1)

0

has a weak solution.

Theorem 2.2 Suppose that assumption (H2) holds, and let (β¦π , β± π , Fπ , ππ , π΅ π , π π ), π = 1, 2, be two weak solutions of mean-field SDE (2.1). Then (π΅ 1 , π 1 ) and (π΅ 2 , π 2 ) have the same law under their respective probability measures, i.e., π1(π΅ 1 ,π 1 ) = π2(π΅ 2 ,π 2 ) . 14 / 44

Case 1: Uniqueness in law of weak solutions Sketch of the proof: For π β πΆ([0, π ]; R), π β π«1 (R), we define ΜοΈπ(π , πΒ·β§π  , π) = π β1 (π , πΒ·β§π  )π(π , πΒ·β§π  , π), and we introduce β§ β«οΈ π‘ βͺ π π βͺ ΜοΈπ(π , π π , ππ π )ππ , π‘ β [0, π ], βͺ β¨ ππ‘ = π΅π‘ + Β·β§π  ππ  0 β«οΈ β«οΈ π βͺ 1 π ΜοΈ βͺ π π π π π ΜοΈ βͺ π(π , πΒ·β§π  , ππ π )ππ΅π  β |π(π , πΒ·β§π  , πππ π )|2 ππ }, β© πΏπ = exp{β π  π  2 0 0 (2.4) π = 1, 2. Then from the Girsanov Theorem we know that (ππ‘π )π‘β[0,π ] is an ΜοΈ π = πΏπ ππ , π = 1, 2, respectively. Fπ -B.M. under the probability measure π π

From (H2), for each π, we have a unique strong solution π π of the SDE β«οΈ π‘ π π(π , πΒ·β§π  )πππ π , π‘ β [0, π ]. (2.5) ππ‘π = π0π + 0 15 / 44

Case 1: Uniqueness in law of weak solutions It is by now standard that β a meas. and non-anticipating function Ξ¦ : [0, π ] Γ R Γ πΆ([0, π ]; R) β R not depending on π = 1, 2, s.t. ΜοΈ π -a.s. (and, ππ -a.s.), π = 1, 2. (2.6) ππ‘π = Ξ¦π‘ (π0π , π π ), π‘ β [0, π ], π β«οΈ π‘ Then from (2.4) that ππ‘π = π΅π‘π + 0 ΜοΈπ(π , Ξ¦Β·β§π  (π0π , π π ), πππ π )ππ , π = 1, 2. π  Hence, putting π (π , πΒ·β§π  ) = ΜοΈπ(π , πΒ·β§π  , π1 ), (π , π) β [0, π ] Γ πΆ([0, π ]; ππ

R), from (2.4) and (2.6) we have β«οΈ π‘ β§ 1 1 βͺ βͺ π = π΅ + π (π , Ξ¦Β·β§π  (π01 , π 1 ))ππ , π‘ β [0, π ], β¨ π‘ π‘ 0 β«οΈ π‘ βͺ βͺ ΜοΈπ‘2 + β© ππ‘2 = π΅ π (π , Ξ¦Β·β§π  (π02 , π 2 ))ππ , π‘ β [0, π ], 0

where, π‘ β [0, π ], β«οΈ π‘ (οΈ )οΈ ΜοΈπ(π , Ξ¦Β·β§π  (π 2 , π 2 ), π2 2 ) β ΜοΈπ(π , Ξ¦Β·β§π  (π 2 , π 2 ), π1 1 ) ππ . ΜοΈ 2 = π΅ 2 + π΅ π‘ π‘ 0 0 ππ  ππ  0

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Case 1: Uniqueness in law of weak solutions Β― : [0, π ] Γ R Γ πΆ([0, π ]; R) β R meas. s.t., for both π΅ 1 , π΅ ΜοΈ 2 , Hence, βΞ¦ Β― π‘ (π 1 , π 1 ) and π΅ ΜοΈ 2 = Ξ¦ Β― π‘ (π 2 , π 2 ), π‘ β [0, π ]. π΅π‘1 = Ξ¦ 0 π‘ 0

(2.8)

Now we define β§ ΜοΈ 2π‘ ππ΅π‘2 , π‘ ΜοΈ 2π‘ = β(ΜοΈπ(π , Ξ¦Β·β§π  (π02 , π 2 ), π2 2 ) β ΜοΈπ(π , Ξ¦Β·β§π  (π02 , π 2 ), π1 1 ))πΏ β¨ ππΏ ππ  ππ  β© πΏ ΜοΈ 20 = 1. (2.9) ΜοΈ 2 is an Brownian motion From the Girsanov Theorem we know that π΅ ΜοΈ 2 = πΏ ΜοΈ 2 π2 . Moreover, putting under the probability measure π π

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Case 1: Uniqueness in law of weak solutions

β§ β«οΈ π β«οΈ π βͺ ΜοΈ 2 = exp{β π (π , Ξ¦Β·β§π  (π 2 , π 2 ))ππ 2 + 1 |π (π , Ξ¦Β·β§π  (π 2 , π 2 ))|2 ππ }, β¨πΏ π 0 π  0 20 0 βͺ β© Β―2 ΜοΈ 2 π ΜοΈ 2 , π =πΏ π

(2.10) we have that (ππ‘1 )π‘β[0,π ]

(ππ‘2 )π‘β[0,π ]

is a B.M. under both ΜοΈ 1 . is a B.M. under π

ΜοΈ 2 π

and

Β― 2, π

while

On the other hand, since π is bounded and meas., we can prove that ΜοΈ : R Γ πΆ([0, π ]; R) β R, s.t. β a meas. function Ξ¦ ΜοΈ 0π , π π ) = Ξ¦(π

β«οΈ

π

π (π , Ξ¦Β·β§π  (π0π , π π ))πππ π , ππ -a.s., π = 1, 2.

0

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Case 1: Uniqueness in law of weak solutions

Therefore, recalling the definition of πΏ1π and (2.10), we have β§ β«οΈ π β«οΈ βͺ 1 π 1 1 1 1 βͺ βͺ |π (π , Ξ¦Β·β§π  (π01 , π 1 ))|2 ππ }, β¨ πΏπ = exp{β π (π , Ξ¦Β·β§π  (π0 , π ))πππ  + 2 0 0 β«οΈ π β«οΈ π βͺ 1 βͺ βͺ ΜοΈ 2 = exp{β π (π , Ξ¦Β·β§π  (π 2 , π 2 ))ππ 2 + |π (π , Ξ¦Β·β§π  (π02 , π 2 ))|2 ππ }, β©πΏ π 0 π  2 0 0 (2.11) ΜοΈ and we see that β a meas. function Ξ¦ : R Γ πΆ([0, π ]; R) β R, s.t. ΜοΈ 01 , π 1 ), π1 -a.s., and πΏ ΜοΈ 2π = Ξ¦(π ΜοΈ 02 , π 2 ), π2 -a.s. (and, π Β― 2 -a.s.). πΏ1π = Ξ¦(π (2.12)

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Case 1: Uniqueness in law of weak solutions Consequently, as π0π is β±0π -measurable, π = 1, 2 and π1π 1 = π2π 2 , 0

0

from (2.8), (2.10), (2.11) and (2.12) we have that, for all bounded measurable function πΉ : πΆ([0, π ]; Rπ )2 β R, 1 Β― 1 , π 1 ), π 1 )] πΈπ1 [πΉ (π΅ 1 , π 1 )] = πΈπΜοΈ1 [ πΉ (Ξ¦(π 0 1 ΜοΈ Ξ¦(π , π 1) 0

= πΈπΒ― 2 [

1 ΜοΈ 2 , π 2 ) Ξ¦(π 0

ΜοΈ 2 , π 2 )]. Β― 02 , π 2 ), π 2 )] = πΈ ΜοΈ2 [πΉ (π΅ πΉ (Ξ¦(π π

That is, ΜοΈ 2 π1(π΅ 1 ,π 1 ) = π ΜοΈ 2 ,π 2 ) . (π΅

(2.13)

Taking into account (2.6), we have ΜοΈ 2 π1(π΅ 1 ,π 1 ,π 1 ) = π ΜοΈ 2 ,π 2 ,π 2 ) , (π΅

(2.14)

ΜοΈ 2 2 . and, in particular, π1π 1 = π π 20 / 44

Case 1: Uniqueness in law of weak solutions On the other hand, we can prove β«οΈ ΜοΈ 2 2 , π2 2 )2 β€ πΆ π  π(π1 (π1 1 , π2 2 )2 )ππ; βπ1 (π1π 1 , π2π 2 )2 = π1 (π 0 π π π π π

π

π

π

π

π

β The continuity of π  β π1 (π1π 1 , π2π 2 ). π

π

Putting π’(π ) := π1 (π1π 1 , π2π 2 ), π  β [0, π ], then we have from above, π  β«οΈ π π  π’(π )2 β€ πΆ 0 π(π’(π)2 )ππ, 0 β€ π  β€ π‘ β€ π . β«οΈ ππ’ From (H2)-(iii), 0+ π(π’) = +β, it follows from Bihariβs inequality that π’(π ) = 0, for any π  β [0, π ], that is, π1π 1 = π2π 2 , π  β [0, π ]. Thus, from π  π  ΜοΈ 2 = π΅ 2 , πΏ ΜοΈ 2 = 1, and, consequently, (2.7) and (2.9) it follows that π΅ π 2 2 2 2 ΜοΈ ΜοΈ π = π . Then, π = π 2 2 2 , and from (2.14) ΜοΈ 2 ,π 2 ,π 2 ) (π΅

(π΅ ,π ,π )

π1(π΅ 1 ,π 1 ,π 1 ) = π2(π΅ 2 ,π 2 ,π 2 ) . This implies, in particular, π1(π΅ 1 ,π 1 ) = π2(π΅ 2 ,π 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 πΜοΈ on (πΆ([0, π ]; R), β¬(πΆ([0, π ]; R))) is a solution to the local martingale problem associated with πβ² , if for every π β πΆ 1,2 ([0, π ]ΓR; R), ππ‘π

β«οΈ := π (π‘, π¦(π‘)) β π (0, π¦(0)) β

π‘

(ππ  + πβ² )π (π , π¦(π ))ππ , π‘ β [0, π ], (3.1)

0

is a continuous local martingale w.r.t (Fπ¦ , πΜοΈ), where π¦ = (π¦(π‘))π‘β[0,π ] is the coordinate process on πΆ([0, π ]; R), the considered filtration Fπ¦ = (β±π‘π¦ )π‘β[0,π ] is that generated by π¦ = (π¦(π‘))π‘β[0,π ] and augmented by all πΜοΈ-null sets, and πβ² is defined by, π¦ β πΆ([0, π ]; R), 1 πβ² π (π , π¦) = π(π , π¦)ππ₯ π (π , π¦(π )) + π 2 (π , π¦)ππ₯2 π (π , π¦(π )). 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 ΜοΈ πΜοΈ, π ΜοΈ β±, ΜοΈ F, ΜοΈ , π) to the following The existence of a weak solution (β¦, functional SDE with given initial distribution π on β¬(R): β«οΈ ππ‘ = π +

π‘

β«οΈ

0

π‘

ΜοΈπ  , π‘ β [0, π ], π(π , πΒ·β§π  )ππ

π(π , πΒ·β§π  )ππ  + 0

is equivalent to the existence of a solution πΜοΈ to the local martingale problem (3.1) associated with πβ² defined by (3.2), with πΜοΈπ¦(0) = π. The both solutions are related by πΜοΈ = πΜοΈ β π β1 , i.e., the probability measure πΜοΈ is the law of the weak solution π on (πΆ([0, π ]; R), β¬(πΆ([0, π ]; R))).

23 / 44

Case 2: Preliminaries Recall the definition of the derivative of π : π«2 (R) β R w.r.t probability measure π β π«2 (R) (in the sense of P.L.Lions)(P.L.Lionsβ lectures at Coll`ege de France, also see the notes of Cardaliaguet).

Definition 3.2 (i) πΜοΈ : πΏ2 (β¦, β±, π ; R) β R is FrΒ΄echet differentiable at π β πΏ2 (β¦, β±, π ), if β a linear continuous mapping π·πΜοΈ(π)(Β·) β πΏ(πΏ2 (β¦, β±, π ; R); R), s.t. πΜοΈ(π + π)βπΜοΈ(π) = π·πΜοΈ(π)(π) + π(|π|πΏ2 ), with |π|πΏ2 β 0 for π β πΏ2 (β¦, β±, π ). (ii) π : π«2 (R) β R is differentiable at π β π«2 (R), if for πΜοΈ(π) := π (ππ ), π β πΏ2 (β¦, β±, π ; R), there is some π β πΏ2 (β¦, β±, π ; R) with ππ = π such that πΜοΈ : πΏ2 (β¦, β±, π ; R) β R is FrΒ΄echet differentiable in π. 24 / 44

Case 2: Preliminaries From Rieszβ Representation Theorem there exists a π -a.s. unique variable π β πΏ2 (β¦, β±, π ; R) such that π·πΜοΈ(π)(π) = (π, π)πΏ2 = πΈ[ππ], for all π β πΏ2 (β¦, β±, π ; R). P.L. Lions proved that there is a Borel function β : R β R such that π = β(π), π -a.e., and function β depends on π only through its law ππ . Therefore, π (ππ ) β π (ππ ) = πΈ[β(π) Β· (π β π)] + π(|π β π|πΏ2 ), π β πΏ2 (β¦, β±, π ; R).

Definition 3.3 We call ππ π (ππ , π¦) := β(π¦), π¦ β R, the derivative of function π : π«2 (R) β R at ππ , π β πΏ2 (β¦, β±, π ; R). Remark: ππ π (ππ , π¦) is only ππ (ππ¦)-a.e. uniquely determined. 25 / 44

Case 2: Preliminaries Definition 3.4 We say that π β πΆ 1 (π«2 (R)), if for all π β πΏ2 (β¦, β±, π ; R) there exists a ππ -modification of ππ π (ππ , .), also denoted by ππ π (ππ , .), such that ππ π : π«2 (R) Γ R β R is continuous w.r.t the product topology generated by the 2-Wasserstein metric over π«2 (R) and the Euclidean norm over R, and we identify this modified function ππ π as the derivative of π . The function π is said to belong to πΆπ1,1 (π«2 (R)), if π β πΆ 1 (π«2 (R)) is s.t. ππ π : π«2 (R) Γ R β R is bounded and Lipschitz continuous, i.e., there exists some constant πΆ β₯ 0 such that (i) |ππ π (π, π₯)| β€ πΆ, π β π«2 (R), π₯ β R; (ii) |ππ π (π, π₯)βππ π (πβ² , π₯β² )|β€πΆ(π2 (π, πβ² )+|π₯βπ₯β² |), π, πβ² β π«2 (R), π₯, π₯β² β R. 26 / 44

Case 2: Preliminaries

Definition 3.5 We say that π β πΆ 2 (π«2 (R)), if π β πΆ 1 (π«2 (R)) and ππ π (π, .) : R β R is differentiable, and its derivative ππ¦ ππ π : π«2 (R)ΓRβR β R is continuous, for every π β π«2 (R). Moreover, π β πΆπ2,1 (π«2 (R)), if π β πΆ 2 (π«2 (R))

βοΈ

πΆπ1,1 (π«2 (R)) and its

derivative ππ¦ ππ π : π«2 (R) Γ R β R β R is bounded and Lipschitzcontinuous. Remark: πΆπ2,1 (R Γ π«2 (R)), πΆπ1,2,1 ([0, π ]ΓRΓπ«2 (R); R) are similarly defined.

27 / 44

Case 2: Preliminaries Now we can give our It^ oβs formula.

Theorem 3.1 Let π = (ππ  ), πΎ = (πΎπ  ), π = (ππ  ), π½ = (π½π  ) R-valued adapted stochastic processes, such that β«οΈ π 3 (i) There exists a constant π > 6 s.t. πΈ[( 0 (|ππ  |π + |ππ  |π )ππ ) π ] < +β; β«οΈ π (ii) 0 (|πΎπ  |2 + |π½π  |)ππ  < +β, P-a.s. Let πΉ β πΆπ1,2,1 ([0, π ] Γ R Γ π«2 (R)). Then, for the It^ o processes π‘

β«οΈ ππ‘ = π0 +

π‘

β«οΈ

ππ  ππ , π‘ β [0, π ], π0 β πΏ2 (β¦, β±0 , π ),

ππ  πππ  + 0

β«οΈ ππ‘ = π0 +

0

π‘

β«οΈ πΎπ  πππ  +

0

π‘

π½π  ππ , π‘ β [0, π ], π0 β πΏ2 (β¦, β±0 , π ),

0 28 / 44

Case 2: Preliminaries Theorem 3.1 (continued) we have πΉ (π‘, ππ‘ , πππ‘ ) β πΉ (0, π0 , ππ0 ) β«οΈ π‘ (οΈ 1 = ππ πΉ (π, ππ , πππ ) + ππ¦ πΉ (π, ππ , πππ )π½π + ππ¦2 πΉ (π, ππ , πππ )πΎπ2 2 0 )οΈ 1 2 Β― Β― Β― Β― + πΈ[(ππ πΉ )(π, ππ , πππ , ππ )ππ + ππ§ (ππ πΉ )(π, ππ , πππ , ππ )Β― ππ ] ππ 2 β«οΈ π‘ + ππ¦ πΉ (π, ππ , πππ )πΎπ πππ , π‘ β [0, π ]. 0

Β― Β―π, π Here (π, Β― ) denotes an independent copy of (π, π, π), defined on a P.S. Β― β±, Β― πΒ― ). The expectation πΈ[Β·] Β― on (β¦, Β― β±, Β― πΒ― ) concerns only r.v. endowed (β¦, with the superscriptΒ―. 29 / 44

Case 2: Preliminaries (H3) The coefficients (π, π) β πΆπ1,2,1 ([0, π ] Γ R Γ π«2 (R); R Γ R).

Theorem 3.2 (Buckdahn, Li, Peng and Rainer, 2014) Let Ξ¦ β πΆπ2,1 (R Γ π«2 (R)), then under assumption (H3) the following PDE: β§ 1 βͺ 0 = ππ‘ π (π‘, π₯, π) + ππ₯ π (π‘, π₯, π)π(π₯, π) + ππ₯2 π (π‘, π₯, π)π 2 (π₯, π) βͺ βͺ βͺ 2 βͺ β«οΈ βͺ βͺ βͺ βͺ + (ππ π )(π‘, π₯, π, π¦)π(π¦, π)π(ππ¦) βͺ βͺ βͺ Rβ«οΈ β¨ 1 ππ¦ (ππ π )(π‘, π₯, π, π¦)π 2 (π¦, π)π(ππ¦), + βͺ βͺ 2 βͺ R βͺ βͺ βͺ βͺ βͺ (π‘, π₯, π) β [0, π ) Γ R Γ π«2 (R); βͺ βͺ βͺ βͺ β© π (π, π₯, π) = Ξ¦(π₯, π), (π₯, π) β R Γ π«2 (R). has a unique classical solution π (π‘, π₯, π) β πΆπ1,2,1 ([0, π ] Γ R Γ π«2 (R); R). 30 / 44

Case 2: Existence of a weak solution

Let π and π satisfy the following assumption: (H4) π, π : [0, π ] Γ R Γ π«2 (R) β R are continuous and bounded. We want to study weak solution of the following mean-field SDE: β«οΈ π‘ β«οΈ π‘ π(π , ππ  , πππ  )ππ΅π  , π‘ β [0, π ], ππ‘ = π + π(π , ππ  , πππ  )ππ  0

(3.3)

0

where π β πΏ2 (β¦, β±0 , π ; R) obeys a given distribution law ππ = π β π«2 (R) and (π΅π‘ )π‘β[0,π ] is a B.M. under the probability measure π.

31 / 44

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

Definition 3.6 A probability measure πΜοΈ on (πΆ([0, π ]; R), β¬(πΆ([0, π ]; R))) is a solution to the local martingale problem (resp., martingale problem) associated with ΜοΈ if for every π β πΆ 1,2 ([0, π ] Γ R; R) (resp., π β πΆ 1,2 ([0, π ] Γ R; R)), the π, π

process β«οΈ

π

πΆ (π‘, π¦, π) := π (π‘, π¦(π‘)) β π (0, π¦(0)) β

π‘ (οΈ

)οΈ ΜοΈ (π , π¦(π ), π(π ))ππ , (ππ  + π)π

0

(3.4) is a continuous local

(Fπ¦ , πΜοΈ)-martingale

(resp., continuous

(Fπ¦ , πΜοΈ)-

martingale), 32 / 44

Case 2: Existence of a weak solution

Definition 3.6 (continued) where π(π‘) = πΜοΈπ¦(π‘) is the law of the coordinate process π¦ = (π¦(π‘))π‘β[0,π ] on πΆ([0, π ]; R) at time π‘, the filtration Fπ¦ is that generated by π¦ and completed, and πΜοΈ is defined by ΜοΈ )(π , π¦, π) := ππ¦ π (π , π¦)π(π , π¦, π) + 1 ππ¦2 π (π , π¦)π 2 (π , π¦, π), (ππ 2

(3.5)

ΜοΈ )(π , π¦(π ), π(π )) abbreviates (π , π¦, π) β [0, π ] Γ R Γ π«2 (R). Here ((ππ  + π΄)π ΜοΈ )(π , π¦(π ), π(π )) := (ππ  π )(π , π¦(π )) + (π΄π ΜοΈ )(π , π¦(π ), π(π )). ((ππ  + π΄)π

33 / 44

Case 2: Existence of a weak solution

Proposition 3.1 ΜοΈ π, π΅, π) to equation (3.3) with ΜοΈ β±, ΜοΈ F, The existence of a weak solution (β¦, initial distribution π on β¬(R) is equivalent to the existence of a solution πΜοΈ to the local martingale problem (3.4) associated with πΜοΈ defined by (3.5), with πΜοΈπ¦(0) = π.

34 / 44

Case 2: Existence of a weak solution

Lemma 3.2 Let the probability measure πΜοΈ on (πΆ([0, π ]; R), β¬(πΆ([0, π ]; R))) be a ΜοΈ Then, for the solution to the local martingale problem associated with π. second order differential operator (οΈ

β«οΈ ΜοΈ ππ )(π , π¦, π) := (ππ )(π , π¦, π) + (ππ π )(π , π¦, π, π§)π(π , π§, π)π(ππ§) R β«οΈ 1 + ππ§ (ππ π )(π , π¦, π, π§)π 2 (π , π§, π)π(ππ§), 2 R (3.6)

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Case 2: Existence of a weak solution Lemma 3.2 (continued) applying to functions π β πΆ 1,2 ([0, π ] Γ R Γ π«2 (R); R) we have that, for every such π β πΆ 1,2 ([0, π ] Γ R Γ π«2 (R); R), the process πΆ π (π‘, π¦, π) :=π (π‘, π¦(π‘), π(π‘)) β π (0, π¦(0), π(0)) β«οΈ π‘ β (ππ  + π)π (π , π¦(π ), π(π ))ππ , π‘ β [0, π ],

(3.7)

0

is a continuous local (Fπ¦ , πΜοΈ)-martingale, where π(π‘) = πΜοΈπ¦(π‘) is the law of the coordinate process π¦ = (π¦(π‘))π‘β[0,π ] on πΆ([0, π ]; R) at time π‘, the filtration Fπ¦ is that generated by π¦ and completed. Moreover, if π βπΆ 1,2,1 ([0, π ]ΓRΓπ«2 (R); R), this process πΆ π is an (Fπ¦ , πΜοΈ)-martingale. π

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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 ΜοΈ π, ΜοΈ β±, ΜοΈ F, ΜοΈ π΅, π). (β¦, Remark 2. If π, π : [0, π ] Γ πΆ([0, π ]; R) Γ π«2 (πΆ([0, π ]; R)) β R are bounded and continuous, then the following mean-field SDE β«οΈ π‘ β«οΈ π‘ ππ‘ = π + π(π , πΒ·β§π  , ππΒ·β§π  )ππ  + π(π , πΒ·β§π  , ππΒ·β§π  )ππ΅π  , π‘ β [0, π ], 0

0

(1.1) πΏ2 (β¦, β±

where π β 0 , π ) obeys a given distribution law ππ = π, has a weak ΜοΈ π, ΜοΈ β±, ΜοΈ F, ΜοΈ π, π΅). solution (β¦, 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 β πβ¬(R) = {π | π : R β R bounded Borel-measurable function} a determining class on R, if for any two finite measures π1 and π2 on β«οΈ β«οΈ β¬(R), Rπ π(π₯)π1 (ππ₯) = Rπ π(π₯)π2 (ππ₯) for all π β π implies π1 = π2 . Remark: The class πΆ0β (R) is a determining class on R.

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Case 2: Uniqueness in law of weak solutions Theorem 3.4 For given π β πΆ0β (R), we consider the Cauchy problem π π£(π‘, π₯, π) = ππ£(π‘, π₯, π), (π‘, π₯, π) β [0, π ] Γ R Γ π«2 (R), ππ‘ π£(0, π₯, π) = π (π₯), π₯ β R,

(3.8)

where β«οΈ ΜοΈ ππ£(π‘, π₯, π) = (ππ£)(π‘, π₯, π) +

(ππ π£)(π‘, π₯, π, π’)π(π‘, π’, π)π(ππ’) β«οΈ 1 + ππ§ (ππ π£)(π‘, π₯, π, π’)π 2 (π‘, π’, π)π(ππ’), 2 R 1 ΜοΈ (ππ£)(π‘, π₯, π) = ππ¦ π£(π‘, π₯, π)π(π‘, π₯, π) + ππ¦2 π£(π‘, π₯, π)π 2 (π‘, π₯, π), 2 R

(π‘, π₯, π) β [0, β) Γ R Γ π«2 (R). 39 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 (continued) We suppose that, for all π β πΆ0β (R), (3.8) has a solution βοΈ π£π β πΆπ ([0, β) Γ R Γ π«2 (R)) πΆπ1,2,1 ((0, β) Γ R Γ π«2 (R)). Then, the local martingale problem associated with πΜοΈ (Recall Definition 3.6) and with the initial condition πΏπ₯ 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, π = 1, 2 (β¦π , β± π , Fπ , ππ , π΅ π , π π ), of SDE (3.3), we have π1π 1 = π2π 2 . 40 / 44

Uniqueness in law of weak solutions Sketch of proof of Theorem 3.4: Let π > 0, denote by π¦=(π¦(π‘))π‘β[0,π ] the coordinate process on πΆ([0, π ]; R). Let π 1 and π 2 be two arbitrary solutions of the local martingale problem associated with πΜοΈ and initial π condition π₯ β R: ππ¦(0) = πΏπ₯ , π = 1, 2.

Consequently, due to Lemma 3.2, for any π β πΆπ1,2,1 ([0, π ] Γ R Γ π«2 (R)), πΆ

π

(π‘, π¦, ππ¦π )

:=

π π(π‘, π¦(π‘), ππ¦(π‘) )βπ(0, π₯, πΏπ₯ )β

β«οΈ 0

π‘ π (ππ  +π)π(π , π¦(π ), ππ¦(π ) )ππ ,

(3.9) π π -martingale,

πΆ0β (R),

π = 1, 2, π‘ β [0, π ]. For given π β let βοΈ 1,2,1 π£π β πΆπ ([0, π ] Γ R Γ π«2 (R)) πΆπ ((0, π ) Γ R Γ π«2 (R)) be a solution of

is a

the Cauchy problem (3.8). 41 / 44

Uniqueness in law of weak solutions Then putting π(π‘, π§, π) := π£π (π βπ‘, π§, π), π‘ β [0, π ], π§ β R, π β π«2 (R), defines a function π of class βοΈ πΆπ ([0, π ] Γ R Γ π«2 (R)) πΆπ1,2,1 ((0, π ) Γ R Γ π«2 (R)) which satisfies ππ  π(π , π§, π)+ππ(π , π§, π) = 0, π(π, π§, π) = π (π§), (π , π§, π)β[0, π ]ΓRΓπ«2 (R). From (3.9) we see that {πΆ π (π , π¦, ππ¦π ), π  β [0, π ]} is an (Fπ¦ , π π )β«οΈ martingale. Hence, for πΈ π [Β·] = Ξ©π (Β·)ππ π , π πΈ π [π (π¦(π ))] = πΈ π [π(π, π¦(π ), ππ¦(π ) )] = π(0, π₯, πΏπ₯ ), π₯ β R, π = 1, 2,

that is πΈ 1 [π (π¦(π ))] = πΈ 2 [π (π¦(π ))], for all π β πΆ0β (R). Combining this 1 = π 2 , for every π‘ β₯ 0. with the arbitrariness of π β₯ 0, we have that ππ¦(π‘) π¦(π‘) 42 / 44

Uniqueness in law of weak solutions Consequently, π 1 , π 2 are solutions of the same classical martingale problem, associated with πΜοΈ = πΜοΈπ , π = 1, 2, πΜοΈπ π(π‘, π§) = ππ¦ π(π‘, π§)ΜοΈππ (π‘, π§) + ππ¦2 π(π‘, π§)(ΜοΈ π π (π‘, π§))2 , π β πΆ 1,2 ([0, π ] Γ R; R), with the coefficients π ΜοΈ1 = π ΜοΈ2 , ΜοΈπ1 = ΜοΈπ2 (without mean field term), π π π ΜοΈπ (π‘, π§) = π(π‘, π§, ππ¦(π‘) ), ΜοΈππ (π‘, π§) = π(π‘, π§, ππ¦(π‘) ), (π‘, π§) β [0, π ] Γ R, 1 = π 2 , π‘ β [0, π ]. and we have seen that ππ¦(π‘) π¦(π‘)

.............. π 1 = π 2 , i.e., the local martingale problem has at most one solution.  43 / 44

Thank you very much!

θ°’θ°’!

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