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 26429, China. Email: juanli@sdu.edu.cn Based on joint works with Hui Min (Beijing University of Technology) Workshop PDE and Probability Methods for Interactions, Inria, Sophia Antipolis, France, March 3-31, 217. 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 [, T ], ξ L 2 (Ω, F, P ; R d ), X t = ξ + b(s, X s, Q X s )ds + σ(s, X s, Q X s )db s, (1.1) where Q is a probability measure with respect to which B is a B.M. Remark: Q X 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, Méléard, Sznitman, Talay,...); Mean-Field Games, since 26-27 (Lasry, Lions,...); 2) Mean-Field SDEs/FBSDEs and associated nonlocal PDEs: Preliminary works in 29 (AP, SPA); Classical solution of non-linear PDE related with the mean-field SDE: Buckdahn, Peng, Li, Rainer (214); Chassagneux, Crisan, Delarue (214); For the case with jumps: Li, Hao (216); Li (216); Weak solution: Oelschläger(1984), Funaki (1984), Gärtner (1988), Lacker (215), Carmona, Lacker (215), Li, Hui (216, 217)... 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 F. p-wasserstein metric on P p (R) := {μ μ probab. on (R, B(R)) with R x p μ(x) < + }; W p (μ, ν):= inf { ( x p ρ(dxdy) ) 1 p, ρ( R) = μ, ρ(r ) = ν }. R R (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 A six-tuple ( Ω, F, F, Q, B, X) is a weak solution of SDE (1.1), if (i) ( Ω, F, Q) is a complete probability space, and F = { F t } t T is a filtration on ( Ω, F, Q) satisfying the usual conditions. (ii) X = {X t } t T is a continuous, F-adapted R-valued process; B = {B t } t T is an ( F, Q)-BM. (iii) Q{ T ( b(s, X s, Q X s ) + σ(s, X s, Q X 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 (Ω i, F i, F i, Q i, B i, X i ), i = 1, 2, we have Q 1 X = Q 2 1 X, i.e., the two processes X 1 and X 2 have the same law. 2 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 : [, T ] C([, T ]; R) P 1 (R) R is bounded and measurable; (ii) σ : [, T ] C([, T ]; R) R is bounded, measurable, and s.t., for all (t, φ) [, T ] C([, T ]; R), 1/σ(t, φ) is bounded in (t, φ); (iii) (Modulus of continuity) ρ : R + R + increasing, continuous, with ρ(+) = s.t., for all t [, T ], φ C([, T ]; R), μ, ν P 1 (R), b(t, φ t, μ) b(t, φ t, ν) ρ(w 1 (μ, ν)); (iv) L s.t., for all t [, T ], φ, ψ C([, T ]; R), σ(t, φ t ) σ(t, ψ t ) L sup φ s ψ s. s t 9 / 44

Case 1: Existence of a weak solution We want to study weak solutions of the following mean-field SDE: X t = ξ + σ(s, X s )db s + b(s, X s, Q Xs )ds, t [, T ], (2.1) where (B t ) t [,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 ( Ω, F, F, Q, B, X). Proof: Girsanov s Theorem. Schauder s Fixed Point Theorem. 1 / 44

Case 1: Existence of a weak solution Let us give two examples. Example 1. Take diffusion coefficient σ I d and drift coefficient b(s, φ s, μ s ) := b(s, φ s, ψdμ s ), φ C([, T ]), μ P 1 (R), s [, T ]; the function ψ C([, T ]; R) is arbitrarily given but fixed, and Lipschitz. Then our mean-field SDE (2.1) can be written as follows: X t = B t + b(s, X s, E Q [ψ(x s )])ds, t [, T ]. (2.2) Here b : [, T ] C([, T ]) R R is bounded, meas., Lips. in y. Then, the coefficients b and σ satisfy (H1), and from Theorem 2.1, we obtain that the mean-field SDE (2.2) has a weak solution ( Ω, F, F, Q, B, X). 11 / 44

Case 1: Existence of a weak solution Example 2. Take diffusion coefficient σ I d and drift coefficient b(s, φ s, μ s ) := b(s, φ s, y)μ s (dy), φ C([, T ]), μ s P 1 (R), s [, T ], i.e., we consider the following mean-field SDE: X t = B t + R b(s, X s, y)q Xs (dy)ds, t [, T ]. (2.3) Here the coefficient b : [, T ] C([, T ]) R R is bounded, meas. and Lips. in y. Then, the coefficients b and σ satisfy (H1), and from Theorem 2.1 the mean-field SDE (2.3) has a weak solution ( Ω, F, F, Q, B, X). 12 / 44

Case 1: Uniqueness in law of weak solutions Let the functions b and σ satisfy the following assumption (H2): (i) b : [, T ] C([, T ]; R) P 1 (R) R is bounded and measurable; (ii) σ : [, T ] C([, T ]; R) R is bounded and measurable, and 1/σ(t, φ) C, (t, φ) [, T ] C([, T ]; R), for some C R + ; (iii) (Modulus of continuity) There exists a continuous and increasing function ρ : R + R + with du ρ(r) >, for all r >, and + ρ(u) = +, such that, for all t [, T ], φ C([, T ]; R), μ, ν P 1 (R), b(t, φ t, μ) b(t, φ t, ν) 2 ρ(w 1 (μ, ν) 2 ); (iv) L such that, for all t [, T ], φ, ψ C([, T ]; R), σ(t, φ t ) σ(t, ψ t ) L sup φ s ψ s. 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 X t = ξ + b(s, X s, Q Xs )ds + σ(s, X s )db s, t [, T ], (2.1) has a weak solution. Theorem 2.2 Suppose that assumption (H2) holds, and let (Ω i, F i, F i, Q i, 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., Q 1 (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([, T ]; R), μ P 1 (R), we define b(s, φ s, μ) = σ 1 (s, φ s )b(s, φ s, μ), and we introduce Wt i = Bt i + L i T = exp{ T b(s, X i s, Q i X )ds, t [, T ], s i b(s, X i s, Q i X i s )dbi s 1 2 T b(s, X i s, Q i X i s ) 2 ds}, (2.4) i = 1, 2. Then from the Girsanov Theorem we know that (W i t ) t [,T ] is an F i -B.M. under the probability measure Q i = L i T Qi, i = 1, 2, respectively. From (H2), for each i, we have a unique strong solution X i of the SDE X i t = X i + σ(s, X i s)dw i s, t [, T ]. (2.5) 15 / 44

Case 1: Uniqueness in law of weak solutions It is by now standard that a meas. and non-anticipating function Φ : [, T ] R C([, T ]; R) R not depending on i = 1, 2, s.t. X i t = Φ t (X i, W i ), t [, T ], Qi -a.s. (and, Q i -a.s.), i = 1, 2. (2.6) Then from (2.4) that W i t = B i t + t b(s, Φ s (X i, W i ), Q i X i s)ds, i = 1, 2. Hence, putting f(s, φ s ) = b(s, φ s, Q 1 X s ), (s, φ) [, T ] C([, T ]; R), from (2.4) and (2.6) we have Wt 1 = Bt 1 + f(s, Φ s (X, 1 W 1 ))ds, t [, T ], Wt 2 = B t 2 + f(s, Φ s (X 2, W 2 ))ds, t [, T ], where, t [, T ], B t 2 = Bt 2 + ( b(s, Φ s (X, 2 W 2 ), Q 2 X ) b(s, ) s 2 Φ s (X, 2 W 2 ), Q 1 X ) ds. s 1 16 / 44

Case 1: Uniqueness in law of weak solutions Hence, Φ : [, T ] R C([, T ]; R) R meas. s.t., for both B 1, B 2, Bt 1 = Φ t (X, 1 W 1 ) and B t 2 = Φ t (X, 2 W 2 ), t [, T ]. (2.8) Now we define d L 2 t = ( b(s, Φ s (X, 2 W 2 ), Q 2 X ) b(s, s 2 Φ s (X, 2 W 2 ), Q 1 X )) L 2 s 1 t dbt 2, t L 2 = 1. (2.9) From the Girsanov Theorem we know that B 2 is an Brownian motion under the probability measure Q 2 = L 2 T Q2. Moreover, putting 17 / 44

Case 1: Uniqueness in law of weak solutions L 2 T = exp{ Q 2 = L 2 T Q 2, T f(s, Φ s (X 2, W 2 ))dw 2 s + 1 2 T f(s, Φ s (X 2, W 2 )) 2 ds}, we have that (W 2 t ) t [,T ] is a B.M. under both Q 2 and Q 2, while (W 1 t ) t [,T ] is a B.M. under Q 1. (2.1) On the other hand, since f is bounded and meas., we can prove that a meas. function Φ : R C([, T ]; R) R, s.t. Φ(X i, W i ) = T f(s, Φ s (X i, W i ))dw i s, Q i -a.s., i = 1, 2. 18 / 44

Case 1: Uniqueness in law of weak solutions Therefore, recalling the definition of L 1 T and (2.1), we have L 1 T= exp{ L 2 T= exp{ T T f(s, Φ s (X 1, W 1 ))dw 1 s + 1 2 f(s, Φ s (X 2, W 2 ))dw 2 s + 1 2 T T f(s, Φ s (X 1, W 1 )) 2 ds}, f(s, Φ s (X 2, W 2 )) 2 ds}, (2.11) and we see that a meas. function Φ : R C([, T ]; R) R, s.t. L 1 T = Φ(X, 1 W 1 ), Q 1 -a.s., and L 2 T = Φ(X, 2 W 2 ), Q 2 -a.s. (and, Q 2 -a.s.). (2.12) 19 / 44

Case 1: Uniqueness in law of weak solutions Consequently, as X i is F i-measurable, i = 1, 2 and Q1 = Q 2, X 1 X 2 from (2.8), (2.1), (2.11) and (2.12) we have that, for all bounded measurable function F : C([, T ]; R d ) 2 R, E Q 1[F (B 1, W 1 1 )] = E Q1[ Φ(X 1, W 1 ) F ( Φ(X, 1 W 1 ), W 1 )] That is, = E Q2[ 1 Φ(X 2, W 2 ) F ( Φ(X, 2 W 2 ), W 2 )] = E Q2[F ( B 2, W 2 )]. Taking into account (2.6), we have Q 1 (B 1,W 1 ) = Q 2 ( B 2,W 2 ). (2.13) Q 1 (B 1,W 1,X 1 ) = Q 2 ( B 2,W 2,X 2 ), (2.14) and, in particular, Q 1 X 1 = Q 2 X 2. 2 / 44

Case 1: Uniqueness in law of weak solutions On the other hand, we can prove W 1 (Q 1 X 1 s, Q2 X 2 s )2 = W 1 ( Q 2 X 2 s, Q2 X 2 s )2 C s ρ(w 1(Q 1 X 1 r, Q2 X 2 r )2 )dr; The continuity of s W 1 (Q 1 X 1 s, Q2 X 2 s ). Putting u(s) := W 1 (Q 1 X, s 1 Q2 X ), s [, T ], then we have from above, s 2 u(s) 2 C s ρ(u(r)2 )dr, s t T. From (H2)-(iii), + du ρ(u) = +, it follows from Bihari s inequality that u(s) =, for any s [, T ], that is, Q 1 X = s 1 Q2 X, s [, T ]. Thus, from s 2 (2.7) and (2.9) it follows that B 2 = B 2, L2 T = 1, and, consequently, Q 2 = Q 2. Then, Q 2 ( B = 2,W 2,X 2 ) Q2 (B 2,W 2,X 2 ), and from (2.14) Q 1 (B 1,W 1,X 1 ) = Q2 (B 2,W 2,X 2 ). (2.15) This implies, in particular, Q 1 (B 1,X 1 ) = Q2 (B 2,X 2 ). 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 P on (C([, T ]; R), B(C([, T ]; R))) is a solution to the local martingale problem associated with A, if for every f C 1,2 ([, T ] R; R), M f t := f(t, y(t)) f(, y()) ( s + A )f(s, y(s))ds, t [, T ], (3.1) is a continuous local martingale w.r.t (F y, P ), where y = (y(t)) t [,T ] is the coordinate process on C([, T ]; R), the considered filtration F y = (F y t ) t [,T ] is that generated by y = (y(t)) t [,T ] and augmented by all P -null sets, and A is defined by, y C([, T ]; R), A f(s, y) = b(s, y) x f(s, y(s)) + 1 2 σ2 (s, y) xf(s, 2 y(s)). (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 The existence of a weak solution ( Ω, F, F, P, W, X) to the following functional SDE with given initial distribution μ on B(R): X t = ξ + b(s, X s )ds + σ(s, X s )d W s, t [, T ], is equivalent to the existence of a solution P to the local martingale problem (3.1) associated with A defined by (3.2), with P y() = μ. The both solutions are related by P = P X 1, i.e., the probability measure P is the law of the weak solution X on (C([, T ]; R), B(C([, T ]; R))). 23 / 44

Case 2: Preliminaries Recall the definition of the derivative of f : P 2 (R) R w.r.t probability measure μ P 2 (R) (in the sense of P.L.Lions)(P.L.Lions lectures at Collège de France, also see the notes of Cardaliaguet). Definition 3.2 (i) f : L 2 (Ω, F, P ; R) R is Fréchet differentiable at ξ L 2 (Ω, F, P ), if a linear continuous mapping D f(ξ)( ) L(L 2 (Ω, F, P ; R); R), s.t. f(ξ + η) f(ξ) = D f(ξ)(η) + o( η L 2), with η L 2 for η L 2 (Ω, F, P ). (ii) f : P 2 (R) R is differentiable at μ P 2 (R), if for f(ξ) := f(p ξ ), ξ L 2 (Ω, F, P ; R), there is some ζ L 2 (Ω, F, P ; R) with P ζ = μ such that f : L 2 (Ω, F, P ; R) R is Fréchet differentiable in ζ. 24 / 44

Case 2: Preliminaries From Riesz Representation Theorem there exists a P -a.s. unique variable θ L 2 (Ω, F, P ; R) such that D f(ζ)(η) = (θ, η) L 2 = E[θη], for all η L 2 (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 P ζ. Therefore, f(p ξ ) f(p ζ ) = E[h(ζ) (ξ ζ)] + o( ξ ζ L 2), ξ L 2 (Ω, F, P ; R). Definition 3.3 We call μ f(p ζ, y) := h(y), y R, the derivative of function f : P 2 (R) R at P ζ, ζ L 2 (Ω, F, P ; R). Remark: μ f(p ζ, y) is only P ζ (dy)-a.e. uniquely determined. 25 / 44

Case 2: Preliminaries Definition 3.4 We say that f C 1 (P 2 (R)), if for all ξ L 2 (Ω, F, P ; R) there exists a P ξ -modification of μ f(p ξ,.), also denoted by μ f(p ξ,.), such that μ f : P 2 (R) R R is continuous w.r.t the product topology generated by the 2-Wasserstein metric over P 2 (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 C 1,1 b (P 2 (R)), if f C 1 (P 2 (R)) is s.t. μ f : P 2 (R) R R is bounded and Lipschitz continuous, i.e., there exists some constant C such that (i) μ f(μ, x) C, μ P 2 (R), x R; (ii) μ f(μ, x) μ f(μ, x ) C(W 2 (μ, μ )+ x x ), μ, μ P 2 (R), x, x R. 26 / 44

Case 2: Preliminaries Definition 3.5 We say that f C 2 (P 2 (R)), if f C 1 (P 2 (R)) and μ f(μ,.) : R R is differentiable, and its derivative y μ f : P 2 (R) R R R is continuous, for every μ P 2 (R). Moreover, f C 2,1 b (P 2 (R)), if f C 2 (P 2 (R)) C 1,1 b (P 2 (R)) and its derivative y μ f : P 2 (R) R R R is bounded and Lipschitzcontinuous. Remark: C 2,1 b (R P 2 (R)), C 1,2,1 b ([, T ] R P 2 (R); R) are similarly defined. 27 / 44

Case 2: Preliminaries Now we can give our It^o s formula. Theorem 3.1 Let σ = (σ s ), γ = (γ s ), b = (b s ), β = (β s ) R-valued adapted stochastic processes, such that (i) There exists a constant q > 6 s.t. E[( T ( σ s q + b s q )ds) 3 q ] < + ; (ii) T ( γ s 2 + β s )ds < +, P-a.s. Let F C 1,2,1 b ([, T ] R P 2 (R)). Then, for the It^o processes X t = X + Y t = Y + σ s dw s + γ s dw s + b s ds, t [, T ], X L 2 (Ω, F, P ), β s ds, t [, T ], Y L 2 (Ω, F, P ), 28 / 44

Case 2: Preliminaries Theorem 3.1 (continued) we have F (t, Y t, P Xt ) F (, Y, P X ) ( = r F (r, Y r, P Xr ) + y F (r, Y r, P Xr )β r + 1 2 2 yf (r, Y r, P Xr )γr 2 + E[( μ F )(r, Y r, P Xr, X r ) b r + 1 ) 2 z( μ F )(r, Y r, P Xr, X r ) σ r] 2 dr + y F (r, Y r, P Xr )γ r dw r, t [, T ]. Here ( X, b, σ) denotes an independent copy of (X, b, σ), defined on a P.S. ( Ω, F, P ). The expectation E[ ] on ( Ω, F, P ) concerns only r.v. endowed with the superscript. 29 / 44

Case 2: Preliminaries (H3) The coefficients (σ, b) C 1,2,1 b ([, T ] R P 2 (R); R R). Theorem 3.2 (Buckdahn, Li, Peng and Rainer, 214) Let Φ C 2,1 b (R P 2 (R)), then under assumption (H3) the following PDE: = t V (t, x, μ) + x V (t, x, μ)b(x, μ) + 1 2 2 xv (t, x, μ)σ 2 (x, μ) + ( 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 V (T, x, μ) = Φ(x, μ), (x, μ) R P 2 (R). (t, x, μ) [, T ) R P 2 (R); has a unique classical solution V (t, x, μ) C 1,2,1 b ([, T ] R P 2 (R); R). 3 / 44

Case 2: Existence of a weak solution Let b and σ satisfy the following assumption: (H4) b, σ : [, T ] R P 2 (R) R are continuous and bounded. We want to study weak solution of the following mean-field SDE: X t = ξ + b(s, X s, Q Xs )ds σ(s, X s, Q Xs )db s, t [, T ], (3.3) where ξ L 2 (Ω, F, P ; R) obeys a given distribution law Q ξ = ν P 2 (R) and (B t ) t [,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 P on (C([, T ]; R), B(C([, T ]; R))) is a solution to the local martingale problem (resp., martingale problem) associated with A, if for every f C 1,2 ([, T ] R; R) (resp., f C 1,2 b ([, T ] R; R)), the process C f (t, y, μ) := f(t, y(t)) f(, y()) ( ( s + A)f ) (s, y(s), μ(s))ds, is a continuous local (F y, P )-martingale (resp., continuous (F y, P )- martingale), (3.4) 32 / 44

Case 2: Existence of a weak solution Definition 3.6 (continued) where μ(t) = P y(t) is the law of the coordinate process y = (y(t)) t [,T ] on C([, T ]; R) at time t, the filtration F y is that generated by y and completed, and A is defined by ( Af)(s, y, ν) := y f(s, y)b(s, y, ν) + 1 2 2 yf(s, y)σ 2 (s, y, ν), (3.5) (s, y, ν) [, T ] R P 2 (R). Here (( s + A)f)(s, y(s), μ(s)) abbreviates (( s + A)f)(s, y(s), μ(s)) := ( s f)(s, y(s)) + ( Af)(s, y(s), μ(s)). 33 / 44

Case 2: Existence of a weak solution Proposition 3.1 The existence of a weak solution ( Ω, F, F, Q, B, X) to equation (3.3) with initial distribution ν on B(R) is equivalent to the existence of a solution P to the local martingale problem (3.4) associated with A defined by (3.5), with P y() = ν. 34 / 44

Case 2: Existence of a weak solution Lemma 3.2 Let the probability measure P on (C([, T ]; R), B(C([, T ]; R))) be a solution to the local martingale problem associated with A. Then, for the second order differential operator ( Af)(s, y, ν) := ( Af)(s, y, ν) + + 1 2 R R ( μ f)(s, y, ν, z)b(s, z, ν)ν(dz) z ( μ f)(s, y, ν, z)σ 2 (s, z, ν)ν(dz), (3.6) 35 / 44

Case 2: Existence of a weak solution Lemma 3.2 (continued) applying to functions f C 1,2 ([, T ] R P 2 (R); R) we have that, for every such f C 1,2 ([, T ] R P 2 (R); R), the process C f (t, y, μ) :=f(t, y(t), μ(t)) f(, y(), μ()) ( s + A)f(s, y(s), μ(s))ds, t [, T ], (3.7) is a continuous local (F y, P )-martingale, where μ(t) = P y(t) is the law of the coordinate process y = (y(t)) t [,T ] on C([, T ]; R) at time t, the filtration F y is that generated by y and completed. Moreover, if f C 1,2,1 b ([, T ] R P 2 (R); R), this process C f is an (F y, P )-martingale. 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 ( Ω, F, F, Q, B, X). Remark 2. If b, σ : [, T ] C([, T ]; R) P 2 (C([, T ]; R)) R are bounded and continuous, then the following mean-field SDE X t = ξ + b(s, X s, Q X s )ds + σ(s, X s, Q X s )db s, t [, T ], (1.1) where ξ L 2 (Ω, F, P ) obeys a given distribution law Q ξ = ν, has a weak solution ( Ω, F, F, Q, X, B). 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 ν 1 and ν 2 on B(R), R φ(x)ν d 1 (dx) = R φ(x)ν d 2 (dx) for all φ C implies ν 1 = ν 2. Remark: The class C (R) is a determining class on R. 38 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 For given f C (R), we consider the Cauchy problem t v(t, x, ν) = Av(t, x, ν), (t, x, ν) [, T ] R P 2(R), v(, 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) R + 1 z ( μ v)(t, x, ν, u)σ 2 (t, u, ν)ν(du), 2 ( Av)(t, x, ν) = y v(t, x, ν)b(t, x, ν) + 1 2 2 yv(t, x, ν)σ 2 (t, x, ν), (t, x, ν) [, ) R P 2 (R). R 39 / 44

Case 2: Uniqueness in law of weak solutions Theorem 3.4 (continued) We suppose that, for all f C (R), (3.8) has a solution v f C b ([, ) R P 2 (R)) C 1,2,1 b ((, ) R P 2 (R)). Then, the local martingale problem associated with A (Recall Definition 3.6) and with the initial condition δ x 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 (Ω i, F i, F i, Q i, B i, X i ), of SDE (3.3), we have Q 1 X 1 = Q 2 X 2. 4 / 44

Uniqueness in law of weak solutions Sketch of proof of Theorem 3.4: Let T >, denote by y=(y(t)) t [,T ] the coordinate process on C([, T ]; R). Let P 1 and P 2 be two arbitrary solutions of the local martingale problem associated with A and initial condition x R: P l y() = δ x, l = 1, 2. Consequently, due to Lemma 3.2, for any g C 1,2,1 b ([, T ] R P 2 (R)), C g (t, y, P l y) := g(t, y(t), P l y(t) ) g(, x, δ x) ( s +A)g(s, y(s), P l y(s) )ds, (3.9) is a P l -martingale, l = 1, 2, t [, T ]. For given f C (R), let v f C b ([, T ] R P 2 (R)) C 1,2,1 b ((, T ) R P 2 (R)) be a solution of the Cauchy problem (3.8). 41 / 44

Uniqueness in law of weak solutions Then putting g(t, z, ν) := v f (T t, z, ν), t [, T ], z R, ν P 2 (R), defines a function g of class C b ([, T ] R P 2 (R)) C 1,2,1 b ((, T ) R P 2 (R)) which satisfies s g(s, z, ν)+ag(s, z, ν) =, g(t, z, ν) = f(z), (s, z, ν) [, T ] R P 2 (R). From (3.9) we see that {C g (s, y, P l y), s [, T ]} is an (F y, P l )- martingale. Hence, for E l [ ] = Ω l ( )dp l, E l [f(y(t ))] = E l [g(t, y(t ), P l y(t ) )] = g(, x, δ x), x R, l = 1, 2, that is E 1 [f(y(t ))] = E 2 [f(y(t ))], for all f C (R). Combining this with the arbitrariness of T, we have that Py(t) 1 = P y(t) 2, for every t. 42 / 44

Uniqueness in law of weak solutions Consequently, P 1, P 2 are solutions of the same classical martingale problem, associated with A = A l, l = 1, 2, A l φ(t, z) = y φ(t, z) b l (t, z) + 2 yφ(t, z)( σ l (t, z)) 2, φ C 1,2 ([, T ] R; R), with the coefficients σ 1 = σ 2, b 1 = b 2 (without mean field term), σ l (t, z) = σ(t, z, Py(t) l ), b l (t, z) = b(t, z, Py(t) l ), (t, z) [, T ] R, and we have seen that Py(t) 1 = P y(t) 2, t [, 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