Weak solutions of mean-field stochastic differential equations
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1 Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. 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, / 44
2 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
3 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
4 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
5 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 (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
6 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
7 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
8 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
9 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
10 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
11 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
12 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
13 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
14 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
15 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
16 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
17 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
18 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
19 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
20 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 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, / 44
21 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 f(s, Φ s (X 2, W 2 ))dw 2 s 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
22 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
23 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
24 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
25 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)) σ2 (s, y) xf(s, 2 y(s)). (3.2) 22 / 44
26 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
27 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
28 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
29 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
30 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
31 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
32 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 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
33 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) 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
34 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
35 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
36 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) 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
37 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
38 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) 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
39 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
40 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
41 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
42 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) yv(t, x, ν)σ 2 (t, x, ν), (t, x, ν) [, ) R P 2 (R). R 39 / 44
43 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
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
45 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
46 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
47 Thank you very much! 谢谢! 44 / 44
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