Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

Regularit y of the V alue F unction in Discoun ted Infinite-Time Mean Field Games Y ongsheng Song ∗ and Zeyu Y ang ∗ Marc h 17, 2026 Abstract In [ 17 ], we in tro duced the discoun ted infinite-time mean field games. Subsequen tly , in [ 18 ], w e studied the connection betw een infinite-time mean field FBSDEs and elliptic master equations. In this pap er, we further inv estigate the regularity of the representativ e pla yer’s v alue function. Sp ecifically , we first pro ve the strong existence and uniqueness, as well as the uniqueness in la w, for an extended class of infinite-time FBSDEs. W e then establish the Lions- differen tiabilit y for the deriv ativ e of the representativ e play er’s v alue function with resp ect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs. Keyw ords. discoun ted infinite-time mean field games, infinite-time FBSDEs, Lions-deriv ative ∗ State Key Lab oratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academ y of Sciences, Beijing 100190, China, and School of Mathematical Sciences, Univ ersity of Chinese Academy of Sciences, Beijing 100049, China. E-mails: yssong@amss.ac.cn (Y. Song), yangzeyu@amss.ac.cn (Z. Y ang). 1 1 In tro duction The study of mean field games was initiated indep enden tly by Lasry-Lions [ 7 – 9 ] and Huang- Malham ´ e-Caines [ 6 ], which is an analysis of limit mo dels for symmetric weakly in teracting ( N + 1) − pla yer differen tial games. W e refer the reader to [ 2 , 3 , 5 ] for a comprehensiv e exp osition on the sub ject. F orward-bac kw ard stochastic differen tial equations (FBSDEs) serv e as a p o w erful tool for the study of mean field games. The inv estigation of general nonlinear BSDEs w as pioneered b y P ardoux and P eng [ 12 , 13 ] in the early 1990s. [ 16 ] studied the infinite-time FBSDEs and established connections with quasilinear elliptic PDEs. Recen tly , [ 1 ] extended this framework to the McKean-Vlasov FBSDEs. In this pap er, we establish the strong existence and uniqueness, as well as uniqueness in law, for a broader class of infinite-time FBSDEs, and emplo y it to prov e the Lions-differen tiability of the v alue function’s deriv ativ e for the represen tative play er in the discoun ted infinite-time mean field games. In the recen t w ork [ 17 ], we prop osed the discoun ted infinite-time mean field games, which extends the traditional framework to infinite-time case. Within this framework, we in tro duce the follo wing t wo systems of infinite-time FBSDEs:            d X ξ t = ∂ y H ( X ξ t , L X ξ t , Y ξ t )d t + d B t , d Y ξ t = − h ∂ x H ( X ξ t , L X ξ t , Y ξ t ) − r Y ξ t i d t + Z ξ t d B t , X ξ 0 = ξ , (1.1)            d X x,ξ t = ∂ y H ( X x,ξ t , L X ξ t , Y x,ξ t )d t + d B t , d Y x,ξ t = − h ∂ x H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r Y x,ξ t i d t + Z x t d B t , X x 0 = x. (1.2) Here r > 0 is the discoun t factor, H ( x, µ, y ) ≜ min a ∈ R [ b ( x, µ, a ) · y + f ( x, µ, a )] , (1.3) and denote b y ˆ α ( x, µ, y ) the unique minimizer. The process X ξ represen ts the state process of the so cial equilibrium, while X x,ξ denotes the state pro cess of the represen tative pla yer with initial state x . In [ 18 ], w e define the v alue function V ( x, µ ) ≜ E  Z + ∞ 0 e − rt f  X x,ξ t , L X ξ t , ˆ α ( X x,ξ t , L X ξ t , Y x,ξ t )  d t  (1.4) of the represen tative play er where L ξ = µ . Under certain conditions, w e prov e that V ( x, µ ) is the viscosity solution to the elliptic master equation: 2 r U ( x, µ ) = H ( x, µ, ∂ x U ( x, µ )) + 1 2 ∂ xx U ( x, µ ) + ˜ E  1 2 ∂ ˜ x ∂ µ U ( x, µ, ˜ ξ ) + ∂ µ U ( x, µ, ˜ ξ ) ∂ y H ( ˜ ξ , µ, ∂ x U ( ˜ ξ , µ ))  . (1.5) Here ∂ x , ∂ xx are standard spatial deriv atives, ∂ µ , ∂ ˜ xµ are W 2 -W asserstein deriv atives, ˜ ξ is a ran- dom v ariable with la w µ and ˜ E is the exp ectation with resp ect to its la w. And w e ha v e the relationship Y x,ξ 0 = ∂ x V ( x, µ ) . (1.6) In [ 17 ], we define V ( x, µ ) ≜ Y x,ξ 0 , and prov e that it’s the viscosity solution to r U ( x, µ ) = ∂ x H ( x, µ, U ( x, µ )) + ∂ y H ( x, µ, U ( x, µ )) · ∂ x U ( x, µ ) + 1 2 ∂ xx U ( x, µ ) + ˜ E  1 2 ∂ ˜ x ∂ µ U ( x, µ, ˜ ξ ) + ∂ µ U ( x, µ, ˜ ξ ) ∂ y H ( ˜ ξ , µ, U ( ˜ ξ , µ ))  . (1.7) In this pap er, we prov e that V ( x, µ ) is Lions-differentiable with resp ect to the measure µ and deriv e the explicit form of its Lions-deriv ativ e. In the finite-time mean field games [ 3 ], the Lions- differen tiability of the v alue function and its deriv ative follow directly from the analysis of finite- time FBSDEs. Explicit expressions for these Lions-deriv ativ es are pro vided by Mou et al. [ 11 ] and Gangb o et al. [ 5 ] using FBSDEs. How ever, applying this conv en tional metho dology to our setting of infinite-time FBSDEs introduces significant c hallenges and fundamental differences. Sp ecifically , the directional deriv ativ e of FBSDEs ( 1.1 ) and ( 1.2 ) with respect to ξ along the direction η can b e represented by the following FBSDEs:                        d δ X ξ ,η t = n δ X ξ ,η t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) + δ Y ξ ,η t ∂ y y H ( X ξ t , L X ξ t , Y ξ t ) + ˜ E F t h ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t, d δ Y ξ ,η t = − n δ X ξ ,η t ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + δ Y ξ ,η t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) − r δ Y ξ ,η t + ˜ E F t h ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t + δ Z ξ ,η t d B t , δ X ξ ,η 0 = η ; (1.8)                              d δ X x,ξ ,η t = n δ X x,ξ ,η t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) + δ Y x,ξ ,η t ∂ y y H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ˜ E F t h ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t, d δ Y x,ξ ,η t = − n δ X x,ξ ,η t ∂ xx H ( X x,ξ t , L X ξ t , Y x,ξ t ) + δ Y x,ξ ,η t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r δY x,ξ ,η t + ˜ E F t h ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t + δ Z x,ξ ,η t d B t , δ X x,ξ ,η 0 = 0 . (1.9) 3 In the con ven tional framework, the Lions-differentiabilit y of V ( x, µ ) with resp ect to µ can b e established by sho wing that the contin uity of δ Y x,ξ ,η 0 in ξ is uniform with resp ect to η , whic h then allows the classical Lions-deriv ative analysis to b e applied. Ho w ev er, this approach is not feasible in our setting. T o ov ercome this difficulty , we adapt the construction of the Lions- deriv ativ e from [ 5 , 11 ] to the infinite-time FBSDE framew ork b y introducing the directional deriv ativ e ∂ µ V ( x, µ, ˜ x ) of V ( x, µ ), whic h satisfies: lim δ → 0 1 δ |V ( x, L ξ + δη ) − V ( x, L ξ ) | = E [ ∂ µ V ( x, L ξ , ξ ) · η ] . (1.10) Finally , b y proving that ∂ µ V ( x, µ, ˜ x ) is b ounded and con tinuous, we demonstrate that ∂ µ V ( x, µ, ˜ x ) is indeed the Lions-deriv ativ e of V ( x, µ ). This pap er is organized as follo ws: in section 2 , w e present the preliminaries of problems in this pap er; in section 3 , we prov e the strong existence and uniqueness, as well as the uniqueness in distribution, for a broader class of infinite-time FBSDEs, in preparation for the subsequen t analysis; in section 4 w e present an explicit construction of the directional deriv ativ e of V ( x, µ ) with resp ect to µ and ultimately establish the Lions-differen tiability of V ( x, µ ). 2 Preliminaries W e will use the filtered probabilit y space (Ω , F , P , F ) endo wed with a Brownian motion B . Its filtration F ≜ ( F t ) t ≥ 0 is augmen ted b y all P -n ull sets and a sufficiently rich sub- σ -algebra F 0 indep enden t of B , such that it can supp ort an y measure on R with finite second momen t. Let (Ω ′ , F ′ , P ′ , F ′ ) be a cop y of the filtered probability space (Ω , F , P , F ) with corresp onding Bro wnian motion B ′ , define the larger filtered probability space b y ˜ Ω ≜ Ω × Ω ′ , ˜ F ≜ F ⊗ F ′ ˜ F = { ˜ F t } t ≥ 0 ≜ {F t ⊗ F ′ t } t ≥ 0 , ˜ P ≜ P ⊗ P ′ , ˜ E ≜ E ˜ P . (2.1) Throughout the paper we will use the probabilit y space (Ω , F , P , F ). Ho wev er, when we deal with the distribution-dep enden t master equation, indep enden t copies of random v ariables or pro cesses are needed. Then w e will tacitly use their extensions to the larger space ( ˜ Ω , ˜ F , ˜ P , ˜ F ). Let P ≜ P ( R ) b e the set of all probabilit y measures on R and let P p ( p ≥ 1) denote the set of µ ∈ P with finite p -th moment. F or any sub- σ -field G ⊂ F and µ ∈ P p , w e define L p ( G ) to b e the set of R -v alued, G -measurable, and p -integrable random v ariables ξ , and L p ( G ; µ ) to b e the set of ξ ∈ L p ( G ) suc h that the law L ξ = µ . F or any µ, ν ∈ P p , w e define the W p –W asserstein distance b et ween them as follows: W p ( µ, ν ) ≜ inf n  E [ | ξ − η | q ]  1 /q : for all ξ ∈ L p ( F ; µ ), η ∈ L p ( F ; ν ) o . 4 W e introduce the W asserstein space and differential calculus on W asserstein space. F or a W 2 - con tinuous functions U : P 2 → R , its W 2 -W asserstein deriv ativ es [ 3 ](also called Lions-deriv ative), tak es the form ∂ µ U : ( µ, ˜ x ) ∈ P 2 × R → R and satisfies: U ( L ξ + η ) − U ( µ ) = E  ⟨ ∂ µ U ( µ, ξ ) , η ⟩  + o ( ∥ η ∥ 2 ) , ∀ ξ ∈ L 2 ( F ; µ ) , η ∈ L 2 ( F ) . (2.2) Let C 0 ( P 2 ) denote the set of W 2 -con tinuous functions U : P 2 → R . F or C 1 ( P 2 ), w e mean the space of functions U ∈ C 0 ( P 2 ) suc h that ∂ µ U exists and is contin uous on P 2 × R , whic h is uniquely determined by ( 2.2 ). Let C 2 , 1 ( R × P 2 ) denote the set of contin uous functions U : R × P 2 → R suc h that ∂ x U, ∂ xx U exist and are joint contin uous on R × P 2 , ∂ µ U, ∂ xµ U, ∂ ˜ xµ U exist and are con tinuous on R × P 2 × R . 3 Solutions to infinite-time FBSDEs 3.1 Strong solutions to infinite-time FBSDEs F or the needs of subsequen t problems, we aim to establish a more general theorem on the existence and uniqueness of solutions for infinite-time FBSDEs. Consider the follo wing form of infinite-time FBSDEs:          d X t = G ( t, ω , X t , Y t , L ( X t ,A t ) )d t + σ d B t , d Y t = − F ( t, ω , X t , Y t , L ( X t ,A t ) )d t + Z t d B t , X 0 = ξ , (3.1) where G, F : R + × Ω × R × R × P 2 ( R 2 ) → R are t wo progressively measurable functions, A t is a giv en adapted square integrable pro cess, σ ∈ R is a constan t and ξ is an F 0 − measurable square in tegrable random v ariable. F or an y ( v t ) ∈ L 2 K , we define the exp onen tially weigh ted L 2 norm ∥ v ∥ 2 K ≜ E  Z ∞ 0 e − K t | v t | 2 d t  . (3.2) F or simplicity , we only solve ( 3.1 ) for one dimensional ( X t , Y t , Z t ) and starting time t 0 = 0 , but our result can b e easily generalized to multidimensional case and arbitrary starting time t 0 > 0. The key idea of our pro of follows [ 1 , 16 ]. Assumption 3.1 Assume that for some c onstant K , th e functions F and G satisfy: (i) F or any L 2 K pr o c esses ( X t , Y t ) , G ( t, ω , X t , Y t , L ( X t ,A t ) ) and F ( t, ω , X t , Y t , L ( X t ,A t ) ) b elong to L 2 K . (ii) Ther e exists a p ositive c onstant ℓ such that for any x, x ′ , y , y ′ ∈ R , and any squar e inte gr able r andom variables X, X ′ , A 5 | G ( t, ω , x, y , L ( X,A ) ) − G ( t, ω , x ′ , y ′ , L ( X ′ ,A ) ) | + | F ( t, ω , x, y , L ( X,A ) ) − F ( t, ω , x ′ , y ′ , L ( X ′ ,A ) ) | ≤ ℓ ( | x − x ′ | + | y − y ′ | + E [ | X − X ′ | 2 ] 1 2 ) . a . s . (3.3) (iii) Ther e exists a c onstant κ > K / 2 , such that for any t ≥ 0 and any squar e inte gr able r andom variables X, X ′ , Y , Y ′ , A , E h − K ˆ X ˆ Y − ˆ X ( F ( t, ω , U ) − F ( t, ω , U ′ )) + ˆ Y ( G ( t, ω , U ) − G ( t, ω , U ′ )) i ≤ − κ E h ˆ X 2 + ˆ Y 2 i , (3.4) wher e ˆ X ≜ X − X ′ , ˆ Y ≜ Y − Y ′ and U ≜ ( X , Y , L ( X,A ) ) , U ′ ≜ ( X ′ , Y ′ , L ( X ′ ,A ) ) . Theorem 3.2 Under Assumption 3.1 , for e ach F 0 -me asur able squar e inte gr able r andom variable ξ , ( 3.1 ) has a unique solution ( X t , Y t , Z t ) in L 2 K . Pro of. First, w e pro v e the uniqueness. Supp ose there exist tw o solutions ( X t , Y t , Z t ), ( X ′ t , Y ′ t , Z ′ t ) in L 2 K to ( 3.1 ), and denote ˆ X ≜ X − X ′ ˆ Y ≜ Y − Y ′ ˆ Z ≜ Z − Z ′ . (3.5) W e choose a sequence of T i → ∞ suc h that E h e − K T i ˆ X T i ˆ Y T i i → 0 . (3.6) Applying Itˆ o’s formula to e − K t ˆ X t ˆ Y t , we get that E h e − K T i ˆ X T i ˆ Y T i i = E  Z T i 0 e − K t  − K ˆ X t ˆ Y t − ˆ X t ( F ( t, ω , X t , Y t , L ( X t ,A t ) ) − F ( t, ω , X ′ t , Y ′ t , L ( X ′ t ,A t ) )) + ˆ Y t ( G ( t, ω , X t , Y t , L ( X t ,A t ) ) − G ( t, ω , X ′ t , Y ′ t , L ( X ′ t ,A t ) ))  dt i ≤ − κ E  Z T i 0 e − K t  ˆ X 2 t + ˆ Y 2 t  d t  . (3.7) Letting T i → ∞ , w e get that ∥ ˆ X ∥ 2 K = ∥ ˆ Y ∥ 2 K = 0 , (3.8) and hence we complete the proof of the uniqueness. Next, we pro ve the existence of solutions, for this purp ose, we use the con tin uity metho d. W e study the following family of infinite-time FBSDEs parametrized by λ ∈ [0 , 1],            d X λ t = h λG ( t, ω , X λ t , Y λ t , L ( X λ t ,A t ) ) − κ (1 − λ ) Y λ t + ϕ t ( ω ) i d t + σ d B t , d Y λ t = − h λF ( t, ω , X λ t , Y λ t , L ( X λ t ,A t ) ) + κ (1 − λ ) X λ t + ψ t ( ω ) i d t + Z λ t d B t , X λ 0 = ξ , ( X λ t , Y λ t , Z λ t ) ∈ L 2 K , (3.9) 6 where ϕ, ψ are t wo arbitrary pro cesses in L 2 K . Note that when λ = 1 , ϕ = ψ = 0, ( 3.9 ) b ecomes ( 3.1 ), and when λ = 0, ( 3.9 ) b ecomes          d X 0 t = ( − κY 0 t + ϕ t ( ω ))d t + σ d B t , d Y 0 t = − ( κX 0 t + ψ t ( ω ))d t + Z 0 t d B t , X 0 0 = ξ . (3.10) It has b een prov ed in ( [ 1 ], Lemma 2.1) that ( 3.10 ) has unique solution ( X 0 , Y 0 , Z 0 ) ∈ L 2 K . No w, supp ose for some λ 0 ∈ [0 , 1), w e hav e that for any F 0 -measurable square in tegrable random v ariable ξ and ϕ, ψ ∈ L 2 K , ( 3.9 ) has a unique solution ( X λ 0 , Y λ 0 , Z λ 0 ) in L 2 K . W e try to find a constan t δ 0 , suc h that for an y δ ∈ [0 , δ 0 ], the FBSDE( 3.9 ) has a unique solution ( X λ 0 + δ , Y λ 0 + δ , Z λ 0 + δ ) in L 2 K for any given ξ , ϕ, ψ . T o do this, we consider the following FBSDE:                      d X t =  λ 0 G ( t, ω , X t , Y t , L ( X t ,A t ) ) − κ (1 − λ 0 ) Y t + δ  G ( t, ω , x t , y t , L ( x t ,A t ) ) + κy t  + ϕ t ( ω )  d t + σ d B t , d Y t = −  λ 0 F ( t, ω , X t , Y t , L ( X t ,A t ) ) + κ (1 − λ 0 ) X t + δ  F ( t, ω , x t , y t , L ( x t ,A t ) ) − κx t  + ψ t ( ω )  d t + Z t d B t , X 0 = ξ . (3.11) F or an y pair ( x t , y t ) in L 2 K , we hav e G ( t, ω , x t , y t , L ( x t ,A t ) ) and F ( t, ω , x t , y t , L ( x t ,A t ) ) b elong to L 2 K according to our hypothesis, then the FBSDE ( 3.11 ) admits a unique solution ( X, Y , Z ) in L 2 K . W e can define a map Φ through Φ : ( x, y ) 7→ ( X , Y ) . (3.12) W e then pro ve that the map Φ is a contraction on L 2 K . T ake another pair ( x ′ t , y ′ t ) in L 2 K and its image ( X ′ t , Y ′ t ). Denote U t = ( X t , Y t , L ( X t ,A t ) ) , u t = ( x t , y t , L ( x t ,A t ) ) , U ′ t = ( X ′ t , Y ′ t , L ( X ′ t ,A t ) ) , u t = ( x ′ t , y ′ t , L ( x ′ t ,A t ) ) , ˆ X t = X t − X ′ t , ˆ Y t = Y t − Y ′ t , ˆ x t = x t − x ′ t , ˆ y t = y t − y ′ t . (3.13) Applying Itˆ o’s formula to e − K t ˆ X t ˆ Y t , we get that 7 E h e − K T ˆ X T ˆ Y T i = λ 0 E  Z T 0 e − K t  − K ˆ X t ˆ Y t − ˆ X t  F ( t, ω , U t ) − F ( t, ω , U ′ t )  + ˆ Y t  G ( t, ω , U t ) − G ( t, ω , U ′ t )   d t  − κ (1 − λ 0 ) E  Z T 0 e − K t  ˆ X 2 t + ˆ Y 2 t  d t  − ( K − λ 0 K ) E  Z T 0 e − K t ˆ X t ˆ Y t d t  + κδ E  Z T 0 e − K t  ˆ X t ˆ x t + ˆ Y t ˆ y t  d t  + δ E  Z T 0 e − K t  − ˆ X t  F ( t, ω , u t ) − F ( t, ω , u ′ t )  + ˆ Y t  G ( t, ω , u t ) − G ( t, ω , u ′ t )   d t  . (3.14) It holds that E  − K ˆ X t ˆ Y t − ˆ X t  F ( t, ω , U t ) − F ( t, ω , U ′ t )  + ˆ Y t  G ( t, ω , U t ) − G ( t, ω , U ′ t )   ≤ − κ E h ˆ X 2 t + ˆ Y 2 t i (3.15) and E h − ˆ X t  F ( t, ω , u t ) − F ( t, ω , u ′ t )  + ˆ Y t  G ( t, ω , u t ) − G ( t, ω , u ′ t )  i ≤ E h | ˆ X t | · | F ( t, ω , u t ) − F ( t, ω , u ′ t ) | + | ˆ Y t | · | G ( t, ω , u t ) − G ( t, ω , u ′ t ) | i ≤ 3 ℓ 2 E h | ˆ X t | 2 + | ˆ Y t | 2 i + 2 ℓ E  | ˆ x t | 2 + | ˆ y t | 2  . (3.16) So we hav e E h e − K T ˆ X T ˆ Y T i ≤ −  κ − K 2 − κδ + 3 l δ 2  E  Z T 0 e − K t  ˆ X 2 t + ˆ Y 2 t  d t  + κδ + 4 l δ 2 E  Z T 0 e − K t  ˆ x 2 t + ˆ y 2 t  d t  . (3.17) W e take δ 0 = 2 κ − K 3 κ + 11 ℓ (3.18) and choose a sequence of T i → ∞ suc h that E h e − K T i ˆ X T i ˆ Y T i i → 0 . (3.19) F or any δ ∈ [0 , δ 0 ], we hav e E  Z ∞ 0 e − K t  ˆ X 2 t + ˆ Y 2 t  d t  ≤ 1 2 E  Z ∞ 0 e − K t  ˆ x 2 t + ˆ y 2 t  d t  . (3.20) Therefore Φ is a contraction. By rep eating this pro cedure for [1 /δ 0 ] many times, w e conclude that there exists a solution to ( 3.9 ) with λ = 1. In particular, we get a L 2 K solution to ( 3.1 ). 8 3.2 Distributional uniqueness A t first we denote b y C ([0 , ∞ ); R ) the space of con tinuous R -v alued functions on [0 , ∞ ) equipp ed with the metric of uniform conv ergence on compacts: d ( ω 1 , ω 2 ) = X n ≥ 0 2 − n sup t ∈ [0 ,n ] min( | ω 1 t − ω 2 t | , 1) . (3.21) Then, let C 0 ([0 , ∞ ); R ) denote the subspace of C ([0 , ∞ ); R ) consisting of functions satisfying ω (0) = 0, which is endo wed with the Wiener measure µ W (the la w of Bro wnian motion). Let B t = σ { ω ( s ) , 0 ≤ s ≤ t } and denote b y ¯ B ∞ the completion of B ∞ with resp ect to µ W . Then ( C 0 ([0 , ∞ ); R ) , ¯ B ∞ , µ W ) forms a complete probability space, and the filtration { ¯ B t , t ≥ 0 } satisfies the usual conditions. No w w e consider t w o functions ϕ, ψ : R × R + × C 0 ([0 , ∞ ); R ) → R m whic h satisfy the follo wing conditions: 1. ϕ, ψ are B ( R ) × B ( R + ) × ¯ B ∞ / B ( R m )-measurable; 2. for an y x ∈ R , ω ∈ C 0 ([0 , ∞ ); R ), we hav e ϕ ( x, · , ω ) , ψ ( x, · , ω ) ∈ C ([0 , ∞ ); R m ); 3. for an y x ∈ R , t ∈ R + , ϕ ( x, t, · ) and ψ ( x, t, · ) are ¯ B t / B ( R m )-measurable. Under the ab o ve conditions, we make an additional assumption that for any η ∈ L 2 ( F 0 ), the adapted pro cesses ( ϕ ( η , t, B ) , ψ ( η , t, B )) t ≥ 0 b elong to L 2 r . Indeed, we will subsequen tly prov e that all the solutions to the FBSDEs admit this representation. W e attempt to prov e the Y amada-W atanabe theorem for a broader class of FBSDEs, stating that pathwise uniqueness implies uniqueness in distribution. Consider the FBSDE of the form:          d X t = G ( t, X t , Y t , L ( X t ,ϕ t ( η ,B )) , ψ t ( η , B ))d t + d B t , d Y t = − F ( t, X t , Y t , L ( X t ,ϕ t ( η ,B )) , ψ t ( η , B ))d t + Z t d B t , X 0 = ξ . (3.22) where ξ , η ∈ L 2 ( F 0 ) and G, F : R + × R × R × P 2 ( R m +1 ) × R × R m → R are t wo measurable functions. W e then give the definitions of strong and weak uniqueness. Definition 3.3 (Strong uniqueness) We say that the str ong uniqueness holds for FBSDE ( 3.22 ) if on any set-up (Ω , F , P , F ) with inputs ( ξ , η , B ) , for any two F -pr o gr essively me asur- able L 2 r thr e e-tuples ( X 1 t , Y 1 t , Z 1 t ) t ≥ 0 , ( X 2 t , Y 2 t , Z 2 t ) t ≥ 0 (3.23) satisfying the FBSDE ( 3.22 ) with the same initial c ondition ( ξ , η ) (up to an exc eptional event), it holds that 9 E  Z ∞ 0 e − rt  | X 1 t − X 2 t | 2 + | Y 1 t − Y 2 t | 2 + | Z 1 t − Z 2 t | 2  d t  = 0 (3.24) Definition 3.4 (W eak uniqueness) F or any two set-ups (Ω 1 , F 1 , P 1 , F 1 ) and (Ω 2 , F 2 , P 2 , F 2 ) with inputs ( ξ 1 , η 1 , B 1 ) and ( ξ 2 , η 2 , B 2 ) , wher e ( ξ 1 , η 1 ) and ( ξ 2 , η 2 ) have the same joint law on R 2 , we say the we ak uniqueness holds for FBSDE ( 3.22 ) if for the L 2 r solutions ( X 1 t , Y 1 t , Z 1 t ) t ≥ 0 and ( X 2 t , Y 2 t , Z 2 t ) t ≥ 0 on c orr esp onding set-ups, the pr o c esses ( X 1 t , Y 1 t , R t 0 Z 1 s d s ) t ≥ 0 and ( X 2 t , Y 2 t , R t 0 Z 2 s d s ) t ≥ 0 have the same distribution. W e use the same scheme as the one dev elop ed b y Y amada and W atanab e to pro v e that path-wise uniqueness of solutions of FBSDE implies uniqueness in the sense of probability la w. Theorem 3.5 Assume that on (Ω , F , P , F ) with inputs ( ξ , η , B ) , the FBSDE ( 3.22 ) has a unique str ong solution ( X t , Y t , Z t ) t ≥ 0 . Then the law of ( X t , Y t , R t 0 Z s d s ) t ≥ 0 only dep ends on L ( ξ ,η ) . Mor e- over, ther e exists a me asur able function Φ : R 2 × R + × C 0 ([0 , ∞ ); R ) → R 3 , satisfying: 1. Φ is B ( R 2 ) × B ( R + ) × ¯ B ∞ / B ( R 3 ) -me asur able; 2. for any ( a, b ) ∈ R 2 , ω ∈ C 0 ([0 , ∞ ); R ) , we have Φ( a, b, · , ω ) ∈ C ([0 , ∞ ); R 3 ) ; and 3. for any ( a, b ) ∈ R 2 , t ∈ R + , Φ( a, b, t, · ) is ¯ B t / B ( R 3 ) -me asur able, such that, for any t ≥ 0 , we have P almost sur ely, ( X t , Y t , Z t 0 Z s d s ) = Φ( ξ , η , t, B ) . (3.25) Pro of. Let us consider tw o filtered probability spaces (Ω i , F i , P i , F i ) with identically distributed inputs ( ξ i , η i , B i ), i = 1 , 2, on eac h of which a solution ( X i t , Y i t , R t 0 Z i s d s ) t ≥ 0 to the FBSDE ( 3.22 ) is defined. Define Ω input ≜ R 2 × C 0 ([0 , ∞ ); R ) , Ω output ≜ C ([0 , ∞ ); R ) × C ([0 , ∞ ); R ) × C ([0 , ∞ ); R ) , Ω canon ≜ Ω input × Ω output . (3.26) Denote b y Q 1 and Q 2 the distribution of ( ξ 1 , η 1 , B 1 t , X 1 t , Y 1 t , R t 0 Z 1 s d s ) t ≥ 0 and ( ξ 2 , η 2 , B 2 t , X 2 t , Y 2 t , R t 0 Z 2 s d s ) t ≥ 0 on Ω canon = Ω input × Ω output , b y Q input the common distribution of the pro cesses ( ξ 1 , η 1 , B 1 t ), ( ξ 2 , η 2 , B 2 t ) on Ω input . Let us now define for i ∈ { 1 , 2 } , q i ( ω input ; F ) : Ω input × B (Ω output ) → [0 , 1] (3.27) as the regular conditional probability for B (Ω output ) given ω input ∈ Ω input (under Q i ). It satisfies: 10 • ∀ ω input ∈ Ω input , q i ( ω input ; · ) is a probabilit y measure on (Ω output , B (Ω output )). • ∀ F ∈ B (Ω output ), the mapping ω input → q i ( ω input ; F ) is B ( R 2 ) ⊗B ( C 0 ([0 , ∞ ) , R ))-measurable. • ∀ F ∈ B (Ω output ) , ∀ G ∈ B (Ω input ): Q i ( G × F ) = Z G q i ( ω input ; F ) Q input (d ω input ) . (3.28) Next, we need a enlarged space (Ω total , G , Q ) to supp ort all processes. Define Ω total ≜ Ω input × Ω output × Ω output , (3.29) and G is the completion of the σ -field B (Ω canon ) ⊗ B (Ω output ) by the collection N of all null sets under the probability measure Q ( G × F 1 × F 2 ) = Z G q 1 ( ω input ; F 1 ) q 2 ( ω input ; F 2 ) Q input (d ω input ) , (3.30) where F 1 , F 2 ∈ B (Ω output ) , G ∈ B (Ω input ). W e observ e that Q ( G × F 1 × Ω output ) = Q 1 ( G × F 1 ) , Q ( G × Ω output × F 2 ) = Q 2 ( G × F 2 ) . Inparticular, we denote b y ( a, b, w , x 1 , y 1 , ζ 1 , x 2 , y 2 , ζ 2 ) the canonical pro cess on Ω total , then ( a, b, w , x 1 , y 1 , ζ 1 ) has distribution Q 1 and ( a, b, w , x 2 , y 2 , ζ 2 ) has distribution Q 2 . W e define ( z i t ) t ≥ 0 , i ∈ { 1 , 2 } b y z i t =      lim n →∞ n  ζ i t − ζ i ( t − 1 n )+  if the limit exists, 0 otherwise. (3.31) Since P i -a.s., R t 0 Z i s d s is absolutely con tinuous on every finite interv al, we kno w Q i -a.s., ζ i t is absolutely contin uous on ev ery finite in terv al. So ζ i t = Z t 0 z i s d s, t ≥ 0 . (3.32) Moreo ver, w e hav e E Q Z ∞ 0 e − rt | z i t | d t ≤ lim n →∞ E Q Z ∞ 0 e − rt | n ( ζ i t − ζ i ( t − 1 n )+ ) | 2 d t = lim n →∞ n 2 E P i Z ∞ 0 e − rt | Z t ( t − 1 n )+ Z i s d s | 2 d t ≤ lim n →∞ n E P i Z ∞ 0 e − rt Z t ( t − 1 n )+ | Z i s | 2 d s d t ≤ E Q i Z ∞ 0 e − rs | Z i s | 2 d s, (3.33) 11 the last inequality follo wing from F ubini’s theorem. Let us now endow (Ω total , G , Q ) with the filtration G , where G = {G t } t ≥ 0 is the complete and righ t-contin uous augumen tation under Q of the canonical filtration H t = { ( a, b, w s , x 1 s , y 1 s , ζ 1 s , x 2 s , y 2 s , ζ 2 s ); 0 ≤ s ≤ t } (3.34) on Ω total . It’s easy to see that ( a, b ) are G 0 -measurable and that ( w s , x 1 s , y 1 s , z 1 s , x 2 s , y 2 s , z 2 s ) t ≥ 0 are {G t } t ≥ 0 -progressiv ely measurable. Moreov er, for i ∈ { 1 , 2 } , Q  ω ∈ Ω total ; ( a, b, w , x i , y i , ζ i ) ∈ A  = Q i  ( ξ i , η i , B i , X i , Y i , Z · 0 Z i s d s ) ∈ A  ; A ∈ B (Ω canon ) . (3.35) Actually , we just ha ve to pro ve that ( w t ) t ≥ 0 is a {G t } t ≥ 0 Bro wnian motion: w e follo w the pro of given in ( [ 4 ], Remark 1.6). Let us firstly define π t : C 0 ([0 , ∞ ); R ) → C 0 ([0 , t ]; R ) , h → h | [0 ,t ] , (3.36) and π ′ t : γ → γ t , h → h | [0 ,t ] , (3.37) where γ = Ω output and γ t = C ([0 , t ]; R ) × C ([0 , t ]; R ) × C ([0 , t ]; R ) . (3.38) Endo wing C ([0 , t ]; R ) and γ t with their b orelian σ -fields, we define K t ≜ σ { π t } , K ′ t ≜ σ { π ′ t } . (3.39) Using the separability of the spaces C 0 ([0 , t ]; R ) and γ t , we see that K t = σ { w s ; 0 ≤ s ≤ t } , (3.40) and that ∀ i ∈ { 1 , 2 } , ∀ A ∈ K ′ t , the set { ( X i , Y i , R · 0 Z i s d s ) ∈ A } belongs to F i t . No w, considering A ∈ K ′ t , we wan t to sho w that, for i ∈ { 1 , 2 } , the map Ω input → [0 , 1] , ( a, b, w ) → q i ( a, b, w ; A ) (3.41) is measurable with respect to the completion of the σ -field B ( R 2 ) ⊗ K t under the probabilit y measure Q input , denoted B ( R 2 ) ⊗ K t . Indeed, let us consider F ∈ B ( R 2 ), G 1 ∈ K t and G 2 ∈ σ { w s − w t ; s ≥ t } . Then, ∀ i ∈ { 1 , 2 } 12 Z I F ( a, b ) I G 1 ( w ) I G 2 ( w ) q i ( ξ , w ; A ) Q input (d ξ d w ) = E P i  I F ( ξ i , η i ) I G 1 ( B i ) I G 2 ( B i ) I A ( X i , Y i , Z · 0 Z i s d s )  = E P i  I F ( ξ i , η i ) I G 1 ( B i ) I A ( X i , Y i , Z · 0 Z i s d s )  E P i  I G 2 ( B i )  = Z I F ( a, b ) I G 1 ( w ) q i ( ξ , w ; A ) Q input (d ξ d w ) Z I G 2 ( w ) Q input (d ξ d w ) . (3.42) Hence, using Exercise (17.10) Chapter V of Rogers and Williams [ 15 ], the map ( a, b, w ) → q i ( a, b, w ; A ) is measurable with resp ect to B ( R 2 ) ⊗ K t . No w w e pro v e that ( w t ) t ≥ 0 is a {G t } t ≥ 0 Bro wnian motion. Let us consider ( A, A ′ ) ∈ ( K ′ t ) 2 , F ∈ B ( R 2 ) , G 1 ∈ K t and G 2 ∈ σ { w s − w t ; s ≥ t } . Then, E Q  I F ( a, b ) I G 1 ( w ) I G 2 ( w ) I A ( x 1 , y 1 , ζ 1 ) I A ′ ( x 2 , y 2 , ζ 2 )  = Z I F ( a, b ) I G 1 ( w ) I G 2 ( w ) q 1 ( ξ , w ; A ) q 2 ( ξ , w ; A ′ ) Q input (d ξ d w ) = Z I F ( a, b ) I G 1 ( w ) q 1 ( ξ , w ; A ) q 2 ( ξ , w ; A ′ ) Q input (d ξ d w ) Z I G 2 ( w ) Q input (d ξ d w ) = E Q  I F ( a, b ) I G 1 ( w ) I A ( x 1 , y 1 , ζ 1 ) I A ′ ( x 2 , y 2 , ζ 2 )  E Q [ I G 2 ( w )] . (3.43) Noting that H t = B ( R 2 ) ⊗ K t ⊗ K ′ t ⊗ K ′ t , we conclude that ( w t ) t ≥ 0 is a {G t } t ≥ 0 Bro wnian motion. A t last, applying the same pro cedure on ( z i t ) t ≥ 0 , i ∈ { 1 , 2 } in ( [ 3 ], V olume I I, Lemma 1.27), w e obtain that Q − a.s. : for all 0 ≤ t ≤ T , i ∈ { 1 , 2 } ,          x i t = a + R t 0 G ( s, x i s , y i s , L ( x i s ,ϕ t ( b,w )) , ψ t ( b, w )) ds + w t , y i t = y i T + R T t F ( s, x i s , y i s , L ( x i s ,ϕ t ( b,w )) , ψ t ( b, w )) ds − R T t z i s d w s , E Q  R ∞ 0 e − rt  | x i t | 2 + | y i t | 2 + | z i t | 2  d t  < ∞ . (3.44) Through the strong uniqueness, w e know that under Q , the pro cesses ( x 1 t , y 1 t , ζ 1 t ) t ≥ 0 and ( x 2 t , y 2 t , ζ 2 t ) t ≥ 0 ha ve the same la w. This implies that, Q input -a.s., ( q 1 ( ω input ; · ) × q 2 ( ω input ; · ))(( x 1 t , y 1 t , ζ 1 t ) t ≥ 0 = ( x 2 t , y 2 t , ζ 2 t ) t ≥ 0 ) = 1 . (3.45) A pro duct measure that is concentrated on the diagonal set is necessarily a Dirac measure at some p oin t Ψ( ω input ) ∈ Ω output , which means, Q input -a.s., q 1 ( ω input ; · ) = q 2 ( ω input ; · ) = δ Ψ( ω input ) . (3.46) Then we hav e, for an y t ≥ 0, i ∈ { 1 , 2 } , P i -a.s., ( X i t ( ω ) , Y i t ( ω ) , Z t 0 Z i s ( ω )d s ) = Ψ t ( ξ i ( ω ) , η i ( ω ) , B i · ( ω )) . (3.47) 13 No w, w e define Φ( a, b, t, w ) ≜ Ψ t ( a, b, w ). Then, Φ is contin uous with respect to t and B ( R 2 ) × ¯ B ∞ - measurable with resp ect to ( a, b, w ), hence it is B ( R 2 ) × B ( R + ) × ¯ B ∞ -measurable with resp ect to ( a, b, t, w ). Moreov er, since for any A ∈ K ′ t , q 1 ( ω input ; A ) is B ( R 2 ) ⊗ K t -measurable, then for ev ery t ≥ 0, Ψ t ( ω input ) is B ( R 2 ) ⊗ K t -measurable; consequently , it clearly follows that for an y ( a, b ) ∈ R 2 , t ∈ R + , Φ( a, b, t, · ) is ¯ B t / B ( R 3 )-measurable. Now we finish the proof. 4 Lions-differen tiabilit y of the v alue function’s deriv ativ e In this section, for the L 2 r solutions of FBSDEs ( 1.1 ) and ( 1.2 ) with some ξ ∈ L 2 ( F 0 ), we define V ( x, µ ) ≜ Y x,ξ 0 . we will prov e that V ( x, µ ) is Lions-differen tiable with resp ect to µ . T o this end, w e first establish the existence of the function ∂ µ V ( x, µ, ˜ x ) : R × P 2 × R → R , whic h satisfies lim δ → 0 1 δ |V ( x, L ξ + δη ) − V ( x, L ξ ) | = E [ ∂ µ V ( x, L ξ , ξ ) · η ] (4.1) for all ξ , η ∈ L 2 ( F 0 ). Referring to the construction of the Lions-deriv ativ e for the parab olic master equation in [ 5 , 11 ], we derive a representation of ∂ µ V via solutions of FBSDEs, such that ∂ µ V ( x, µ, ˜ x ) is uniformly b ounded and jointly con tinuous. Consequently , ∂ µ V ( x, L ξ , ξ ) b ecomes con tinuous in ξ with respect to the L 2 -norm, which ensures the Lions-differen tiability of V with resp ect to the measure µ ; the deriv ative ∂ µ V ( x, µ, ˜ x ) is then iden tified as the Lions-deriv ative. W e require the following stronger conditions in this section, and it is straigh tforward to v erify that they subsume the Assumption 5.1 in [ 18 ]. Therefore b oth FBSDEs ( 1.1 ) and ( 1.2 ) admit unique L 2 r solutions. Assumption 4.1 (i) H ( x, µ, y ) has at most quadr atic gr owth. ∂ x H , ∂ y H , ∂ xx , ∂ xy H , ∂ y y H , ∂ µ H ( x, µ, y , ˜ x ) , ∂ xµ H ( x, µ, y , ˜ x ) , ∂ y µH ( x, µ, y , ˜ x ) exist and ar e Lipschitz c ontinuous. (ii) Ther e exist c onstants λ 1 , λ 2 > 0 such that − λ 1 + 2 λ 2 < − r / 2 and ∂ y y H ( x, µ, y ) ≤ − λ 1 , ∂ xx H ( x, µ, y ) ≥ λ 1 , | ∂ xµ H ( x, µ, y , ˜ x ) | ≤ λ 2 , | ∂ y µ H ( x, µ, y , ˜ x ) | ≤ λ 2 . (4.2) (iii) Ther e exist a c onstant λ 3 > 0 such that | ∂ xx H ( x, µ, y ) | ≤ λ 3 , | ∂ y y H ( x, µ, y ) | ≤ λ 3 | ∂ xy H ( x, µ, y ) | ≤ λ 3 . (4.3) 4.1 Existence of directional deriv ativ e F or arbitrary ξ , η ∈ L 2 ( F 0 ) , x ∈ R , w e introduce the follo wing FBSDEs: 14                        d δ X ξ ,η t = n δ X ξ ,η t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) + δ Y ξ ,η t ∂ y y H ( X ξ t , L X ξ t , Y ξ t ) + ˜ E F t h ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t, d δ Y ξ ,η t = − n δ X ξ ,η t ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + δ Y ξ ,η t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) − r δ Y ξ ,η t + ˜ E F t h ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t + δ Z ξ ,η t d B t , δ X ξ ,η 0 = η ; (4.4)                              d δ X x,ξ ,η t = n δ X x,ξ ,η t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) + δ Y x,ξ ,η t ∂ y y H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ˜ E F t h ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t, d δ Y x,ξ ,η t = − n δ X x,ξ ,η t ∂ xx H ( X x,ξ t , L X ξ t , Y x,ξ t ) + δ Y x,ξ ,η t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r δY x,ξ ,η t + ˜ E F t h ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t io d t + δ Z x,ξ ,η t d B t , δ X x,ξ ,η 0 = 0 . (4.5) Under Assumption 4.1 , it’s clear that the abov e FBSDEs admit unique solutions in L 2 r . Similar to Theorem 5.2 in [ 18 ], we can pro ve that lim δ → 0     1 δ  X ξ + δη − X ξ  − δ X ξ ,η     r = 0 , lim δ → 0     1 δ  Y ξ + δη − Y ξ  − δ Y ξ ,η     r = 0 , lim δ → 0     1 δ  X x,ξ + δη − X x,ξ  − δ X x,ξ ,η     r = 0 , lim δ → 0     1 δ  Y x,ξ + δη − Y x,ξ  − δ Y x,ξ ,η     r = 0 . (4.6) And most imp ortan tly , lim δ → 0     1 δ ( Y x,ξ + δη 0 − Y x,ξ 0 ) − δ Y x,ξ ,η 0     = 0 . (4.7) Lemma 4.2 Fix x ∈ R , ξ ∈ L 2 ( F 0 ) , the mapping η → δ Y x,ξ ,η 0 is a b ounde d line ar functional on the Hilb ert sp ac e L 2 ( F 0 ) . Pro of. By the linearit y of the FBSDE ( 4.4 ) and ( 4.5 ), it is straigh tforw ard to see that the mapping is linear. Th us, it suffices to pro ve its b oundedness. Supp ose that E [ η 2 ] ≤ 1. Applying Itˆ o’s form ula to e − rt δ X ξ ,η t δ Y ξ ,η t and integating it from 0 to infinity , we can get that − E [ η δ Y ξ ,η 0 ] = E  Z ∞ 0 e − rt  ( δ Y ξ ,η t ) 2 ∂ y y H ( X ξ t , L X ξ t , Y ξ t ) − ( δ X ξ ,η t ) 2 ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + δ Y ξ ,η t · ˜ E F t h ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t i − δ X ξ ,η t · ˜ E F t h ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t i  d t  ≤ − r 2 E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  . (4.8) 15 Therefore, E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  ≤ 2 r E [ η δ Y ξ ,η 0 ] ≤ ϵ E [( δ Y ξ ,η 0 ) 2 ] + 1 ϵr 2 . (4.9) Applying Itˆ o’s formula to e − rt ( δ Y ξ ,η t ) 2 , we can get that E [( δ Y ξ ,η 0 ) 2 ] = E  Z ∞ 0 e − rt  2 δ X ξ ,η t δ Y ξ ,η t ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + 2( δ Y ξ ,η t ) 2 ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) + 2 δ Y ξ ,η t · ˜ E F t h ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) δ ˜ X ξ ,η t i − r ( δY ξ ,η t ) 2 − ( δ Z ξ ,η t ) 2  d t  ≤ ( λ 2 + 3 λ 3 ) E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  . (4.10) Com bining it with ( 4.9 ) and taking ϵ = 1 2( λ 2 +3 λ 3 ) , we can get that E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  ≤ C 1 , (4.11) where C 1 > 0 is a constant. A t last, we apply the same procedure to e − rt δ X x,ξ ,η t δ Y x,ξ ,η t and e − rt ( δ Y ξ ,η t ) 2 . W e can get that E  Z ∞ 0 e − rt  ( δ X x,ξ ,η t ) 2 + ( δ Y x,ξ ,η t ) 2  d t  ≤ C 2 E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  (4.12) and ( δ Y x,ξ ,η 0 ) 2 ≤ C 3 E  Z ∞ 0 e − rt  ( δ X x,ξ ,η t ) 2 + ( δ Y x,ξ ,η t ) 2  d t  + C 4 E  Z ∞ 0 e − rt  ( δ X ξ ,η t ) 2 + ( δ Y ξ ,η t ) 2  d t  , (4.13) where C 2 , C 3 , C 4 are all p ositive constan ts. Th us, we ha ve pro ved that δ Y x,ξ ,η 0 is b ounded for fixed ( x, ξ ) and all η such that E [ η 2 ] ≤ 1. Corollary 4.3 The fol lowing mappings ar e jointly c ontinuous: 1. L 2 ( F 0 ) × L 2 ( F 0 ) → L 2 r × L 2 r × L 2 ( F 0 ) : ( ξ , η ) → ( δ X ξ ,η , δ Y ξ ,η , δ Y ξ ,η 0 ) , 2. R × L 2 ( F 0 ) × L 2 ( F 0 ) → L 2 r × L 2 r × R : ( x, ξ , η ) → ( δ X x,ξ ,η , δ Y x,ξ ,η , δ Y x,ξ ,η 0 ) . Pro of. Fix some ( x, ξ , η ) ∈ R × L 2 ( F 0 ) × L 2 ( F 0 ) and consider a sequence { ( x n , ξ n , η n ) ∈ R × L 2 ( F 0 ) × L 2 ( F 0 ); n ≥ 1 } suc h that lim n →∞  | x n − x | + E [ | ξ n − ξ | 2 ] + E [ | η n − η | 2 ]  = 0 . (4.14) 16 Without loss of generality , w e may assume that { η n } is b ounded in L 2 ( F 0 ). By Lemma 4.2 , all solutions to the FBSDEs that app ear in the subsequen t proof are uniformly b ounded. Denote X n = δ X ξ n ,η n − δ X ξ ,η , Y n = δ Y ξ n ,η n − δ Y ξ ,η , Z n = δ Z ξ n ,η n − δ Z ξ ,η . (4.15) By applying Itˆ o’s formula to e − rt X n t Y n t and e − rt ( Y n t ) 2 (instead of e − rt δ X ξ ,η t δ Y ξ ,η t and e − rt ( δ Y ξ ,η t ) 2 in the pro of of Lemma 4.2 ), and combining it with the pro of of Theorem 5.2 in [ 18 ] together with the Dominated Conv ergence Theorem, w e obtain: lim n →∞ E  ( Y n 0 ) 2 + Z ∞ 0 e − rt  ( X n t ) 2 + ( Y n t ) 2  d t  = 0 . (4.16) Hence, the contin uity (1) is prov ed. The con tinuit y (2) can b e prov ed similarly . Since the mapping η → δ Y x,ξ ,η 0 is a b ounded linear functional on the Hilb ert space L 2 ( F 0 ), w e can find a random v ariable D x,ξ ∈ L 2 ( F 0 ), such that: δ Y x,ξ ,η 0 = E [ D x,ξ · η ] . (4.17) Whereas the framework in [ 3 ] successfully deriv es the Lions-differentiabilit y of V ( t, x, µ ) with resp ect to µ for parabolic master equations and finite-time FBSDEs by establishing the contin uit y of D x,ξ in ξ , our framew ork encounters a significan t h urdle. Sp ecifically , the contin uity result in Corollary 4.3 is merely p oin twise, and the contin uity of δ Y x,ξ ,η 0 with resp ect to ξ lacks uniformit y o ver η . This fundamental difference prompts our inv estigation in to an alternativ e methodology . Next, we will prov e that for an y x ∈ R , D x,ξ can b e expressed as f ( ξ ), where f is a Borel function dep ends only on the distribution of ξ . This implies the existence of the directional deriv ativ e of V ( x, µ ). Theorem 4.4 Ther e exists a function ∂ µ V ( x, µ, ˜ x ) : R × P 2 × R → R , such that lim δ → 0 1 δ |V ( x, L ξ + δη ) − V ( x, L ξ ) | = E [ ∂ µ V ( x, L ξ , ξ ) · η ] (4.18) for any ξ , η ∈ L 2 ( F 0 ) . Pro of. Fix x ∈ R , we already kno w that lim δ → 0 1 δ |V ( x, L ξ + δη ) − V ( x, L ξ ) | = δ Y x,ξ ,η 0 = E [ D x,ξ · η ] . (4.19) F or some ξ ∈ L 2 ( F 0 ), we w an t to show that D x,ξ = f ( ξ ) for some Borel function f . W e will prov e this b y contradiction. Assume that D x,ξ  = E [ D x,ξ | ξ ], tak e η 1 = D x,ξ − E [ D x,ξ | ξ ]. Construct a random v ariable η 2 suc h that, conditional on ξ , η 1 and η 2 are indep enden t and identically distributed (i.i.d.). On the one hand, 17 δ Y x,ξ ,η 1 0 = E [ D x,ξ · η 1 ] = E [( η 1 ) 2 ] > 0 . (4.20) On the other hand, δ Y x,ξ ,η 2 0 = E [ D x,ξ · η 2 ] = E [ η 1 η 2 ] + E [ E [ D x,ξ | ξ ] η 2 ] = E [ E [ η 1 | ξ ] E [ η 2 | ξ ]] + E [ E [ D x,ξ | ξ ] E [ η 2 | ξ ]] =0 . (4.21) Ho wev er, by the w eak uniqueness of FBSDEs, w e kno w that δ Y x,ξ ,η 1 0 = δ Y x,ξ ,η 2 0 . W e therefore conclude that D x,ξ = E [ D x,ξ | ξ ] , a.s. (4.22) W e already know that D x,ξ = f ( ξ ). Next, we will pro v e that f depends only on the distribu- tion of ξ . Select ξ 1 and ξ 2 suc h that L ξ 1 = L ξ 2 = µ , and D x,ξ 1 = f 1 ( ξ 1 ) , D x,ξ 2 = f 2 ( ξ 2 ). Then, for any g ∈ C b ( R ), take η 1 = g ( ξ 1 ) and η 2 = g ( ξ 2 ). W e ha ve δ Y x,ξ 1 ,η 1 0 = δ Y x,ξ 2 ,η 2 0 , which implies Z R f 1 ( x ) g ( x ) µ (d x ) = Z R f 2 ( x ) g ( x ) µ (d x ) , ∀ g ∈ C b ( R ) . (4.23) Then we know f 1 ( x ) = f 2 ( x ) , µ -a.e. (4.24) No w w e can conclude that, for an y ( x, µ ) ∈ R × P 2 , there exists a function f x,µ ( ˜ x ) defined on the supp ort of µ , suc h that D x,ξ = f x,µ ( ξ ) for all ξ ∈ L 2 ( F 0 ; µ ). Then w e can define ∂ µ V ( x, µ, ˜ x ) ≜ f x,µ ( ˜ x ) (4.25) for some version of f x,µ . This satisfies the conditions of the theorem. It should be noted that although w e hav e pro v ed that ∂ µ V ( x, µ, ˜ x ) satisfies ( 4.18 ), this only demonstrates the differentiabilit y of V with resp ect to the measure in the sense of directional deriv ativ e, which is still far from constituting a Lions-deriv ativ e. Similarly , for the v alue function V ( x, µ ) = E  Z ∞ 0 e − rt F ( X x,ξ t , L X ξ t , Y x,ξ t )d t  (4.26) where F ( x, µ, y ) ≜ f ( x, µ, ˆ α ( x, y )) = H ( x, µ, y ) − y ∂ y H ( x, µ, y ), it can also b e shown that: lim δ → 0 1 δ | V ( x, L ξ + δη ) − V ( x, L ξ ) | = E  Z ∞ 0 e − rt  ∂ x F ( X x,ξ t , L X ξ t , Y x,ξ t ) · δ X x,ξ ,η t + ∂ y F ( X x,ξ t , L X ξ t , Y x,ξ t ) · δ Y x,ξ ,η t + ˜ E F t h ∂ µ F ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) · δ ˜ X ξ ,η t i  d t  . (4.27) 18 By the same tec hnique as b efore, one can prov e the existence of a function ∂ µ V ( x, µ, ˜ x ) satisfying lim δ → 0 1 δ | V ( x, L ξ + δη ) − V ( x, L ξ ) | = E [ ∂ µ V ( x, L ξ , ξ ) · η ] (4.28) for all ξ , η ∈ L 2 ( F 0 ). Next, w e aim to select a sufficiently regular v ersion of ∂ µ V such that ∂ µ V ( x, L ξ , ξ ) is contin uous with resp ect to ξ . This allows us to strengthen the Gˆ ateaux deriv ativ e to the F r´ ec het deriv ative, thereb y V ( x, L ξ + η ) − V ( x, L ξ ) = E [ ∂ µ V ( x, L ξ , ξ ) · η ] + o ( ∥ η ∥ L 2 ) . (4.29) This demonstrates that ∂ µ V is the Lions-deriv ativ e of V . 4.2 Represen tation of the Lions-deriv ative First we consider the case that ξ is discrete: p i = P ( ξ = x i ) , i = 1 , 2 , · · · , n. (4.30) Then we hav e δ Y x,ξ ,I { ξ = x i } 0 = E [ ∂ µ V ( x, L ξ , ξ ) · I { ξ = x i } ] = p i · ∂ µ V ( x, L ξ , x i ) . (4.31) Th us, w e can determine the v alues of ∂ µ V ( x, L ξ , · ) on the support of L ξ . T o further in vestigate δ Y x,ξ ,I { ξ = x i } 0 , we introduce the follo wing FBSDEs: 19                                                                                          d ∇ X ξ ,x i t = n ∇ X ξ ,x i t ∂ xy H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t ) + ∇ Y ξ ,x i t ∂ y y H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t ) + p i ˜ E F t h ∇ ˜ X ξ ,x i t ∂ y µ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ y µ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X ξ t ) io d t, d ∇ X ξ ,x i ,⋆ t = n ∇ X ξ ,x i ,⋆ t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) + ∇ Y ξ ,x i ,⋆ t ∂ y y H ( X ξ t , L X ξ t , Y ξ t ) + ˜ E F t h ∇ ˜ X ξ ,x i t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) i · I { ξ  = x i } o d t, d ∇ Y ξ ,x i t = − n ∇ X ξ ,x i t ∂ xx H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t ) + ∇ Y ξ ,x i t ∂ xy H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t ) − r ∇ Y ξ ,x i t + p i ˜ E F t h ∇ ˜ X ξ ,x i t ∂ xµ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ xµ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X ξ t ) io d t + ∇ Z ξ ,x i t d B t , d ∇ Y ξ ,x i ,⋆ t = − n ∇ X ξ ,x i ,⋆ t ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + ∇ Y ξ ,x i ,⋆ t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) − r ∇ Y ξ ,x i ,⋆ t + ˜ E F t h ∇ ˜ X ξ ,x i t ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) i · I { ξ  = x i } o d t + ∇ Z ξ ,x i ,⋆ t d B t , ∇ X ξ ,x i 0 = 1 , ∇ X ξ ,x i ,⋆ 0 = 0 . (4.32)                                            d ∇ µ X x,ξ ,x i t = n ∇ µ X x,ξ ,x i t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ µ Y x,ξ ,x i t ∂ y y H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ˜ E F t h ∇ ˜ X ξ ,x i ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) io d t, d ∇ µ Y x,ξ ,x i t = − n ∇ µ X x,ξ ,x i t ∂ xx H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ µ Y x,ξ ,x i t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r ∇ µ Y x,ξ ,x i t + ˜ E F t h ∇ ˜ X ξ ,x i t ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) io d t + ∇ µ Z x,ξ ,x i t d B t , ∇ µ X x,ξ ,x i 0 = 0 . (4.33) Theorem 4.5 Under Assumption 4.1 and the c ondition that ξ ∈ L 2 ( F 0 ) is a discr ete r andom variable satisfying ( 4.30 ), b oth FBSDEs ( 4.32 ) and ( 4.33 ) admit unique solutions in L 2 r . And we have the r elationship: δ Y x,ξ ,I { ξ = x i } t = p i · ∇ µ Y x,ξ ,x i t . (4.34) This me ans 20 ∂ µ V ( x, L ξ , x i ) = ∇ µ Y x,ξ ,x i 0 . (4.35) Pro of. F or the FBSDE ( 4.32 ), we denote X ≜ ( ∇ X ξ ,x i , ∇ X ξ ,x i ,⋆ ) T , Y ≜ ( ∇ Y ξ ,x i , ∇ Y ξ ,x i ,⋆ ) T , (4.36) thereb y reform ulating it as a tw o-dimensional linear FBSDE in terms of the pair ( X , Y ). Under Assumption 4.1 , it can b e easily shown that this FBSDE admits a unique solution ( X , Y ) in L 2 r ([0 , ∞ ); R 2 ). Next, w e attempt to verify the follo wing relation: δ Φ ξ ,I { ξ = x i } = ∇ Φ ξ ,x i I { ξ = x i } + p i ∇ Φ ξ ,x i ,⋆ , Φ ∈ { X , Y , Z } . (4.37) Since the solutions to all equations hav e b een explicitly obtained and all the equations are linear, w e can verify by direct substitution. Noticing that Φ ξ = n X i =1 Φ x i ,ξ I { ξ = x i } , Φ ∈ { X , Y , Z } , (4.38) w e ha ve ∇ X ξ ,x i t I { ξ = x i } · ∂ xy H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t ) + p i ∇ X ξ ,x i ,⋆ t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) =( ∇ X ξ ,x i t I { ξ = x i } + p i ∇ X ξ ,x i ,⋆ t ) ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) . (4.39) Moreo ver, the pro cesses ( ∇ X ξ ,x i , ∇ Y ξ ,x i , ∇ Z ξ ,x i ) are indep enden t of ξ , then w e ha ve I { ξ = x i } · p i ˜ E F t h ∇ ˜ X ξ ,x i t ∂ y µ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ y µ H ( X x i ,ξ t , L X ξ t , Y x i ,ξ t , ˜ X ξ t ) i = I { ξ = x i } · ˜ E F t h ( ∇ ˜ X ξ ,x i t I { ˜ ξ = x i } + p i ∇ ˜ X ξ ,x i ,⋆ t ) ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) i (4.40) and p i ˜ E F t h ∇ ˜ X ξ ,x i t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) i · I { ξ  = x i } = ˜ E F t h ( ∇ ˜ X ξ ,x i t I { ˜ ξ = x i } + p i ∇ ˜ X ξ ,x i ,⋆ t ) ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) i · I { ξ  = x i } . (4.41) Com bining all the relations abov e, w e can v erify Equation ( 4.37 ). By substituting the solution of FBSDE ( 4.32 ) into FBSDE ( 4.33 ), it is straigh tforw ard to conclude that FBSDE ( 4.33 ) admits a unique L 2 r solution. Observing that ˜ E F t h δ ˜ X ξ ,I { ξ = x i } t ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) i = ˜ E F t h ∇ ˜ X ξ ,x i I { ˜ ξ = x i } · ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x i ,ξ t ) + p i ∇ ˜ X ξ ,x i ,⋆ ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) i = p i ˜ E F t h ∇ ˜ X ξ ,x i ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x i ,ξ t ) + ∇ ˜ X ξ ,x i ,⋆ ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) i , (4.42) 21 w e obtain δ Φ x,ξ ,I { ξ = x i } = p i ∇ µ Φ x,ξ ,x i , Φ ∈ { X , Y , Z } . (4.43) P articularly , δ Y x,ξ ,I { ξ = x i } 0 = p i · ∇ µ Y x,ξ ,x i 0 . (4.44) Com bined with ( 4.31 ), we can get that: ∂ µ V ( x, L ξ , x i ) = ∇ µ Y x,ξ ,x i 0 . (4.45) Next, w e appro ximate the absolutely con tinuous distribution by the discrete ones, thereb y obtaining a version of ∂ µ V ( x, µ, · ) when µ is absolutely con tin uous. T o this end, for any ( x, ξ , ˜ x ) ∈ R × L 2 ( F 0 ) × R , we introduce the follo wing three FBSDEs:                d ∇ X x,ξ t = h ∇ X x,ξ t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ Y x,ξ t ∂ y y H ( X x,ξ t , L X ξ t , Y x,ξ t ) i d t, d ∇ Y x,ξ t = − h ∇ X x,ξ t ∂ xx H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ Y x,ξ t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r ∇ Y x,ξ t i d t + ∇ Z x,ξ t d B t , ∇ X x,ξ 0 = 1 . (4.46)                                            d ∇ X ξ ,x t = n ∇ X ξ ,x t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) + ∇ Y ξ ,x t ∂ y y H ( X ξ t , L X ξ t , Y ξ t ) + ˜ E F t h ∇ ˜ X x,ξ t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X x,ξ t ) + ∇ ˜ X ξ ,x t ∂ y µ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) io d t, d ∇ Y ξ ,x t = − n ∇ X ξ ,x t ∂ xx H ( X ξ t , L X ξ t , Y ξ t ) + ∇ Y ξ ,x t ∂ xy H ( X ξ t , L X ξ t , Y ξ t ) − r ∇ Y ξ ,x t + ˜ E F t h ∇ ˜ X x,ξ t ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X x,ξ t ) + ∇ ˜ X ξ ,x t ∂ xµ H ( X ξ t , L X ξ t , Y ξ t , ˜ X ξ t ) io d t + ∇ Z ξ ,x t d B t , ∇ X ξ ,x 0 = 0 . (4.47) 22                                            d ∇ µ X x,ξ , ˜ x t = n ∇ µ X x,ξ , ˜ x t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ µ Y x,ξ , ˜ x t ∂ y y H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ˜ E F t h ∇ ˜ X ˜ x,ξ t ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x,ξ t ) + ∇ ˜ X ξ , ˜ x t ∂ y µ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) io d t, d ∇ µ Y x,ξ , ˜ x t = − n ∇ µ X x,ξ , ˜ x t ∂ xx H ( X x,ξ t , L X ξ t , Y x,ξ t ) + ∇ µ Y x,ξ , ˜ x t ∂ xy H ( X x,ξ t , L X ξ t , Y x,ξ t ) − r ∇ µ Y x,ξ , ˜ x t + ˜ E F t h ∇ ˜ X ˜ x,ξ t ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X x,ξ t ) + ∇ ˜ X ξ , ˜ x t ∂ xµ H ( X x,ξ t , L X ξ t , Y x,ξ t , ˜ X ξ t ) io d t + ∇ µ Z x,ξ , ˜ x t d B t , ∇ µ X x,ξ , ˜ x 0 = 0 . (4.48) Lemma 4.6 F or any ( x, ξ , ˜ x ) ∈ R × L 2 ( F 0 ) × R , the FBSDEs ( 4.46 ) , ( 4.47 ) , and ( 4.48 ) admit unique solutions in L 2 r . Mor e over, the fol lowing mappings ar e uniformly b ounde d and jointly c ontinuous: 1. ( x, ξ ) → ( ∇ X x,ξ , ∇ Y x,ξ , ∇ Y x,ξ 0 ) for the solution of ( 4.46 ) , 2. ( x, ξ ) → ( ∇ X ξ ,x , ∇ Y ξ ,x , ∇ Y ξ ,x 0 ) for the solution of ( 4.47 ) , 3. ( x, ξ , ˜ x ) → ( ∇ µ X x,ξ , ˜ x , ∇ µ Y x,ξ , ˜ x , ∇ µ Y x,ξ , ˜ x 0 ) for the solution of ( 4.48 ) . Pro of. Under Assumption 4.1 , the existence and uniqueness of solutions to these three FB- SDEs is straigh tforward. Moreo ver, the uniform b oundedness of their solutions can b e readily established b y following a reasoning similar to the pro of of Lemma 4.2 , and the contin uity of the solutions with resp ect to their initial v alues can b e established b y following a pro of analogous to that in Corollary 4.3 . Remark 4.7 If we define ψ ( x, µ, ˜ x ) ≜ ∇ µ Y x,ξ , ˜ x 0 for some ξ ∈ L 2 ( F 0 ; µ ) , then ψ is uniformly b ounde d and jointly c ontinuous on R × P 2 × R . No w, w e consider the case that µ is absolutely contin uous and choose some ξ ∈ L 2 ( F 0 ; µ ). F or each n ≥ 1, set x n i ≜ i n , △ n i ≜ [ i n , i + 1 n ) , i ∈ Z . (4.49) F or any x ∈ R , let i n ( x ) b e the index function suc h that x ∈ △ n i n ( x ) . Denote ξ n ≜ n 2 − 1 X i = − n 2 x n i · I △ n i ( ξ ) − n 2 I ( −∞ , − n 2 ) ( ξ ) + n 2 I [ n 2 , ∞ ) ( ξ ) . (4.50) It’s clear that lim n →∞ ξ n = ξ , a.s., lim n →∞ E [ | ξ n − ξ | 2 ] = 0, and th us lim n →∞ W 2 ( L ξ n , L ξ ) = 0. F or any η ∈ L 2 ( F 0 ), by stability of FBSDE ( 4.5 ), we ha ve that 23 E [ ∂ µ V ( x, µ, ξ ) η ] = δ Y x,ξ ,η 0 = lim n →∞ δ Y x,ξ n ,η 0 . (4.51) The follo wing lemma reveals the limiting relationship b et ween FBSDEs ( 4.32 )-( 4.33 ) and FBSDEs ( 4.46 )-( 4.48 ), whic h provides a to ol for our further study of δ Y x,ξ n ,η 0 . Its pro of differs sligh tly from that of Lemma 4.6 ; hence, we provide a detailed demonstration here. Lemma 4.8 F or e ach ( x, ξ , ˜ x ) ∈ R × L 2 ( F 0 ) × R such that L ξ is absolutely c ontinuous, and let { ξ n ; n ≥ 1 } b e define d as in ( 4.50 ). We have that, as n → ∞ , the fol lowing limits hold (the c onver genc e of the sto chastic pr o c ess is in the L 2 r sense): (1) ( ∇ X ξ n , i n ( ˜ x ) n , ∇ Y ξ n , i n ( ˜ x ) n , ∇ Y ξ n , i n ( ˜ x ) n 0 ) → ( ∇ X ˜ x,ξ , ∇ Y ˜ x,ξ , ∇ Y ˜ x,ξ 0 ); (4.52) (2) ( ∇ X ξ n , i n ( ˜ x ) n ,⋆ , ∇ Y ξ n , i n ( ˜ x ) n ,⋆ , ∇ Y ξ n , i n ( ˜ x ) n ,⋆ 0 ) → ( ∇ X ξ , ˜ x , ∇ Y ξ , ˜ x , ∇ Y ξ , ˜ x 0 ); (4.53) (3) ( ∇ µ X x,ξ n , i n ( ˜ x ) n , ∇ µ Y x,ξ n , i n ( ˜ x ) n , ∇ µ Y x,ξ n , i n ( ˜ x ) n 0 ) → ( ∇ µ X x,ξ , ˜ x , ∇ Y x,ξ , ˜ x , ∇ Y x,ξ , ˜ x 0 ) . (4.54) Pro of. F or each ˜ x ∈ R , we hav e ( L ξ n , i n ( ˜ x ) /n ) → ( L ξ , ˜ x ) as n → ∞ , then we ha v e ( X ξ n , Y ξ n ) → ( X ξ , Y ξ ) , ( X i n ( ˜ x ) n ,ξ n , Y i n ( ˜ x ) n ,ξ n ) → ( X ˜ x,ξ , Y ˜ x,ξ ) (4.55) in L 2 r . An essential observ ation is that the solutions of FBSDEs ( 4.32 )-( 4.33 ) and ( 4.46 )-( 4.48 ) are uniformly b ounded in L 2 r for all initial v alues, which is imp ortan t to the subsequen t pro of. Denote X n = ∇ X ξ n , i n ( ˜ x ) n − ∇ X ˜ x,ξ , Y n = ∇ Y ξ n , i n ( ˜ x ) n − ∇ X ˜ x,ξ , Z n = ∇ Z ξ n , i n ( ˜ x ) n − ∇ X ˜ x,ξ , (4.56) then they satisfy:                                                                    d X n t =  X n t ∂ xy H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) + Y n t ∂ y y H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) + X ˜ x,ξ t H n xy + Y ˜ x,ξ t H n y y + p n ˜ E F t  ∇ ˜ X ξ n , i n ( ˜ x ) n t ∂ y µ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X i n ( ˜ x ) n ,ξ n t ) + ∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t ∂ y µ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X ξ n t )  d t, d Y n t = −  X n t ∂ xx H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) + Y n t ∂ xy H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) − r Y n t + X ˜ x,ξ t H n xx + Y ˜ x,ξ t H n xy + p n i ˜ E F t  ∇ ˜ X ξ n , i n ( ˜ x ) n t ∂ xµ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X i n ( ˜ x ) n ,ξ n t ) + ∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t ∂ xµ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X ξ n t )  d t + Z n t d B t , X n 0 = 0 , (4.57) 24 where H n xx ≜ ∂ xx H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) − ∂ xx H ( X ˜ x,ξ t , L X ξ t , Y ˜ x,ξ t ) , H n xy ≜ ∂ xy H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) − ∂ xy H ( X ˜ x,ξ t , L X ξ t , Y ˜ x,ξ t ) , H n y y ≜ ∂ y y H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) − ∂ y y H ( X ˜ x,ξ t , L X ξ t , Y ˜ x,ξ t ) , p n ≜ P ( ξ n = i n ( ˜ x ) n ) = P ( ξ ∈ △ n i n ( ˜ x ) ) . (4.58) Applying Itˆ o’s formula to e − rt X n t Y n t , we get that 0 = E  Z ∞ 0 e − rt  − ( X n t ) 2 ∂ xx H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) + ( Y n t ) 2 ∂ y y H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t ) − X n t ( X ˜ x,ξ t H n xx + Y ˜ x,ξ t H n xy ) + Y n t ( X ˜ x,ξ t H n xy + Y ˜ x,ξ t H n y y ) − p n X n t ˜ E F t  ∇ ˜ X ξ n , i n ( ˜ x ) n t ∂ xµ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X i n ( ˜ x ) n ,ξ n t ) + ∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t ∂ xµ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X ξ n t )  + p n Y n t ˜ E F t  ∇ ˜ X ξ n , i n ( ˜ x ) n t ∂ y µ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X i n ( ˜ x ) n ,ξ n t ) + ∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t ∂ y µ H ( X i n ( ˜ x ) n ,ξ n t , L X ξ n t , Y i n ( ˜ x ) n ,ξ n t , ˜ X ξ n t )  d t  . (4.59) Using the conditions in Assumption 4.1 , w e hav e that E  Z ∞ 0 e − rt  ( X n t ) 2 + ( Y n t ) 2  d t  ≤ 2 r E  Z ∞ 0 e − rt  − X n t ( X ˜ x,ξ t H n xx + Y ˜ x,ξ t H n xy ) + Y n t ( X ˜ x,ξ t H n xy + Y ˜ x,ξ t H n y y ) + p n λ 2 ( | X n t | + | Y n t | ) ˜ E F t  |∇ ˜ X ξ n , i n ( ˜ x ) n t | + |∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t |  d t  . (4.60) F rom the uniform b oundedness of X n and Y n and the Dominated Conv ergence Theorem, we conclude that: lim n →∞ E  Z ∞ 0 e − rt  − X n t ( X ˜ x,ξ t H n xx + Y ˜ x,ξ t H n xy ) + Y n t ( X ˜ x,ξ t H n xy + Y ˜ x,ξ t H n y y )  d t  = 0 . (4.61) Since L ξ is absolutely contin uous, w e hav e p n → 0, and thus: lim n →∞ E  Z ∞ 0 e − rt  p n ( | X n t | + | Y n t | ) ˜ E F t  |∇ ˜ X ξ n , i n ( ˜ x ) n t | + |∇ ˜ X ξ n , i n ( ˜ x ) n ,⋆ t |  d t  = 0 . (4.62) By combining all the ab o ve relations, we arrive at the follo wing con vergence result: lim n →∞ E  Z ∞ 0 e − rt  ( X n t ) 2 + ( Y n t ) 2  d t  = 0 . (4.63) 25 Applying Itˆ o’s formula to e − rt ( Y n t ) 2 , we can easily obtain Y n 0 → 0, th us finishing the pro of of (1). The proofs for (2) and (3) follow analogously . The follo wing theorem provides a b ounded and con tinuous representation of ∂ µ V ( x, µ, ˜ x ). This implies that V ( x, µ ) is Lions-differentiable with resp ect to µ . Theorem 4.9 Define ψ ( x, µ, ˜ x ) ≜ ∇ µ Y x,ξ , ˜ x 0 for some ξ ∈ L 2 ( F 0 ; µ ) , then for any ( x, µ ) ∈ R × P 2 , ψ ( x, µ, · ) is a version of ∂ µ V ( x, µ, · ) . Pro of. F or an y ( x, µ ), w e first consider the case where µ is absolutely contin uous. T ake ξ ∈ L 2 ( F 0 ; µ ) and construct { ξ n ; n ≥ 1 } as in ( 4.50 ). F or an y b ounded and Lipsc hitz con tinuous function φ , we ha v e { φ ( ξ n ) } conv erges to ξ in L 2 . So w e can get that E [ ∂ µ V ( x, µ, ξ ) φ ( ξ )] = δ Y x,ξ ,φ ( ξ ) 0 = lim n →∞ δ Y x,ξ n ,φ ( ξ n ) 0 = lim n →∞ n 2 X i = − n 2 φ ( x n i ) δ Y x,ξ n ,I { ξ n = x n i } 0 = lim n →∞ n 2 X i = − n 2 φ ( x n i ) ∇ µ Y x,ξ n ,x n i 0 P ( ξ n = x n i ) = lim n →∞ E h φ ( ξ n ( ω )) ∇ µ Y x,ξ n ,ξ n ( ω ) 0 i = E h φ ( ξ ( ω )) ∇ µ Y x,ξ ,ξ ( ω ) 0 i . (4.64) The last equalit y holds due to the conv ergence of φ ( ξ n ( ω )) ∇ µ Y x,ξ n ,ξ n ( ω ) 0 to φ ( ξ ( ω )) ∇ µ Y x,ξ ,ξ ( ω ) 0 in probability and the Bounded Conv ergence Theorem. Then we hav e ∂ µ V ( x, µ, · ) = ψ ( x, µ, · ) , µ -a.e. (4.65) whenev er µ is absolutely con tinuous. No w w e consider the general case. Fix an arbitrary ( µ, ξ ). One can easily construct { ξ n ; n ≥ 1 } suc h that L ξ n is absolutely contin uous, and E [ | ξ n − ξ | 2 ] → 0. 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