Weak Solutions to Vlasov-McKean Equations under Lyapunov-Type Conditions
aa r X i v : . [ m a t h . P R ] N ov Weak Solutions to Vlasov-McKean Equations underLyapunov-Type Conditions
Sima Mehri ∗†‡
Wilhelm Stannat ∗ November 19, 2019
Abstract
We present a Lyapunov type approach to the problem of existence and uniqueness ofgeneral law-dependent stochastic differential equations. In the existing literature mostresults concerning existence and uniqueness are obtained under regularity assumptionsof the coefficients w.r.t the Wasserstein distance. Some existence and uniqueness re-sults for irregular coefficients have been obtained by considering the total variationdistance. Here we extend this approach to the control of the solution in some weightedtotal variation distance, that allows us now to derive a rather general weak uniquenessresult, merely assuming measurability and certain integrability on the drift coefficientand some non-degeneracy on the dispersion coefficient. We also present an abstractweak existence result for the solution of law-dependent stochastic differential equationswith merely measurable coefficients, based on an approximation with law-dependentstochastic differential equations with regular coefficients under Lyapunov type assump-tions.
Keywords:
Vlasov-McKean equations; Girsanov theorem; existence and uniqueness ofweak solution; Lyapunov method; weighted total variation.
The purpose of this paper is to provide general existence and uniqueness results for thesolution of Vlasov-McKean equations, and more general law-dependent stochastic differentialequations, using a Lyapunov approach. The existence and uniqueness of solutions of Vlasov-McKean equations under global Lipschitz conditions is well-known. Surprisingly, uniquenessfails under local Lipschitz assumptions (see [12]). However, in these counterexamples, thenoise is degenerate (in fact zero). As the following Example of uniqueness with merelymeasurable coefficients shows, the situation changes, if the noise becomes non-degenerate. ∗ Institut f¨ur Mathematik, Technische Universit¨at Berlin, D-10623 Berlin, Germany † Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran ‡ The work of this author was supported by the Hilda Geiringer Scholarship awarded by the BerlinMathematical School xample 1.1. On the complete probability space (Ω , F , ( F t ) t ≥ , P ) with real valued ( F t ) t ≥ -Wiener process ( W t ) t ≥ on R , consider the following Vlasov-McKean equation ( d X t = E ( h ( X t )) d t + d W t X = ξ (1) with measurable h satisfying the growth condition | h ( x ) | ≤ Ce x T for some T > . Let µ := P ◦ ξ − be absolutely continuous with continuous differentiable density, and define φ h ( t, x ) := Z R Z R √ πt h ( x + x + w ) e − w t d wµ (d x )= Z R Z R √ πt h ( x + w ) e − ( w − x )22 t d wµ (d x ) . Then for t < T , x φ h ( t, x ) is continuous differentiable, hence locally Lipschitz continuous.Let X t = ξ + g ( t ) + W t be a solution of (1) , then g ′ ( t )d t + d W t = d X t = E ( h ( X t )) d t + d W t = E ( h ( ξ + g ( t ) + W t )) d t + d W t = Z R Z R √ πt h ( x + g ( t ) + w ) e − w t d wµ (d x )d t + d W t = φ h ( t, g ( t ))d t + d W t . So g : [0 , T ) → R is the unique solution to the equation g ′ ( t ) = φ h ( t, g ( t )) , with initial value g (0) = 0 . Therefore equation (1) has a unique strong solution on [0 , T ) . Hence there is a considerable interest in relaxing the assumptions on the coefficients ofVlasov-McKean equations. Strong well-posedness of Vlasov-McKean equation with H¨olderdrift and Lipschitz dispersion coefficient has been obtained in [3]. Strong existence anduniqueness of solutions to the Vlasov-McKean equation under one-sided Lipschitz continuityfor the drift and Lipschitz continuous dispersion coefficient have been obtained in [4]. Thepaper [13] considers strong well-posedness of distribution dependent stochastic differentialequations with one-sided Lipschitz continuous drift and Lipschitz-continuous dispersion coef-ficients, [6] generalizes the latter result to path-distribution dependent stochastic differentialequations.Weak existence and strong uniqueness of solutions to the Vlasov-McKean equation withcontinuous coefficients have been obtained with the help of a Lyapunov method in [5]. Therecent preprint [11] proves weak and strong well-posedness of the solutions of Vlasov-McKeanequations under non-degeneracy assumptions on the noise term with even non-regular driftof at most linear growth.Existence and uniqueness of weak solutions of Vlasov-McKean equations have been ob-tained in [9], with regularity of the coefficients w.r.t. the total variation distance. [1] obtainsexistence and uniqueness of weak and strong solutions of Vlasov-McKean equations withadditive noise and drift coefficients that can be decomposed into bounded measurable partand a part that is Lipschitz continuous w.r.t. the Kantorovich distance.2he paper [7] contains an existence result of a weak solution of a distribution-dependentstochastic differential equation with merely measurable coefficients based on an approxima-tion with stochastic differential equations with Lipschitz continuous coefficients. This resultrequires uniform boundedness of the diffusion term.In the present paper now, we will extend the result for the existence of weak solutions toVlasov-McKean equations with measurable coefficients and uniformly non-degenerate andmerely integrable diffusion matrix (see the Theorem 3.1). The abstract conditions in thistheorem can be verified with the help of a Lyapunov type growth condition on the coefficientsin Theorem 3.4. Sufficient conditions, in terms of the coefficients only, are presented inCorollary 3.5.We also obtain a corresponding uniqueness result for weak solutions of functional law-dependent stochastic differential equations under Lyapunov type growth conditions on thecoefficients (see Corollary 2.5), based on an abstract stability result for weak solutions w.r.t.a weighted total variation distance (see Theorem 2.4). Two sets of sufficient conditions interms of the coefficients are presented in Example 2.6. Our uniqueness results generalizethe corresponding result obtained in [11] not only w.r.t. the general law-dependence butalso w.r.t. the more general growth conditions. In [11], only linear growth is allowed.Stability results for Vlasov-McKean equations w.r.t. weighted total variation distances havebeen obtained previously in the references [2, 10], using an analytic approach, that cannot,however, cover general functional law-dependent stochastic differential equations consideredin the present work.
Let M be the space of signed measures on (cid:0) R d , B ( R d ) (cid:1) . Given a measurable function φ : R d → (0 , ∞ ), we define the φ -weighted total variation of µ ∈ M by k µ k φ := Z R d φ ( y ) | µ | (d y )Here | µ | denotes the total variation measure associated with µ . For a continuous function φ , this norm is lower semi-continuous with respect to the weak topology by the followingLemma. Lemma 2.1.
Let φ : R d → (0 , ∞ ) be continuous and assume that the sequence of signedmeasures ( µ n ) n ∈ N in M converges weakly to the measure µ and assume that φ ∈ L ( | µ | ) .Then k µ k φ ≤ lim inf n →∞ k µ n k φ . Proof.
Using the Hahn decomposition theorem we can find a measurable subset A ∈ B ( R d )such that | µ | = µ A − µ A c , where µ A ( B ) = µ ( B ∩ A ) and µ A c ( B ) = µ ( B ∩ A c ), B ∈ B ( R d ),are finite nonnegative Borel measures. Let ε > φ ∈ L ( | µ | ) we can find R > k µ k φ ≤ Z R d ( φ ( y ) ∧ R ) ( A ( y ) − A c ( y )) µ (d y ) + ε. φ ∧ R ) d µ is a finite Borel measure we can find a continuous function ψ : R d → [ − , Z R d ( φ ( y ) ∧ R ) ( A ( y ) − A c ( y )) µ (d y ) ≤ Z R d ( φ ( y ) ∧ R ) ψ ( y ) µ (d y ) + ε . Consequently, k µ k φ ≤ Z R d ( φ ( y ) ∧ R ) ψ ( y ) µ (d y ) + 2 ε ≤ lim n →∞ Z R d ( φ ( y ) ∧ R ) ψ ( y ) µ n (d y ) + 2 ε ≤ lim inf n →∞ k µ n k φ + 2 ε. Since ε >
T, τ >
0. Let M be the Borel σ -algebra induced by the weak topology on M . Letus define M T := { µ : [ − τ, T ] → M ; µ is B ([ − τ, T ]) / M -measurable } . Let ( W t ) t ≥ be the standard Brownian motion on R d . We consider the nonlinear equation d X t = b ( t, X, µ )d t + σ ( t, X )d W t , t ∈ [0 , T ] ,X t = ξ t , t ∈ [ − τ, ,µ ∈ M T , µ s = L ( X s ) , where L ( X s ) denotes the law of X s , s ∈ [ − τ, T ] (2)with initial condition ξ ∈ C ([ − τ, R d ), independent of ( W t ) t ≥ , where b ≡ σ ˜ b and ( ˜ b : [0 , T ] × C ([ − τ, T ] , R d ) × M T → R d ,σ : [0 , T ] × C ([ − τ, T ] , R d ) → R d × d are measurable functions and adapted, i.e. ˜ b ( t, x, µ ) and σ ( t, x ) depend only on the path of x and µ on [ − τ, t ]. This equation is called a Vlasov-McKean equation. Definition 2.2.
We say that equation (2) has a weak solution on [0 , T ] with initial distri-bution Ξ on C ([ − τ, , R d ) if there exist a probability space (cid:0) Ω , F , ( F t ) t ≥ , P (cid:1) , an ( F t ) t ≥ -Wiener process ( W t ) t ≥ on R d , an F -measurable random variable ξ ∈ C ([ − τ, , R d ) withthe law Ξ , and an ( F t ) t ≥ -adapted stochastic process X ∈ C ([ − τ, T ] , R d ) such that X t = ξ (0) + Z t b ( s, X, µ )d s + Z t σ ( s, X )d W s , t ∈ [0 , T ] ,X t = ξ t , t ∈ [ − τ, ,µ ∈ M T , µ s = L ( X s ) , (3) which requires that the integrals are well defined, i.e., Z T | b ( s, X, µ ) | + | σ ( s, X ) | d s < ∞ , P -a.s. (4)4 emark 2.3. Note that by Levy’s theorem on characterization of Brownian motion, forany ( F t ) t ≥ -Wiener process ( W t ) t ≥ , W t − W s is independent of F s . Specially ( W t ) t ≥ isindependent of F , that means in Definition 2.2, ξ is in fact independent of ( W t ) t ≥ . We will first state an abstract uniqueness result for the weak solution to the Vlasov-McKean equation (2) in the following theorem, that is based on an estimate of the distanceof the laws of two weak solutions with different drift and same dispersion coefficient w.r.t.the weighted total variation distance introduced above.
Theorem 2.4.
Suppose that equation ( d X t = σ ( t, X )d W t , t ∈ [0 , T ] ,X t = ξ t , t ∈ [ − τ, , (5) has a unique strong solution on the probability space (cid:0) Ω , F , ( F t ) t ≥ , P (cid:1) for some F -measurablerandom variable ξ ∈ C ([ − τ, , R d ) . Let ˜ b , ˜ b : [0 , T ] × C ([ − τ, T ] , R d ) → R d be such that for i = 1 , , Z T (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ) (cid:12)(cid:12)(cid:12) d s < ∞ , P - a.s. (6) and ˜ b i ( t, x ) depends only to the path of x on [ − τ, t ] . Let X ( i ) , i = 1 , , defined on theprobability spaces (cid:18) Ω ( i ) , F ( i ) , (cid:16) F ( i ) t (cid:17) t ≥ , Q ( i ) (cid:19) be weak solutions to the equations ( d X ( i ) t = b i ( t, X ( i ) )d t + σ ( t, X ( i ) )d W ( i ) t , t ∈ [0 , T ] X ( i ) t = ξ ( i ) t , t ∈ [ − τ, , (7) where b i ≡ σ ˜ b i , and ξ ( i ) is independent of W ( i ) and has the same law as ξ . Assume that for i = 1 , , X ( i ) satisfies for j = 1 , , Z T (cid:12)(cid:12)(cid:12) ˜ b j ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s < ∞ , Q ( i ) - a.s. (8) If µ ( i ) t , i = 1 , denotes the law of X ( i ) t , then for any continuous function φ : R d → (0 , ∞ ) (cid:13)(cid:13)(cid:13) µ (1) t − µ (2) t (cid:13)(cid:13)(cid:13) φ ≤ X i =1 E Q ( i ) (cid:20) φ (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) s, X ( i ) (cid:1) − ˜ b (cid:0) s, X ( i ) (cid:1)(cid:12)(cid:12)(cid:12) d s (cid:21) + X i =1 (cid:16) E Q ( i ) h φ (cid:16) X ( i ) t (cid:17)i(cid:17) / (cid:18) E Q ( i ) (cid:20)Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) s, X ( i ) (cid:1) − ˜ b (cid:0) s, X ( i ) (cid:1)(cid:12)(cid:12)(cid:12) d s (cid:21)(cid:19) / . (9) In addition, let b i ( t, x ) := b (cid:0) t, x, µ ( i ) (cid:1) and assume that there exist measurable functions ϕ : [0 , T ] → C ( R d , (0 , ∞ )) and ψ : [0 , T ] × C ([ − τ, T ] , R d ) → [0 , ∞ ) and an increasing positivevalued function g with R + g ( u ) d u = ∞ such that for every µ, ν ∈ M T with µ | [ − τ, = ν | [ − τ, , (cid:12)(cid:12)(cid:12) ˜ b ( t, x, µ ) − ˜ b ( t, x, ν ) (cid:12)(cid:12)(cid:12) ≤ ψ ( t, x ) g / sup s ∈ [0 ,t ] k µ s − ν s k ϕ s ! , (10)5 hen Q (1) ◦ (cid:0) X (1) (cid:1) − = Q (2) ◦ (cid:0) X (2) (cid:1) − provided that Z T sup t ∈ [ s,T ] ( X i =1 E Q ( i ) (cid:20) ϕ t (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) u, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) u, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d u (cid:21) ·· E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17) ψ ( s, X ( i ) ) i + X i =1 E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17)i · E Q ( i ) (cid:2) ψ ( s, X ( i ) ) (cid:3) ) d s < ∞ (11) for i = 1 , .Proof. Let X be the unique strong solution to the following equation ( d X t = σ ( t, X )d W t , t ∈ [0 , T ] ,X t = ξ t , t ∈ [ − τ, . Using Girsanov transformation, it turns out that equation (7) has at most one weak solutionsatisfying (8). Let us define the stopping time τ n as τ n := inf (cid:26) t ≥ , min i =1 , Z t (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ) (cid:12)(cid:12)(cid:12) d s > n (cid:27) . Then the following process for i = 1 , M ( i ) t ∧ τ n := exp (cid:18)Z t ∧ τ n ˜ b i ( s, X ) · d W s − Z t ∧ τ n (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ) (cid:12)(cid:12)(cid:12) d s (cid:19) , t ∈ [0 , T ] . Let P i,n be the probability measure with densityd P i,n d P (cid:12)(cid:12)(cid:12)(cid:12) F T = M ( i ) T ∧ τ n . By Girsanov theorem, the process˜ W ( i ) t ∧ τ n = W t ∧ τ n − Z t ∧ τ n ˜ b i ( s, X )d s, t ∈ [0 , T ] , with respect to the probability measure P i,n for i = 1 ,
2, is a standard Brownian motion on R d until time τ n and we have X t ∧ τ n = ξ + Z t ∧ τ n b i ( s, X )d s + Z t ∧ τ n σ ( s, X )d ˜ W ( i ) s . Let ζ ( i ) n := inf (cid:26) t ≥ , min j =1 , Z t (cid:12)(cid:12)(cid:12) ˜ b j ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s > n (cid:27) . Then if we defined Q i,n d Q ( i ) (cid:12)(cid:12)(cid:12)(cid:12) F ( i ) T ∧ ζ ( i ) n := exp − Z T ∧ ζ ( i ) n ˜ b i ( s, X ( i ) ) · d W ( i ) s − Z T ∧ ζ ( i ) n (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s ! W ( i ) t ∧ ζ ( i ) n = W ( i ) t ∧ ζ ( i ) n + Z t ∧ ζ ( i ) n ˜ b i ( s, X ( i ) )d s, t ∈ [0 , T ] , with respect to the probability measure Q i,n , for i = 1 ,
2, is a standard Brownian motion in R d until time ζ ( i ) n and we have that X ( i ) t ∧ ζ ( i ) n = ξ ( i )0 + Z t ∧ ζ ( i ) n σ ( s, X ( i ) )d ¯ W ( i ) s , and (cid:16) X ( i ) t (cid:17) − τ ≤ t ≤ w.r.t Q i,n has the same law as ξ . Since equation (5) has a unique strongsolution, there exists a measurable function F : C (cid:0) [ − τ, , R d (cid:1) × C (cid:0) [0 , T ] , R d (cid:1) → C (cid:0) [ − τ, T ] , R d (cid:1) such that X = F ( ξ, W ) and similarly X ( i ) ·∧ ζ ( i ) n = F (cid:16) ξ ( i ) , ¯ W ( i ) ·∧ ζ ( i ) n (cid:17) . Hence for − τ ≤ t ≤ t ≤· · · ≤ t m ≤ T , Q ( i ) h(cid:16) X ( i ) t ∧ ζ ( i ) n , . . . , X ( i ) t m ∧ ζ ( i ) n (cid:17) ∈ Γ i = Z Ω ( i ) exp Z T ∧ ζ ( i ) n ˜ b i ( s, X ( i ) ) · d ¯ W ( i ) s − Z T ∧ ζ ( i ) n (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s ! ·· (cid:26)(cid:18) X ( i ) t ∧ ζ ( i ) n ,...,X ( i ) tm ∧ ζ ( i ) n (cid:19) ∈ Γ (cid:27) d Q i,n = Z Ω exp (cid:18)Z T ∧ τ n ˜ b i ( s, X ) · d W s − Z T ∧ τ n (cid:12)(cid:12)(cid:12) ˜ b i ( s, X ) (cid:12)(cid:12)(cid:12) d s (cid:19) ·· { ( X t ∧ τn ,...,X tm ∧ τn ) ∈ Γ } d P = P i,n (cid:2)(cid:0) X t ∧ τ n , . . . , X t m ∧ τ n (cid:1) ∈ Γ (cid:3) . By taking the limit of n → ∞ , we get that the law of X ·∧ τ n with respect to P i,n convergesweakly to the law of X ( i ) with respect to Q ( i ) since Q ( i ) (cid:16) sup n ≥ ζ ( i ) n ≥ T (cid:17) = 1. Let us definethe function φ ε ( y ) := φ ( y )1 + εφ ( y ) . Using Lemma 2.1, applied to the bounded function φ ε ∈ L ( | µ (1) t − µ (2) t | ), we obtain that (cid:13)(cid:13)(cid:13) µ (1) t − µ (2) t (cid:13)(cid:13)(cid:13) φ = Z R d φ ( y ) (cid:12)(cid:12)(cid:12) µ (1) t − µ (2) t (cid:12)(cid:12)(cid:12) (d y )= lim ε ց Z R d φ ε ( y ) (cid:12)(cid:12)(cid:12) µ (1) t − µ (2) t (cid:12)(cid:12)(cid:12) (d y ) ≤ lim inf ε ց lim inf n →∞ Z R d φ ε ( y ) (cid:12)(cid:12)(cid:12) ( P ,n ) ◦ (cid:0) X t ∧ τ n (cid:1) − − ( P ,n ) ◦ (cid:0) X t ∧ τ n (cid:1) − (cid:12)(cid:12)(cid:12) (d y ) . (12)7or A ∈ B (cid:0) R d (cid:1) , we have (cid:12)(cid:12)(cid:12) ( P ,n ) ◦ (cid:0) X t ∧ τ n (cid:1) − − ( P ,n ) ◦ (cid:0) X t ∧ τ n (cid:1) − (cid:12)(cid:12)(cid:12) ( A )= sup m ≥ ⊔ mi =1 A i = A m X i =1 (cid:12)(cid:12) P ,n (cid:0) X t ∧ τ n ∈ A i (cid:1) − P ,n (cid:0) X t ∧ τ n ∈ A i (cid:1)(cid:12)(cid:12) = sup m ≥ ⊔ mi =1 A i = A m X i =1 (cid:12)(cid:12)(cid:12)(cid:12)Z Ω (cid:16) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:17) { X t ∧ τn ∈ A i } d P (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup m ≥ ⊔ mi =1 A i = A m X i =1 Z Ω (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12) · { X t ∧ τn ∈ A i } d P = Z Ω (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12) · { X t ∧ τn ∈ A } d P , where ⊔ mi =1 A i means the disjoint union of Borel measurable sets A i , 1 ≤ i ≤ m . Therefore (cid:13)(cid:13)(cid:13) µ (1) t − µ (2) t (cid:13)(cid:13)(cid:13) φ ≤ lim inf ε ց lim inf n →∞ E P h φ ε (cid:0) X t ∧ τ n (cid:1) (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12)i . By using the inequality | e x − e y | ≤ | x − y | ( e x + e y ), we get (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12) ≤ ( M (1) t ∧ τ n + M (2) t ∧ τ n ) N t ∧ τ n where N t := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t h ˜ b ( s, X ) − ˜ b ( s, X ) i · d W s − Z t (cid:20)(cid:12)(cid:12)(cid:12) ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) (cid:21) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t h ˜ b ( s, X ) − ˜ b ( s, X ) i · d ˜ W (1) s − Z t (cid:20)(cid:12)(cid:12)(cid:12) ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) (cid:21) d s + Z t h ˜ b ( s, X ) − ˜ b ( s, X ) i · ˜ b ( s, X )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t h ˜ b ( s, X ) − ˜ b ( s, X ) i · d ˜ W (1) s + 12 Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ) − ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and also similarly N t = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t h ˜ b ( s, X ) − ˜ b ( s, X ) i · d ˜ W (2) s − Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ) − ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .
8o using Cauchy-Schwartz inequality, we get E P h φ ε (cid:0) X t ∧ τ n (cid:1) (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12)i ≤ X i =1 E P (cid:20) φ ε (cid:0) X t ∧ τ n (cid:1) M ( i ) t ∧ τ n Z t ∧ τ n (cid:12)(cid:12)(cid:12) ˜ b ( s, X ) − ˜ b ( s, X ) (cid:12)(cid:12)(cid:12) d s (cid:21) + X i =1 (cid:16) E P h φ ε (cid:0) X t ∧ τ n (cid:1) M ( i ) t ∧ τ n i(cid:17) / E P " M ( i ) t ∧ τ n (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ τ n h ˜ b ( s, X ) − ˜ b ( s, X ) i · d ˜ W ( i ) s (cid:12)(cid:12)(cid:12)(cid:12) / ≤ X i =1 E Q ( i ) " φ ε (cid:16) X ( i ) t ∧ ζ ( i ) n (cid:17) Z t ∧ ζ ( i ) n (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s + X i =1 (cid:16) E Q ( i ) h φ ε (cid:16) X ( i ) t ∧ ζ ( i ) n (cid:17)i(cid:17) / E Q ( i ) "Z t ∧ ζ ( i ) n (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s / . Since φ ε is bounded and continuous, we havelim inf n →∞ E P h φ ε (cid:0) X t ∧ τ n (cid:1) (cid:12)(cid:12)(cid:12) M (1) t ∧ τ n − M (2) t ∧ τ n (cid:12)(cid:12)(cid:12)i ≤ X i =1 E Q ( i ) (cid:20) φ ε (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s (cid:21) + X i =1 (cid:16) E Q ( i ) h φ ε (cid:16) X ( i ) t (cid:17)i(cid:17) / (cid:18) E Q ( i ) (cid:20)Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s (cid:21)(cid:19) / ≤ X i =1 E Q ( i ) (cid:20) φ (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s (cid:21) + X i =1 (cid:16) E Q ( i ) h φ (cid:16) X ( i ) t (cid:17)i(cid:17) / (cid:18) E Q ( i ) (cid:20)Z t (cid:12)(cid:12)(cid:12) ˜ b ( s, X ( i ) ) − ˜ b ( s, X ( i ) ) (cid:12)(cid:12)(cid:12) d s (cid:21)(cid:19) / Therefore by (12), we get inequality (9). Let us now turn to the case where ˜ b i ( s, x ) = b ( s, x, µ ( i ) ). First we square both sides of (9) with φ = ϕ t and then we substitute inequality910) in (9) in the following calculation, (cid:13)(cid:13)(cid:13) µ (1) t − µ (2) t (cid:13)(cid:13)(cid:13) ϕ t ≤ C X i =1 (cid:18) E Q ( i ) (cid:20) ϕ t (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) s, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) s, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d s (cid:21)(cid:19) + C X i =1 E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17)i · E Q ( i ) (cid:20)Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) s, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) s, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d s (cid:21) ≤ C X i =1 E Q ( i ) (cid:20) ϕ t (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) s, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) s, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d s (cid:21) · E Q ( i ) " ϕ t (cid:16) X ( i ) t (cid:17) Z t ψ ( s, X ( i ) ) g sup u ∈ [0 ,s ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! d s + C X i =1 E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17)i · E Q ( i ) "Z t ψ ( s, X ( i ) ) g sup u ∈ [0 ,s ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! d s . Then for the function H ( t, s ) := C X i =1 E Q ( i ) (cid:20) ϕ t (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) u, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) u, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d u (cid:21) ·· E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17) ψ ( s, X ( i ) ) i + C X i =1 E Q ( i ) h ϕ t (cid:16) X ( i ) t (cid:17)i · E Q ( i ) (cid:2) ψ ( s, X ( i ) ) (cid:3) , we have (cid:13)(cid:13)(cid:13) µ (1) t − µ (2) t (cid:13)(cid:13)(cid:13) ϕ t ≤ Z t H ( t, s ) g sup u ∈ [0 ,s ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! d s. Now define h ( s ) := sup u ∈ [ s,T ] H ( u, s ). The assumption (11) implies that h is integrable andon the other hand,sup u ∈ [0 ,t ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ≤ Z t h ( s ) g sup u ∈ [0 ,s ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! d s. Now consider the function F ( t ) := Z t h ( s ) g sup u ∈ [0 ,s ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! d s . Since sup u ∈ [0 ,t ] (cid:13)(cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13)(cid:13) ϕ u ≤ F ( t ) and g is increasing, we have F ′ ( t ) = h ( t ) g sup u ∈ [0 ,t ] (cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13) ϕ u ! ≤ h ( t ) g ( F ( t ))10nd therefore Z F ( t )0 g ( u ) d u = Z t F ′ ( s ) g ( F ( s )) d s ≤ Z t h ( s )d s < ∞ . Since R + g ( u ) d u = ∞ , F ( t ) must be zero and hence sup u ∈ [0 ,t ] (cid:13)(cid:13)(cid:13) µ (1) u − µ (2) u (cid:13)(cid:13)(cid:13) ϕ u ≡
0. Since ϕ is positive, this implies µ (1) t = µ (2) t , for all t ∈ [0 , T ]. Therefore P ,n = P ,n and since P i,n ◦ (cid:0) X ·∧ τ n (cid:1) − converges weakly to Q ( i ) ◦ (cid:0) X ( i ) (cid:1) − , we get Q (1) ◦ (cid:0) X (1) (cid:1) − = Q (2) ◦ (cid:0) X (2) (cid:1) − . Corollary 2.5.
Let b ( t, x, µ ) := Z [ − τ, Z R d β ( t, s, x, y ) µ t + s (d y ) κ (d s ) where β ≡ σ ˜ β and ˜ β : [0 , ∞ ) × [ − τ, × C (cid:0) [ − τ, T ] , R d (cid:1) × R d → R d ,σ : [0 , ∞ ) × C (cid:0) [ − τ, T ] , R d (cid:1) → R d × d are measurable functions and κ is a probability measure on [ − τ, . Assume that ( d X t = σ ( t, X )d W t , t ∈ [0 , T ] ,X t = ξ t , t ∈ [ − τ, , has a unique strong solution. Suppose there exist a function V ∈ C , (cid:0) [ − τ, T ] × R d , [0 , ∞ ) (cid:1) and measurable functions ϕ : [ − τ, T ] → C ( R d , (0 , ∞ )) and η : [ − τ, T ] × R d → [0 , ∞ ) suchthat for all x ∈ C (cid:0) [ − τ, T ] , R d (cid:1) and all y ∈ R d the following properties hold:(C1) ∂ t V ( t, x t ) + h∇ V ( t, x t ) , β ( t, s, x, y ) i + tr (cid:0) σ T ( t, x )D V ( t, x t ) σ ( t, x ) (cid:1) ≤ CV ( t, x t ) , (C2) (cid:12)(cid:12)(cid:12) ˜ β ( t, s, x, y ) (cid:12)(cid:12)(cid:12) ≤ Cϕ ( t + s, y ) η ( t + s, x t + s ) , (C3) η ( t, y ) + ϕ ( t, y ) ≤ CV ( t, y ) , (C4) sup s ∈ [ − τ, E V ( s, ξ s ) < ∞ .Then uniqueness holds for the weak solution to Vlasov-McKean equation (2) in the sense ofDefinition 2.2 with initial value ξ .Proof. Let X (1) t and X (2) t be two solutions to the Vlasov-McKean equation (2) with laws µ (1) t and µ (2) t . We want to prove that the assumptions of Theorem 2.4 hold with the function ϕ and ψ ( t, x ) := C Z − τ η ( t + s, x t + s ) κ (d s ) , g ( u ) = u.
11e have for µ, ν ∈ M T with µ | [ − τ, = ν | [ − τ, , | b ( t, x, µ ) − b ( t, x, ν ) | ≤ Z − τ Z R d Cϕ ( t + s, y ) η ( t + s, x t + s ) | µ t + s − ν t + s | (d y ) κ (d s ) ≤ C Z − τ η ( t + s, x t + s ) k µ t + s − ν t + s k ϕ t + s κ (d s ) ≤ ψ ( t, x ) sup u ∈ [0 ,t ] k µ u − ν u k ϕ u All expectations and integrals in Theorem 2.4 are finite via (C3) provided thatsup s ∈ [ − τ,T ] E V (cid:0) s, X ( i ) s (cid:1) < ∞ . We have by inequality (C1), e − Ct V (cid:16) t, X ( i ) t (cid:17) = V (0 , ξ ) + Z t e − Cs h − CV (cid:0) s, X ( i ) s (cid:1) + ∂ t V (cid:0) s, X ( i ) s (cid:1) + (cid:28) ∇ V (cid:0) s, X ( i ) s (cid:1) , ˜ E Z − τ β (cid:16) s, u, X ( i ) , ˜ X ( i ) s + u (cid:17) κ (d u ) (cid:29) + 12 tr (cid:0) σ T (cid:0) s, X ( i ) (cid:1) D V (cid:0) s, X ( i ) s (cid:1) σ (cid:0) s, X ( i ) (cid:1)(cid:1) i d s + M t ≤ V (0 , ξ ) + M t , where, according to (4), M t := Z t e − Cs (cid:10) ∇ V (cid:0) s, X ( i ) s (cid:1) , σ (cid:0) s, X ( i ) (cid:1) d W s (cid:11) , t ≥ σ n ↑ ∞ as n → ∞ such that ( M t ∧ σ n ) t ≥ is martingale. By Fatou’s lemma, E h e − Ct V (cid:16) t, X ( i ) t (cid:17)i ≤ lim inf n →∞ E h e − C ( t ∧ σ n ) V (cid:16) t ∧ σ n , X ( i ) t ∧ σ n (cid:17)i ≤ E V (0 , ξ ) . This implies sup t ∈ [ − τ,T ] E V (cid:16) t, X ( i ) t (cid:17) ≤ e CT sup s ∈ [ − τ, E V ( s, ξ s ) < ∞ . Hence we have by (C2)and (C3) and locally boundedness of η that for x ∈ C ([ − τ, T ] , R d ), Z T (cid:12)(cid:12)(cid:12) ˜ b (cid:0) t, x, µ ( i ) (cid:1)(cid:12)(cid:12)(cid:12) d t ≤ Z T Z [ − τ, Z R d (cid:12)(cid:12)(cid:12) ˜ β ( t, s, x, y ) (cid:12)(cid:12)(cid:12) µ ( i ) t + s (d y ) κ (d s )d t ≤ C Z T Z [ − τ, Z R d ϕ ( t + s, y ) η ( t + s, x t + s ) µ ( i ) t + s (d y ) κ (d s )d t ≤ C Z T Z [ − τ, E h V ( t + s, X ( i ) t + s ) i η ( t + s, x t + s ) κ (d s )d t ≤ C sup t ∈ [ − τ,T ] E h V ( t, X ( i ) t ) i Z T Z [ − τ, η ( t + s, x t + s ) κ (d s )d t < ∞ . (13)12o the conditions (6) and (8) in Theorem 2.4 hold. The right hand side of inequality (11)has the following bound, Z T sup t ∈ [ s,T ] ( X i =1 E (cid:20) ϕ t (cid:16) X ( i ) t (cid:17) Z t (cid:12)(cid:12)(cid:12) ˜ b (cid:0) u, X ( i ) , µ (1) (cid:1) − ˜ b (cid:0) u, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) d u (cid:21) ·· E h ϕ t (cid:16) X ( i ) t (cid:17) ψ ( s, X ( i ) ) i + X i =1 E h ϕ t (cid:16) X ( i ) t (cid:17)i · E (cid:2) ψ ( s, X ( i ) ) (cid:3) ) d s ≤ Z T sup t ∈ [ s,T ] ( X i =1 E h ϕ t (cid:16) X ( i ) t (cid:17)i · (cid:0) E (cid:2) ψ ( s, X ( i ) ) (cid:3)(cid:1) / ·· E "(cid:18) Z T (cid:18)(cid:12)(cid:12)(cid:12) ˜ b (cid:0) u, X ( i ) , µ (1) (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ b (cid:0) u, X ( i ) , µ (2) (cid:1)(cid:12)(cid:12)(cid:12) (cid:19) d u (cid:19) / + X i =1 E h ϕ t (cid:16) X ( i ) t (cid:17)i · E Q ( i ) (cid:2) ψ ( s, X ( i ) ) (cid:3) ) d s Hence by (13) to prove inequality (11), it suffices to show that for i = 1 , t ∈ [0 ,T ] E " ϕ t (cid:16) X ( i ) t (cid:17) + (cid:18)Z [ − τ, η (cid:16) t + s, X ( i ) t + s (cid:17) κ (d s ) (cid:19) < ∞ which is obvious by sup t ∈ [ − τ,T ] E V ( t, X ( i ) t ) < ∞ and (C3). Example 2.6.
Assume the equation d X t = σ ( t, X )d W t , X t = ξ t , t ∈ [ − τ, has unique strong solution for a locally bounded measurable function σ : [0 , T ] × C (cid:0) [ − τ, T ] , R d (cid:1) → R d × d . Let b ( t, x, µ ) := Z R d β ( t, x, y ) µ t (d y ) where β ≡ σ ˜ β for a measurable function ˜ β : [0 , T ] × C (cid:0) [ − τ, T ] , R d (cid:1) × R d → R d . Assumethat there exist α ≥ and p ∈ [0 , such that one of the following assumptions holds for all x ∈ C (cid:0) [ − τ, T ] , R d (cid:1) and y ∈ R d , | x t | (cid:0) h x t , β ( t, x, y ) i + | σ ( t, x ) | (cid:1) + ( α − (cid:12)(cid:12) σ T ( t, x ) x t (cid:12)(cid:12) ≤ C (1 + | x t | ) , (cid:12)(cid:12)(cid:12) ˜ β ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ C (1 + | y | α/ )(1 + | x t | α/ ) , E [ | ξ | α ] < ∞ ; | x t | (cid:0) h x t , β ( t, x, y ) i + | σ ( t, x ) | (cid:1) + ( αp | x | p + p − (cid:12)(cid:12) σ T ( t, x ) x t (cid:12)(cid:12) ≤ C (1 + | x t | − p ) , (cid:12)(cid:12)(cid:12) ˜ β ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ C exp( α | y | p + α | x t | p ) , E [exp( α | ξ | p )] < ∞ . hen the assumptions of Corollary 2.5 hold with κ = δ (the Dirac measure at point zero)and ϕ ( y ) := 1 + | y | α/ ,η ( y ) := 1 + | y | α/ ,V ∈ C (cid:0) R d , [0 , ∞ ) (cid:1) such that V ( y ) = 1 + | y | α for | y | ≥ , in the case 1 and ϕ ( y ) := exp( α | y | p ) ,η ( y ) := exp( α | y | p ) ,V ∈ C (cid:0) R d , [0 , ∞ ) (cid:1) such that V ( y ) = exp( α | y | p ) for | y | ≥ , in the case 2. In particular the solution to the Vlasov-McKean equation (2) with initial value ξ is weakly unique. We first show an abstract theorem on the existence of weak solutions to Vlasov-McKeanequations with measurable coefficients by approximating the respective equation with moreregular coefficients. We then present explicit Lyapunov type assumptions on the coefficientsthat imply the assumptions made in the abstract approximation result.
Theorem 3.1.
Let b, σ : [0 , ∞ ) × R d × R d → R d , R d × d be measurable and locally bounded.Consider the equation d X t = ˜ E b ( t, X t , ˜ X t )d t + ˜ E σ ( t, X t , ˜ X t )d W t (14) with initial value X = ξ ∈ L (cid:0) Ω , F , P ; R d (cid:1) . Here ˜ X t is an independent copy of X t . Assumethat there exist sequences of measurable functions b n , σ n : [0 , ∞ ) × R d × R d → R d , R d × d , n ∈ N such that for all t ∈ [0 , T ] , the functions ( x, y ) b n ( t, x, y ) , σ n ( t, x, y ) are continuous andequation d X nt = ˜ E b n ( t, X nt , ˜ X nt )d t + ˜ E σ n ( t, X nt , ˜ X nt )d W t (15) with initial value X n = ξ has a weak solution satisfying sup t ∈ [0 ,T ] n ∈ N E ˜ E h(cid:12)(cid:12)(cid:12) b n ( t, X nt , ˜ X nt ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) σ n ( t, X nt , ˜ X nt ) (cid:12)(cid:12)(cid:12) q i < ∞ (16) for some q > . Assume one of the following hypotheses hold: Case A:
For every t ∈ [0 , T ] , the mappings ( x, y ) b ( t, x, y ) , σ ( t, x, y ) are continuous and forevery R > , b n ( t, · , · ) → b ( t, · , · ) , σ n ( t, · , · ) → σ ( t, · , · ) as n → ∞ in C ( B R × B R ) . ase B: The function ( t, x ) sup n ∈ N ˜ E b n ( t, x, ˜ X nt ) is locally bounded and for every R > , lim inf n →∞ (cid:2) inf (cid:8) h T σ n ( t, x, y ) σ Tn ( t, x, y ) h : | h | = 1; t ∈ [0 , T ]; | x | , | y | ≤ R (cid:9)(cid:3) > , (17) and also b n → b and σ n → σ as n → ∞ in L d +2 ([0 , T ] × B R × B R , λ ) .Here B R is the ball with radius R centered at the origin and λ is the Lebesgue measure on R d +1 . Then equation (14) has a weak solution on [0 , T ] . We will use the following lemma in the proof of case B, which is a consequence of theKrylov’s estimate (see Theorem 2.2.4 in [8]).
Lemma 3.2.
Consider the probability space (cid:0) Ω , F , ( F t ) t ≥ , P (cid:1) and an ( F t ) t ≥ -Wiener process ( W t ) t ≥ on R d . Let Z ( t ) = R t f ( t, ω )d t + R t g ( t, ω )d W t be an Itˆo process on R d where f, g : [0 , T ] × Ω → R d , R d × d are F t -adapted stochastic processes. Let us denote the exit timeof Z from domain D ⊂ R d by τ D , i.e., τ D := inf { t ≥ Z ( t ) / ∈ D } . Assume that there exist constants K and δ such that for all ( t, ω ) ∈ [0 , T ] × Ω with theproperty t < τ D ( ω ) , the following inequalities hold | f ( t, ω ) | ≤ K, inf | h | =1 h T g ( t, ω ) g T ( t, ω ) h ≥ δ Then there exists a constant N δ,K,d,D depending only on δ, K, d and the diameter of the region D such that for any measurable function u : [0 , T ] × D → R , E (cid:20)Z T ∧ τ D u ( t, Z ( t ))d t (cid:21) ≤ N δ,K,d,D (cid:18)Z [0 ,T ] × D | u ( t, x ) | d +1 d t d x (cid:19) d +1 . Proof of Theorem 3.1.
First we prove tightness of distributions of X n on C (cid:0) [0 , T ] , R d (cid:1) . Us-ing X nt − X ns = Z ts ˜ E b n (cid:16) u, X nu , ˜ X nu (cid:17) d u + Z ts ˜ E σ n (cid:16) u, X nu , ˜ X nu (cid:17) d W u and Burkholder-Davis-Gundy inequality, it follows that E | X nt − X ns | q ≤ q − E (cid:12)(cid:12)(cid:12)(cid:12)Z ts ˜ E b n (cid:16) u, X nu , ˜ X nu (cid:17) d u (cid:12)(cid:12)(cid:12)(cid:12) q + 2 q − E (cid:12)(cid:12)(cid:12)(cid:12)Z ts ˜ E σ n (cid:16) u, X nu , ˜ X nu (cid:17) d W u (cid:12)(cid:12)(cid:12)(cid:12) q ≤ q − | t − s | q − E Z ts ˜ E (cid:12)(cid:12)(cid:12) b n (cid:16) u, X nu , ˜ X nu (cid:17)(cid:12)(cid:12)(cid:12) q d u + 2 q − C E (cid:20)Z ts (cid:12)(cid:12)(cid:12) ˜ E σ n (cid:16) u, X nu , ˜ X nu (cid:17)(cid:12)(cid:12)(cid:12) d u (cid:21) q/ ≤ q − | t − s | q − E Z ts ˜ E (cid:12)(cid:12)(cid:12) b n (cid:16) u, X nu , ˜ X nu (cid:17)(cid:12)(cid:12)(cid:12) q d u + 2 q − C | t − s | q − E Z ts ˜ E (cid:12)(cid:12)(cid:12) σ n (cid:16) u, X nu , ˜ X nu (cid:17)(cid:12)(cid:12)(cid:12) q d u ≤ C | t − s | q . q >
2, the laws of X n in the space of C (cid:0) [0 , T ] , R d (cid:1) are tight and there exist somesubsequence X n k which converges in law to some law µ on C (cid:0) [0 , T ] , R d (cid:1) . According toSkorokhod’s theorem, there exist random variables say (cid:16) Y n k , ˜ Y n k (cid:17) given on some probabil-ity space (Ω , F , P ) with the same distribution as (cid:16) X n k , ˜ X n k (cid:17) converging to some randomvariable (cid:16) Y, ˜ Y (cid:17) having distribution µ ⊗ µ . Let us define M n k t := Y n k t − Z t ˜ E b n k ( s, Y n k s , ˜ Y n k s )d s. ( M n k t ) t ≥ is a martingale with quadratic variation N n k t := Z t ˜ E σ n k ( s, Y n k s , ˜ Y n k s ) ˜ E σ Tn k ( s, Y n k s , ˜ Y n k s )d s. We have by Burkholder-Davis-Gundy inequality and (16) thatsup k ∈ N E | M n k t | q ≤ C sup k ∈ N E | N n k t | q/ ≤ C T . Let us also define M t := Y t − Z t ˜ E b ( s, Y s , ˜ Y s )d t, and N t := Z t ˜ E σ ( s, Y s , ˜ Y s ) ˜ E σ T ( s, Y s , ˜ Y s )d s. If we can show that M n k t → M t and N n k t → N t in probability, then we have for boundedcontinuous function F : C ([0 , s ] , R d ) → R and v, u ∈ R d E (cid:2) h M t − M s , v i F ( Y | [0 ,s ] ) (cid:3) = lim k →∞ E (cid:2) h M n k t − M n k s , v i F ( Y n k | [0 ,s ] ) (cid:3) = 0and also E (cid:2)(cid:0) h M t − M s , v i h M t − M s , u i − v T N t u (cid:1) F ( Y | [0 ,s ] ) (cid:3) = lim k →∞ E (cid:2)(cid:0) h M n k t − M n k s , v i h M n k t − M n k s , u i − v T N n k t u (cid:1) F ( Y n k | [0 ,s ] ) (cid:3) = 0So M t is a martingale with quadratic variation N t and the proof is completed by using themartingale representation theorem. Now we continue the proof for each set of assumptionsseparately. Case A:
For Θ ∈ { b, σ } , we have (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ( t, Y n k t , ˜ Y n k t ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Θ( t, Y n k t , ˜ Y n k t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) . Y n k t , ˜ Y n k t ) tends to ( Y t , ˜ Y t ) almost surely as k → ∞ , it is a boundedsequence in R d almost surely and the right hand side of inequality above tends to zero as k → ∞ . So by uniform integrability, we get the convergence of M n k t → M t and N n k t → N t in L as k → ∞ . Case B:
Let τ n k ( R ) := inf n t ≥ | Y n k t | ∨ (cid:12)(cid:12)(cid:12) ˜ Y n k t (cid:12)(cid:12)(cid:12) > R o , τ ( R ) := inf n t ≥ | Y t | ∨ (cid:12)(cid:12)(cid:12) ˜ Y t (cid:12)(cid:12)(cid:12) > R o , and ¯ τ ( R ) := lim inf k →∞ τ n k ( R ). Since (cid:16) Y n k , ˜ Y n k (cid:17) tends to ( Y, ˜ Y ) in C ([0 , T ] , R d ), ¯ τ ( R ) ≤ τ ( R ). We have E " sup t ∈ [0 ,T ] | X nt | ≤ C E (cid:0) | ξ | (cid:1) + CT E Z T ˜ E (cid:12)(cid:12)(cid:12) b n (cid:16) t, X nt , ˜ X nt (cid:17)(cid:12)(cid:12)(cid:12) d t + C E (cid:20)Z T ˜ E (cid:12)(cid:12)(cid:12) σ n (cid:16) t, X nt , ˜ X nt (cid:17)(cid:12)(cid:12)(cid:12) d t (cid:21) ≤ C T . So the stopping times τ n k ( R ) satisfylim R →∞ lim sup k →∞ P ⊗ ˜ P ( τ n k ( R ) < T ) = 0 . (18)We have P ⊗ ˜ P (cid:18)Z T (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t > δ (cid:19) ≤ P ⊗ ˜ P ( T > τ n k ( R ) ∧ ¯ τ ( R ))+ P ⊗ ˜ P (cid:18) T ≤ τ n k ( R ) ∧ ¯ τ ( R ); Z T (cid:12)(cid:12)(cid:12) Θ n k ( t, Y t , ˜ Y t ) − Θ n k ( t, Y n k t , ˜ Y n k t ) (cid:12)(cid:12)(cid:12) d t > δ/ (cid:19) + P ⊗ ˜ P (cid:18) T ≤ τ n k ( R ) ∧ ¯ τ ( R ); Z T (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ n k ( t, Y n k t , ˜ Y n k t ) (cid:12)(cid:12)(cid:12) d t > δ/ (cid:19) + P ⊗ ˜ P (cid:18) T ≤ τ n k ( R ) ∧ ¯ τ ( R ); Z T (cid:12)(cid:12)(cid:12) Θ n k ( t, Y t , ˜ Y t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t > δ/ (cid:19) = I + I + I + I , say.Now observe that I ≤ P ⊗ ˜ P ( τ n k ( R ) < T ) + P ⊗ ˜ P (¯ τ ( R ) < T ) ≤ P ⊗ ˜ P ( τ n k ( R ) < T ) + lim sup l →∞ P ⊗ ˜ P ( τ n l ( R ) < T ) . From (18) we obtain that lim R →∞ lim sup k →∞ I = 0.By continuity of Θ n k , it is clear that for fixed k , the second term, i.e. I tends to zero as k → ∞ . To take the limit of I and I , we use Lemma 3.2. Since ( t, x ) sup n ∈ N ˜ E b n ( t, x, ˜ Y nt )17s locally bounded, for t ≤ T ∧ τ n k ( R ), sup k ∈ N ˜ E b n k (cid:16) t, Y n k t , ˜ Y n k t (cid:17) is bounded. Inequality (17)implies that there exists K R ∈ N such that for all k ≥ K R ,inf t ∈ [0 ,T ∧ τ nk ( R )] | h |≤ h T σ n k (cid:16) t, Y n k t , ˜ Y n k t (cid:17) σ Tn k (cid:16) t, Y n k t , ˜ Y n k t (cid:17) h ≥ ε ( R, T ) > . Therefore the conditions of Lemma 3.2 for Itˆo process (cid:16) Y n k t , ˜ Y n k t (cid:17) and the exit time τ n k ( R )hold for all k ≥ K R and there exists a constant C ( R, T ) such that, I ≤ δ E ˜ E Z T ∧ τ nk ( R )0 (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ n k ( t, Y n k t , ˜ Y n k t ) (cid:12)(cid:12)(cid:12) d t ! ≤ C ( R, T ) (cid:18)Z [0 ,T ] × B R × B R (cid:12)(cid:12)(cid:12) Θ n k ( t, x, y ) − Θ n k ( t, x, y ) (cid:12)(cid:12)(cid:12) d +2 d t d x d y (cid:19) d +1 → , which tends to zero as k, k → ∞ since Θ n → Θ in L d +2 loc as n → ∞ . Let w ∈ C (cid:0) R d × R d , R (cid:1) be compactly supported with 1 B R × B R ≤ w ≤
1. Then I ≤ δ E ˜ E Z T ∧ ¯ τ ( R )0 w ( Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) Θ n k ( t, Y t , ˜ Y t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t. Since continuous functions are dense in L ([0 , T ] × B R × B R , µ ) ∩ L d +2 ([0 , T ] × B R × B R , λ ) , where λ is the Lebesgue measure and µ is the following finite Borel measure, µ ( A ) := E ˜ E Z T { ( t,Y t , ˜ Y t ) ∈ A } w ( Y t , ˜ Y t )d t, we can find for every ε >
0, a continuous function g on [0 , T ] × R d × R d such that (cid:18) E ˜ E Z T w ( Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) Θ n k ( t, Y t , ˜ Y t ) − Θ( t, Y t , ˜ Y t ) − g ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t (cid:19) / + (cid:18)Z T Z B R Z B R (cid:12)(cid:12)(cid:12) Θ n k ( t, x, y ) − Θ( t, x, y ) − g ( t, x, y ) (cid:12)(cid:12)(cid:12) d +2 d x d y d t (cid:19) d +2 ≤ ε. So ( δI ) / ≤ E ˜ E Z T ∧ ¯ τ ( R )0 w ( Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) g ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t ! / + ε = (cid:18) E ˜ E Z T { t< ¯ τ ( R ) } w ( Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) g ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t (cid:19) / + ε ≤ lim inf l →∞ (cid:18) E ˜ E Z T { t ≤ τ nl ( R ) } w ( Y n l t , ˜ Y n l t ) (cid:12)(cid:12)(cid:12) g ( t, Y n l t , ˜ Y n l t ) (cid:12)(cid:12)(cid:12) d t (cid:19) / + ε. l ∈ N ,( δI ) / ≤ E ˜ E Z T ∧ τ nl ( R )0 w ( Y n l t , ˜ Y n l t ) (cid:12)(cid:12)(cid:12) g ( t, Y n l t , ˜ Y n l t ) (cid:12)(cid:12)(cid:12) d t ! / + 2 ε Then by Lemma 3.2, we have( δI ) / ≤ C ( R, T ) | g | L d +2 ([0 ,T ] × B R × B R ,λ ) + 2 ε ≤ C ( R, T ) (cid:18)(cid:12)(cid:12)(cid:12) Θ n k − Θ (cid:12)(cid:12)(cid:12) L d +2 ([0 ,T ] × B R × B R ,λ ) + ε (cid:19) + 2 ε So, I also tends to zero as k → ∞ . Hence Z T (cid:12)(cid:12)(cid:12) Θ n k ( t, Y n k t , ˜ Y n k t ) − Θ( t, Y t , ˜ Y t ) (cid:12)(cid:12)(cid:12) d t → , as k → ∞ and therefore M n k t → M t and N n k t → N t almost surely. Remark 3.3.
The proof of Theorem 3.1 is shorter than the proof of weak existence theoremin [11] because in case B, we estimated b, σ in the smaller space L d +2 instead of L d +1 andalso we used the representation theorem for martingales. In fact Theorem 3.1 is more generalthan the weak existence result stated in [11] and to prove that, it is enough to approximate b, σ in the space L d +2 instead of L d +1 . Theorem 3.4.
Let b, σ : [0 , ∞ ) × R d × R d → R d , R d × d be measurable. Consider the equation d X t = ˜ E b ( t, X t , ˜ X t )d t + ˜ E σ ( t, X t , ˜ X t )d W t (19) with initial value X = ξ . Assume that there exists a convex function V ∈ C (cid:0) R d , [0 , ∞ ) (cid:1) such that for some q > ,(H1) h∇ V ( x ) , b ( t, x, y ) i + tr (cid:0) σ T ( t, x, y )D V ( x ) σ ( t, x, y ) (cid:1) < CV ( x ) , (H2) | b ( t, x, y ) | q + | σ ( t, x, y ) | q < V ( x ) V ( y ) , (H3) E V ( ξ ) + E | ξ | < ∞ . Also assume that ( x, y ) b ( t, x, y ) , σ ( t, x, y ) are continuous or σ , for every R > , satisfies inf t ∈ [0 ,T ] , | x |
12 tr (cid:0) σ T D V ( x ) σ (cid:1) is convex. Since V ∈ C ( R d , [0 , ∞ )), for an arbitrary ε >
0, there exists r n > h∇ V ( x ) , b n,r n ( t, z ) i + 12 tr (cid:0) σ Tn,r n ( t, z ) D V ( x ) σ n,r n ( t, z ) (cid:1) ≤ ψ (cid:16) zn (cid:17) " (cid:28) ∇ V ( x ) , Z R d b ( t, z + ˜ z ) r dn φ ( r n ˜ z )d˜ z (cid:29) + 12 tr (cid:18)Z R d σ T ( t, z + ˜ z ) r dn φ ( r n ˜ z )d˜ z D V ( x ) Z R d σ ( t, z + ˜ z ) r dn φ ( r n ˜ z )d˜ z (cid:19) ≤ ψ (cid:16) zn (cid:17) Z R d h h∇ V ( x ) , b ( t, z + ˜ z ) i + 12 tr (cid:0) σ T ( t, z + ˜ z ) D V ( x ) σ ( t, z + ˜ z ) (cid:1) i r dn φ ( r n ˜ z )d˜ z ≤ ψ (cid:16) zn (cid:17) Z R d h h∇ V ( x + ˜ x ) , b ( t, z + ˜ z ) i + 12 tr (cid:0) σ T ( t, z + ˜ z ) D V ( x + ˜ x ) σ ( t, z + ˜ z ) (cid:1) i r dn φ ( r n ˜ z )d˜ z + ε ≤ Cψ (cid:16) zn (cid:17) Z R d V ( x + ˜ x ) r dn φ ( r n ˜ z )d˜ z + ε ≤ CV ( x ) + ε The same argument implies | b n,r n ( t, x, y ) | q + | σ n,r n ( t, x, y ) | q < V ( x ) V ( y ) + ε, Let us take r n > r n → ∞ as n → ∞ . It is clear that b n = b n,r n and σ n = σ n,r n are bounded and globally Lipschitz. Therefore there exists a unique solutionto d X nt = ˜ E b ( t, X nt , ˜ X nt )d t + ˜ E σ ( t, X nt , ˜ X nt )d W t (20)with any arbitrary F measurable initial value X n = ξ . To show that the assumptions ofTheorem 3.1 hold, it is sufficient to prove that sup n ∈ N E V ( X nt ) < C T for all t ∈ [0 , T ]. We20ave by convexity of V and (H1), e − Ct V ( X nt ) = V ( ξ ) + Z t e − Cs " D ∇ V ( X ns ) , ˜ E b n (cid:16) s, X ns , ˜ X ns (cid:17)E + 12 tr (cid:16) ˜ E σ Tn (cid:16) s, X ns , ˜ X ns (cid:17) D V ( X ns ) ˜ E σ n (cid:16) s, X ns , ˜ X ns (cid:17)(cid:17) − CV ( X ns ) d s + M t ≤ V ( ξ ) + ˜ E Z t e − Cs " D ∇ V ( X ns ) , b n (cid:16) s, X ns , ˜ X ns (cid:17)E + 12 tr (cid:16) σ Tn (cid:16) s, X ns , ˜ X ns (cid:17) D V ( X ns ) σ n (cid:16) s, X ns , ˜ X ns (cid:17)(cid:17) − CV ( X ns ) d s + M t ≤ V ( ξ ) + Z t εe − Cs d s + M t = V ( ξ ) + εC (cid:0) − e − Ct (cid:1) + M t where M t is a local martingale starting from zero. Let τ m ↑ ∞ be a corresponding localizingsequence. Then by Fatou’s lemma, e − Ct E V ( X nt ) ≤ lim inf m →∞ E (cid:2) e − C ( t ∧ τ m ) V ( X nt ∧ τ m ) (cid:3) = E V ( ξ ) + εC (cid:0) − e − Ct (cid:1) , and therefore, E V ( X nt ) ≤ e Ct E V ( ξ ) + εC (cid:0) e Ct − (cid:1) . By Theorem 3.4, there exist a weak solution to (19) like X t and some subsequence X n k whichconverges in law to X on C ([0 , T ] , R d ) as k → ∞ . Hence E V ( X t ) ≤ e Ct E V ( ξ ). Corollary 3.5.
Let b, σ : [0 , ∞ ) × R d × R d → R d , R d × d be measurable. Consider the equation d X t = ˜ E b ( t, X t , ˜ X t )d t + ˜ E σ ( t, X t , ˜ X t )d W t (21) with initial value X = ξ . Here ˜ X t is an independent copy of X t . Suppose that one of thefollowing assumptions holds for q > :1. Assume that there exists α ≥ such that | x | (cid:0) h x, b ( t, x, y ) i + | σ ( t, x, y ) | (cid:1) + ( α − (cid:12)(cid:12) σ T ( t, x, y ) x (cid:12)(cid:12) ≤ C (1 + | x | ) , | b ( t, x, y ) | q + | σ ( t, x, y ) | q ≤ C (1 + | x | α )(1 + | y | α ) , and E (cid:0) | ξ | α ∨ (cid:1) < ∞ .2. Assume that there exist p ∈ [1 , and α > such that | x | (cid:0) h x, b ( t, x, y ) i + | σ ( t, x, y ) | (cid:1) + ( αp | x | p + p − (cid:12)(cid:12) σ T ( t, x, y ) x (cid:12)(cid:12) ≤ C (1 + | x | − p ) , | b ( t, x, y ) | q + | σ ( t, x, y ) | q ≤ C exp( α | x | p + α | y | p ) , and E [exp( α | ξ | p )] < ∞ .Also assume that ( x, y ) b ( t, x, y ) , σ ( t, x, y ) are continuous or σ is symmetric and uniformlypositive definite, i.e., inf s,x,y inf | λ | =1 λ T σ ( s, x, y ) λ > . Then equation (21) has a weak solution. cknowledgments This work is part of the first author’s PhD thesis jointly at Sharif University of Technologyand Technical University of Berlin under the supervision of Professor Bijan Z. Zangeneh,Professor Michael Scheutzow and Professor Wilhelm Stannat. She wishes to thank hersupervisors for their support, encouragement and guidance.
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