Errata: Mean field games with common noise

Abstract

This note corrects Lemma 3.7 in our paper [1]. The main results of the paper remain correct as stated. This note corrects an error in [1, Lemma 3.7]. The lemma is not correct as stated, and the first conclusion must instead be stated as a hypothesis. This erratum corrects the statement of the lemma and then shows that the additional hypothesis is satisfied in each of the three applications of the lemma later in the paper. The main results of the paper remain unchanged. The error in [1, Lemma 3.7] is at the end of the “first step” of the proof. Specifically, the last equation before the “second step” (lines 5-6 of page 3769) is not accurate, because the preceding equation was proven only for all F t -measurable functions φ1(μ), not for all F μ T -measurable functions. We rewrite the lemma as follows, stating equivalence between its two claims as well as a third and often more convenient form: Lemma 3.7*. Let P ∈ Pp(Ω) such that (B,W ) is a Wiener process with respect to the filtration (F t )t∈[0,T ] under P , and define ρ := P ◦ (ξ,B,W, μ)−1. Suppose that (1) and (3) of Definition 3.4 are satisfied, that P (X0 = ξ) = 1, and that the state equation (3.3) holds under P . The following are equivalent: (A) For P ◦ μ−1-almost every ν ∈ Pp(X ), it holds that (Wt)t∈[0,T ] is an (F W,Λ,X t )t∈[0,T ]-Wiener process under ν. (B) Under P , F T ∨ F ξ,W,Λ t is independent of σ{Ws −Wt : s ∈ [t, T ]} for every t ∈ [0, T ). (C) P is an MFG pre-solution Proof. (A⇒ C): Let Q = P ◦ (ξ,B,W, μ,Λ)−1. Assuming (A) holds, the second and third steps of the original proof [1, Lemma 3.7] are correct and show that Q ∈ A(ρ). As all of the other defining properties of an MFG pre-solution hold by assumption, we deduce (C). (C ⇒ B): Note that (C) entails that FΛ t is conditionally independent of F ξ,B,W,μ T given F ξ,B,W,μ t under P , for every t ∈ [0, T ). Fix t ∈ [0, T ), and fix bounded functions φt, ψT , ψt, and ht+ such that φt(Λ) is FΛ t -measurable, ψT (B,μ) is F B,μ T -measurable, ψt(ξ,W ) is F ξ,W t -measurable, and ht (W ) is σ{Ws −Wt : s ∈ [t, T ]}-measurable. The conditional independence yields E [ φt(Λ)| F T ] = E [ φt(Λ)| F t ] , a.s. The independence of ξ, (B,μ), and W easily implies that F T ∨F ξ,W t is independent of σ{Ws− Wt : s ∈ [t, T ]}, and we deduce E [φt(Λ)ψT (B,μ)ψt(ξ,W )ht+(W )] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W )ht+(W ) ] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W ) ] E [ht+(W )] = E [φt(Λ)ψT (B,μ)ψt(ξ,W )]E [ht+(W )] .