On the mild It\^o formula in Banach spaces

The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \&R\"ockner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild It\^o formula so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingale differences) Banach spaces. This generalization is especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs.


Introduction
The standard Itô formula for finite dimensional Itô processes has been generalized in the literature to infinite dimensions so that it is applicable to Itô processes with values in infinite dimensional Hilbert or Banach spaces; see Theorem 2.4 in Brzeźniak, Van Neerven, Veraar & Weiss [1]. This infinite dimensional generalization of the standard Itô formula is, however, typically not applicable to a solution (or a numerical approximation) of a stochastic partial differential equation (SPDE) as solutions of SPDEs are often only solutions in the mild or weak sense, which are not Itô processes on the considered state space of the SPDE. To overcome this lack of regularity of solutions of SPDEs, Da Prato et al. proposed in Theorem 1 in [2] (see also [5,Section 5]) an alternative formula which Da Prato et al. refer to as a mild Itô formula. The mild Itô formula in Theorem 1 in [2] is (even in finite dimensions) different to the standard Itô formula but it applies to the class of Hilbert space valued mild Itô processes which is a rather general class of Hilbert space valued stochastic processes that includes standard Itô processes as well as mild solutions and numerical approximations of semilinear SPDEs as special cases. In this work we generalize the mild Itô formula so that it is applicable to mild Itô processes which take values in UMD (unconditional martingale differences) Banach spaces with type 2; see Definition 3.1 in Subsection 3.2, see Theorem 3.5 in Subsection 3.4, and see Corollary 3.8 in Subsection 3.4 below. This generalization of the mild Itô formula is especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs. In Section 2 below we also briefly review a few well-known results for Nemytskii and multiplication operators in Banach spaces (see Proposition 2.6, Proposition 2.10, and Corollary 2.11 in Section 2 below) which provide natural examples for the possibly nonlinear test function appearing in the mild Itô formula in Corollary 3.8 in Subsection 3.4 below.

Notation
Throughout this article the following notation is frequently used. Let

Stochastic partial differential equations in Banach spaces
In this section we recall a few well-known results for SPDEs on UMD Banach spaces. In particular, Proposition 2.6 below provides natural examples for the possibly nonlinear test function appearing in the mild Itô formula in Corollary 3.8 in Subsection 3.4 below.
Lemma 2.2. Consider the notation in Subsection 1.1, let (U, ·, · U , · U ) be a separable R-Hilbert space, let (V, · V ) and (V, · V ) be R-Banach spaces, and let β ∈ L (2) (V, V). Then (i) it holds for all A 1 , A 2 ∈ γ(U, V ) and all orthonormal sets U ⊆ U of U that there exists a unique v ∈ V such that inf I⊆U, (ii) it holds for all orthonormal bases U 1 , and (iv) it holds for all orthonormal sets U ⊆ U of U that Then Proof of Lemma 2.4. Observe that Lemma 2.3 and the assumption that ensure that for every strictly increasing function n : N → N there exists a strictly increasing function m : N → N such that The assumption that φ is continuous hence shows that for every strictly increasing function n : N → N there exists a strictly increasing function m : N → N such that Proof of Corollary 2.5. Observe that the assumption that lim sup n→∞ Ω |d(f n , f 0 )| p dν = 0 and Hölder's inequality ensure that lim sup n→∞ Ω |d(f n , f 0 )| min{p,1} dν = 0. Hence, we obtain that This allows us to apply Lemma 2.4 to obtain that The fact that the function [0, ∞) ∋ x → |x| q ∈ [0, ∞) is continuous and again Lemma 2.4 hence show that The proof of Corollary 2.5 it thus completed.

Regular test functions
Proposition 2.6. Consider the notation in Subsection 1.1, let k, l, d, n ∈ N, p ∈ [1, ∞), q ∈ (np, ∞), let O ∈ B(R d ) be a bounded set, let f : R k → R l be an n-times continuously differentiable function with globally bounded derivatives, and let F : Then (i) it holds that F is n-times continuously Fréchet differentiable with globally bounded derivatives, and Proof of Proposition 2.6. Throughout this proof we assume w.l.o.g. that λ R d (O) > 0. We claim that for all m ∈ {1, 2, . . . , n} it holds (a) that F is m-times Fréchet differentiable and (b) that for all v, u 1 , . . . , u m ∈ L q (λ O ; R k ) it holds that We now prove item (a) and item (b) by induction on m ∈ {1, 2, . . . , n}. For the base case m = 1 we note that Minkowski's integral inequality and Hölder's inequality show that for all v, h ∈ L q (λ O ; R k ), ε ∈ (0, ∞) it holds that Next observe that Corollary 2.5 (with (Ω, This, the fact that sup This together with Hölder's inequality and (24) implies that for all Hölder's inequality hence shows that for This demonstrates that F is Fréchet differentiable and that for all v, h ∈ L q (λ O ; R k ) it holds that This proves item (a) and item (b) in the base case m = 1. For the induction step N ∩ [0, n−1] ∋ m → m+1 ∈ {1, 2, . . . , n} assume that there exists a natural number m ∈ N∩ [0, n−1] such that item (a) and item (b) hold for m = m. Next observe that Minkowski's integral inequality and Hölder's inequality show that for all v, h, u 1 . . . , u m ∈ L q (λ O ; R k ), ε ∈ (0, q p(1+m) − 1) it holds that Moreover, note that Corollary 2.5 (with (Ω, Hölder's inequality therefore shows that for all The induction hypothesis hence implies that F (m) is Fréchet differentiable and that for all v, h, u 1 , . . . , u m ∈ L q (λ O ; R k ) it holds that This establishes item (a) and item (b) in the case m + 1. Induction thus completes the proof of item (a) and item (b).
In the next step we observe that Hölder's inequality ensures that for all m ∈ This establishes that F (n) is continuous. Combining this with item (a) and item (b) proves item (i) and item (ii). Next note that Hölder's inequality shows that for all This and item (ii) imply that for all m ∈ {1, 2, Hence, we obtain that for all m ∈ {1, 2, . . . , n}, r ∈ [mp, ∞) it holds that This proves item (iii). In the next step we observe that (37) assures that for all m ∈ {1, 2, . . . , n}, r, This and item (ii) establish that for all m ∈ {1, 2, . . . , n}, r, This proves item (iv). Item (v) is an immediate consequence of item (iv). The proof of Proposition 2.6 is thus completed.
Lemma 2.8. Consider the notation in Subsection 1. which satisfies for all v ∈ L max{p,4} (λ (0,1) d ; R), u ∈ L 4 (λ (0,1) d ; R) that and (ii) it holds that Proof of Lemma 2.8. Throughout this proof let be the function which satisfies for all v ∈ L max{p,4} (λ (0,1) Hölder's inequality hence ensures that for all Combining this and the Sobolev embedding theorem with the fact that proves that for all v ∈ L max{p,4} (λ (0,1) d ; R) it holds that This implies that there exists a unique function M : and This, in turn, assures that there exists a unique bounded linear operator which satisfies for all v ∈ L max{p,4} (λ (0,1) d ; R) that and B L(L p (λ (0,1) ;R),L(H,H β )) ≤ sup Combining (56) λ (0,1) ; R), ·, · L 2 (λ (0,1) ;R) , · L 2 (λ (0,1) ;R) ), (V, · V ) = (L p (λ (0,1) ; R), · L p (λ (0,1) ;R) ), let A : D(A) ⊆ H → H be the Laplacian with Dirichlet boundary conditions on H, let (H r , ·, · Hr , · Hr ), r ∈ R, be a family of interpolation spaces associated to −A, let A : D(A) ⊆ V → V be the Laplacian with Dirichlet boundary conditions on V , and let (V r , · Vr ), r ∈ R, be a family of interpolation spaces associated to −A. Then (i) there exists a unique continuous function ι : Proof of Lemma 2.9. Throughout this proof let ϕ ∈ L(H −ε , H) be the unique bounded linear operator which satisfies for all v ∈ H that and let φ ∈ L(V, V β ) be the unique bounded linear operator which satisfies for all Observe that Lemma 2.7 and the assumption that β Note that item (a) assures that there exist functions Φ : H → V and ι : and Observe that item (b) and item (c) establish that Φ ∈ γ(H, V ) and Combining this, the fact that ϕ ∈ L(H −ε , H), and the fact that φ ∈ L(V, V β ) with Lemma 2.1 ensures that ι ∈ γ(H −ε , V β ) and Next note that the fact that ∀ v ∈ V, t ∈ [0, ∞) : e tA v = e tA v, e.g., [ Hence, we obtain for all v ∈ V that This and (68) complete the proof of Lemma 2.9.

Mild stochastic calculus in Banach spaces
In this section we generalize the machinery in [5, Section 5] from separable Hilbert spaces to separable UMD Banach spaces with type 2.

Mild Itô processes
Definition 3.1 (Mild Itô process). Consider the notation in Subsection 1.1, let (V , · V ), (V, · V ), and (V , · V ) be separable UMD R-Banach spaces with type 2 which satisfy V ⊆ V ⊆V continuously and densely, let (U, ·, · U , · U ) be a separable R-Hilbert space, let t 0 ∈ [0, ∞), T ∈ (t 0 , ∞), let (Ω, F , P) be a probability space with a normal filtration F = (F t ) t∈[t 0 ,T ] , and let (W t ) t∈[t 0 ,T ] be an Id U -cylindrical (Ω, F , P, F)-Wiener process. Then we say that X is a mild Itô process on (Ω, F , P, F, W, (V , · V ), (V, · V ), (V , · V )) with evolution family S, mild drift Y , and mild diffusion Z (we say that X is a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z, we say that X is a mild Itô process) if and only if it holds Next observe that Definition 3.1 ensures for all t ∈ (t 0 , T ) that Hence, we obtain for all t ∈ [t 0 , T ) that Combining this and (104) shows that for all t ∈ [t 0 , T ) it holds that Moreover, observe that for all stochastic processes A, B : [0, T ] × Ω →V with continuous sample paths which satisfy ∀ t ∈ [t 0 , T ) : Combining this with (107) completes the proof of Lemma 3.2. (ii) it holds that P(X T = X T ) = 1, The assumption thatX has continuous sample paths, the fact that X is F/B(V )-adapted, and the fact that ∀ t ∈ [t 0 , T ) : P(X t = S t,T X t ) = 1 establish item (i). Moreover, note that the assumption that X is a mild Itô process proves item (iii). In addition, observe that the assumption that ∀ t ∈ [t 0 , T ) : P X t = S t,T X t = 1 implies that for all t ∈ [t 0 , T ) it holds that (iii) it holds that P( T r ( ∂ ∂t ϕ)(s, S s,T X s ) V ds < ∞) = 1,

Standard Itô formula
(v) it holds for all ω ∈ Ω, s ∈ [r, T ] that there exists a unique v ∈ V such that sup I⊆U, and (vii) it holds that . Note that Lemma 3.3 ensures thatX is an F/B(V )-adapted stochastic process with continuous sample paths which satisfies for all t ∈ [t 0 , T ] that Moreover, the assumption that ϕ ∈ C 1,2 ([r, T ] ×V , V), the assumption thatX : [t 0 , T ] × Ω →V has continuous sample paths, and the fact that ∀ t ∈ [t 0 , T ] : P( and Combining this with, e.g., Lemma 3.1 in [6] proves items (i)-(iv Combining this with, e.g., Lemma 3.1 in [6], the fact that ∀ t ∈ [t 0 , T ) : P X t = S t,T X t = 1, and the fact that ∀ t ∈ [t 0 , T ] : P( u∈U ϕ 0,2 (s,X s )(S s,T Z s u, S s,T Z s u) = u∈U ϕ 0,2 (s, S s,T X s )(S s,T Z s u, S s,T Z s u)) = 1 shows that item (v) holds, that item (vi) holds, and that for all t ∈ [r, T ] it holds that This implies item (vii). The proof of Theorem 3.5 is thus completed. Definition 3.6 (Extended mild Kolmogorov operators). Assume the setting in Subsection 3.1, let S : ∠ → L(V ,V ) be a B(∠)/S(V ,V )-measurable function which satisfies for all t 1 , t 2 , t 3 ∈ [t 0 , T ] with t 1 < t 2 < t 3 that S t 2 ,t 3 S t 1 ,t 2 = S t 1 ,t 3 , and let (t 1 , t 2 ) ∈ ∠. Then we denote by L S t 1 ,t 2 : C 2 (V , V) → C(V ×V × γ(U,V ), V) the function which satisfies for all ϕ ∈ C 2 (V , V), x ∈ V , y ∈V , z ∈ γ(U,V ) that The next corollary of Theorem 3.5 specialises Theorem 3.5 to the case where r = t 0 and where the test function (ϕ(t, x)) t∈[t 0 ,T ], x∈V ∈ C 1,2 ([t 0 , T ] ×V , V) depends on x ∈V only.
Moreover, the fact that This establishes that P lim n→∞ τ n = T = 1.
In addition, note that Lemma 3.3 and the assumption that the set ϕ([S t 0 ,T X t 0 ] P,B(V ) + τ t 0 S s,T Y s ds + τ t 0 S s,T Z s dW s ) V : F-stopping time τ : Ω → [t 0 , T ] is uniformly Pintegrable ensure that the set { ϕ(X τn ) V : n ∈ N} is uniformly P-integrable. Equation (133) hence shows that for all n ∈ N it holds that E ϕ(X τn ) V + E τn 0 ϕ ′ (S s,T X s )S s,T Z s ) 2 γ(U,V) ds < ∞.
The fact that for all n ∈ N it holds that τ n is an F-stopping time thus allows us to apply Corollary 3.9 to obtain that for all n ∈ N it holds that The triangle inequality hence proves that This together with (136), item (ii) of Lemma 3.3, and the uniform P-integrability of { ϕ(X τn ) V : n ∈ N} assures (132). The proof of Proposition 3.10 is thus completed. (ii) it holds that E ϕ(X T ½ {X T ∈V } ) V + ϕ(S t 0 ,T X t 0 ) V < ∞, and (iii) it holds that Proof of Proposition 3.11. Throughout this proof letX : [t 0 , T ]×Ω →V be a stochastic process with continuous sample paths which satisfies ∀ t ∈ [t 0 , T ) : P X t = S t,T X t = 1 (cf. Lemma 3.2) and let Z : [t 0 , T ] × Ω →V be a stochastic process with continuous sample paths which satisfies for all t ∈ [t 0 , T ] that Observe that Lemma 3.3 implies that for all t ∈ [t 0 , T ] it holds P-a.s. that Moreover, e.g., the Burkholder-Davis-Gundy type inequality in Van Neerven et al. [9,Theorem 4.7] shows that there exists a real number The assumption that E S t 0 ,T X t 0 Combining this with Proposition 3.10 completes the proof of Proposition 3.11.