It\^o Formula for Processes Taking Values in Intersection of Finitely Many Banach Spaces

Motivated by applications to SPDEs we extend the It\^o formula for the square of the norm of a semimartingale $y(t)$ from Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the case \begin{equation*} \sum_{i=1}^m \int_{(0,t]} v_i^{\ast}(s)\,dA(s) + h(t)=:y(t)\in V \quad \text{$dA\times \mathbb{P}$-a.e.}, \end{equation*} where $A$ is an increasing right-continuous adapted process, $v_i^{\ast}$ is a progressively measurable process with values in $V_i^{\ast}$, the dual of a Banach space $V_i$, $h$ is a cadlag martingale with values in a Hilbert space $H$, identified with its dual $H^{\ast}$, and $V:=V_1\cap V_2 \cap \ldots \cap V_m$ is continuously and densely embedded in $H$. The formula is proved under the condition that $\|y\|_{V_i}^{p_i}$ and $\|v_i^\ast\|_{V_i^\ast}^{q_i}$ are almost surely locally integrable with respect to $dA$ for some conjugate exponents $p_i, q_i$. This condition is essentially weaker than the one which would arise in application of the results in Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the semimartingale above.


Introduction
Itô formula for the square of the norm is an essential tool in the study of stochastic evolution equations of the type dv(t) = A(t, v(t)) dt + k B k (t, v(t)) dW k (t), (1.1) where (W k ) ∞ k=1 is a sequence of independent Wiener processes, and A(t, ·) and B k (t, ·) are (possibly random nonlinear) operators on a separable real Banach space V , with values in a Banach space V ′ and a Hilbert space H respectively, such that V ֒→ H ֒→ V ′ with continuous and dense embeddings. We assume there is a constant K such that (v, h) ≤ K v V h V ′ for all v ∈ V and h ∈ H. This means that for the linear mapping Ψ : H → H * , which identifies H with its dual H * via the inner product in H, we have Ψ (h) V * ≤ K h V ′ . Therefore, since H is dense in V ′ , Ψ can be extended to a continuous mapping from V ′ into V * , the dual of V . It is assumed that this extension is one-toone from V ′ into V * . Thus an initial value problem for equation ( holds for all t ∈ [0, T ], where ·, · denotes the duality pairing of V * and V . This formula is used in proofs of existence and uniqueness theorems for PDEs, see e.g., [2] and [13]. A generalisation of it, a "stochastic energy equality", i.e., an Itô formula for the square of the H-norm of y, was first presented in Pardoux [14], and was used to obtain existence and uniqueness theorems for SPDEs. The proof of it in [14] was not separated from the theory of SPDEs developed there. A proof, not bound to the theory of SPDEs, was given in Krylov and Rozovskii [12], and then this stochastic energy equality was generalised in Gyöngy and Krylov [6] to V * -valued semimartingales y of the form where A is an adapted nondecreasing cadlag process and h is an H-valued cadlag martingale. This generalisation is used in Gyöngy [7] to extend the theory of SPDEs developed in [14] and [12] to SPDEs driven by random orthogonal measures and Lévy martingales, written in the form dv(t) = A(t, v(t)) dA(t) + B(t, v(t)) dM (t) (1.4) with cadlag (quasi left-continuous) martingales M with values in a Hilbert space.
In the present paper we are interested in stochastic energy equalities which can be applied to SPDEs (1.4) when A is of the form A = A 1 +A 2 +· · ·+A m and the operators A i have different analytic and growth properties. This means, for some Banach spaces V i and V ′ i , such that with a constant R and a process g, locally integrable with respect to dA, one has for all t In the special case when A(t) = t and M is a Wiener process the above situation was considered in [14], and a related stochastic energy equality was also presented there. Our main result, Theorem 2.1 generalises the results on stochastic energy equalities from [14] and [6]. We prove it by adapting the method of the proof of the main theorem in [6].
In the present paper we consider a semimartingale y of the form (1.3) such that dA × P-almost everywhere y takes values in V = V 1 ∩ . . . ∩ V m , where V i are Banach spaces (over R) such that V with the norm · := m i=1 · Vi is continuously and densely embedded in H.
i -valued progressively measurable processes. We prove that y is almost surely cadlag as a process with values in H and for |y| 2 H an Itô formula holds under the assumption that y pi Vi and v * i qi V * i are almost surely locally integrable with respect to dA for some conjugate exponents p i , q i . See Section 2 for precise formulation of the main theorem. To apply the result of [6] to y given by (1.3), one needs the local integrability (with respect to dA) of We note that in the context of stochastic evolution equations it is possible to prove Itô formulae for more general functions (satisfying appropriate differentiability assumptions), see again Pardoux [14], Krylov [11], [10], [9], Da Prato, Jentzen and Röckner [4], as well as Dareiotis and Gyöngy [1]. The Itô formula for the square of the norm is used in particular to establish a priori estimates as well as uniqueness and existence of solutions of stochastic evolution equations. The more general Itô formula can then be used to study finer properties of solutions of stochastic evolution equations, for example the maximum principle.
For general theory of SPDEs in the variational setting we refer the reader to Krylov and Rozovskii [12], Prévôt and Röckner [15] and Rozovskii [16].

Main Results
For i = 1, . . . , m let (V i , · Vi ) be real Banach spaces with duals (V * i , · V * i ). Let V denote the vector space V 1 ∩ · · · ∩ V m with the norm · := · V1 + · · · + · Vm . Then clearly, V is a Banach space. Assume that it is separable and is continuously and densely embedded in a Hilbert space (H, | · |), which is identified with its dual H * by the help of the inner product (·, ·) in H. Thus we have where H * ֒→ V * is the adjoint of the embedding V ֒→ H. We use the notation ·, · for the duality pairing between V and V * . Note that if v * ∈ V * i for some i, then its restriction to V belongs to A complete probability space (Ω, F , P) together with an increasing family of σ-algebras (F t ) t≥0 , F t ⊂ F will be used throughout the paper. Moreover it is assumed that the usual conditions are satisfied: s>t F s = F t and F 0 contains all subsets of P-null sets of F . We use the notation B(R + ) for the σ-algebra of Borel subsets of R + = [0, ∞), and for a real-valued increasing B(R + ) ⊗ F -measurable process (A(t)) t≥0 the notation dA × P stands for the measure defined on B(R + ) ⊗ F by Let h = (h(t)) t≥0 be an H-valued locally square integrable martingale that is cadlag (continuous from the right with left-hand limits) in the strong topology on H. Its quadratic variation process is denoted by [h], and h denotes the unique predictable process starting from zero such that |h| 2 − h is a local martingale. Furthermore let A be a real-valued nondecreasing adapted cadlag process starting from zero. Finally let v = (v(t)) t≥0 be a V -valued progressively measurable process and for i = 1, . . . , m let v * i = (v * i ) t≥0 be V * ivalued processes such that ϕ, v * i are progressively measurable for any ϕ ∈ V . Notice that v is also progressively measurable as a process with values inV i , the closure in V i -norm of the linear hull of {v(t) : t ≥ 0, ω ∈ Ω}.
The following theorem is the main result of this paper.
Theorem 2.1 Let τ be a stopping time. Suppose that for all ϕ ∈ V and for dA × P almost all (ω, t) such that t ∈ (0, τ (ω)) we have Then there isΩ ⊂ Ω with P(Ω) = 1 and an H-valued cadlag processṽ such that the following statements hold.
Consider now a situation where the assumptions on h and A are as above but m = 1 and regarding v and v * : Then v * V * ≤ 1 and so v,v * andĀ satisfy the conditions on v, v * and A, respectively, with p 1 = 1 and q 1 = ∞. If (2.2) holds for all ϕ ∈ V and for dA × P almost all (ω, t) such that t ∈ (0, τ (ω)) then Applying Theorem 2.1 then means that we have all of its conclusions withv * andĀ in place of v * and A respectively. In particular, we get Hence we see that Theorem 2.1 is a generalisation of the main theorem in Gyöngy and Krylov [6].
Remark 2.1 One might think that Theorem 2.1 follows from the main theorem in [6] by considering the process v * = i v * i as a process with values in V * . However, taking into account that for any (see for example Gajewski, Gröger and Zacharias [3, Chapter 1, Theorem 5.13]), one can show that the local integrability condition in [6] for is not implied by our assumption (2.1). Thus the main theorem in [6] is not applicable in our situation.
We consider the following motivating example.
Example 2.1 Consider the stochastic partial differential equation Here W is a Wiener process (finite or infinite dimensional depending on the choice of f ), (Z, Σ) is a measurable space and q(ds, dz) a stochastic martingale measure on [0, ∞) × Z. See, for example, Gyöngy and Krylov [5] for detailed definition. We take D to be a bounded Lipschitz domain in R d . It is natural to assume that a solution u should be such that u p1 Lp 2 (D) are almost surely locally integrable. To apply the result in Gyöngy and Krylov [6] one could try to take V : . The dual of V can be identified with the linear space One would then need to show that u V ∇(|∇u| p1−2 ∇u) + |u| p2−2 u V * is locally integrable. To ensure this in general we need, in particular, that is locally integrable, which we may not have if p 1 < p 2 . Thus one cannot apply the Itô formula from Gyöngy and Krylov. On the other hand it is easy to check that the assumptions of Theorem 2.1 are satisfied.
An application of the above Itô's formula to SPDEs driven by Wiener processes is given in [14] (Chapter 2, Example 5.1) and in [8]. Further examples can be found in [13, Chapter 2, Section 1.7].

Preliminaries
Lemma 3.1 For r ∈ [0, ∞) let β(r) := inf{t ≥ 0 : A(t) ≥ r} and let x(t) be a real valued process that is locally integrable with respect to dA for all ω ∈ Ω. Then i) β(r) is a stopping time (not necessarily finite) for every r ∈ [0, ∞), ii) x(β(r)) dr, x(β(r)) dr iv) If 0 = r n 0 < r n 1 < · · · < r n k < · · · is an increasing sequence of decompositions of [0, ∞) such that sup k |r n k+1 − r n k | → 0 as n → ∞ then for every This Lemma is proved in Gyöngy and Krylov [6, Lemma 1]. Let κ (j) n for j = 1, 2 and integers n ≥ 1 denote the functions defined by The following lemma is known and the authors believe is due to Doob.
Then there exists a subsequence n k → ∞ such that for dt-almost all t ∈ [0, 1] By change of variables and changing the order of integration Note that by the shift invariance of the Lebesgue measure Therefore by Lebesgue's theorem on dominated convergence for almost all t ∈ [0, 1], and the statement of the lemma follows.
The following lemma is proved in Gyöngy and Krylov [6,Lemma 3].
Then for any ε > 0 P sup

Proof of the Main Result
The following standard steps, as in Krylov and Rozovskii [12], allow us to work under more convenient assumptions without any loss of generality.
1) We note that τ can be assumed to be a bounded stopping time. Indeed if we prove Theorem 2.1 under this assumption then we can extend it to unbounded stopping times by considering τ ∧ n and letting n → ∞. In fact using a non-random time change we may assume that τ ≤ 1. 2) Recall the processes η i from assumption (2.1), and set t≥0 denote a nondecreasing cadlag adapted process such that almost surely It is not difficult to see that such a process Q i exists, we can take, e.g., the adapted right-continuous modification of the process dA-ess sup s≤t η i (s), i.e., lim n→∞ dA-ess sup s≤t+1/n η i (s).
Let (e j ) j∈N ⊂ V be an orthonormal basis in H and define with c k := max 1≤i≤m j≤k |e j | 2 Vi and w k := Π k h, where Π k denotes the orthogonal projection of H onto its subspace spanned by (e i ) k i=1 . We may and will assume, without loss of generality, that r and h are bounded. Indeed, imagine we have proved Theorem 2.1 under this assumption. Consider τ n := inf{t ≥ 0 : r(t) ≥ n}.
Indeed let v n := n −1 v, A n := n −1 A and h n := n −1 h. Then r n , defined analogously to r in (4.1) but with v, A and h replaced by v n , A n and h n respectively, satisfies r n (t) ≤ n −1 r(t) ≤ 1. We thus get (2.3) and (2.4) with v, A and h replaced by v n , A n and h n respectively. We can now multiply by n and n 2 to obtain the desired conclusions.
Now we proceed to prove Theorem 2.1 under the assumption that τ ≤ 1, r(t) ≤ 1 and h t ≤ 1 for t ∈ [0, ∞). Our approach is the same as in Gyöngy and Krylov [6]. The idea is to approximate v by simple processes whose jumps happen at stopping times where equation (2.2) holds. But (2.2) only holds for every ϕ ∈ V and dA×P almost all (t, ω) ∈ 0, τ , and thus it is not immediately clear how to choose an appropriate piecewise constant approximation to v.
We see that due to (4.7) there is Ω ′′ ⊂ Ω ′ such that P(Ω ′′ ) = 1 and Moreover, since h is cadlag, for all ω ∈ Ω ′′ we have for t ≥ 0, where the integrals are defined as weak* integrals. Recall that v * = i v * is a V * -valued such that v * (t), ϕ is a progressively measurable process for every ϕ ∈ V , and Therefore z (1) and z (2) are well-defined V * -valued processes such that z (1) , ϕ and z (2) , ϕ are left-continuous and right-continuous adapted processes, respectively.
Proposition 4.4 There isΩ ⊂ Ω ′′ with P(Ω) = 1 such that for a subsequence n ′ and for every ω ∈Ω |v * (s)| 2 ∆A(s) dA(s) for t ∈ I(ω) and ω ∈Ω. To this end set Then for all ω ∈ Ω ′′ , t > 0 and ϕ ∈ V with r n j given by (4.4). Consequently, taking also into account (4.2) of Proposition 4.1 we have Ω ′′′ ⊂ Ω ′′ and a subsequence n ′ such that the first three limits are zero for ω ∈ Ω ′′′ . Taking the limit along the subsequence n ′ in (4.17) we see that K n ′ (t) converges for ω ∈ Ω ′′′ and t ∈ I(ω) to some K(t), and From this point onwards we will always consider only the subsequence n ′ but we will keep writing n to ease notation. Our task is now to identify K(t). We note that, using Parseval's identity, Hence, using Lemma 3.1, Parseval's identity and (4.14), we get To obtain an upper bound we use first the identity together with the definition of g to get For j = 0 we split J and notice that Using Π k , the orthogonal projection of H onto the space spanned by (e j ) k j=1 ⊂ V , we have J Notice that Similarly, taking into accountṽ(0) = h(0), for J and J (22) 0 Thus from (4.19) we get for every n, k ∈ N. As n → ∞ we see that where we use the notation v * (s) = i v * i (s). By Hölder's inequality, taking into account r(t) ≤ 1, we have | v * i (s), w Note that by Fatou's lemma and the martingale property of h Eξ k (t) ≤ E|(I − Π k )h(1)| 2 → 0 as k → ∞.
Now we can finish the proof of Theorem 2.1 by noting that by the above proposition |ṽ(t)| 2 is a cadlag process, and since by Proposition 4.3 the process v is H-valued and weakly cadlag, it follows by identity (4.23) thatṽ is an Hvalued cadlag process.