Abstract
Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case
where A is an increasing right-continuous adapted process, \(v_i^{*}\) is a progressively measurable process with values in \(V_i^{*}\), the dual of a Banach space \(V_i\), h is a cadlag martingale with values in a Hilbert space H, identified with its dual \(H^{*}\), and \(V:=V_1\cap V_2 \cap \cdots \cap V_m\) is continuously and densely embedded in H. The formula is proved under the condition that \(\Vert y\Vert _{V_i}^{p_i}\) and \(\Vert v_i^*\Vert _{V_i^*}^{q_i}\) 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 Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the semimartingale above.
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1 Introduction
Itô formula for the square of the norm is an essential tool in the study of stochastic evolution equations of the type
where \((W^k)_{k=1}^{\infty }\) is a sequence of independent Wiener processes, and \({\mathbb {A}}(t,\cdot )\) and \({\mathbb {B}}_k(t,\cdot )\) 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 \(V\hookrightarrow H\hookrightarrow V'\) with continuous and dense embeddings. We assume there is a constant K such that \((v,h)\le K\Vert v\Vert _V\Vert h\Vert _{V'}\) for all \(v\in V\) and \(h\in H\). This means that for the linear mapping \(\varPsi :H\rightarrow H^{*}\), which identifies H with its dual \(H^{*}\) via the inner product in H, we have \(\Vert \varPsi (h)\Vert _{V^{*}}\le K\Vert h\Vert _{V'}\). Therefore, since H is dense in \(V'\), \(\varPsi \) can be extended to a continuous mapping from \(V'\) into \(V^{*}\), the dual of V. It is assumed that this extension is one-to-one from \(V'\) into \(V^{*}\). Thus an initial value problem for Eq. (1.1) can be viewed as
with the \(V^{*}\)-valued process \(v^{*}(t):={\mathbb {A}}(t,v(t))\) and \(H\equiv H^{*}\)-valued process
where \(h_0\) is a given initial value and the equality (1.2) in \(V^{*}\) is required \( dt\times {\mathbb {P}}\) almost everywhere. In the special case \(B_k=0\) for every k, and nonrandom \(h_0\) and A, i.e., in the case
it is well-known that when \(v\in L_p([0,T], V)\), \(v^{*}\in L_q([0,T], V^{*})\) for \(T>0\) and conjugate exponents p and q, then there is \(u\in C([0,T],H)\) such that \(u=v\) for dt-almost all \(t\in [0,T]\) and the “energy equality”
holds for all \(t\in [0,T]\), where \(\langle \cdot ,\cdot \rangle \) denotes the duality pairing of \(V^{*}\) and V. This formula is used in proofs of existence and uniqueness theorems for PDEs, see e.g., [3] 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
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 \({\mathbb {A}}\) is of the form \({\mathbb {A}}={\mathbb {A}}_1+{\mathbb {A}}_2+\cdots +{\mathbb {A}}_m\) and the operators \({\mathbb {A}}_i\) 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
for all \(w\in V\), \(q_i=p_i/(p_i-1)\) with (possibly) different exponents \(p_i\ge 1\), which for \(p_i=1\) means that \(\Vert {\mathbb {A}}_i(t,w)\Vert _{V'_i}\) is bounded by a constant.
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 \(dA \times {\mathbb {P}}\)-almost everywhere y takes values in \(V=V_1\cap \ldots \cap V_m\), where \(V_i\) are Banach spaces (over \({\mathbb {R}}\)) such that V with the norm \(\Vert \cdot \Vert := \sum _{i=1}^m \Vert \cdot \Vert _{V_i}\) is continuously and densely embedded in H. The process \(v^*\) in (1.3) is of the form \(v^*= \sum _{i=1}^m v_i^*\), where \(v_i^*\) are \(V_i^*\)-valued progressively measurable processes. We prove that y is almost surely cadlag as a process with values in H and for \(|y|_H^2\) an Itô formula holds under the assumption that \(\Vert y\Vert _{V_i}^{p_i}\) and \(\Vert v_i^*\Vert _{V_i^*}^{q_i}\) are almost surely locally integrable with respect to dA for some conjugate exponents \(p_i, q_i\). See Sect. 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
which, in general, is not satisfied under our assumptions. See Remark 2.1 and Example 2.1.
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 [9,10,11], Da Prato et al. [1], as well as Dareiotis and Gyöngy [2]. 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].
2 Main results
For \(i=1,\ldots ,m\) let \((V_i,\Vert \cdot \Vert _{V_i})\) be real Banach spaces with duals \((V_i^*,\Vert \cdot \Vert _{V_i^*})\). Let V denote the vector space \(V_1\cap \cdots \cap V_m\) with the norm \(\Vert \cdot \Vert := \Vert \cdot \Vert _{V_1} + \cdots + \Vert \cdot \Vert _{V_m}\). Then clearly, V is a Banach space. Assume that it is separable and is continuously and densely embedded in a Hilbert space \((H, |\cdot |)\), which is identified with its dual \(H^{*}\) by the help of the inner product \((\cdot ,\cdot )\) in H. Thus we have
where \(H^{*}\hookrightarrow V^{*}\) is the adjoint of the embedding \(V\hookrightarrow H\). We use the notation \(\langle \cdot , \cdot \rangle \) for the duality pairing between V and \(V^{*}\). Note that if \(v^{*}\in V_i^{*}\) for some i, then its restriction to V belongs to \(V^{*}\) and \(|\langle v^{*},v\rangle |\le \Vert v^{*}\Vert _{V^{*}_i}\Vert v\Vert _{V_i}\) for all \(v\in V\). Note also that \(\langle v^{*},v\rangle =(h,v)\) for for all \(v\in V\) when \(v^{*}=h\in H\).
A complete probability space \((\varOmega , {\mathcal {F}}, {\mathbb {P}})\) together with an increasing family of \(\sigma \)-algebras \(({\mathcal {F}}_t)_{t\ge 0}\), \({\mathcal {F}}_t \subset {\mathcal {F}}\) will be used throughout the paper. Moreover it is assumed that the usual conditions are satisfied: \(\bigcap _{s> t} {\mathcal {F}}_s = {\mathcal {F}}_t\) and \({\mathcal {F}}_0\) contains all subsets of \({\mathbb {P}}\)-null sets of \({\mathcal {F}}\). We use the notation \({\mathcal {B}}({\mathbb {R}}_+)\) for the \(\sigma \)-algebra of Borel subsets of \({\mathbb {R}}_+=[0,\infty )\), and for a real-valued increasing \({\mathcal {B}}({\mathbb {R}}_+)\otimes {\mathcal {F}}\)-measurable process \((A(t))_{t\ge 0}\) the notation \(dA\times {\mathbb {P}}\) stands for the measure defined on \({\mathcal {B}}({\mathbb {R}}_+)\otimes {\mathcal {F}}\) by
Let \(h=(h(t))_{t\ge 0}\) 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 \(\langle h\rangle \) denotes the unique predictable process starting from zero such that \(|h|^2-\langle h\rangle \) is a local martingale. Furthermore let A be a real-valued nondecreasing adapted cadlag process starting from zero. Finally let \(v=(v(t))_{t\ge 0}\) be a V-valued progressively measurable process and for \(i=1,\ldots ,m\) let \(v^{*}_i=(v^{*}_i)_{t\ge 0}\) be \(V_i^*\)-valued processes such that \(\langle \varphi ,v_i^{*}\rangle \) are progressively measurable for any \(\varphi \in V\). Notice that v is also progressively measurable as a process with values in \({\bar{V}}_i\), the closure in \(V_i\)-norm of the linear hull of \( \{v(t):t\ge 0, \omega \in \varOmega \}. \)
Let there be \(p_i \in [1,\infty )\) and \(q_i=p_i/(p_i-1)\in (1,\infty ]\), where, as usual, \(1/0:=\infty \). Assume that for each \(i=1,2,\ldots ,m\) and \(T>0\)
for some progressively measurable process \(\eta _i\) such that \(\Vert v_i^{*}\Vert _{V_i^{*}}\le \eta _i\) for \(dA\times {\mathbb {P}}\) almost everywhere, where for \(q_i=\infty \) the second expression means
the essential supremum (with respect to dA) of \(\eta _i\) over [0, T].
The following theorem is the main result of this paper.
Theorem 2.1
Let \(\tau \) be a stopping time. Suppose that for all \(\varphi \in V\) and for \(dA\times {\mathbb {P}}\) almost all \((\omega , t)\) such that \(t\in (0,\tau (\omega ))\) we have
Then there is \({\tilde{\varOmega }} \subset \varOmega \) with \({\mathbb {P}}({\tilde{\varOmega }}) = 1\) and an H-valued cadlag process \({{\tilde{v}}}\) such that the following statements hold.
-
(i)
For \(dA\times {\mathbb {P}}\) almost all \((t, \omega )\) satisfying \(t \in (0,\tau (\omega ))\) we have \({\tilde{v}}=v\).
-
(ii)
For all \(\omega \in {\tilde{\varOmega }}\) and \(t\in [0,\tau (\omega ))\) we have
$$\begin{aligned} ({\tilde{v}}(t),\varphi ) = \sum _{i=1}^m \int _{(0,t]}\langle v^{*}_i(s),\varphi \rangle \,dA(s) + h(t)\varphi \quad \text {for all }\, \varphi \in V. \end{aligned}$$(2.3) -
(iii)
For all \(\omega \in {\tilde{\varOmega }}\) and \(t\in [0, \tau (\omega ))\)
$$\begin{aligned} \begin{aligned} |{\tilde{v}}(t)|^2&= |h(0)|^2 + 2\sum _{i=1}^m \int _{(0,t]}\langle v^{*}_i(s),v(s)\rangle \,dA(s) + 2\int _{(0,t]}( {\tilde{v}}(s-)\,dh(s)) \\&\quad - \int _{(0,t]} |v^{*}(s)|^2 \varDelta A(s) dA(s) + [h]_t, \end{aligned} \end{aligned}$$(2.4)where \(v^{*}(t):=\sum _{i=1}^m v^{*}_i(t)\in H\) for \(\varDelta A(t) > 0\).
Consider now a situation where the assumptions on h and A are as above but \(m=1\) and regarding v and \(v^{*}:=v^{*}_1\) we know that \(\Vert v(t)\Vert \), \(\Vert v^{*}(t)\Vert _{V^*}\) and \(\Vert v(t)\Vert \Vert v^{*}(t)\Vert _{V^*}\) are almost surely locally integrable with respect to dA(t). Let
Then \(\Vert \bar{v}^{*}\Vert _{V^*} \le 1\) and so v, \(\bar{v}^{*}\) and \({\bar{A}}\) satisfy the conditions on v, \(v^{*}\) and A, respectively, with \(p_1 = 1\) and \(q_1 = \infty \). If (2.2) holds for all \(\varphi \in V\) and for \(dA\times {\mathbb {P}}\) almost all \((\omega , t)\) such that \(t\in (0,\tau (\omega ))\) then
Applying Theorem 2.1 then means that we have all of its conclusions with \(\bar{v}^{*}\) and \({\bar{A}}\) 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 \(v^{*}=\sum _iv^{*}_i\) as a process with values in \(V^{*}\). However, taking into account that for any \(w^{*}\in V^{*}\)
(see for example Gajewski, Gröger and Zacharias [4, 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), \((Z,\varSigma )\) is a measurable space and q(ds, dz) a stochastic martingale measure on \([0,\infty )\times Z\). See, for example, Gyöngy and Krylov [5] for detailed definition. We take \({\mathscr {D}}\) to be a bounded Lipschitz domain in \(\mathbb {R}^d\).
It is natural to assume that a solution u should be such that \(\Vert u\Vert _{W^{1}_{p_1}({\mathscr {D}})}^{p_1}\) and \(\Vert u\Vert _{L_{p_2}({\mathscr {D}})}^{p_2}\) are almost surely locally integrable. To apply the result in Gyöngy and Krylov [6] one could try to take \(V := W^{1}_{p_1}({\mathscr {D}}) \cap L_{p_2}({\mathscr {D}})\) with the norm \(\Vert \cdot \Vert _V = \Vert \cdot \Vert _{W^{1}_{p_1}({\mathscr {D}})} +\Vert \cdot \Vert _{L_{p_2}({\mathscr {D}})}\). The dual of V can be identified with the linear space
equipped with the norm
One would then need to show that \(\Vert u\Vert _V \, \Vert \nabla (|\nabla u|^{p_1-2}\nabla u)+|u|^{p_2-2} u\Vert _{V^*}\) 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].
3 Preliminaries
Lemma 3.1
For \(r\in [0,\infty )\) let \(\beta (r) := \inf \{ t\ge 0: A(t) \ge r\}\) and let x(t) be a real valued process that is locally integrable with respect to dA for all \(\omega \in \varOmega \). Then
-
(i)
\(\beta (r)\) is a stopping time (not necessarily finite) for every \(r\in [0,\infty )\),
-
(ii)
$$\begin{aligned} \begin{aligned} \int _{(0,t]} x(s)\,dA(s)&= \int _{(0,A(t)]} x(\beta (r))\, dr, \\ \int _{(0,t)} x(s)\,dA(s)&= \int _{(0,A(t-)]} x(\beta (r)) \,dr \end{aligned} \end{aligned}$$
for every \(t\in [0,\infty )\),
-
(iii)
$$\begin{aligned} A(\beta (t)-) - A(\beta (s)) \le t-s \end{aligned}$$
for every \(s,t \in [0,\infty )\).
-
(iv)
If \(0 = r^n_0< r^n_1< \cdots< r^n_k < \cdots \) is an increasing sequence of decompositions of \([0,\infty )\) such that \(\sup _{k}|r^n_{k+1} - r^n_k| \rightarrow 0\) as \(n\rightarrow \infty \) then for every \(t\ge 0\) and \(\omega \in \varOmega \)
$$\begin{aligned} \sum _k\left| X(\tau ^n_{k+1}\wedge t) - X(\tau ^n_k \wedge t)\right| ^2 \rightarrow \sum _{s\le t} \left| X(s)\right| ^2 \left| \varDelta A(s)\right| ^2 \end{aligned}$$as \(n\rightarrow \infty \), where \(X(t):= \int _{(0,t]}x(s)dA(s)\) and \(\tau ^n_k := \beta (r^n_k)\).
This Lemma is proved in Gyöngy and Krylov [6, Lemma 1].
Let \(\kappa _n^{(j)}\) for \(j=1,2\) and integers \(n\ge 1\) denote the functions defined by
The following lemma is known and the authors believe is due to Doob.
Lemma 3.2
For integers \(i\ge 1\) let \((X_i,\Vert \cdot \Vert _{X_i})\) be Banach spaces, and let \(p_i \in [1,\infty )\). Let \(x_i: \mathbb {R}\times \varOmega \rightarrow X_i\) be \( {\mathscr {B}}(\mathbb {R}) \otimes {\mathcal {F}}\) Bochner-measurable such that \(x_i(r)= 0\) for \(r\notin [0,1]\) and
Then there exists a subsequence \(n_k \rightarrow \infty \) such that for dt-almost all \(t\in [0,1]\)
for \(j=1,2\) and all \(i\ge 1\).
Proof
Let \((c_i)_{i=1}^{\infty }\) be a sequence of positive numbers such that
By change of variables and changing the order of integration
Note that by the shift invariance of the Lebesgue measure
for \(s\in (0,1)\), \(i\ge 1\), and
Therefore by Lebesgue’s theorem on dominated convergence
Hence for a subsequence \(n_k\rightarrow \infty \)
for almost all \(t\in [0,1]\), and the statement of the lemma follows. \(\square \)
The following lemma is proved in Gyöngy and Krylov [6, Lemma 3].
Lemma 3.3
Let \((\xi _n)_{n\in {\mathbb {N}}}\) be a sequence of H-valued predictable processes. Suppose
and
Then for any \(\varepsilon > 0\)
as \(n\rightarrow \infty \).
4 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 \(\tau \) 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 \(\tau \wedge n\) and letting \(n\rightarrow \infty \). In fact using a non-random time change we may assume that \(\tau \le 1\).
-
2.
Recall the processes \(\eta _i\) from assumption (2.1), and set
$$\begin{aligned} Q_i(t)=\left( \int _{(0,t]} \eta _i^{q_i}(s)\,dA(s)\right) ^{1/q_i}\quad t\ge 0 \end{aligned}$$when \(q_i<\infty \), and for \(q_i=\infty \) let \(Q_i=(Q_i(t))_{t\ge 0}\) denote a nondecreasing cadlag adapted process such that almost surely
$$\begin{aligned} dA\text {-ess sup}_{s\le t}\eta _i(s)\le Q_i(t)\quad \text { for all }t\ge 0. \end{aligned}$$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 \(dA-\text {ess sup}_{s\le t}\eta _i(s)\), i.e.,
$$\begin{aligned} \lim _{n\rightarrow \infty }dA\text {-ess sup}_{s\le t+1/n}\eta _i(s). \end{aligned}$$Let \((e^j)_{j\in {\mathbb {N}}} \subset V\) be an orthonormal basis in H and define
$$\begin{aligned} \begin{aligned}&r(t):=|h(0)|+A(t) + \sum _{i=1}^m\left( \int _{(0,t]} \Vert v(s)\Vert _{V_i}^{p_i} dA(s)\right) ^{1/p_i} \\&\quad + \sum _{i=1}^mQ_i(t) + \sum _{i=1}^m\sum _{k\in {\mathbb {N}}} 2^{-c_k} \left( \int _{(0,t]}\Vert w_k(s)\Vert ^{p_i}_{V_i} dA(s)\right) ^{1/p_i}, \end{aligned} \end{aligned}$$(4.1)with \(c_k:=\max _{1\le i\le m}\sum _{j\le k}|e_j|^2_{V_i}\) and \(w_k := \varPi ^k h\), where \(\varPi ^k\) denotes the orthogonal projection of H onto its subspace spanned by \((e_i)_{i=1}^k\). We may and will assume, without loss of generality, that r and \(\langle h \rangle \) are bounded. Indeed, imagine we have proved Theorem 2.1 under this assumption. Consider
$$\begin{aligned} \tau _n := \inf \left\{ t \ge 0: r(t) \ge n\right\} . \end{aligned}$$Then \(\tau _n\) is a stopping time and \(\tau _n\rightarrow \infty \) for \(n\rightarrow \infty \). Since \(\langle h\rangle \) is a predictable process starting from 0, there is an increasing sequence of stopping times \(\sigma _n\) such that \(\sigma _n\rightarrow \infty \) and \(\langle h\rangle _t\le n\) for \(t\in [0,\sigma _n]\). Therefore \(\tau _n \wedge \sigma _n \wedge \tau \rightarrow \tau \) as \(n\rightarrow \infty \), and for fixed n we get \(r(t)\le n\) for \(t\in (0,\tau _n\wedge \sigma _n)\) and \(\langle h \rangle _t \le n\) for \(t\in [0,\tau _n\wedge \sigma _n]\). Thus we get (2.3) and (2.4) for the stopping time \(\tau _n \wedge \sigma _n \wedge \tau \) in place of \(\tau \). Letting \(n\rightarrow \infty \) provides (2.3) and (2.4) for \(\tau \). Thus we may assume that there is \(n\ge 1\) such that \(r(t)\le n\) for \(t\in (0,\tau )\) and \(\langle h\rangle _t\le n\) for \(t\in [0,\tau ]\). Moreover, by taking \(h\mathbf{1}_{|h(0)|<n}\), \(v\mathbf{1}_{|h(0)|<n}\) and \(A\mathbf{1}_{|h(0)|<n}\) in place of h, v and A, respectively, and then taking \(n\rightarrow \infty \), we may assume that \(r(t)\le n\) for \(t\in [0,\tau )\) and \(\langle h\rangle _t\le n\) for \(t\in [0,\tau ]\). Furthermore, we can define \(A(t) := A(\tau -)\), \(h(t)=h(\tau )\), \(v(t) = 0\) and \(v_i^{*}(t) = 0\) for \(t\ge \tau \). Then \(r(t) \le n\) and \(\langle h \rangle _t \le n\) for \(t\in [0,\infty )\).
-
3.
Finally, we can assume that \(r(t) \le 1\) for \(t\in [0,\tau )\) and \(\langle h \rangle _t\le 1\) for \(t\in [0,\tau ]\). 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 \(r_n(t) \le n^{-1}r(t) \le 1\). 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 \(\tau \le 1\), \(r(t) \le 1\) and \(\langle h \rangle _t \le 1\) for \(t\in [0,\infty )\). 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 Eq. (2.2) holds. But (2.2) only holds for every \(\varphi \in V\) and \(dA\times {\mathbb {P}}\) almost all \((t,\omega ) \in \rrbracket 0,\tau \llbracket \), and thus it is not immediately clear how to choose an appropriate piecewise constant approximation to v. Here and later on for stopping times \(\tau \) the notation \(\rrbracket 0,\tau \llbracket \) means the stochastic interval \(\{(t,\omega ):t\in (0,\tau (\omega )),\omega \in \varOmega \}\).
Proposition 4.1
There is a nested sequence of random partitions of \([0,\infty ]\),
with stopping times \(\tau ^n_j\), \(j=1,\ldots ,N(n)+1\), such that for every \(\omega \in \varOmega \) either \(\tau ^n_j(\omega ) < \tau (\omega )\) or \(\tau ^n_j(\omega ) = \infty \), and such that the following statements hold.
-
(1)
There is \(\varOmega ' \subset \varOmega \) such that \({\mathbb {P}}(\varOmega ') = 1\) and with
$$\begin{aligned} I(\omega ):=\left\{ \tau ^n_j(\omega ):n\in {\mathbb {N}}, j=1,\ldots ,N(n)\right\} \cap (0,\infty ) \end{aligned}$$we have (2.2) satisfied for every \(\omega \in \varOmega '\), \(t\in I(\omega )\) and \(\varphi \in V\). Moreover, if \(\varDelta A(t)>0\) for some \(t>0\) and \(\omega \in \varOmega '\), then \(t\in I(\omega )\). Furthermore, if \(0\le s<t\) and \((s,t]\cap I(\omega )=\emptyset \), then \(A(s)=A(t)\).
-
(2)
For \(l\in \{1,2\}\), \(i=1,\ldots ,m\) and for all \(k\ge 1\)
$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty } {\mathbb {E}}\int _{(0,\infty )} \Vert v(s) - v^{(l)}_n(s)\Vert _{V_i}^{p_i}\,dA(s) = 0,\\ \lim _{n\rightarrow \infty } {\mathbb {E}}\int _{(0,\infty )} \Vert w_k(s) - w^{(l)}_{kn}(s)\Vert ^{p_i}_{V_i}\,dA(s) = 0, \end{aligned} \end{aligned}$$(4.2)where
$$\begin{aligned} v_n^{(1)}(t):=\sum _{j=1}^{N(n)}v(\tau _j^n)\mathbf{1}_{[\tau _j^n,\tau _{j+1}^n)}(t), \quad v_n^{(2)}(t):=\sum _{j=0}^{N(n)}v(\tau _{j+1}^n)\mathbf{1}_{(\tau _j^n,\tau _{j+1}^n]}(t), \end{aligned}$$and \(w^{(l)}_{kn}\) is defined analogously from \(w_k=\varPi ^kh\).
Proof
Since V is separable there is \(\{\varphi _i\}_{i\in {\mathbb {N}}}\subset V\) which is dense in V. For each \(\varphi _i\) there is an exceptional set \(D_i \in [0,\infty )\times \varOmega \) such that (2.2) holds for \((t,\omega )\in \rrbracket 0,\tau \llbracket \setminus D_i\) and \((dA\times {\mathbb {P}})(D_i)=0\). Let \(D = \bigcup _{i\in {\mathbb {N}}} D_i\). Then \((dA \times {\mathbb {P}})(D)=0\) and (2.2) holds for all \(\varphi \in V\) and all \((t,\omega )\in \rrbracket 0,\tau \llbracket \setminus D\). Now using Lemma 3.1 and the Fubini theorem
From this we see that for dr almost all \(r\in (0,\infty )\) there is \(\varOmega (r) \subset \varOmega \) with \({\mathbb {P}}(\varOmega (r)) = 1\) such that for any \(\omega \in \varOmega (r)\) either \(r > A(\tau (\omega ),\omega )\) or \(\beta (r,\omega ) < \tau (\omega )\) and for \(t=\beta (r)\) and for all \(\varphi \in V\)
By virtue of Lemma 3.2 there is a nested sequence of decompositions of [0, 1],
such that \(\lim _{n\rightarrow \infty } \max _i |r^n_{j+1} - r^n_j| = 0\), and
for all \(i=1,\ldots ,m\), all \(k\in {\mathbb {N}}\) and \(l=1,2\), where \(\kappa _n^{(1)}(r) = r^n_j\) if \(r\in [r^n_j,r^n_{j+1})\) and \(\kappa _n^{(2)}(r) = r^n_{j+1}\) if \(r\in (r^n_j,r^n_{j+1}]\).
Now let \(\varOmega ' := \bigcap _{n\in {\mathbb {N}}} \left( \varOmega (r^n_0)\cap \ldots \cap \varOmega (r^n_{N(n)+1})\right) \), \(\tau ^n_j := \beta (r^n_j)\), and
Then \({\mathbb {P}}(\varOmega ') = 1\) and
is a nested sequence of random partitions of (0, 1) by stopping times \(\tau ^n_j\) such that statement (1) holds. To prove (2) we notice that, just like in [6], for \(r\in (r_j^n,r^n_{j+1}]\)
Thus with appropriate sets \(S_n\in {\mathcal {B}}(\mathbb {R})\times {\mathcal {F}}\)
Hence due to (4.5) and Lemma 3.1 we obtain the first equality in (4.2) for \(l=2\), \(i=1,\ldots ,m\) and for all \(k\in {\mathbb {N}}\). The rest of (4.2) is obtained similarly. \(\square \)
Proposition 4.2
For every \(n\in {\mathbb {N}}\), every \(\omega \in \varOmega '\) and every \(\tau ^n_j(\omega ) \in I(\omega )\)
where \({\bar{v}}_n(s)=0\) for \(s\in [0,\tau ^n_1]\) and \({\bar{v}}_n(s)=v(\tau ^n_j)\) for \(s\in (\tau ^n_j,\tau ^n_{j+1}]\) for \(j=1,\ldots ,N(n)\). Moreover,
Proof
Let \(\omega \in \varOmega '\) and \(t,t' \in I(\omega )\) and \(t' \ge t\). Clearly,
which by statement (1) of Proposition 4.1 gives
Hence by the identity
we have
By (1) in Proposition 4.1 again
which by the identity \(2(h(t),v(t)) = -|v(t)-h(t)|^2 + |v(t)|^2 + |h(t)|^2\) gives
Summing up for \(k=1,\ldots ,j-1\) equations (4.8) with \(t'=\tau ^n_{k+1}\), \(t=\tau ^n_{k}\), and adding to it Eq. (4.9) with \(t=\tau ^n_1\), we obtain (4.6). Form (4.6) we have
Clearly
and by Doob’s inequality and \(\langle h\rangle \le 1\),
Since h is a martingale,
By Hölder’s inequality and \(\sum _iQ_i\le 1\) we have
which by virtue of (4.2) is finite. Hence, taking also into account \({\mathbb {E}}|h(0)|^2\le 1\) we have
which immediately yields (4.7), provided
To show (4.10) note that due to (4.9), for every \(n\in {\mathbb {N}}\) and \(j=1,\ldots ,N(n)+1\), we get
since \(\tau \le 1\) and \(r(t) \le 1\) for all \(t\in [0,\infty )\). For \(i=1,\ldots ,m\)
where \(r^n_k\) are given by (4.4), \(d_n := \min _{k=1,\ldots ,N(n)}|r^n_{k+1}-r^n_k|>0\) and
which due to (4.5) is finite. Hence by virtue of (4.11) we have (4.10), which completes the proof of (4.7). \(\square \)
We see that due to (4.7) there is \(\varOmega '' \subset \varOmega '\) such that \({\mathbb {P}}(\varOmega '') = 1\) and
Moreover, since h is cadlag, for all \(\omega \in \varOmega ''\) we have
Define
for \(t\ge 0\), where the integrals are defined as weak* integrals. Recall that \(v^{*}=\sum _iv^{*}\) is a \(V^{*}\)-valued such that \(\langle v^{*}(t),\varphi \rangle \) is a progressively measurable process for every \(\varphi \in V\), and
Therefore \(z^{(1)}\) and \(z^{(2)}\) are well-defined \(V^{*}\)-valued processes such that \(\langle z^{(1)}, \varphi \rangle \) and \(\langle z^{(2)}, \varphi \rangle \) are left-continuous and right-continuous adapted processes, respectively.
In what follows we use the notation \(\varDelta ^w f(t):=f(t)-\text {w-lim}_{s\nearrow t}f(s)\) for H-valued functions f, when the weak limit from the left exists at t.
Proposition 4.3
Let \(z^{(l)}\), \(l\in \{1,2\}\) be given by (4.13).
-
1.
If \(\omega \in \varOmega ''\) and \(t\in (0,\infty )\) then \(z^{(l)}(t)\in H\) for \(l\in \{1,2\}\). Moreover
$$\begin{aligned} \sup _{t\in (0,\infty )} |z^{(l)}(t)| < \infty \,\,\,\, \forall \omega \in \varOmega '',\,\, l \in \{1,2\}. \end{aligned}$$ -
2.
Let \({\tilde{v}}\) be given by
$$\begin{aligned} {\tilde{v}}(t) := \chi _{\varOmega ''} z^{(2)}(t) + h(t). \end{aligned}$$Then \({\tilde{v}}\) is a H-valued adapted and weakly cadlag process such that \(v(t) = {\tilde{v}}(t)\) for all \(t\in I(\omega )\) and \(\omega \in \varOmega ''\). Moreover
$$\begin{aligned} \sup _{t\in (0,\infty )} |{\tilde{v}}(t)| < \infty \,\,\,\, \forall \omega \in \varOmega ''. \end{aligned}$$ -
3.
If \(\omega \in \varOmega ''\) then for all \(t\in (0,\tau (\omega ))\)
$$\begin{aligned} \varDelta ^w ({\tilde{v}}-h)(t) = (\varDelta A)(t)\sum _{i=1}^m v_i^{*}(t). \end{aligned}$$(4.14)
Proof
Fix \(\omega \in \varOmega ''\). If \(t\in I(\omega )\) then for all \(\varphi \in V\)
and hence \(z^{(2)}(t) \in H\). Consider now the situation when \(t\in (0,\tau (\omega )] \setminus I(\omega )\). Let \(\bar{I}^l(\omega )\) denote the left-closure of the set \(I(\omega )\). If \(t\in \bar{I}^l(\omega )\setminus I(\omega )\) then \(\varDelta A(t)=0\) by Proposition 4.1, and there is a sequence \((t_n)_{n\in {\mathbb {N}}} \subset I(\omega )\) such that \(t_n \nearrow t\). Moreover, due to (4.12) there is a subsequence \(t_{n'}\nearrow t\) such that \(v(t_{n'}) - h(t_{n'})\) converges weakly in H to some \(\xi \in H\). Hence for all \(\varphi \in V\)
which implies \(z^{(2)}(t)=\xi \in H\). If \(t\in (0,\infty )\setminus \bar{I}^l(\omega )\), then there is \(s \in \{0\}\cup \bar{I}^l(\omega )\) such that \(s<t\) and \((s,t] \cap I(\omega ) = \emptyset \). So \(\int _{(s,t]} v_i^{*}(s)\,dA(s) = 0\) and \(z^{(2)}(t) = z^{(2)}(s) \in H\). Of course if \(t=0\) then \(z^{(2)}(t) = 0 \in H\). Finally, due to (4.12),
Now we consider \(z^{(1)}(t)\) for \(t\in (0,\infty )\). Take \((t_n)_{n\in {\mathbb {N}}}\) such that \(t_n < t\) and \(t_n \nearrow t\) as \(n\rightarrow \infty \). From (4.15) we know that \(\sup _{n\in {\mathbb {N}}} |z^{(2)}(t_n)|^2 < \infty \) and so there is a subsequence \(t_{n'}\nearrow t\) such that \(z^{(2)}(t_n)\) converges weakly in H to some \(\xi \in H\). Thus for any \(\varphi \in V\)
Hence \(z^{(1)}(t) = \xi \in H\), and due to (4.15)
By construction \({\tilde{v}}\) is weakly cadlag. Due to (4.15) for \(\omega \in \varOmega ''\)
We note that for any \(\varphi \in V\) the real valued random variable
is \({\mathcal {F}}_t\)-measurable. Hence, since H is separable, \({\tilde{v}}(t)\) is \({\mathcal {F}}_t\)-measurable by the Pettis theorem. Finally notice that
for all \(\varphi \in V\) and \(\omega \in \varOmega ''\). Hence on \(\varOmega ''\)
\(\square \)
Let
Then from (4.6) it follows that for every \(\omega \in \varOmega ''\) and \(t:=\tau ^n_j(\omega ) \in I(\omega )\)
where
In order to let \(n\rightarrow \infty \) in the above equation we first rewrite it as
by noticing that
To perform the limit procedure we use the following two propositions.
Proposition 4.4
There is \({{\tilde{\varOmega }}} \subset \varOmega ''\) with \({\mathbb {P}}({{\tilde{\varOmega }}}) = 1\) such that for a subsequence \(n'\) and for every \(\omega \in {{\tilde{\varOmega }}}\)
as \(n'\rightarrow \infty \). Moreover,
Proof
Set \(\xi (t) := {\tilde{v}}(t-) - h(t-)\) and \(\xi _n(t):={\tilde{v}}_n(t)-h_n(t)\). By Lemma 3.3, taking into account that by Proposition 4.3 on \(\varOmega ''\)
and that V is dense in H, we have
if we show that almost surely
To this end set
Then for all \(\omega \in \varOmega ''\), \(t>0\) and \(\varphi \in V\)
with \(r^n_j\) given by (4.4). Consequently, taking also into account (4.2) of Proposition 4.1 we have \(\varOmega '''\subset \varOmega ''\) and a subsequence \(n'\) such that the first three limits are zero for \(\omega \in \varOmega '''\). Taking the limit along the subsequence \(n'\) in (4.17) we see that \(K_{n'}(t)\) converges for \(\omega \in \varOmega '''\) and \(t\in I(\omega )\) 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
with
For \(j\ne 0\) we split \(J^{(2)}_j=J^{(21)}_j-J^{(22)}_j\) with
and notice that
Using \(\varPi _k\), the orthogonal projection of H onto the space spanned by \((e_j)_{j=1}^k\subset V\), we have
with
Notice that
and
Similarly, taking into account \({\tilde{v}}(0)=h(0)\), for \(J^{(2)}_0\) we have
where
and
with
Thus from (4.19) we get
with
for every \(n,k\in {\mathbb {N}}\). As \(n\rightarrow \infty \) we see that
where we use the notation \(v^{*}(s)=\sum _iv^{*}_i(s)\). By Hölder’s inequality, taking into account \(r(t)\le 1\), we have
and similarly,
for all integers \(k\ge 1\) and \(i=1,2,\ldots ,m\). Thus
for every \(k\in N\), where
Note that by Fatou’s lemma and the martingale property of h
Note also that for each \(\omega \in \varOmega \) and \(t\in (0,\infty )\) we have \(\xi _k \ge \xi _{k+1}\) and \(\xi _k \ge 0\). Thus there exists a set \(\varOmega ''''\subset \varOmega \) with \(P(\varOmega '''')=1\) such that for every \(t\in [0,\infty )\) and \(\omega \in \varOmega ''''\) we have \(\xi _k(t) \rightarrow 0\). Letting here \(k\rightarrow \infty \) in (4.20) we obtain
which together with (4.18) gives
for \(\omega \in {\tilde{\varOmega }}:=\varOmega '''\cap \varOmega ''''\) and \(t\in I(\omega )\). \(\square \)
Proposition 4.5
For \(\omega \in {{\tilde{\varOmega }}}\)
for \(t\in [0,\tau (\omega ))\).
Proof
Let \(\omega \in {\tilde{\varOmega }}\) be fixed and let \(t\in [0,\tau (\omega ))\). To ease notation we use \(n\rightarrow \infty \) in place of the subsequence \(n'\rightarrow \infty \) defined in the previous proposition. If \(t\in I(\omega )\), then by virtue of the previous proposition taking \(n\rightarrow \infty \) in (4.17) we obtain
Hence using the Itô formula for Hilbert space valued processes
we get (4.21) for \(t\in I(\omega )\). If \(t\in \bar{I}^l(\omega )\setminus I(\omega )\), then for sufficiently large n there is \(j=j(n)\) such that \(t_n:=\tau ^{n}_j(\omega )\in I(\omega )\) and \(t_n\nearrow t\) for \(n\rightarrow \infty \). Using the algebraic relationship
with \(s:=t_n\), \(r:=t_l\), and since (4.21) holds for every \(t\in I(\omega )\), we get
for \(n>l\). Moreover
Hence by (4.22)
Since h is cadlag we have
By the previous proposition we get
and
via \(r(t)\le 1\) and Hölder’s inequality. Thus
and so the sequence \(({\tilde{v}}(t_n))_{n\in {\mathbb {N}}}\) converges strongly to some \(\xi \) in H. Moreover since \({\tilde{v}}\) is weakly cadlag and \(t_n \nearrow t\), we conclude that \(\xi ={\tilde{v}}(t-)\). Hence using (4.21) with \(t_n\) in place of t, and letting \(n\rightarrow \infty \) we obtain
for \(t\in I(\omega )\setminus {\bar{I}}^{l}(\omega )\), and so for this t we get also (4.21) by taking into account that \(\varDelta A(t)=0\). If \(t \in (0,\tau (\omega )) \setminus \bar{I}^l(\omega )\), then there is \(t'\in \{0\} \cup \bar{I}^l(\omega )\) such that \(t'<t\) and \((t',t] \cap I(\omega ) = \emptyset \). Thus \(dA(s) = 0\) for \(s\in (t',t]\), and so \({\tilde{v}}(s)-{\tilde{v}}(t') = h(s)-h(t')\). Hence applying (4.21) with \(t:=t'\), and the formula
together with the Itô formula for Hilbert space valued martingales,
we obtain (4.21) for the t under consideration. \(\square \)
Now we can finish the proof of Theorem 2.1 by noting that by the above proposition \(|{\tilde{v}}(t)|^2\) is a cadlag process, and since by Proposition 4.3 the process \({\tilde{v}}\) is H-valued and weakly cadlag, it follows by identity (4.23) that \({\tilde{v}}\) is an H-valued cadlag process.
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The authors are sincerely grateful to the anonymous referees. Their corrections and valuable suggestions helped improve the presentation of the paper.
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Gyöngy, I., Šiška, D. Itô formula for processes taking values in intersection of finitely many Banach spaces. Stoch PDE: Anal Comp 5, 428–455 (2017). https://doi.org/10.1007/s40072-017-0093-6
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DOI: https://doi.org/10.1007/s40072-017-0093-6