11institutetext:  István Gyöngy 22institutetext: School of Mathematics,
University of Edinburgh
22email<EMAIL_ADDRESS>33institutetext: David Šiška 44institutetext:
School of Mathematics, University of Edinburgh
44email<EMAIL_ADDRESS>
# Itô Formula for Processes Taking Values in Intersection of Finitely Many
Banach Spaces††thanks: This an electronic reprint of the article published in
Stochastics and Partial Differential Equations: Analysis and Computations
available at http://dx.doi.org/10.1007/s40072-017-0093-6. It may differ from
the published version in typographical detail.

István Gyöngy  David Šiška

(Submitted for publication: 6th September 2016, First Online: 17th March 2017.
ⓒ The Author(s) 2017. This article is an open access publication.)

###### Abstract

Motivated by applications to SPDEs we extend the Itô formula for the square of
the norm of a semimartingale $y(t)$ from gyongy:krylov:on:stochastic:II  to
the case

$\sum_{i=1}^{m}\int_{(0,t]}v_{i}^{\ast}(s)\,dA(s)+h(t)=:y(t)\in
V\quad\text{$dA\times\mathbb{P}$-a.e.},$

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 gyongy:krylov:on:stochastic:II  to the semimartingale above.

###### Keywords:

Stochastic evolution equations, Stochastic partial differential equations,
Itô’s formula, Energy equality

##  1 Introduction

Itô formula for the square of the norm is an essential tool in the study of
stochastic evolution equations of the type

$dv(t)=\mathbb{A}(t,v(t))\,dt+\sum_{k}\mathbb{B}_{k}(t,v(t))\,dW^{k}(t),$
(1.1)

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 $V^{\prime}$ and a Hilbert space $H$ respectively, such that
$V\hookrightarrow H\hookrightarrow V^{\prime}$ with continuous and dense
embeddings. We assume there is a constant $K$ such that $(v,h)\leq
K\|v\|_{V}\|h\|_{V^{\prime}}$ for all $v\in V$ and $h\in H$. This means that
for the linear mapping $\Psi:H\to H^{\ast}$, which identifies $H$ with its
dual $H^{\ast}$ via the inner product in $H$, we have
$\|\Psi(h)\|_{V^{\ast}}\leq K\|h\|_{V^{\prime}}$. Therefore, since $H$ is
dense in $V^{\prime}$, $\Psi$ can be extended to a continuous mapping from
$V^{\prime}$ into $V^{\ast}$, the dual of $V$. It is assumed that this
extension is one-to-one from $V^{\prime}$ into $V^{\ast}$. Thus an initial
value problem for equation (1.1) can be viewed as

$v(t)=\int_{0}^{t}v^{\ast}(s)\,ds+h(t)=:y(t)$ (1.2)

with the $V^{\ast}$-valued process $v^{\ast}(t):=\mathbb{A}(t,v(t))$ and
$H\equiv H^{\ast}$-valued process

$h(t):=h_{0}+\sum_{k}\int_{0}^{t}\mathbb{B}^{k}(s,v(s))\,dW^{k}(s),$

where $h_{0}$ is a given initial value and the equality (1.2) in $V^{\ast}$ 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

$v(t)=h_{0}+\int_{0}^{t}v^{\ast}(s)\,ds,\quad dt\text{-a.e.},$

it is well-known that when $v\in L_{p}([0,T],V)$, $v^{\ast}\in
L_{q}([0,T],V^{\ast})$ 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”

$|u(t)|_{H}^{2}=|h_{0}|_{H}^{2}+2\int_{0}^{t}\langle
v^{\ast}(s),v(s)\rangle\,ds$

holds for all $t\in[0,T]$, where $\langle\cdot,\cdot\rangle$ denotes the
duality pairing of $V^{\ast}$ and $V$. This formula is used in proofs of
existence and uniqueness theorems for PDEs, see e.g., Evans  and Lions . 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
pardoux:thesis , and was used to obtain existence and uniqueness theorems for
SPDEs. The proof of it in pardoux:thesis  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 krylov:rozovskii:stochastic , and then this stochastic
energy equality was generalised in Gyöngy and Krylov
gyongy:krylov:on:stochastic:II  to $V^{\ast}$-valued semimartingales $y$ of
the form

$y(t)=\int_{(0,t]}v^{\ast}(s)\,dA(s)+h(t),$ (1.3)

where $A$ is an adapted nondecreasing cadlag process and $h$ is an $H$-valued
cadlag martingale. This generalisation is used in Gyöngy gyongy  to extend the
theory of SPDEs developed in pardoux:thesis  and krylov:rozovskii:stochastic
to SPDEs driven by random orthogonal measures and Lévy martingales, written in
the form

$dv(t)=\mathbb{A}(t,v(t))\,dA(t)+\mathbb{B}(t,v(t))\,dM(t)$ (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 $\mathbb{A}$ 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,

$\mathbb{A}_{i}(t,\cdot):V_{i}\to V_{i}^{\prime}\quad i=1,2,\ldots,m$

for some Banach spaces $V_{i}$ and $V^{\prime}_{i}$, such that with a constant
$R$ and a process $g$, locally integrable with respect to $dA$, one has for
all $t$

$\|\mathbb{A}_{i}(t,w)\|_{V^{\prime}_{i}}\leq|g_{t}|^{1/q_{i}}+R\|w\|_{V_{i}}^{p_{i}-1}$

for all $w\in V$, $q_{i}=p_{i}/(p_{i}-1)$ with (possibly) different exponents
$p_{i}\geq 1$, which for $p_{i}=1$ means that
$\|\mathbb{A}_{i}(t,w)\|_{V^{\prime}_{i}}$ 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 pardoux:thesis , and a related stochastic energy
equality was also presented there. Our main result, Theorem 2.1 generalises
the results on stochastic energy equalities from pardoux:thesis  and
gyongy:krylov:on:stochastic:II . We prove it by adapting the method of the
proof of the main theorem in gyongy:krylov:on:stochastic:II .

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
$\|\cdot\|:=\sum_{i=1}^{m}\|\cdot\|_{V_{i}}$ is continuously and densely
embedded in $H$. The process $v^{\ast}$ in (1.3) is of the form
$v^{\ast}=\sum_{i=1}^{m}v_{i}^{\ast}$, where $v_{i}^{\ast}$ are
$V_{i}^{\ast}$-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 $\|y\|_{V_{i}}^{p_{i}}$ and
$\|v_{i}^{\ast}\|_{V_{i}^{\ast}}^{q_{i}}$ are almost surely locally integrable
with respect to $dA$ for some conjugate exponents $p_{i},q_{i}$. See Section 2
for precise formulation of the main theorem. To apply the result of
gyongy:krylov:on:stochastic:II  to $y$ given by (1.3), one needs the local
integrability (with respect to $dA$) of

$\|y\|_{V}\|v^{\ast}\|_{V^{\ast}}=\left(\|y\|_{V_{1}}+\cdots+\|y\|_{V_{m}}\right)\|v_{1}^{\ast}+\cdots+v_{m}^{\ast}\|_{V^{\ast}},$

which, in general, is not satisfied under our assumptions. See Remark 1 and
Example 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 pardoux:thesis , Krylov
krylov:ito_fla , K2010 , K , Da Prato, Jentzen and Röckner
daprato:jentzen:rockner , as well as Dareiotis and Gyöngy dareiotis:gyongy .
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 krylov:rozovskii:stochastic , Prévôt and Röckner PR  and
Rozovskii R .

##  2 Main Results

For $i=1,\ldots,m$ let $(V_{i},\|\cdot\|_{V_{i}})$ be real Banach spaces with
duals $(V_{i}^{*},\|\cdot\|_{V_{i}^{*}})$. Let $V$ denote the vector space
$V_{1}\cap\cdots\cap V_{m}$ with the norm
$\|\cdot\|:=\|\cdot\|_{V_{1}}+\cdots+\|\cdot\|_{V_{m}}$. 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
$H^{\ast}$ by the help of the inner product $(\cdot,\cdot)$ in $H$. Thus we
have

$V\hookrightarrow H\equiv H^{\ast}\hookrightarrow V^{\ast},$

where $H^{\ast}\hookrightarrow V^{\ast}$ is the adjoint of the embedding
$V\hookrightarrow H$. We use the notation $\langle\cdot,\cdot\rangle$ for the
duality pairing between $V$ and $V^{\ast}$. Note that if $v^{\ast}\in
V_{i}^{\ast}$ for some $i$, then its restriction to $V$ belongs to $V^{\ast}$
and $|\langle v^{\ast},v\rangle|\leq\|v^{\ast}\|_{V^{\ast}_{i}}\|v\|_{V_{i}}$
for all $v\in V$. Note also that $\langle v^{\ast},v\rangle=(h,v)$ for for all
$v\in V$ when $v^{\ast}=h\in H$.

A complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ together with
an increasing family of $\sigma$-algebras $(\mathcal{F}_{t})_{t\geq 0}$,
$\mathcal{F}_{t}\subset\mathcal{F}$ 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
$(A(t))_{t\geq 0}$ the notation $dA\times\mathbb{P}$ stands for the measure
defined on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{F}$ by

$(dA\times\mathbb{P})(F)=\mathbb{E}\int_{0}^{\infty}{\bf 1}_{F}\,dA(t),\quad
F\in\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{F}.$

Let $h=(h(t))_{t\geq 0}$ 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 $v=(v(t))_{t\geq 0}$ be a $V$-valued progressively measurable
process and for $i=1,\ldots,m$ let $v^{\ast}_{i}=(v^{\ast}_{i})_{t\geq 0}$ be
$V_{i}^{*}$-valued processes such that $\langle\varphi,v_{i}^{\ast}\rangle$
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 $\\{v(t):t\geq
0,\omega\in\Omega\\}.$

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
$T>0$

$\int_{0}^{T}\|v(t)\|_{V_{i}}^{p_{i}}\,dA(t)<\infty,\quad\left(\int_{0}^{T}\eta_{i}^{q_{i}}(t)\,dA(t)\right)^{1/q_{i}}<\infty,$
(2.1)

for some progressively measurable process $\eta_{i}$ such that
$\|v_{i}^{\ast}\|_{V_{i}^{\ast}}\leq\eta_{i}$ for $dA\times\mathbb{P}$-almost
everywhere, where for $q_{i}=\infty$ the second expression means

$\text{$dA$-ess}\,\sup_{t\leq T}\eta_{i}(t),$

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

$(v(t),\varphi)=\sum_{i=1}^{m}\int_{(0,t]}\langle
v^{\ast}_{i}(s),\varphi\rangle\,dA(s)+(h(t),\varphi).$ (2.2)

Then there is $\tilde{\Omega}\subset\Omega$ with
$\mathbb{P}(\tilde{\Omega})=1$ and an $H$-valued cadlag process $\tilde{v}$
such that the following statements hold.

  1. (i)

For $dA\times\mathbb{P}$ almost all $(t,\omega)$ satisfying
$t\in(0,\tau(\omega))$ we have $\tilde{v}=v$.

  2. (ii)

For all $\omega\in\tilde{\Omega}$ and $t\in[0,\tau(\omega))$ we have

$(\tilde{v}(t),\varphi)=\sum_{i=1}^{m}\int_{(0,t]}\langle
v^{\ast}_{i}(s),\varphi\rangle\,dA(s)+h(t)\varphi\quad\textrm{for all
}\,\varphi\in V.$ (2.3)

  3. (iii)

For all $\omega\in\tilde{\Omega}$ and $t\in[0,\tau(\omega))$

$\begin{split}|\tilde{v}(t)|^{2}=&|h(0)|^{2}+2\sum_{i=1}^{m}\int_{(0,t]}\langle
v^{\ast}_{i}(s),v(s)\rangle\,dA(s)+2\int_{(0,t]}(\tilde{v}(s-)\,dh(s))\\\
&-\int_{(0,t]}|v^{\ast}(s)|^{2}\Delta A(s)dA(s)+[h]_{t},\end{split}$ (2.4)

where $v^{\ast}(t):=\sum_{i=1}^{m}v^{\ast}_{i}(t)\in H$ for $\Delta A(t)>0$.

Consider now a situation where the assumptions on $h$ and $A$ are as above but
$m=1$ and regarding $v$ and $v^{\ast}:=v^{\ast}_{1}$ we know that $\|v(t)\|$,
$\|v^{\ast}(t)\|_{V^{*}}$ and $\|v(t)\|\|v^{\ast}(t)\|_{V^{*}}$ are almost
surely locally integrable with respect to $dA(t)$. Let

$\bar{v}^{\ast}(t):=\frac{v^{\ast}(t)}{1+\|v^{\ast}(t)\|_{V^{*}}}\,\,\,\textrm{and}\,\,\,\bar{A}(t):=\int_{(0,t]}(1+\|v^{\ast}(t)\|_{V^{*}})dA(t).$

Then $\|\bar{v}^{\ast}\|_{V^{*}}\leq 1$ and so $v$, $\bar{v}^{\ast}$ and
$\bar{A}$ satisfy the conditions on $v$, $v^{\ast}$ 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

$(v(t),\varphi)=\sum_{i=1}^{m}\int_{(0,t]}\langle\bar{v}^{\ast}(s),\varphi\rangle\,d\bar{A}(s)+(h(t),\varphi).$

Applying Theorem 2.1 then means that we have all of its conclusions with
$\bar{v}^{\ast}$ and $\bar{A}$ in place of $v^{\ast}$ and $A$ respectively. In
particular, we get

$\begin{split}|\tilde{v}(t)|^{2}=&|h(0)|^{2}+2\int_{(0,t]}\langle\bar{v}^{\ast}(s),v(s)\rangle\,d\bar{A}(s)+2\int_{(0,t]}(\tilde{v}(s-),dh(s))\\\
&-\int_{(0,t]}\left|\bar{v}^{\ast}(s)\right|^{2}\Delta\bar{A}(s)d\bar{A}(s)+[h]_{t}\\\
=&|h(0)|^{2}+2\int_{(0,t]}\langle
v^{\ast}(s),v(s)\rangle\,dA(s)+2\int_{(0,t]}(\tilde{v}(s-),dh(s))\\\
&-\int_{(0,t]}\left|v^{\ast}(s)\right|^{2}\Delta
A(s)\,dA(s)+[h]_{t}.\end{split}$

Hence we see that Theorem 2.1 is a generalisation of the main theorem in
Gyöngy and Krylov gyongy:krylov:on:stochastic:II .

######  Remark 1.

One might think that Theorem 2.1 follows from the main theorem in
gyongy:krylov:on:stochastic:II  by considering the process
$v^{\ast}=\sum_{i}v^{\ast}_{i}$ as a process with values in $V^{\ast}$.
However, taking into account that for any $w^{\ast}\in V^{\ast}$

$\|w^{\ast}\|_{V^{\ast}}=\inf\Big{\\{}\max_{i=1,\ldots,m}\|w_{i}^{\ast}\|_{V^{\ast}_{i}}:w^{\ast}=\sum_{i=1}^{m}w_{i}^{\ast},w_{i}^{\ast}\in
V_{i}^{\ast}\Big{\\}}$

(see for example Gajewski, Gröger and Zacharias (ggz, , Chapter 1, Theorem
5.13)), one can show that the local integrability condition in
gyongy:krylov:on:stochastic:II  for

$\|v\|_{V}\|v^{\ast}\|_{V^{\ast}}=(\|v\|_{1}+\cdots+\|v\|_{m})\|v^{\ast}\|_{V^{\ast}}$

is not implied by our assumption (2.1). Thus the main theorem in
gyongy:krylov:on:stochastic:II  is not applicable in our situation.

We consider the following motivating example.

######  Example 1.

Consider the stochastic partial differential equation

$\begin{split}du=&\left[\nabla(|\nabla u|^{p_{1}-2}\nabla
u)+|u|^{p_{2}-2}u\right]dt\\\ &+f(u,\nabla
u)\,dW+\int_{Z}g(u)\,q(dt,dz)\,\,\textrm{in}\,\,\mathscr{D}\times(0,T).\end{split}$

Here $W$ is a Wiener process (finite or infinite dimensional depending on the
choice of $f$), $(Z,\Sigma)$ is a measurable space and $q(ds,dz)$ a stochastic
martingale measure on $[0,\infty)\times Z$. See, for example, Gyöngy and
Krylov gyongy:krylov:on:stochastic:I  for detailed definition. We take
$\mathscr{D}$ to be a bounded Lipschitz domain in $\R^{d}$.

It is natural to assume that a solution $u$ should be such that
$\|u\|_{W^{1}_{p_{1}}(\mathscr{D})}^{p_{1}}$ and
$\|u\|_{L_{p_{2}}(\mathscr{D})}^{p_{2}}$ are almost surely locally integrable.
To apply the result in Gyöngy and Krylov gyongy:krylov:on:stochastic:II  one
could try to take $V:=W^{1}_{p_{1}}(\mathscr{D})\cap L_{p_{2}}(\mathscr{D})$
with the norm
$\|\cdot\|_{V}=\|\cdot\|_{W^{1}_{p_{1}}(\mathscr{D})}+\|\cdot\|_{L_{p_{2}}(\mathscr{D})}$.
The dual of $V$ can be identified with the linear space

$V^{*}=\\{f=f_{1}+f_{2}:f_{1}\in W^{1}_{p_{1}}(\mathscr{D})^{*},f_{2}\in
L_{p_{2}}(\mathscr{D})^{*}\\}$

equipped with the norm

$\begin{split}\|f\|_{V*}=\inf\\{\max(&\|f_{1}\|_{W^{1}_{p_{1}}(\mathscr{D})^{*}},\|f_{2}\|_{L_{p_{2}}(\mathscr{D})^{*}}):\\\
&f=f_{1}+f_{2},f_{1}\in W^{1}_{p_{1}}(\mathscr{D})^{*},\,\,f_{2}\in
L_{p_{2}}(\mathscr{D})^{*}\\}.\end{split}$

One would then need to show that $\|u\|_{V}\,\|\nabla(|\nabla
u|^{p_{1}-2}\nabla u)+|u|^{p_{2}-2}u\|_{V^{*}}$ is locally integrable. To
ensure this in general we need, in particular, that

$\|u\|_{W^{1}_{p_{1}}(\mathscr{D})}\,\||u|^{p_{2}-2}u\|_{L_{p_{2}}(\mathscr{D})^{*}}=\|u\|_{W^{1}_{p_{1}}(\mathscr{D})}\,\|u\|^{p_{2}-1}_{L_{p_{2}}(\mathscr{D})}$

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 pardoux:thesis  (Chapter 2, Example 5.1) and in GSS . Further
examples can be found in (Lions, , Chapter 2, Section 1.7).

##  3 Preliminaries

######  Lemma 1.

For $r\in[0,\infty)$ let $\beta(r):=\inf\\{t\geq 0:A(t)\geq r\\}$ and let
$x(t)$ be a real valued process that is locally integrable with respect to
$dA$ for all $\omega\in\Omega$. Then

  1. i)

$\beta(r)$ is a stopping time (not necessarily finite) for every
$r\in[0,\infty)$,

  2. ii)

$\begin{split}\int_{(0,t]}x(s)\,dA(s)&=\int_{(0,A(t)]}x(\beta(r))\,dr,\\\
\int_{(0,t)}x(s)\,dA(s)&=\int_{(0,A(t-)]}x(\beta(r))\,dr\end{split}$

for every $t\in[0,\infty)$,

  3. iii)

$A(\beta(t)-)-A(\beta(s))\leq t-s$

for every $s,t\in[0,\infty)$.

  4. 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
$\sup_{k}|r^{n}_{k+1}-r^{n}_{k}|\to 0$ as $n\to\infty$ then for every $t\geq
0$ and $\omega\in\Omega$

$\sum_{k}\left|X(\tau^{n}_{k+1}\wedge t)-X(\tau^{n}_{k}\wedge
t)\right|^{2}\to\sum_{s\leq t}|X(s)|^{2}|\Delta A(s)|^{2}$

as $n\to\infty$, 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 (gyongy:krylov:on:stochastic:II, ,
Lemma 1).

Let $\kappa_{n}^{(j)}$ for $j=1,2$ and integers $n\geq 1$ denote the functions
defined by

$\kappa^{(1)}_{n}(t)=2^{-n}\lfloor
2^{n}t\rfloor,\quad\kappa^{(2)}_{n}(t)=2^{-n}\lceil 2^{n}t\rceil$

The following lemma is known and the authors believe is due to Doob.

######  Lemma 2.

For integers $i\geq 1$ let $(X_{i},\|\cdot\|_{X_{i}})$ be Banach spaces, and
let $p_{i}\in[1,\infty)$. Let $x_{i}:\R\times\Omega\to X_{i}$ be
$\mathscr{B}(\R)\otimes\mathcal{F}$ Bochner-measurable such that $x_{i}(r)=0$
for $r\notin[0,1]$ and

$\alpha_{i}:=\mathbb{E}\int_{0}^{1}\|x_{i}(r)\|_{X_{i}}^{p_{i}}\,dr<\infty.$

Then there exists a subsequence $n_{k}\to\infty$ such that for $dt$-almost all
$t\in[0,1]$

$\mathbb{E}\int_{(0,1]}\|x_{i}(r)-x_{i}(\kappa^{(j)}_{n_{k}}(r-t)+t)\|_{X_{i}}^{p_{i}}\,dr\to
0\,\,\textrm{ as }\,\,k\to\infty$

for $j=1,2$ and all $i\geq 1$.

###### Proof.

Let $(c_{i})_{i=1}^{\infty}$ be a sequence of positive numbers such that

$\sum_{i=1}^{\infty}c_{i}2^{p_{i}}\alpha_{i}<\infty.$

By change of variables and changing the order of integration

$I_{n}:=\sum_{i=1}^{\infty}c_{i}\int_{0}^{1}\mathbb{E}\int_{0}^{1}\|x_{i}(r)-x_{i}(\kappa^{(j)}_{n}(r-t)+t)\|_{X_{i}}^{p_{i}}\,dr\,dt$
$\leq\sum_{i=1}^{\infty}c_{i}\mathbb{E}\int_{-1}^{1}\int_{0}^{1}\|x_{i}(s+t)-x_{i}(\kappa^{(j)}_{n}(s)+t)\|^{p_{i}}_{X_{i}}\,dt\,ds.$

Note that by the shift invariance of the Lebesgue measure

$J_{in}(s):=\int_{0}^{1}\|x_{i}(s+t)-x_{i}(\kappa^{(j)}_{n}(s)+t)\|^{p_{i}}_{X_{i}}\,dt\to
0\,(a.s.)$

for $s\in(0,1)$, $i\geq 1$, and

$\sum_{i=1}^{\infty}c_{i}|J_{in}(s)|\leq\sum_{i=1}^{\infty}c_{i}2^{p_{i}-1}\left(\int_{0}^{1}\|x_{i}(s+t)\|^{p_{i}}_{X_{i}}\,dt+\int_{0}^{1}\|x_{i}(\kappa_{n}(s)+t)\|^{p_{i}}_{X_{i}}\,dt\right)$
$\leq\sum_{i=1}^{\infty}{c_{i}}2^{p_{i}}\int_{0}^{1}\|x_{i}(t)\|^{p_{i}}_{X_{i}}\,dt.$

Therefore by Lebesgue’s theorem on dominated convergence

$I_{n}=\int_{0}^{1}\left(\sum_{i=1}^{\infty}c_{i}\mathbb{E}\int_{0}^{1}\|x_{i}(r)-x_{i}(\kappa^{(j)}_{n}(r-t)+t)\|_{X_{i}}^{p_{i}}\,dr\right)\,dt\to
0.$

Hence for a subsequence $n_{k}\to\infty$

$\sum_{i=1}^{\infty}c_{i}\mathbb{E}\int_{0}^{1}\|x_{i}(r)-x_{i}(\kappa^{(j)}_{n}(r-t)+t)\|_{X_{i}}^{p_{i}}\,dr\to
0$

for almost all $t\in[0,1]$, and the statement of the lemma follows. ∎

The following lemma is proved in Gyöngy and Krylov
(gyongy:krylov:on:stochastic:II, , Lemma 3).

######  Lemma 3.

Let $(\xi_{n})_{n\in\mathbb{N}}$ be a sequence of $H$-valued predictable
processes. Suppose

$\mathbb{P}\left[\sup_{n\in\mathbb{N},t\leq T}|\xi_{n}(t)|<\infty\right]=1$

and

$\mathbb{P}\left[\forall t\leq T,\,\,\forall\varphi\in
H\,\,\,\lim_{n\to\infty}(\xi_{n}(t),\varphi)=0\right]=1.$

Then for any $\varepsilon>0$

$\mathbb{P}\left[\sup_{t\leq
T}\left|\int_{(0,t]}(\xi_{n}(s),dh(s))\right|>\varepsilon\right]\to 0$

as $n\to\infty$.

##  4 Proof of the Main Result

The following standard steps, as in Krylov and Rozovskii
krylov:rozovskii:stochastic , allow us to work under more convenient
assumptions without any loss of generality.

  1. 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 $n\to\infty$. In fact
using a non-random time change we may assume that $\tau\leq 1$.

  2. 2)

Recall the processes $\eta_{i}$ from assumption (2.1), and set

$Q_{i}(t)=\left(\int_{(0,t]}\eta_{i}^{q_{i}}(s)\,dA(s)\right)^{1/q_{i}}\quad
t\geq 0$

when $q_{i}<\infty$, and for $q_{i}=\infty$ let $Q_{i}=(Q_{i}(t))_{t\geq 0}$
denote a nondecreasing cadlag adapted process such that almost surely

$\text{$dA$-ess\,sup}_{s\leq t}\eta_{i}(s)\leq Q_{i}(t)\quad\text{ for all
$t\geq 0$}.$

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
$\text{$dA$-ess\,sup}_{s\leq t}\eta_{i}(s)$, i.e.,

$\lim_{n\to\infty}\text{$dA$-ess\,sup}_{s\leq t+1/n}\eta_{i}(s).$

Let $(e^{j})_{j\in\mathbb{N}}\subset V$ be an orthonormal basis in $H$ and
define

$\begin{split}&r(t):=|h(0)|+A(t)+\sum_{i=1}^{m}\left(\int_{(0,t]}\|v(s)\|_{V_{i}}^{p_{i}}dA(s)\right)^{1/p_{i}}\\\
&+\sum_{i=1}^{m}Q_{i}(t)+\sum_{i=1}^{m}\sum_{k\in\mathbb{N}}2^{-c_{k}}\left(\int_{(0,t]}\|w_{k}(s)\|^{p_{i}}_{V_{i}}dA(s)\right)^{1/p_{i}},\end{split}$
(4.1)

with $c_{k}:=\max_{1\leq i\leq m}\sum_{j\leq k}|e_{j}|^{2}_{V_{i}}$ and
$w_{k}:=\Pi^{k}h$, where $\Pi^{k}$ 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

$\tau_{n}:=\inf\\{t\geq 0:r(t)\geq n\\}.$

Then $\tau_{n}$ is a stopping time and $\tau_{n}\to\infty$ for $n\to\infty$.
Since $\langle h\rangle$ is a predictable process starting from $0$, there is
an increasing sequence of stopping times $\sigma_{n}$ such that
$\sigma_{n}\to\infty$ and $\langle h\rangle_{t}\leq n$ for
$t\in[0,\sigma_{n}]$. Therefore $\tau_{n}\wedge\sigma_{n}\wedge\tau\to\tau$ as
$n\to\infty$, and for fixed $n$ we get $r(t)\leq n$ for
$t\in(0,\tau_{n}\wedge\sigma_{n})$ and $\langle h\rangle_{t}\leq n$ 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
$n\to\infty$ provides (2.3) and (2.4) for $\tau$. Thus we may assume that
there is $n\geq 1$ such that $r(t)\leq n$ for $t\in(0,\tau)$ and $\langle
h\rangle_{t}\leq n$ for $t\in[0,\tau]$. Moreover, by taking $h{\bf
1}_{|h(0)|<n}$, $v{\bf 1}_{|h(0)|<n}$ and $A{\bf 1}_{|h(0)|<n}$ in place of
$h$, $v$ and $A$, respectively, and then taking $n\to\infty$, we may assume
that $r(t)\leq n$ for $t\in[0,\tau)$ and $\langle h\rangle_{t}\leq n$ for
$t\in[0,\tau]$. Furthermore, we can define $A(t):=A(\tau-)$, $h(t)=h(\tau)$,
$v(t)=0$ and $v_{i}^{\ast}(t)=0$ for $t\geq\tau$. Then $r(t)\leq n$ and
$\langle h\rangle_{t}\leq n$ for $t\in[0,\infty)$.

  3. 3)

Finally, we can assume that $r(t)\leq 1$ for $t\in[0,\tau)$ and $\langle
h\rangle_{t}\leq 1$ 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 $r_{n}(t)\leq n^{-1}r(t)\leq 1$. 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 $\tau\leq 1$,
$r(t)\leq 1$ and $\langle h\rangle_{t}\leq 1$ for $t\in[0,\infty)$. Our
approach is the same as in Gyöngy and Krylov gyongy:krylov:on:stochastic:II .
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
$\varphi\in V$ 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 $\\{(t,\omega):t\in(0,\tau(\omega)),\omega\in\Omega\\}$.

######  Proposition 1.

There is a nested sequence of random partitions of $[0,\infty]$,

$0=\tau^{n}_{0}<\tau^{n}_{1}\leq\tau^{n}_{2}\leq\cdots\leq\tau^{n}_{N(n)+1}=\infty,$

with stopping times $\tau^{n}_{j}$, $j=1,\ldots,N(n)+1$, such that for every
$\omega\in\Omega$ either $\tau^{n}_{j}(\omega)<\tau(\omega)$ or
$\tau^{n}_{j}(\omega)=\infty$, and such that the following statements hold.

  1. (1)

There is $\Omega^{\prime}\subset\Omega$ such that
$\mathbb{P}(\Omega^{\prime})=1$ and with

$I(\omega):=\\{\tau^{n}_{j}(\omega):n\in\mathbb{N},j=1,\ldots,N(n)\\}\cap(0,\infty)$

we have (2.2) satisfied for every $\omega\in\Omega^{\prime}$, $t\in I(\omega)$
and $\varphi\in V$. Moreover, if $\Delta A(t)>0$ for some $t>0$ and
$\omega\in\Omega^{\prime}$, then $t\in I(\omega)$. Furthermore, if $0\leq s<t$
and $(s,t]\cap I(\omega)=\emptyset$, then $A(s)=A(t)$.

  2. (2)

For $l\in\\{1,2\\}$, $i=1,\ldots,m$ and for all $k\geq 1$

$\begin{split}\lim_{n\to\infty}\mathbb{E}\int_{(0,\infty)}\|v(s)-v^{(l)}_{n}(s)\|_{V_{i}}^{p_{i}}\,dA(s)=0,\\\
\lim_{n\to\infty}\mathbb{E}\int_{(0,\infty)}\|w_{k}(s)-w^{(l)}_{kn}(s)\|^{p_{i}}_{V_{i}}\,dA(s)=0,\end{split}$
(4.2)

where

$v_{n}^{(1)}(t):=\sum_{j=1}^{N(n)}v(\tau_{j}^{n}){\bf
1}_{[\tau_{j}^{n},\tau_{j+1}^{n})}(t),\quad
v_{n}^{(2)}(t):=\sum_{j=0}^{N(n)}v(\tau_{j+1}^{n}){\bf
1}_{(\tau_{j}^{n},\tau_{j+1}^{n}]}(t),$

and $w^{(l)}_{kn}$ is defined analogously from $w_{k}=\Pi^{k}h$.

###### 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
$D_{i}\in[0,\infty)\times\Omega$ 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 1 and
the Fubini theorem

$\begin{split}0&=\mathbb{E}\int_{(0,\tau)}\chi_{D}(s)\,dA(s)=\mathbb{E}\int_{(0,A(\tau-)]}\chi_{D}(\beta(r))\,dr\\\
&=\int_{(0,\infty)}\mathbb{P}(r\leq A(\tau),(\beta(r),\omega)\in
D)\,dr.\end{split}$

From this we see that for $dr$ almost all $r\in(0,\infty)$ there is
$\Omega(r)\subset\Omega$ with $\mathbb{P}(\Omega(r))=1$ such that for any
$\omega\in\Omega(r)$ either $r>A(\tau(\omega),\omega)$ or
$\beta(r,\omega)<\tau(\omega)$ and for $t=\beta(r)$ and for all $\varphi\in V$

$(v(t),\varphi)=\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)+(h(t),\varphi).$ (4.3)

By virtue of Lemma 2 there is a nested sequence of decompositions of $[0,1]$,

$0=r^{n}_{0}<r^{n}_{1}<\cdots<r^{n}_{N(n)+1}=1,$ (4.4)

such that $\lim_{n\to\infty}\max_{i}|r^{n}_{j+1}-r^{n}_{j}|=0$, and

$\begin{split}\lim_{n\to\infty}\mathbb{E}\int_{0}^{1}\|v(\beta(r))-v(\beta(\kappa_{n}^{(l)}(r)))\|_{V_{i}}^{p_{i}}\,dr=0,\\\
\lim_{n\to\infty}\mathbb{E}\int_{0}^{1}\|w_{k}(\beta(r))-w_{k}(\beta(\kappa_{n}^{(l)}(r)))\|^{p_{i}}_{V_{i}}\,dr=0\end{split}$
(4.5)

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
$\Omega^{\prime}:=\bigcap_{n\in\mathbb{N}}\left(\Omega(r^{n}_{0})\cap\ldots\cap\Omega(r^{n}_{N(n)+1})\right)$,
$\tau^{n}_{j}:=\beta(r^{n}_{j})$, and

$I(\omega):=\\{\tau^{n}_{i}(\omega):n\in\mathbb{N},i=1,\ldots,N(n)\\}\cap(0,\infty).$

Then $\mathbb{P}(\Omega^{\prime})=1$ and

$0=\tau^{n}_{0}<\tau^{n}_{1}\leq\tau^{n}_{2}\leq\cdots\leq\tau^{n}_{N(n)+1}=\infty,\quad
n=1,2,\ldots,$

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 gyongy:krylov:on:stochastic:II , for
$r\in(r_{j}^{n},r^{n}_{j+1}]$

$v_{n}^{(2)}(\beta(r))=\left\\{\begin{array}[]{lll}v(\beta(r^{n}_{j+1}))=v(\beta(\kappa^{(2)}_{n}(r)))&\text{if}&\beta(r^{n}_{j})<\beta(r)\\\
v(\beta(r^{n}_{j}))=v(\beta(\kappa^{(1)}_{n}(r)))&\text{if}&\beta(r^{n}_{j})=\beta(r).\end{array}\right.$

Thus with appropriate sets $S_{n}\in\mathcal{B}(\R)\times\mathcal{F}$

$v^{(2)}(\beta(r))={\bf 1}_{S_{n}}(r)v(\beta(\kappa^{(2)}_{n}(r)))-(1-{\bf
1}_{S_{n}})(r))v(\beta(\kappa^{(1)}_{n}(r))).$

Hence due to (4.5) and Lemma 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. ∎

######  Proposition 2.

For every $n\in\mathbb{N}$, every $\omega\in\Omega^{\prime}$ and every
$\tau^{n}_{j}(\omega)\in I(\omega)$

$\begin{split}|v(\tau^{n}_{j})|^{2}=&|h(0)|^{2}+2\sum_{i=1}^{m}\int_{(0,\tau^{n}_{j}]}\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)\rangle\,dA(s)\\\
&+2\int_{(0,\tau^{n}_{j}]}\bar{v}_{n}(s)\,dh(s)+2(h(0),h(\tau^{n}_{1})-h(0))\\\
&+\sum_{k=0}^{j-1}|h(\tau^{n}_{k+1})-h(\tau^{n}_{k})|^{2}-|v(\tau^{n}_{1})-h(\tau^{n}_{1})|^{2}\\\
&-\sum_{k=1}^{j-1}|v(\tau^{n}_{k+1})-v(\tau^{n}_{k})-(h(\tau^{n}_{k+1})-h(\tau^{n}_{k}))|^{2},\end{split}$
(4.6)

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,

$\mathbb{E}\sup_{t\in I}|v(t)|^{2}<\infty.$ (4.7)

###### Proof.

Let $\omega\in\Omega^{\prime}$ and $t,t^{\prime}\in I(\omega)$ and
$t^{\prime}\geq t$. Clearly,

$|v(t^{\prime})|^{2}-|v(t)|^{2}=2(v(t^{\prime}),v(t^{\prime})-v(t))-|v(t^{\prime})-v(t)|^{2},$

which by statement (1) of Proposition 1 gives

$\begin{split}&|v(t^{\prime})|^{2}-|v(t)|^{2}\\\
&=2\sum_{i=1}^{m}\int_{(t,t^{\prime}]}\langle
v_{i}^{\ast}(s),v(t^{\prime})\rangle\,dA(s)+2(h(t^{\prime})-h(t),v(t^{\prime}))-|v(t^{\prime})-v(t)|^{2}.\end{split}$

Hence by the identity

$\begin{split}&2(h(t^{\prime})-h(t),v(t^{\prime})-v(t))\\\
&=-|v(t^{\prime})-v(t)-(h(t^{\prime})-h(t))|^{2}+|v(t^{\prime})-v(t)|^{2}+|h(t^{\prime})-h(t)|^{2},\end{split}$

we have

$\begin{split}|v(t^{\prime})|^{2}&-|v(t)|^{2}=2\sum_{i=1}^{m}\int_{(t,t^{\prime}]}\langle
v_{i}^{\ast}(s),v(t^{\prime})\rangle dA(s)+2(v(t),h(t^{\prime})-h(t))\\\
&+|h(t^{\prime})-h(t)|^{2}-|v(t^{\prime})-v(t)-(h(t^{\prime})-h(t))|^{2}.\end{split}$
(4.8)

By (1) in Proposition 1 again

$2|v(t)|^{2}=2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v(t)\rangle\,dA(s)+2(h(t),v(t)),$

which by the identity $2(h(t),v(t))=-|v(t)-h(t)|^{2}+|v(t)|^{2}+|h(t)|^{2}$
gives

$|v(t)|^{2}=2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v(t)\rangle\,\,dA(s)+|h(t)|^{2}-|v(t)-h(t)|^{2}.$ (4.9)

Summing up for $k=1,\ldots,j-1$ equations (4.8) with
$t^{\prime}=\tau^{n}_{k+1}$, $t=\tau^{n}_{k}$, and adding to it equation (4.9)
with $t=\tau^{n}_{1}$, we obtain (4.6). Form (4.6) we have

$\begin{split}\mathbb{E}\max_{1\leq j\leq
N(n)}|v(\tau^{n}_{j})|^{2}\leq&2\mathbb{E}|h(0)|^{2}+2\mathbb{E}\sum_{i=1}^{m}\int_{(0,\tau]}|\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)\rangle|\,dA(s)\\\ &+2\mathbb{E}\max_{1\leq
j\leq N(n)}\left|\int_{(0,\tau^{n}_{j}]}\bar{v}_{n}(s)\,dh(s)\right|\\\
&+2\mathbb{E}\sum_{k=0}^{N(n)}|h(\tau^{n}_{k+1})-h(\tau^{n}_{k})|^{2}.\end{split}$

Clearly

$2\mathbb{E}\max_{1\leq j\leq
N(n)}\left|\int_{(0,\tau^{n}_{j}]}\bar{v}_{n}(s)\,dh(s)\right|\leq
16+\frac{1}{16}\mathbb{E}\sup_{t\geq
0}\left|\int_{(0,t]}\bar{v}_{n}(s)\,dh(s)\right|^{2},$

and by Doob’s inequality and $\langle h\rangle\leq 1$,

$\mathbb{E}\sup_{t\geq
0}\left|\int_{(0,t]}\bar{v}_{n}(s)\,dh(s)\right|^{2}\leq
4\mathbb{E}\int_{0}^{\infty}|\bar{v}_{n}(s)|^{2}d\langle h\rangle_{s}\leq
4\mathbb{E}\max_{1\leq j\leq N(n)}|v(\tau^{n}_{j})|^{2}.$

Since $h$ is a martingale,

$\mathbb{E}\sum_{k=0}^{N(n)}|h(\tau^{n}_{k+1})-h(\tau^{n}_{k})|^{2}\leq\mathbb{E}|h(1)|^{2}=\mathbb{E}\langle
h\rangle(1)\leq 1.$

By Hölder’s inequality and $\sum_{i}Q_{i}\leq 1$ we have

$\sum_{i=1}^{m}\mathbb{E}\int_{(0,\tau]}|\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)\rangle|\,dA(s)$ $\leq\sum_{i}\sup_{n\geq
1}\left(\mathbb{E}\int_{(0,\tau]}\|v^{(2)}_{n}(s)\|_{V_{i}}^{p_{i}}\,dA(s)\right)^{\tfrac{1}{p_{i}}}=:c\,,$

which by virtue of (4.2) is finite. Hence, taking also into account
$\mathbb{E}|h(0)|^{2}\leq 1$ we have

$\mathbb{E}\max_{1\leq j\leq N(n)}|v(\tau^{n}_{j})|^{2}\leq
22+2c+\frac{1}{4}\mathbb{E}\max_{1\leq j\leq N(n)}|v(\tau^{n}_{j})|^{2},$

which immediately yields (4.7), provided

$\mathbb{E}\max_{1\leq j\leq N(n)}|v(\tau^{n}_{j})|^{2}<\infty.$ (4.10)

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

$\begin{split}\mathbb{E}|v(\tau^{n}_{j})|^{2}\leq&\mathbb{E}|h(\tau^{n}_{j})|^{2}+2\mathbb{E}\sum_{i=1}^{m}\int_{(0,\tau^{n}_{j}]}\langle
v_{i}^{\ast}(s),v(\tau^{n}_{j})\rangle dA(s)\\\
\leq&\mathbb{E}|h(0)|^{2}+2\mathbb{E}\sum_{i=1}^{m}Q_{i}(\tau)\left(\int_{(0,\tau]}\|v(\tau^{n}_{j})\|_{V_{i}}^{p_{i}}dA(s)\right)^{\tfrac{1}{p_{i}}}\\\
\leq&1+2\sum_{i}\mathbb{E}\|v(\tau^{n}_{j})\|_{V_{i}},\end{split}$ (4.11)

since $\tau\leq 1$ and $r(t)\leq 1$ for all $t\in[0,\infty)$. For
$i=1,\ldots,m$

$\begin{split}&\mathbb{E}\|v(\tau^{n}_{j})\|^{p_{i}}_{V_{i}}\leq\mathbb{E}\sup_{s\in[0,\infty)}\|v^{(2)}_{n}(s)\|_{V_{i}}^{p_{i}}\leq\mathbb{E}\sup_{r\in(0,1]}\|v^{(2)}_{n}(\beta(r))\|_{V_{i}}^{p_{i}}\\\
&\leq
2^{p_{i}-1}\sum_{l=1}^{2}\mathbb{E}\sup_{r\in(0,1]}\|v(\beta(\kappa^{(l)}_{n}(r)))\|_{V_{i}}^{p_{i}}\\\
&\leq
2^{p_{i}-1}\sum_{l=1}^{2}\mathbb{E}\sum_{k=0}^{N(n)}\frac{1}{r^{n}_{k+1}-r^{n}_{k}}\int_{r^{n}_{k}}^{r^{n}_{k+1}}\|v(\beta(\kappa^{(l)}_{n}(r)))\|_{V_{i}}^{p_{i}}\,dr\\\
&\leq\frac{2^{p_{i}-1}}{d_{n}}\sum_{l=1}^{2}\mathbb{E}\int_{0}^{1}\|v(\beta(\kappa^{(l)}_{n}(r)))\|_{V_{i}}^{p_{i}}\,dr<2^{p_{i}}\frac{c_{i}}{d_{n}},\\\
\end{split}$

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

$c_{i}:=\max_{l}\sup_{n}\sum_{i=1}^{m}\mathbb{E}\int_{0}^{1}\|v(\beta(\kappa^{(l)}_{n}(r)))\|_{V_{i}}^{p_{i}}\,dr,$

which due to (4.5) is finite. Hence by virtue of (4.11) we have (4.10), which
completes the proof of (4.7). ∎

We see that due to (4.7) there is
$\Omega^{\prime\prime}\subset\Omega^{\prime}$ such that
$\mathbb{P}(\Omega^{\prime\prime})=1$ and

$\sup_{t\in I(\omega)}|v(t)|^{2}<\infty\quad\text{for all
$\omega\in\Omega^{\prime\prime}$}.$

Moreover, since $h$ is cadlag, for all $\omega\in\Omega^{\prime\prime}$ we
have

$\sup_{t\in I(\omega)}|v(t)-h(t)|^{2}<\infty.$ (4.12)

Define

$z^{(1)}(t):=\int_{(0,t)}\sum_{i=1}^{m}v_{i}^{\ast}(s)\,dA(s),\quad
z^{(2)}(t):=\int_{(0,t]}\sum_{i=1}^{m}v_{i}^{\ast}(s)\,dA(s),$ (4.13)

for $t\geq 0$, where the integrals are defined as weak* integrals. Recall that
$v^{\ast}=\sum_{i}v^{\ast}$ is a $V^{\ast}$-valued such that $\langle
v^{\ast}(t),\varphi\rangle$ is a progressively measurable process for every
$\varphi\in V$, and

$\int_{(0,t]}|\langle
v^{\ast}(s),\varphi\rangle|\,dA(s)\leq\sum_{i}\int_{(0,t]}|\langle
v_{i}^{\ast}(s),\varphi\rangle|\,dA(s)$
$\leq\sum_{i}|\varphi|_{V_{i}}\int_{(0,t]}\eta_{i}(s)\,dA(s)\leq|\varphi|_{V}\sum_{i}\int_{(0,t]}\eta_{i}(s)\,dA(s)<\infty.$

Therefore $z^{(1)}$ and $z^{(2)}$ are well-defined $V^{\ast}$-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
$\Delta^{w}f(t):=f(t)-\text{w-lim}_{s\nearrow t}f(s)$ for $H$-valued functions
$f$, when the weak limit from the left exists at $t$.

######  Proposition 3.

Let $z^{(l)}$, $l\in\\{1,2\\}$ be given by (4.13).

  1. 1.

If $\omega\in\Omega^{\prime\prime}$ and $t\in(0,\infty)$ then $z^{(l)}(t)\in
H$ for $l\in\\{1,2\\}$. Moreover

$\sup_{t\in(0,\infty)}|z^{(l)}(t)|<\infty\,\,\,\,\forall\omega\in\Omega^{\prime\prime},\,\,l\in\\{1,2\\}.$

  2. 2.

Let $\tilde{v}$ be given by

$\tilde{v}(t):=\chi_{\Omega^{\prime\prime}}z^{(2)}(t)+h(t).$

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
$\omega\in\Omega^{\prime\prime}$. Moreover

$\sup_{t\in(0,\infty)}|\tilde{v}(t)|<\infty\,\,\,\,\forall\omega\in\Omega^{\prime\prime}.$

  3. 3.

If $\omega\in\Omega^{\prime\prime}$ then for all $t\in(0,\tau(\omega))$

$\Delta^{w}(\tilde{v}-h)(t)=(\Delta A)(t)\sum_{i=1}^{m}v_{i}^{\ast}(t).$
(4.14)

###### Proof.

Fix $\omega\in\Omega^{\prime\prime}$. If $t\in I(\omega)$ then for all
$\varphi\in V$

$\left(v(t)-h(t),\varphi\right)=\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s),$

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 $\Delta A(t)=0$ by Proposition 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 $t_{n^{\prime}}\nearrow t$ such
that $v(t_{n^{\prime}})-h(t_{n^{\prime}})$ converges weakly in $H$ to some
$\xi\in H$. Hence for all $\varphi\in V$

$\begin{split}(\xi,\varphi)&=\lim_{n^{\prime}\to\infty}(v(t_{n^{\prime}})-h(t_{n^{\prime}}),\varphi)\\\
&=\lim_{n^{\prime}\to\infty}\sum_{i=1}^{m}\int_{(0,t_{n^{\prime}}]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)=\sum_{i=1}^{m}\int_{(0,t)}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)\\\ &=\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s),\end{split}$

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 $\int_{(s,t]}v_{i}^{\ast}(s)\,dA(s)=0$ 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),

$\sup_{t\in(0,\infty)}|z^{(2)}(t)|^{2}=\sup_{t\in(0,\infty)}|v(t)-h(t)|^{2}<\infty.$
(4.15)

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
$n\to\infty$. From (4.15) we know that
$\sup_{n\in\mathbb{N}}|z^{(2)}(t_{n})|^{2}<\infty$ and so there is a
subsequence $t_{n^{\prime}}\nearrow t$ such that $z^{(2)}(t_{n})$ converges
weakly in $H$ to some $\xi\in H$. Thus for any $\varphi\in V$

$\begin{split}&(\xi,\varphi)=\lim_{n^{\prime}\to\infty}(z^{(2)}(t_{n^{\prime}}),\varphi)\\\
&=\lim_{n^{\prime}\to\infty}\sum_{i=1}^{m}\int_{(0,t_{n^{\prime}}]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)=\sum_{i=1}^{m}\int_{(0,t_{n^{\prime}})}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)=\langle
z^{(1)}(t),\varphi\rangle.\end{split}$

Hence $z^{(1)}(t)=\xi\in H$, and due to (4.15)

$\sup_{t\in(0,\infty)}|z^{(1)}(t)|^{2}\leq\sup_{t\in(0,\infty)}|z^{(2)}(t)|^{2}<\infty.$

By construction $\tilde{v}$ is weakly cadlag. Due to (4.15) for
$\omega\in\Omega^{\prime\prime}$

$\sup_{t\in(0,\infty)}|\tilde{v}(t)|^{2}\leq\sup_{t\in(0,\infty)}|z^{(2)}(t)|^{2}+\sup_{t\in(0,\infty)}|h(t)|^{2}<\infty.$

We note that for any $\varphi\in V$ the real valued random variable

$(\tilde{v}(t),\varphi)=\chi_{\Omega^{\prime\prime}}\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)+(h(t),\varphi)$

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

$\Delta((\tilde{v}-h)(t),\varphi)=\sum_{i=1}^{m}\langle
v_{i}^{\ast}(t),\varphi\rangle(\Delta A)(t)$

for all $\varphi\in V$ and $\omega\in\Omega^{\prime\prime}$. Hence on
$\Omega^{\prime\prime}$

$\Delta^{w}(\tilde{v}-h)(t)=\sum_{i=1}^{m}v_{i}^{\ast}(t)(\Delta A)(t)\,.$

∎

Let

$\tilde{v}_{n}(t):=\tilde{v}(\tau^{n}_{j})\,\,\,\textrm{and}\,\,\,\,h_{n}(t):=h(\tau^{n}_{j})\,\,\,\,\textrm{for}\,\,\,\,t\in(\tau^{n}_{j},\tau^{n}_{j+1}],\,\,\,\,j=0,1,\ldots,N(n).$

Then from (4.6) it follows that for every $\omega\in\Omega^{\prime\prime}$ and
$t:=\tau^{n}_{j}(\omega)\in I(\omega)$

$\begin{split}|\tilde{v}(t)|^{2}=&|h(0)|^{2}+2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)\rangle\,dA(s)\\\
&+2\int_{(0,t]}(\tilde{v}_{n}(s),dh(s))+\sum_{k=0}^{j-1}|h(\tau^{n}_{k+1})-h(\tau^{n}_{k})|^{2}-K_{n}(t),\end{split}$
(4.16)

where

$K_{n}(t):=\sum_{k:\tau^{n}_{k+1}\leq
t}^{j-1}|\tilde{v}(\tau^{n}_{k+1})-\tilde{v}(\tau^{n}_{k})-(h(\tau^{n}_{k+1})-h(\tau^{n}_{k}))|^{2}.$

In order to let $n\to\infty$ in the above equation we first rewrite it as

$\begin{split}|\tilde{v}(t)|^{2}=&2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)\rangle\,dA(s)\\\
&+2\int_{(0,t]}(\tilde{v}_{n}(s)-h_{n}(s),dh(s))+|h(t)|^{2}-K_{n}(t)\\\
\end{split}$ (4.17)

by noticing that

$2\int_{(0,\tau^{n}_{j}]}(h_{n}(s),dh(s))=|h(\tau^{n}_{j})|^{2}-|h(0)|^{2}-\sum_{k=0}^{j-1}|h(\tau^{n}_{k+1})-h(\tau^{n}_{j})|^{2}.$

To perform the limit procedure we use the following two propositions.

######  Proposition 4.

There is $\tilde{\Omega}\subset\Omega^{\prime\prime}$ with
$\mathbb{P}(\tilde{\Omega})=1$ such that for a subsequence $n^{\prime}$ and
for every $\omega\in\tilde{\Omega}$

$\begin{split}&\int_{(0,\infty)}\|v(s)-v^{(l)}_{n^{\prime}}(s)\|_{V_{i}}^{p_{i}}\,dA(s)\to
0\,\,\,\,(l=1,2),\\\
&\int_{(0,\infty)}\|w_{k}(s)-w^{(l)}_{kn^{\prime}}(s)\|^{p_{i}}_{V_{i}}\,dA(s)\to
0\,\,\,\,(l=1,2;k\in\mathbb{N}),\\\
&\sup_{t\in(0,\infty)}\left|\int_{(0,t]}(\tilde{v}_{n^{\prime}}(s)-h_{n^{\prime}}(s),dh(s))-\int_{(0,t]}(\tilde{v}(s-)-h(s-),dh(s))\right|\to
0\end{split}$

as $n^{\prime}\to\infty$. Moreover,

$K_{n^{\prime}}(t)\to\int_{(0,t]}|v^{\ast}(s)|^{2}\Delta
A(s)\,dA(s)\quad\text{for $t\in I(\omega)$ and $\omega\in\tilde{\Omega}$}.$

###### Proof.

Set $\xi(t):=\tilde{v}(t-)-h(t-)$ and $\xi_{n}(t):=\tilde{v}_{n}(t)-h_{n}(t)$.
By Lemma 3, taking into account that by Proposition 3 on
$\Omega^{\prime\prime}$

$\sup_{n}\sup_{t\in(0,\infty)}|\xi(t)-\xi_{n}(t)|\leq\sup_{t\in(0,\infty)}|z^{(1)}(t)|<\infty,$

and that $V$ is dense in $H$, we have

$\sup_{t\geq 0}\left|\int_{(0,t]}(\xi(s)-\xi_{n}(s),dh(s))\right|\to
0\quad\text{in probability as $n\to\infty$},$

if we show that almost surely

$\lim_{n\to\infty}(\xi(t)-\xi_{n}(t),\varphi)=0\quad\text{for all $t>0$ and
$\varphi\in V$}.$

To this end set

$v_{i}^{\ast}:=\int_{(\tau^{n}_{j},t)}v_{i}^{\ast}(s)\,dA(s)\in V_{i}^{\ast}.$

Then for all $\omega\in\Omega^{\prime\prime}$, $t>0$ and $\varphi\in V$

$\begin{split}&(\xi(t)-\xi_{n}(t),\varphi)=\langle\xi(t)-\xi_{n}(t),\varphi\rangle=\bigg{\langle}\sum_{i=1}^{m}v_{i}^{\ast},\varphi\bigg{\rangle}=\sum_{i=1}^{m}\langle
v_{i}^{\ast},\varphi\rangle\\\ &=\sum_{i=1}^{m}\int_{(\tau^{n}_{j},t)}\langle
v_{i}^{\ast}(s),\varphi\rangle\,dA(s)\leq\sum_{i=1}^{m}\|\varphi\|_{V_{i}}\int_{(\tau^{n}_{j},t)}\|v_{i}^{\ast}(s)\|_{V_{i}^{*}}\,dA(s)\\\
&\leq\max_{j=1,\ldots,N(n)}\sum_{i=1}^{m}\|\varphi\|_{V_{i}}\left(A(\tau^{n}_{j+1})-A(\tau^{n}_{j})\right)^{\frac{1}{p_{i}}}Q_{i}(\tau^{n}_{j+1})\\\
&\leq\max_{j=1,\ldots,N(n)}\sum_{i=1}^{m}\|\varphi\|_{V_{i}}\left|r^{n}_{j+1}-r^{n}_{j}\right|^{\frac{1}{p_{i}}}\to
0\,\,\,\textrm{as}\,\,\,n\to\infty,\end{split}$

with $r^{n}_{j}$ given by (4.4). Consequently, taking also into account (4.2)
of Proposition 1 we have
$\Omega^{\prime\prime\prime}\subset\Omega^{\prime\prime}$ and a subsequence
$n^{\prime}$ such that the first three limits are zero for
$\omega\in\Omega^{\prime\prime\prime}$. Taking the limit along the subsequence
$n^{\prime}$ in (4.17) we see that $K_{n^{\prime}}(t)$ converges for
$\omega\in\Omega^{\prime\prime\prime}$ and $t\in I(\omega)$ to some $K(t)$,
and

$\begin{split}|\tilde{v}(t)|^{2}=&2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v(s)\rangle\,dA(s)\\\
&+2\int_{(0,t]}(\tilde{v}(s)-h(s),dh(s))+|h(t)|^{2}-K(t).\\\ \end{split}$

From this point onwards we will always consider only the subsequence
$n^{\prime}$ but we will keep writing $n$ to ease notation. Our task is now to
identify $K(t)$. We note that, using Parseval’s identity,

$\begin{split}K_{n}(t)&=\sum_{0\leq\tau^{n}_{j+1}\leq
t}\left|\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1}]}v_{i}^{\ast}(s)\,dA(s)\right|^{2}\\\
&=\sum_{0\leq\tau^{n}_{j+1}\leq
t}\sum_{k\in\mathbb{N}}\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1}]}v_{i}^{\ast}(s)\,dA(s),e_{k}\right)^{2}\\\
&=\sum_{0\leq\tau^{n}_{j+1}\leq
t}\sum_{k\in\mathbb{N}}\left\langle\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1}]}v_{i}^{\ast}(s)\,dA(s),e_{k}\right\rangle^{2}\\\
&=\sum_{0\leq\tau^{n}_{j+1}\leq
t}\sum_{k\in\mathbb{N}}\left|\int_{(\tau^{n}_{j},\tau^{n}_{j+1}]}\sum_{i}\langle
v_{i}^{\ast}(s),e_{k}\rangle\,dA(s)\right|^{2}.\end{split}$

Hence, using Lemma 1, Parseval’s identity and (4.14), we get

$\displaystyle K(t)$
$\displaystyle=\lim_{n\to\infty}K_{n}(t)\geq\sum_{k\in\mathbb{N}}\varliminf_{n\to\infty}\sum_{0\leq\tau^{n}_{j+1}\leq
t}\left|\int_{(\tau^{n}_{j},\tau^{n}_{j+1}]}\sum_{i}\langle
v_{i}^{\ast}(s),e_{k}\rangle\,dA(s)\right|^{2}$
$\displaystyle=\sum_{k\in\mathbb{N}}\sum_{s\leq t}\left|\sum_{i}\langle
v_{i}^{\ast}(s),e_{k}\rangle\Delta
A(s)\right|^{2}=\sum_{k\in\mathbb{N}}\sum_{s\leq
t}\left|(\Delta^{w}(\tilde{v}-h)(s),e_{k})\right|^{2}$
$\displaystyle=\sum_{s\leq t}\left|\sum_{i}v_{i}^{\ast}(s)\right|^{2}|\Delta
A(s)|^{2}.$ (4.18)

To obtain an upper bound we use first the identity

$|x+y|^{2}=y^{2}+2x(y+x)-x^{2}$

together with the definition of $g$ to get

$\begin{split}K_{n}(t)&=\sum_{0\leq\tau^{n}_{j+1}\leq
t}\left|\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s)+\sum_{i}v_{i}^{\ast}(\tau^{n}_{j+1})\Delta
A(\tau^{n}_{j+1})\right|^{2}\\\ &=\sum_{0\leq\tau^{n}_{j+1}\leq
t}(J^{(1)}_{j}+J^{(2)}_{j}-J^{(3)}_{j})\end{split}$ (4.19)

with

$\begin{split}J^{(1)}_{j}:=&\left|\sum_{i}v_{i}^{\ast}(\tau^{n}_{j+1})\right|^{2}|\Delta
A(\tau^{n}_{j+1})|^{2},\\\
J^{(2)}_{j}:=&2\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s),\tilde{v}(\tau^{n}_{j+1})-\tilde{v}(\tau^{n}_{j})-(h(\tau^{n}_{j+1})-h(\tau^{n}_{j}))\right),\\\
J^{(3)}_{j}:=&\left|\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s)\right|^{2}\end{split}$

For $j\neq 0$ we split $J^{(2)}_{j}=J^{(21)}_{j}-J^{(22)}_{j}$ with

$J^{(21)}_{j}:=2\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s),\tilde{v}(\tau^{n}_{j+1})-\tilde{v}(\tau^{n}_{j})\right),$
$J^{(22)}_{j}:=2\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s),h(\tau^{n}_{j+1})-h(\tau^{n}_{j})\right),$

and notice that

$J^{(21)}_{j}=2\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}\langle
v_{i}^{\ast}(s),v(\tau^{n}_{j+1})-v(\tau^{n}_{j})\rangle\,dA(s)$
$=2\sum_{i}\int_{[\tau^{n}_{j},\tau^{n}_{j+1})}\langle
v_{i}^{\ast}(s),v_{n}^{(2)}(s)-v_{n}^{(1)}(s)\rangle\,dA(s).$

Using $\Pi_{k}$, the orthogonal projection of $H$ onto the space spanned by
$(e_{j})_{j=1}^{k}\subset V$, we have

$J^{(22)}_{j}=J^{(22)}_{jk}+\bar{J}^{(22)}_{jk}$

with

$J^{(22)}_{jk}:=2\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s),\Pi_{k}(h(\tau^{n}_{j+1})-h(\tau^{n}_{j}))\right),$
$\bar{J}^{(22)}_{jk}:=2\left(\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}v_{i}^{\ast}(s)\,dA(s),(I-\Pi_{k})(h(\tau^{n}_{j+1})-h(\tau^{n}_{j}))\right).$

Notice that

$J^{(22)}_{jk}=2\sum_{i}\int_{(\tau^{n}_{j},\tau^{n}_{j+1})}\langle
v_{i}^{\ast}(s),\Pi_{k}(h(\tau^{n}_{j+1})-h(\tau^{n}_{j}))\rangle\,dA(s),$
$=2\sum_{i}\int_{[\tau^{n}_{j},\tau^{n}_{j+1})}\langle
v_{i}^{\ast}(s),w^{(2)}_{kn}(s)-w^{(1)}_{kn}(s)\rangle\,dA(s),$

and

$\bar{J}^{(22)}_{jk}\leq
J_{j}^{(3)}+\left|(I-\Pi_{k})(h(\tau^{n}_{j+1})-h(\tau^{n}_{j}))\right|^{2}.$

Similarly, taking into account $\tilde{v}(0)=h(0)$, for $J^{(2)}_{0}$ we have

$J^{(2)}_{0}=2\left(\sum_{i}\int_{(0,\tau^{n}_{1})}v_{i}^{\ast}(s)\,dA(s),\tilde{v}(\tau^{n}_{1})-h(\tau^{n}_{1})\right)=J^{(21)}_{0}-J^{(22)}_{0},$

where

$J^{(21)}_{0}:=2\sum_{i}\int_{(0,\tau^{n}_{1})}\langle
v_{i}^{\ast}(s),\tilde{v}(\tau^{n}_{1})\rangle\,dA(s)$
$=2\sum_{i}\int_{(0,\tau^{n}_{1})}\langle
v_{i}^{\ast}(s),v^{(2)}(s)-v^{(1)}(s)\rangle\,dA(s),$

and

$J^{(22)}_{0}:=2\left(\sum_{i}\int_{(0,\tau^{n}_{1})}v_{i}^{\ast}(s)\,dA(s),h(\tau^{n}_{1})\right)=J^{(22)}_{0k}+\bar{J}^{(22)}_{0k}$

with

$J^{(22)}_{0k}:=\sum_{i}\int_{(0,\tau^{n}_{1})}\langle
v_{i}^{\ast}(s),w^{(2)}(s)-w^{(1)}(s)\rangle\,dA(s),$
$\bar{J}^{(22)}_{0k}:=2\left(\sum_{i}\int_{(0,\tau^{n}_{1})}v_{i}^{\ast}(s)\,dA(s),(I-\Pi_{k})h(\tau^{n}_{1})\right)$
$\leq J_{0}^{(3)}+\left|(I-\Pi_{k})h(\tau^{n}_{1})\right|^{2}.$

Thus from (4.19) we get

$\begin{split}K_{n}(t)\leq&\sum_{0\leq\tau^{n}_{j+1}\leq
t}|\sum_{i}v_{i}^{\ast}(\tau^{n}_{j+1})|^{2}|\Delta A(\tau^{n}_{j+1})|^{2}\\\
&+2\sum_{i}\int_{(0,t)}\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)-v^{(1)}_{n}(s)\rangle\,dA(s)\\\
&-2\sum_{i}\int_{(0,t)}\langle
v_{i}^{\ast}(s),w^{(2)}_{nk}(s)-w^{(1)}_{nk}(s)\rangle\,dA(s)+\xi_{nk}(t)\end{split}$

with

$\xi_{nk}(t):=\sum_{j=1}^{N(n)}\left|(I-\Pi_{k})(h(\tau^{n}_{j+1}\wedge
t)-h(\tau^{n}_{j}\wedge t))\right|^{2}+\left|(I-\Pi_{k})h(\tau^{n}_{1}\wedge
t)\right|^{2}$

for every $n,k\in\mathbb{N}$. As $n\to\infty$ we see that

$\sum_{0\leq\tau^{n}_{j+1}\leq
t}|\sum_{i}v_{i}^{\ast}(\tau^{n}_{j+1})|^{2}|\Delta
A(\tau^{n}_{j+1})|^{2}\to\sum_{0<s\leq t}|v^{\ast}(s)|^{2}|\Delta A(s)|^{2},$

where we use the notation $v^{\ast}(s)=\sum_{i}v^{\ast}_{i}(s)$. By Hölder’s
inequality, taking into account $r(t)\leq 1$, we have

$\varlimsup_{n\to\infty}\int_{(0,t)}|\langle
v_{i}^{\ast}(s),v^{(2)}_{n}(s)-v^{(1)}_{n}(s)\rangle|\,dA(s)$
$\leq\varlimsup_{n\to\infty}\left(\int_{(0,t)}\|v^{(2)}_{n}(s)-v^{(1)}_{n}(s)\|_{V_{i}}^{p_{i}}\,dA(s)\right)^{\tfrac{1}{p_{i}}}=0$

and similarly,

$\varlimsup_{n\to\infty}\int_{(0,t)}|\langle
v_{i}^{\ast}(s),w^{(2)}_{nk}(s)-w^{(1)}_{nk}(s)\rangle|\,dA(s)=0$

for all integers $k\geq 1$ and $i=1,2,\ldots,m$. Thus

$K(t)=\varliminf_{n\to\infty}K_{n}(t)\leq\sum_{s\leq
t}|{v}^{\ast}(s)|^{2}|\Delta A(s)|^{2}+\xi_{k}(t)$ (4.20)

for every $k\in N$, where

$\xi_{k}(t):=\varliminf_{n\to\infty}\left(\sum_{j=1}^{N(n)}\left|(I-\Pi_{k})(h(\tau^{n}_{j+1}\wedge
t)-h(\tau^{n}_{j}\wedge t))\right|^{2}+\left|(I-\Pi_{k})h(\tau^{n}_{1}\wedge
t)\right|^{2}\right).$

Note that by Fatou’s lemma and the martingale property of $h$

$\begin{split}&\mathbb{E}\xi_{k}(t)\leq\mathbb{E}|(I-\Pi_{k})h(1)|^{2}\to
0\,\,\,\textrm{as}\,\,\,k\to\infty.\end{split}$

Note also that for each $\omega\in\Omega$ and $t\in(0,\infty)$ we have
$\xi_{k}\geq\xi_{k+1}$ and $\xi_{k}\geq 0$. Thus there exists a set
$\Omega^{\prime\prime\prime\prime}\subset\Omega$ with
$P(\Omega^{\prime\prime\prime\prime})=1$ such that for every $t\in[0,\infty)$
and $\omega\in\Omega^{\prime\prime\prime\prime}$ we have $\xi_{k}(t)\to 0$.
Letting here $k\to\infty$ in (4.20) we obtain

$K(t)\leq\int_{(0,t]}|{v}^{\ast}(s)|^{2}\Delta A(s)\,dA(s),$

which together with (4.18) gives

$K(t)=\int_{(0,t]}|v^{\ast}(s)|^{2}\Delta A(s)\,dA(s)$

for
$\omega\in\tilde{\Omega}:=\Omega^{\prime\prime\prime}\cap\Omega^{\prime\prime\prime\prime}$
and $t\in I(\omega)$. ∎

######  Proposition 5.

For $\omega\in\tilde{\Omega}$

$\begin{split}&|\tilde{v}(t)|^{2}=|h(0)|^{2}+2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v(s)\rangle\,dA(s)\\\
&+2\int_{(0,t]}(\tilde{v}(s-),dh(s))-\int_{(0,t]}\left|\sum_{i=1}^{m}v_{i}^{\ast}(s)\right|^{2}\Delta
A(s)\,dA(s)+[h]_{t}\end{split}$ (4.21)

for $t\in[0,\tau(\omega))$.

###### Proof.

Let $\omega\in\tilde{\Omega}$ be fixed and let $t\in[0,\tau(\omega))$. To ease
notation we use $n\to\infty$ in place of the subsequence $n^{\prime}\to\infty$
defined in the previous proposition. If $t\in I(\omega)$, then by virtue of
the previous proposition taking $n\to\infty$ in (4.17) we obtain

$\displaystyle|\tilde{v}(t)|^{2}=$
$\displaystyle|h(t)|^{2}+2\sum_{i=1}^{m}\int_{(0,t]}\langle
v_{i}^{\ast}(s),v(s)\rangle\,dA(s)+2\int_{(0,t]}(\tilde{v}(s-)-h(-),dh(s))$
$\displaystyle-\int_{(0,t]}\left|\sum_{i=1}^{m}v_{i}^{\ast}(s)\right|^{2}\Delta
A(s)\,dA(s).$

Hence using the Itô formula for Hilbert space valued processes

$|h(t)|^{2}=|h(0)|^{2}+2\int_{(0,t]}(h(s-),dh(s))+[h]_{t},$ (4.22)

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
$n\to\infty$. Using the algebraic relationship

$|\tilde{v}(s)-\tilde{v}(r)|^{2}=|\tilde{v}(s)|^{2}-|\tilde{v}(r)|^{2}-2\left(\tilde{v}(r),\tilde{v}(s)-\tilde{v}(r)\right),$
(4.23)

with $s:=t_{n}$, $r:=t_{l}$, and since (4.21) holds for every $t\in
I(\omega)$, we get

$\begin{split}&|\tilde{v}(t_{n})-\tilde{v}(t_{l})|^{2}=2\sum_{i=1}^{m}\int_{(t_{l},t_{n}]}\langle
v_{i}^{\ast}(s),v(s)\rangle\,dA(s)+2\int_{(t_{l},t_{n}]}(\tilde{v}(s-),dh(s))\\\
&-\int_{(t_{l},t_{n}]}\left|\sum_{i=1}^{m}v_{i}^{\ast}(s)\right|^{2}\Delta
A(s)\,dA(s)+[h]_{t_{n}}-[h]_{t_{l}}-2(\tilde{v}(t_{l}),\tilde{v}(t_{n})-\tilde{v}(t_{l}))\end{split}$

for $n>l$. Moreover

$\begin{split}&2(\tilde{v}(t_{l}),\tilde{v}(t_{n})-\tilde{v}(t_{l}))\\\
&=2\sum_{i=1}^{m}\int_{(t_{l},t_{n}]}\langle v_{i}^{\ast}(s),v(t_{l})\rangle
dA(s)+2(\tilde{v}(t_{l}),h(t_{n})-h(t_{l})).\end{split}$

Hence by (4.22)

$\begin{split}|\tilde{v}(t_{n})-\tilde{v}(t_{l})|^{2}=&2\sum_{i=1}^{m}\int_{(t_{l},t_{n}]}\langle
v_{i}^{\ast}(s),v(s)-v(t_{l})\rangle dA(s)\\\
&+2\int_{(t_{l},t_{n}]}(\tilde{v}(s-)-h(s-)-(\tilde{v}(t_{l})-h(t_{l})),dh(s))\\\
&-\int_{(t_{l},t_{n}]}\left|\sum_{i=1}^{m}v_{i}^{\ast}(s)\right|^{2}\Delta
A(s)dA(s)+|h(t_{n})-h(t_{l})|^{2}\\\
&=:2I^{1}_{ln}+2I^{2}_{ln}-I^{3}_{ln}+I^{4}_{ln}.\end{split}$

Since $h$ is cadlag we have

$\lim_{l\to\infty}\sup_{n>l}I^{4}_{ln}=\lim_{l\to\infty}\sup_{n>l}|h(t_{n})-h(t_{l})|^{2}=0.$

By the previous proposition we get

$\begin{split}&\lim_{l\to\infty}\sup_{n>l}|I^{2}_{ln}|=\lim_{l\to\infty}\sup_{n>l}\left|\int_{(t_{l},t_{n}]}(\tilde{v}(s-)-h(s-)-(\tilde{v}_{l}(s)-h_{l}(s)),dh(s))\right|\\\
&\leq
2\lim_{l\to\infty}\sup_{t\in(0,\infty)}\left|\int_{(0,t]}(\tilde{v}(s-)-h(s-)-(\tilde{v}_{l}(s)-h_{l}(s)),dh(s))\right|=0,\end{split}$

and

$\begin{split}&\lim_{l\to\infty}\sup_{n>l}|I^{1}_{ln}|\leq\lim_{l\to\infty}\sum_{i=1}^{m}\int_{(0,\infty)}\|v_{i}^{\ast}(s)\|_{V_{i}^{*}}\|v(s)-v^{(1)}_{l}(s)\|_{V_{i}}\,dA(s)=0,\end{split}$

via $r(t)\leq 1$ and Hölder’s inequality. Thus

$\lim_{l\to\infty}\sup_{n>l}|\tilde{v}(t_{n})-\tilde{v}(t_{l})|^{2}=0,$

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 $n\to\infty$ we obtain

$\begin{split}&|\tilde{v}(t-)|^{2}=|h(0)|^{2}+2\sum_{i=1}^{m}\int_{(0,t)}\langle
v_{i}^{\ast}(s),v(s)\rangle\,dA(s)\\\
&+2\int_{(0,t)}(\tilde{v}(s-),dh(s))-\int_{(0,t)}\left|\sum_{i=1}^{m}v_{i}^{\ast}(s)\right|^{2}\Delta
A(s)\,dA(s)+[h]_{t-}\end{split}$

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 $\Delta A(t)=0$. If
$t\in(0,\tau(\omega))\setminus\bar{I}^{l}(\omega)$, then there is
$t^{\prime}\in\\{0\\}\cup\bar{I}^{l}(\omega)$ such that $t^{\prime}<t$ and
$(t^{\prime},t]\cap I(\omega)=\emptyset$. Thus $dA(s)=0$ for
$s\in(t^{\prime},t]$, and so
$\tilde{v}(s)-\tilde{v}(t^{\prime})=h(s)-h(t^{\prime})$. Hence applying (4.21)
with $t:=t^{\prime}$, and the formula

$|\tilde{v}(t)|^{2}-|\tilde{v}(t^{\prime})|^{2}=2(\tilde{v}(t^{\prime}),\tilde{v}(t)-\tilde{v}(t^{\prime}))+|\tilde{v}(t)-\tilde{v}(t^{\prime})|^{2}$

together with the Itô formula for Hilbert space valued martingales,

$|h(t)-h(t^{\prime})|^{2}=2\int_{(t^{\prime},t]}(h(s-)-h(t^{\prime}),dh(s))+[h]_{t}-[h]_{t^{\prime}},$

we obtain (4.21) for the $t$ under consideration. ∎

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
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.

### Acknowledgements

The authors are sincerely grateful to the anonymous referees. Their
corrections and valuable suggestions helped improve the presentation of the
paper.

### Open Access

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## References

  * (1) K. Dareiotis and I. Gyöngy. A Comparison Principle for Stochastic Integro-Differential Equations. Potential Anal. 41:4, 1203–1222, 2014.
  * (2) L. C. Evans. Partial Differential Equations. AMS, Providence, Rhode Island, 1998.
  * (3) H. Gajewski, K. Gröger and K. Zacharias. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974.
  * (4) G. Da Prato, A. Jentzen and M. Röckner. A mild Itô formula for SPDEs. arXiv:math. 2016.
  * (5) I. Gyöngy and N. V. Krylov. On Stochastic Equations with Respect to Semimartingales I. Stochastics, 4(1):1–21, 1980.
  * (6) I. Gyöngy and N. V. Krylov.  On Stochastic Equations with Respect to Semimartingales II. Itô Formula in Banach Spaces.  Stochastics, 6(3):153–173, 1982.
  * (7) I. Gyöngy. On Stochastic Equations with Respect to Semimartingales III. Stochastics, 7(4):231–254, 1982.
  * (8) I. Gyöngy, S. Sabanis and D. Šiška. Convergence of tamed Euler schemes for a class of stochastic evolution equations. Stoch. Partial Differ. Equ. Anal. Comput., 4(2):225–245, 2016.
  * (9) N.V. Krylov. Maximum principle for SPDEs and its applications in Stochastic differential equations: theory and applications, Interdisciplinary Mathematical Sciences, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
  * (10) N.V. Krylov. Itô’s formula for the $L_{p}$-norm of stochastic $W_{p}^{1}$-valued processes. Probab. Theory Relat. Fields, 147:583–605, 2010.
  * (11) N. V. Krylov. A relatively short proof of Itô’s formula for SPDEs and its applications. Stoch PDE: Anal Comp, 1:152–174, 2013.
  * (12) N. V. Krylov and B. L. Rozovskii. Stochastic evolution equations. J. Sov. Math., 16:1233–1277, 1981.
  * (13) J. L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris, 1969.
  * (14) E. Pardoux. Équations aux dériveés partielles stochastiques non linéaries monotones. Étude des solution forte de type Ito. These Université de Paris Sud, Orsay, 1975.
  * (15) C. Prévôt and M. Röckner. A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007.
  * (16) B.L. Rozovskii. Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht, 1990.