On Itô formulas for jump processes

Abstract

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of $L_p$-valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.

1 Introduction

This is a review paper on some Itô formulas in finite- and infinite-dimensional spaces. First we consider finite-dimensional Itô–Lévy processes, which are $\mathbb {R}^M$-valued stochastic processes $X=(X_t)_{t\ge 0}$ given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures. They play important roles in modelling stochastic phenomena when jumps may occur at random times; see, for example, [4, 5]. Chain rules, called Itô formulas, for their transformations $\phi (X_t)$ by sufficiently smooth functions $\phi$ are basic tools in the investigations of stochastic phenomena modelled by Itô–Lévy processes; see, for example, [13] and the references therein. It is therefore important to have Itô formulas for large classes of processes X and functions $\phi$. Note that classical Itô’s formula (2.4) holds only under some restrictive conditions, which are not satisfied in important applications, for example in applications to filtering theory of partially observed jump diffusions. Therefore, we revisit the chain rule (2.4) for finite-dimensional Itô–Lévy processes, discuss its limitations and derive formula (2.12) from it, which corresponds to a well-known Itô formula for general semimartingales, and is valid without restrictive conditions on the Itô–Lévy processes X and on the functions $\phi$.

In the second part of the paper, we discuss infinite-dimensional generalisations of the Itô formula (2.12) from the point of view of applications in stochastic PDEs (SPDEs). In the theory of parabolic SPDEs, arising in nonlinear filtering theory, the solutions $v=v_t(x)$ of SPDEs have the stochastic differentials

$$\begin{aligned} dv_t(x)=\left( f^0_t(x)+\sum _{i=1}^d\tfrac{\partial }{\partial x^i} f^i_t(x)\right) \,dt +\sum _{r}g^{r}_t(x)\,dm_t^r \end{aligned}$$

(1.1)

with appropriate random functions $f^{\alpha }$ and $g^r$ of $t\in [0,T]$ and $x=(x^1,\ldots ,x^d)\in \mathbb {R}^d$, and a sequence of martingales $(m^i)_{i=1}^{\infty }$. This stochastic differential is understood in a weak sense, i.e. for each smooth function $\varphi$ with compact support on $\mathbb {R}^d$ we have the stochastic differential

$$\begin{aligned} d(v_t,\varphi )=\left( (f^0_t,\varphi )-\sum _i\left( f^i_t,\tfrac{\partial }{\partial x^i}\varphi \right) \right) \,dt +\sum _{r}(g^{r}_t,\varphi )\,dm_t^r, \end{aligned}$$

where (u, v) denotes the Lebesgue integral over $\mathbb {R}^d$ of the product uv for functions u and v of $x\in \mathbb {R}^d$. In the $L_2$-theory of SPDEs $f^{\alpha }$ and $g^r$ are $L_2(\mathbb {R}^d,\mathbb {R})$-valued functions of $(\omega ,t)$, satisfying appropriate measurability conditions, and to get ‘a priori estimates’, a suitable formula for $|v|^2_{L_2}$ plays crucial roles. Such a formula in an abstract setting was first obtained in [18] when $(m^i)_{i=1}^{\infty }$ is a sequence of independent Wiener processes. The proof in [18] is connected with the theory of SPDEs developed in [18]. A direct proof was given in [17], which was generalised in [8] to the case of square integrable martingales $m=(m^i)$. A nice short proof was presented in [15], and further generalisations can be found, for example, in [9, 19]. The above results on Itô formula are used in the $L_2$-theory of linear and nonlinear SPDEs to obtain existence, uniqueness and regularity results under various assumptions; see, for example, [7, 17,18,19,20]. To have a similar tool for studying solvability, uniqueness and regularity problems for solutions in $L_p$-spaces for $p\ne 2$ one should establish a suitable formula for $|v_t|^p_{L_p}$, which was first achieved in Krylov [14] for $p\ge 2$ when $(m^{i})_{i=1}^{\infty }$ is a sequence of independent Wiener processes.

In Sect. 3, we present a generalisation of the main result from Krylov [14] to the case when the stochastic differential of $v_t$ is of the form

$$\begin{aligned} dv_t(x)=\left( f^0_t(x)+\sum _{i=1}^d\tfrac{\partial }{\partial x^i} f^i_t(x)\right) \,dt +\sum _{r}g^{r}_t(x)\,dw_t^r+\int _{Z}h_t(z,x)\tilde{\pi }(dz,dt), \end{aligned}$$

(1.2)

where $\tilde{\pi }(dz,dt)$ is a Poisson martingale measure with a $\sigma$-finite characteristic measure $\mu$ on a measurable space $(Z,\mathcal {Z})$ and h is a function on $\Omega \times [0,T]\times Z\times \mathbb {R}^d$. This is Theorem 3.1, which is a slight generalisation of Theorem 2.2 on Itô’s formula from [10] for $|v_t|^p_{L_p}$ for $p\ge 2$. We prove it by adapting ideas and methods from Krylov [14]. In particular, we use the finite-dimensional Itô’s formula (2.19) for $|v^{\varepsilon }_t(x)|^p$ for each $x\in \mathbb {R}^d$, where $v_t^{\varepsilon }$ is an approximation of $v_t$ obtained by smoothing it in x. Hence, we integrate both sides of the formula for $|v^{\varepsilon }_t(x)|^p$ over $\mathbb {R}^d$, change the order of deterministic and stochastic integrals, integrate by parts in terms containing derivatives of smooth approximations of $f^i$, and finally, we let $\varepsilon \rightarrow 0$. Though the idea of the proof is simple, there are several technical difficulties to implement it. We sketch the proof of Theorem 3.1 in Sect. 3, further details of the proof can be found in [10]. Theorem 3.1 plays a crucial role in proving existence, uniqueness and regularity results in [11] for solutions to stochastic integro-differential equations. In [11], instead of a single random field $v_t(x)$ we have to deal with a system of random fields $v^i_t(x)$ for $i=1,2,\ldots ,M$, and we need estimates for $||\sum _{i}|v^i|^2|^{1/2}|_{L_p}$. This is why in Theorem 3.1 we consider a system of random fields $v^i$, $i=1,2,\ldots ,M$.

There are known theorems in the literature on Itô’s formula for semimartingales with values in separable Banach spaces; see, for example, [3, 21,22,23,24]. In some directions, these results are more general than Theorem 3.1, but they do not cover it. In [3, 22], only continuous semimartingales are considered and their differential does not contain $D_if^i\,dt$ terms. In [21, 23, 24], semimartingales containing stochastic integrals with respect to Poisson random measures and martingale measures are considered, but they do not contain terms corresponding to $D_if^i$. Thus, the Itô formula in these papers cannot be applied to $|v_t|_{L_p}^p$ when the stochastic differential $d v_t$ is given by (1.2).

In conclusion, we present some notions and notation. All random elements are given on a fixed complete probability space $(\Omega ,\mathcal {F},P)$ equipped with a right-continuous filtration $(\mathcal {F}_t)_{t\ge 0}$ such that $\mathcal {F}_0$ contains all P-zero sets of $\mathcal {F}$. The $\sigma$-algebra of the predictable subsets of $\Omega \times [0,\infty )$ is denoted by $\mathcal {P}$. We are given a sequence $w=(w^1_t,w^2_t,\ldots )_{t\ge 0}$ of $\mathcal {F}_t$-adapted independent Wiener processes $w^r=(w^r_t)_{t\ge 0}$, such that $w_t-w_s$ is independent of $\mathcal {F}_s$ for any $0\le s\le t$. For an integer $m\ge 1$, we are given also a sequence of independent Poisson random measures $\pi ^k(dz,dt)$ on $[0,\infty )\times Z^k$, with intensity measure $\mu ^k(dz)\,dt$ for $k=1,2,\ldots , m$, where $\mu ^k$ is a $\sigma$-finite measure on a measurable space $(Z^k,\mathcal {Z}^k)$ with a countably generated $\sigma$-algebra $\mathcal {Z}^k$. We assume that the process $\pi ^k_t(\Gamma ):=\pi ^k(\Gamma \times (0,t])$, $t\ge 0$, is $\mathcal {F}_t$-adapted and $\pi ^k_t(\Gamma )-\pi ^k_s(\Gamma )$ is independent of $\mathcal {F}_s$ for any $0\le s\le t$ and $\Gamma \in \mathcal {Z}^k$ such that $\mu ^k(\Gamma )<\infty$. We use the notation $\tilde{\pi }^k(dz,dt)=\pi ^k(dz,dt)-\mu ^k(dz)dt$ for the compensated Poisson random measure and set $\tilde{\pi }^k_t(\Gamma )=\tilde{\pi }^k(\Gamma \times (0,t])=\pi _t^k(\Gamma )-t\mu ^k(\Gamma )$ for $t\ge 0$ and $\Gamma \in \mathcal {Z}$ such that $\mu ^k(\Gamma )<\infty$. If $m=1$, then we write $\pi$, $\tilde{\pi }$, Z, $\mathcal {Z}$ and $\mu$ in place of $\pi ^1$, $\tilde{\pi }^1$, $Z^1$, $\mathcal {Z}^1$ and $\mu ^1$, respectively. For basic results concerning stochastic integrals with respect to Poisson random measures and Poisson martingale measures, we refer to [1, 12, 16].

Let $M>0$ be an integer. The space of sequences $\nu =(\nu ^{1},\nu ^{2},\ldots )$ of vectors $\nu ^{k}\in \mathbb {R}^{M}$ with finite norm

$$\begin{aligned} |\nu |_{\ell _{2} }=\left( \sum _{k=1}^{\infty }|\nu ^k|^{2}\right) ^{1/2} \end{aligned}$$

is denoted by $\ell _2=\ell _2(\mathbb {R}^M)$ and by $l_2$ when $M=1$. We use the notation $D_i$ to denote the ith derivative, i.e.

$$\begin{aligned} D_i=\frac{\partial }{\partial x_i},\quad i=1,2,\ldots ,M. \end{aligned}$$

For vectors v from Euclidean spaces, |v| means the Euclidean norm of v. The space of smooth functions with compact support in $\mathbb {R}^M$ is denoted by $C^{\infty }_0(\mathbb {R}^M)$. For integers $k\ge 1$, the notation $C^k(\mathbb {R}^M)$ means the space of functions on $\mathbb {R}^M$ whose derivatives up to order k exist and are continuous, and $C_b^k(\mathbb {R}^M)$ denotes the space of functions on $\mathbb {R}^M$ whose derivatives up to order k are bounded continuous functions. When we talk about the derivatives up to order k of a function f, then among these derivatives we always consider the ‘ zeroth-order derivative’ of f, i.e. f itself.

2 Itô formulas in finite dimensions

We consider an $\mathbb {R}^M$-valued semimartingale $X=(X^1_t,\ldots ,X^M_t)_{t\ge 0}$ given by

$$\begin{aligned}&X_t=X_0+\int _0^tf_s\,ds +\int _0^tg_s^{r}\,dw_s^r\nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z^k}{\bar{h}}^k_s(z)\,\pi ^k(dz,ds) +\sum _{k=1}^m\int _0^t\int _{Z^k}h^k_s(z)\,\tilde{\pi }^k(dz,ds), \quad \text {for }t\ge 0, \end{aligned}$$

(2.1)

where $X_0$ is an $\mathbb {R}^M$-valued $\mathcal {F}_0$-measurable random variable, $f=(f^i_t)_{t\ge 0}$ and $g=(g^{ir}_t)_{t\ge 0}$ are predictable processes with values in $\mathbb {R}^M$ and $\ell _2=\ell _2(\mathbb {R}^M)$, respectively, ${\bar{h}}^k=({\bar{h}}_t^{ik}(z))_{t\in [0,T]}$ and $h^k=(h_t^{ik}(z))_{t\ge 0}$ are $\mathbb {R}^M$-valued $\mathcal {P}\otimes \mathcal {Z}$-measurable functions on $\Omega \times \mathbb {R}_+\times Z$ for every $k=1,2,\ldots ,m$, such that almost surely for every $k=1,2,\ldots ,m$

$$\begin{aligned} {\bar{h}}_t^{ik}(z)h_t^{jk}(z)=0 \quad \text {for}\quad i,j=1,2,\ldots ,M,\quad \text {for all}\;t\ge 0\quad \text {and}\quad z\in Z, \end{aligned}$$

(2.2)

and

$$\begin{aligned}&\sum _{k=1}^m\left( \int _0^T\int _{Z_k}|{\bar{h}}^k_t(z)|\,\pi ^k(dz,dt)+ \int _0^T\int _{Z_k}|h^k_t(z)|^2\mu ^k(dz)\,dt\right)<\infty ,\quad \nonumber \\&\quad \int _0^T|f_t|+|g_t|^2_{\ell _2}\,dt<\infty \end{aligned}$$

(2.3)

for every $T>0$. Here and later on, unless otherwise indicated, the summation convention with respect to repeated integer-valued indices is used, i.e. $g^r_s\,dw^r_s$ means $\sum _rg^r_s\,dw^r_s$.

The following Itô’s formula is well known for $m=1$.

Theorem 2.1

Let conditions (2.2) and (2.3) hold and assume there is a constant K such that $|h^k|\le K$ for all $(\omega ,t,z)\in \Omega \times \mathbb {R}_{+}\times Z$ and $k=1,2,\ldots ,m$. Then, for any $\phi \in C^2(\mathbb {R}^M)$, the process $(\phi (X_t))_{t\ge 0}$ is a semimartingale such that

$$\begin{aligned} \phi (X_t)&=\phi (X_0) +\int _0^tf^i_sD_i\phi (X_s)+\tfrac{1}{2}g_s^{ir}g_s^{jr}D_{i}D_{j}\phi (X_s)\,ds +\int _0^tg^{ir}_sD_i\phi (X_s)\,dw^r_s \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}\phi (X_{s-}+{\bar{h}}^k_s(z))-\phi (X_{s-})\,\pi ^k(dz,ds) \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}\phi (X_{s-}+h^k_s(z))-\phi (X_{s-})\,\tilde{\pi }^k(dz,ds) \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k} \left( \phi (X_s+h^k_s(z))-\phi (X_s)-h^{ik}_s(z)D_i\phi (X_s) \right) \,\mu ^k(dz)\,ds \end{aligned}$$

(2.4)

holds almost surely for all $t\ge 0$.

Proof

This theorem with a finite-dimensional Wiener process $w=(w^1,\ldots ,w^{d_1})$ in place of an infinite sequence of independent Wiener processes and for $m=1$ is proved, for example, in [12]; see Theorem 5.1 in chapter II. Following this proof with appropriate changes, one can easily prove the above theorem as follows: Since $\mu ^k$ is $\sigma$-finite for $k=1,2,\ldots ,m$, for each k we have an increasing sequence $(Z^k_n)_{n=1}^{\infty }$ of sets $Z_n^k\in \mathcal {Z}^k$ such that $Z^k=\cup _{n=1}^{\infty }Z_n^k$ and $\mu ^k(Z_n^k)<\infty$ for every n. For a fixed integer $n\ge 1$, let $\rho ^k_1<\rho ^k_2<...$ denote the increasing sequence of times where the jumps of $N^k:=(\pi ^k_t(Z_n^k))_{t\ge 0}$ occur. Similarly, let $\tau _1<\tau _2<...$ be the jump times of the process $N=\sum _{k=1}^mN_k$. Then $\rho ^k_i$ and $\tau _i$ are stopping times for every $k=1,2,\ldots ,m$ and $i\ge 1$, and for almost every $\omega \in \Omega$, the set of time points $\{\tau _i(\omega ):i\ge 1\}$ contains all points of discontinuities of $(X^n_t(\omega ))_{t\ge 0}$, where the process $X^n$ is defined by

$$\begin{aligned} X^n_t&=X_0+\int _0^tf_s\,ds +\int _0^tg_s^{r}\,dw_s^r+\sum _{k=1}^m\int _0^t\int _{Z^k}{\bar{h}}^k_s(z)\mathbf{1}_{Z^k_n}(z)\,\pi ^k(dz,ds)\nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z^k}h^k_s(z)\mathbf{1}_{Z^k_n}(z)\,{\pi }^k(dz,ds) -V^n_t \quad \text {for }t\ge 0 \end{aligned}$$

(2.5)

with

$$\begin{aligned} V^n_t:=\sum _{k=1}^m\int _0^t\int _{Z^k}h^k_s(z)\mathbf{1}_{Z^k_n}(z)\,\mu ^k(dz)\,ds. \end{aligned}$$

Clearly, $\phi (X^n_t)=\phi (X^n_0)+A^n_t+B_t^n$ with

$$\begin{aligned} A^n_t=\sum _{i\ge 1}\big (\phi (X^n_{\tau _i\wedge t})-\phi (X^n_{\tau _i\wedge t-})\big ),\quad B^n_t=\sum _{i\ge 1}\big (\phi (X^n_{\tau _i\wedge t-})-\phi (X^n_{\tau _{i-1}\wedge t})\big ), \end{aligned}$$

where we set $\tau _0:=0$ and $X^n_{\tau _i\wedge t-}:=X^n_{\tau _i-}$ for $t\ge \tau _i$ and $X^n_{\tau _i\wedge t-}:=X^n_t$ for $t<\tau _i$. By Itô’s formula for Itô processes, we have

$$\begin{aligned}&\phi (X^n_{\tau _i\wedge t-})-\phi (X^n_{\tau _{i-1}\wedge t}) =\int _{\tau _{i-1}\wedge t}^{\tau _i\wedge t-}D_l\phi (X^n_s)f^l_s +\tfrac{1}{2}D_{jl}\phi (X^n_s)g^{jr}g^{lr}\,ds \\&\quad +\int _{\tau _{i-1}\wedge t}^{\tau _i\wedge t-}D_l\phi (X^n_s)g^{lr}_s\,dw^r_s -\int _{\tau _{i-1}\wedge t}^{\tau _i\wedge t-}D_l\phi (X^n_s)\,dV^n_s, \end{aligned}$$

which gives

$$\begin{aligned} B^n_t&=\int _{0}^tD_l\phi (X^n_s)f^l_s +\tfrac{1}{2}D_{jl}\phi (X^n_s)g^{jr}g^{lr}\,ds\nonumber \\&\quad +\int _{0}^{t}D_l\phi (X^n_s)g^{lr}_s\,dw^r_s -\int _{0}^{t}D_l\phi (X^n_s)\,dV^n_s. \end{aligned}$$

(2.6)

Notice that $\rho ^k_i$ has a density with respect to the Lebesgue measure for $i\ge 1$, and $\rho ^k_i$ and $\rho ^l_j$ are independent for $k\ne l$. Hence, $P(\rho ^k_i=\rho ^l_j)=0$ for $k\ne l$ and positive integers i, j. Consequently, for almost every $\omega \in \Omega$ we have $\{\tau _i(\omega ):i\ge 1\}=\cup _{k=1}^{m}\{\rho ^k_i(\omega ):i\ge 1\}$ such that the sets in the union are almost surely pairwise disjoint. Hence, taking also into account condition (2.2), we get that almost surely

$$\begin{aligned} A^n_t=\sum _{k=1}^m\sum _{i\ge 1} \left( \phi (X^n_{\rho ^k_i\wedge t})-\phi (X^n_{\rho ^k_i\wedge t-})\right) ={\bar{A}}^n_t+{{\tilde{A}}}^n_t \end{aligned}$$

for all $t\ge 0$, where

$$\begin{aligned} {\bar{A}}^n_t&=\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+{\bar{h}}^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\pi ^k(dz,ds), \\ {{\tilde{A}}}^n_t&=\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+h^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\pi ^k(dz,ds) \\&=\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+h^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\tilde{\pi }^k(dz,ds) \\&\quad +\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+h^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\mu ^k(dz)\,ds. \end{aligned}$$

Combining this with (2.6) we get

$$\begin{aligned} \phi (X^n_t)&=\phi (X_0)+\int _{0}^tD_l\phi (X^n_s)f^l_s +\tfrac{1}{2}D_{jl}\phi (X^n_s)g^{jr}g^{lr}\,ds+\int _{0}^{t}D_l\phi (X^n_s)g^{lr}_s\,dw^r_s \\&\quad +\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+{\bar{h}}^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\pi ^k(dz,ds), \\&\quad +\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+h^k_s(z))-\phi (X^n_{s-})\big ) \mathbf{1}_{Z^n_k}(z)\,\tilde{\pi }^k(dz,ds) \\&\quad +\sum _{k=1}^m \int _0^t\int _{Z^k}\big (\phi (X^n_{s-}+h^k_s(z))-\phi (X^n_{s-})\\&\quad -D_l\phi (X^n_s)h^{lk}_s(z)\big ) \mathbf{1}_{Z^n_k}(z)\,\mu ^k(dz)\,ds. \end{aligned}$$

Hence, we can finish the proof by letting $n\rightarrow \infty$ and using standard facts about convergence of Lebesgue integrals and stochastic integrals with respect to Wiener processes and random measures. $\square$

In some publications, only the natural conditions (2.2) and (2.3) are assumed in the formulation of the above theorem, but these conditions are not sufficient for (2.4) to hold, as the following simple example shows.

Example 2.1

Consider a one-dimensional semimartingale $(X_t)_{t\in [0,T]}$ given by (2.1) with $f=0$, $g=0$, ${\bar{h}}=0$ and $h_t(z)=\mathbf{1}_{t>0}t^{-1/4}$, $t\ge 0$, $z\in Z=\mathbb {R}{\setminus }\{0\}$, when $\pi (dz,dt)$ is the measure of jumps of a standard Poisson process and $\tilde{\pi }(dz,dt)=\pi (dz,dt)-\mu (dz)dt$ is its compensated measure, where $\mu =\delta _1$ is the Dirac measure on Z concentrated at 1. Then obviously conditions (2.2) and (2.3) hold, and for $\phi (x)=x^4$, the last integrand in (2.4) is

$$\begin{aligned} |X_{s-}+h_s(z)|^4-|X_{s-}|^4-4X_{s-}^3h_s(z) =\sum _{i=1}^3c_i(s,z) \end{aligned}$$

with

$$\begin{aligned} c_1(s,z)=6|X_{s-}|^2\mathbf{1}_{s>0}s^{-1/2}, \quad c_2(s,z)=4X_{s-}\mathbf{1}_{s>0}s^{-3/4}, \quad c_3(s,z)=\mathbf{1}_{s>0}s^{-1}. \end{aligned}$$

Clearly,

$$\begin{aligned} \int _0^t\int _Z|c_i(s,z)|\,\mu (dz)\,ds<\infty \quad \text {for~} i=1,2, \quad \text {and} \quad \int _0^t\int _Zc_3(s,z)\,\mu (dz)\,ds=\infty \end{aligned}$$

for every $t>0$, which shows that the last integral in (2.4) is infinite. Similarly, one can show that almost surely

$$\begin{aligned} \int _0^t\int _Z(|X_{s-}+h_s(z)|^4-|X_{s-}|^4)^2\,\mu (dz)\,ds=\infty \quad \text {for every }t>0, \end{aligned}$$

which means the stochastic integral with respect to $\tilde{\pi }(dz,ds)$ in (2.4) does not exist.

It is easy to see that the last two integrals in (2.4) are well defined as Itô and Lebesgue integrals, respectively, under the additional boundedness assumption on h. Instead of this extra condition on h, one can make additional assumptions on $\phi$ to ensure that formula (2.4) holds. It is sufficient to assume that the derivatives of $\phi$ up to second order are bounded. Such a condition, however, excludes the applicability of Itô’s formula to power functions $\phi (x)=|x|^p$ for $p\ge 2$. Notice that for any $\phi \in C^2(\mathbb {R}^M)$ the conditions

$$\begin{aligned} \sum _{k=1}^m\int _0^T\int _{Z_k}|\phi (X_s+h^k_s(z))-\phi (X_s)|^2\,\mu ^k(dz)\,ds<\infty \end{aligned}$$

(2.7)

and

$$\begin{aligned} \sum _{k=1}^m\int _0^T\int _{Z_k}|\phi (X_s+h^k_s(z))-\phi (X_s)-h^k_s(z)\nabla \phi (X_s)| \,\mu ^k(dz)\,ds<\infty \quad \mathrm{(a.s.)} \end{aligned}$$

(2.8)

ensure the existence of the last two integrals in (2.4), respectively. Thus, we can expect that under conditions (2.2)–(2.3) and (2.7)–(2.8) formula (2.4) is valid.

Theorem 2.2

Let conditions (2.2)–(2.3) and (2.7)–(2.8) hold. Assume $\phi \in C^2(\mathbb {R}^M)$. Then $\phi (X_t)$ is a semimartingale such that (2.4) holds almost surely for all $t\ge 0$.

Proof

This theorem is a slight generalisation of Theorem 5.2 in [2]. For the convenience of the reader we deduce this theorem from Theorem 2.1 here. For notational simplicity, we assume $m=1$; with additional indices the case $m>1$ can be proved in the same way.

For vectors $a=(a^1,\ldots ,a^M)\in \mathbb {R}^M$ and functions $\phi \in C^2(\mathbb {R}^M)$, we define the functions $I^a\phi$ and $J^a\phi$ by

$$\begin{aligned} I^a\phi (v)=\phi (v+a)-\phi (v),\quad J^a\phi (v)=\phi (v+a)-\phi (v)-a^iD_i\phi (v),\quad v\in \mathbb {R}^M. \end{aligned}$$

(2.9)

Assume first $\phi \in C_b^2(\mathbb {R}^M)$. Approximate h by $h^{(n)}=(h^{1(n)},\ldots ,h^{M(n)})$ and define

$$\begin{aligned} X_t^{(n)}&=X_0+\int _0^tf_s\,ds +\int _0^tg_s^{r}\,dw_s^r +\int _0^t\int _Z\bar{h}_s(z)\,\pi (dz,ds) \\&\quad +\int _0^t\int _Z h^{(n)}_s(z)\,\tilde{\pi }(dz,ds) \end{aligned}$$

for integers $n\ge 1$, where $h_t^{i(n)}=-n\vee h_t^{i}\wedge n$. Then (2.4) holds with $X^{i(n)}_t$ and $h^{i(n)}_t$ in place of $X^i_t$ and $h^i_t$, respectively, for each $i=1,2,\ldots ,M$. Clearly,

$$\begin{aligned} \int _0^T\int _Z|h^{(n)}_s(z)-h_s(z)|^2\,\mu (dz)\,ds\rightarrow 0\quad \mathrm{(a.s.)} \quad \text {for each }T>0, \end{aligned}$$

which implies

$$\begin{aligned} \int _0^t\int _Z h^{(n)}_s(z)\,\tilde{\pi }(dz,ds)\rightarrow \int _0^t\int _Z h_s(z)\,\tilde{\pi }(dz,ds) \end{aligned}$$

in probability uniformly in $t\in [0,T]$. Consequently, for each $T>0$ we have

$$\begin{aligned} \sup _{t\in [0,T]}|X^{(n)}_t-X_t|\rightarrow 0 \end{aligned}$$

in probability. It is easy to see

$$\begin{aligned} \int _0^tf^i_sD_i\phi (X^{(n)}_s)+\tfrac{1}{2}g_s^{ir}g_s^{jr}D_{i}D_{j}\phi (X^{(n)}_s)\,ds&\rightarrow \int _0^tf^i_sD_i\phi (X_s)\\&\quad +\tfrac{1}{2}g_s^{ir}g_s^{jr}D_{i}D_{j}\phi (X_s)\,ds, \\ \int _0^tg^{ir}_sD_i\phi (X^{(n)}_s)\,dw^r_s&\rightarrow \int _0^tg^{ir}_sD_i\phi (X_s)\,dw^r_s, \\ \int _0^t\int _Z I^{\bar{h}_{s}(z)}\phi (X^{(n)}_{s-})\,\pi (dz,ds)&\rightarrow \int _0^t\int _Z I^{\bar{h}_{s}(z)}\phi (X_{s-})\,\pi (dz,ds) \end{aligned}$$

in probability uniformly in $t\in [0,T]$ for $T>0$. Furthermore, by Taylor’s formula we have

$$\begin{aligned} |J^{h^{(n)}_s(z)}\phi (X^{(n)}_s)|&\le \int _0^1 (1-\theta )|h^{i(n)}_s(z)h^{j(n)}_s(z)D_{ij}\phi (X^{(n)}_s+\theta h^{(n)}_s(z))|\,d\theta \\&\le C|h_s(z)|^2,\\ |I^{h^{(n)}_s(z)}\phi (X^{(n)}_s)|^{2}&\le \int _0^1|\nabla \phi (X^{(n)}_s+\theta h^{(n)}_s(z)) h^{(n)}_s(z)|^2\,d\theta \le C|h_s(z)|^2 \end{aligned}$$

with a constant C independent of n. Hence, by Lebesgue’s theorem on dominated convergence for $T>0$ we have

$$\begin{aligned} \int _0^T\int _Z |J^{h^{(n)}_s(z)}\phi (X^{(n)}_s) -J^{h_s(z)}\phi (X_s)|\, \mu (dz)\,ds\rightarrow 0\quad \text {for }T \ge 0 \end{aligned}$$

and

$$\begin{aligned} \int _0^T\int _Z|I^{h_s^{(n)}(z)}\phi (X^{(n)}_s)-I^{h_s(z)}\phi (X_s)|^2\,\mu (dz)\,ds \rightarrow 0 \quad \text {for }T\ge 0 \end{aligned}$$

in probability, which implies

$$\begin{aligned} \int _0^t\int _ZI^{h^{(n)}_s(z)}\phi (X^{(n)}_{s-})\,\tilde{\pi }(dz,ds) \rightarrow \int _0^t\int _Z I^{h_s(z)}\phi (X_{s-})\,\tilde{\pi }(dz,ds) \end{aligned}$$

in probability uniformly in $t\in [0,T]$ for each $T>0$. Hence, letting $n\rightarrow \infty$ in (2.4) with $h^{(n)}$ and $X^{(n)}$ in place of h and X, respectively, we prove the theorem for $\phi \in C^2_b(\mathbb {R}^M)$. For $\phi \in C^2(\mathbb {R}^M)$, we define $\phi _n$ for integers $n\ge 1$ by $\phi _n(x)=\phi (x)\zeta (x/n)$, $x\in \mathbb {R}^M$, where $\zeta$ is a smooth function on $\mathbb {R}^M$ with values in [0, 1] such that $\zeta (x)=1$ for $|x|\le 1$ and $\zeta (x)=0$ for $|x|\ge 2$. Then $\phi _n\in C_b^2(\mathbb {R}^M)$, and therefore, (2.4) holds with $\phi _n$ in place of $\phi$. Thus, it remains to take limit as $n\rightarrow \infty$ for each term in (2.4) with $\phi _n$ in place of $\phi$. Clearly, as $n\rightarrow \infty$, we have

$$\begin{aligned} \phi _n(x)\rightarrow \phi (x),\quad D_i\phi _n(x)\rightarrow D_i\phi (x), \quad D_{ij}\phi _n(x)\rightarrow D_{ij}\phi (x) \end{aligned}$$

uniformly on compact subsets of $\mathbb {R}^M$ for $i,j=1,2,\ldots ,M$. Hence, it is easy to see

$$\begin{aligned} \int _0^tf^i_sD_i\phi _n(X_s)+\tfrac{1}{2}g_s^{ir}g_s^{jr}D_{i}D_{j}\phi _n(X_s)\,ds&\rightarrow \int _0^tf^i_sD_i\phi (X_s)\\&\quad +\tfrac{1}{2}g_s^{ir}g_s^{jr}D_{i}D_{j}\phi (X_s)\,ds \end{aligned}$$

and

$$\begin{aligned} \int _0^tg^{ir}_sD_i\phi _n(X_s)\,dw^r_s \rightarrow \int _0^tg^{ir}_sD_i\phi (X_s)\,dw^r_s \end{aligned}$$

in probability, uniformly in $t\in [0,T]$ as $n\rightarrow \infty$. Using the simple identity

$$\begin{aligned} I^{a}(\varphi \phi )(x)=\phi (x)I^{a}\varphi (x) +\varphi (x+a)I^{a}\phi (x), \quad a,x\in \mathbb {R}^M \end{aligned}$$

with $\varphi =\phi _n$ and $a=h_s(z)$, we get

$$\begin{aligned} |I^{h_s(z)}\phi _n(X_s)-I^{h_s(z)} \phi (X_s)|&\le |\phi (X_s)||I^{h_s(z)}\zeta _n(X_s)|\nonumber \\&\qquad +|1-\zeta _n(X_s+h_s(z))||I^{h_s(z)}\phi (X_s)| \nonumber \\&\le \frac{C}{n}|\phi (X_s)||h_s(z)|\nonumber \\&\qquad +|1-\zeta _n(X_s+h_s(z))||I^{h_s(z)}\phi (X_s)| \nonumber \\&\le \frac{C}{n}|\phi (X_s)||h_s(z)| +|I^{h_s(z)}\phi (X_s)| \end{aligned}$$

(2.10)

with a constant C independent of n, and since $\lim _{n\rightarrow \infty }|1-\zeta _n(X_s+h_s(z))|=0$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }|I^{h_s(z)}\phi _n(X_s)-I^{h_s(z)} \phi (X_s)|=0 \quad \text {for every }(\omega ,s,z). \end{aligned}$$

Hence, by (2.10), taking into account conditions (2.3) and (2.7) on h and $I^{h_s(z)} \phi (X_s)$, we can apply Lebesgue’s theorem on dominated convergence to obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^T\int _Z|I^{h_s(z)}\phi (X_s)-I^{h_s(z)}\phi _n(X_s)|^2\,\mu (dz)\,ds =0 \quad \mathrm{(a.s.)}, \end{aligned}$$

which implies that for $n\rightarrow \infty$ we have

$$\begin{aligned} \int _0^t\int _ZI^{h_s(z)}\phi _n(X_s)\,\tilde{\pi }(dz,ds) \rightarrow \int _0^t\int _ZI^{h_s(z)}\phi (X_s)\,\tilde{\pi }(dz,ds) \end{aligned}$$

in probability uniformly in $t\in [0,T]$ for each $T>0$. Similarly, we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^T\int _Z| I^{\bar{h}_s(z)}\phi _n(X_s)- I^{\bar{h}_s(z)}\phi (X_s)|\,\pi (dz,ds)=0\quad \mathrm{(a.s.)} \end{aligned}$$

for every $T\ge 0$. Using the identity

$$\begin{aligned} J^{a}(\varphi \phi )(x)=\phi (x)J^{a}\varphi (x) +\varphi (x)J^{a}\phi (x)+I^a\varphi (x) I^a\phi (x), \quad a,x\in \mathbb {R}^M \end{aligned}$$

with $\varphi =\phi _n$ and $a=h_s(z)$, we get

$$\begin{aligned}&J^{h_s(z)}\phi (X_s)-J^{h_s(z)}\phi _n(X_s)\\&\quad =(1-\zeta _n(X_s))J^{h_s(z)}\phi (X_s) +\phi (X_s)J^{h_s(z)}\zeta _n(X_s)+I^{h_s(z)}\phi (X_s)I^{h_s(z)}\zeta _n(X_s). \end{aligned}$$

Hence, taking into account $|(1-\zeta _n(X_s))|\le 1$,

$$\begin{aligned}&|J^{h_s(z)}\zeta _n(X_s)| \le \int _0^1(1-\theta )|h^i_s(z)h^j_s(z) D_{ij}\zeta _n(X_s+\theta h_s(z))|\,d\theta \le \frac{C}{n}|h_s(z)|^2, \\&|I^{h_s(z)}\phi (X_s)I^{h_s(z)}\zeta _n(x)| \le \frac{C}{n} |I^{h_s(z)}\phi (X_s)||h_s(z)| \le \frac{C}{n}( |I^{h_s(z)}\phi (X_s)|^2+|h_s(z)|^2) \end{aligned}$$

and $\lim _{n\rightarrow \infty }|(1-\zeta _n(X_s))|=0$, we obtain

$$\begin{aligned}&|J^{h_s(z)}\phi (X_s)-J^{h_s(z)}\phi _n(X_s)| \nonumber \\&\quad \le |J^{h_s(z)}\phi (X_s)|+ \frac{C}{n}(|\phi (X_s)||h_s(z)|^2+ |I^{h_s(z)}\phi (X_s)|^2+|h_s(z)|^2) \end{aligned}$$

(2.11)

with a constant C independent of n, and

$$\begin{aligned} \lim _{n\rightarrow \infty }|J^{h_s(z)}\phi (X_s)-J^{h_s(z)}\phi _n(X_s)|=0\quad \text {for all }(\omega ,s,z). \end{aligned}$$

Thus, by virtue of (2.11) and conditions (2.2), (2.7) and (2.8) on h, $I^h(X_s)$ and $J^h(X_s)$, we can use Lebesgue’s theorem on dominated convergence again to get

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^T\int _Z |J^{h_s(z)}\phi _n(X_s)-J^{h_s(z)}\phi (X_s)|\,\mu (dz)\,ds\quad \mathrm{(a.s.)} \end{aligned}$$

for every $T\ge 0$, which completes the proof of Theorem 2.2. $\square$

Remark 2.1

The above theorem is useful if one can check that conditions (2.7)–(2.8) are satisfied. If $D_i\phi$ and $D_{ij}\phi$ are bounded functions for every $i,j=1,2,\ldots ,M$, then conditions (2.7)–(2.8) are always satisfied, since for every $t>0$

$$\begin{aligned} \int _0^t\int _Z|I^{h_s(z)}\phi (X_s)|^2\,\mu (dz)\,ds&=\int _0^t\int _Z\left| \int _0^1\nabla \phi (X_s+h_s(z)) h_s(z)\,d\theta \right| ^2\,\mu (dz)\,ds\\&\le C\int _0^t\int _Z|h_s(z)|^2\,\mu (dz)\,ds<\infty \quad (\mathrm {a.s.}) \end{aligned}$$

and

$$\begin{aligned} \int _0^t\int _Z |J^{h_s(z)}\phi (X_s)| \,\mu (dz)\,ds&= \int _0^t\int _Z\Bigg |\int _0^1 (1-\theta )h^i_s(z)h^j_s(z)D_{ij}\phi (X_s \\&\quad +\theta h_s(z))d\theta \Bigg |\,\mu (dz)\,ds\\&\le C\int _0^t\int _Z|h_s(z)|^2\mu (dz)ds<\infty \quad \mathrm {(a.s.)} \end{aligned}$$

with a constant C. Thus, by virtue of the above theorem, under the conditions (2.2) and (2.3) Itô formula (2.4) holds if the first- and second-order derivatives of $\phi$ are bounded continuous functions. As Example 2.1 shows, Theorem 2.2 is not applicable to $\phi (x)=|x|^p$ for $p\ge 2$.

Next we formulate an Itô formula which holds under the natural conditions (2.2)–(2.3).

Theorem 2.3

Let conditions (2.2) and (2.3) hold, and let $\phi$ from $C^2(\mathbb {R}^M)$. Then $\phi (X_t)$ is a semimartingale such that

$$\begin{aligned} \phi (X_t)&= \phi (X_0)+ \int _0^tD_i\phi (X_s)g_s^{ir}\,dw_s^r +\int _0^tD_i\phi (X_s)f^i_s\nonumber \\&\quad + \tfrac{1}{2}D_iD_j\phi (X_s)g_s^{ir}g_s^{jr}\,ds\nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}\phi (X_{s-}+{\bar{h}}^k_s(z))-\phi (X_{s-})\,\pi ^k(dz,ds)\nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}D_i\phi (X_{s-})h^{ik}_s(z)\,\tilde{\pi }^k(dz,ds) \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k} \phi (X_{s-}+h^k_s(z))-\phi (X_{s-})\nonumber \\&\quad -D_i\phi (X_{s-})h^{ik}_s(z)\,\pi ^k(dz,ds) \end{aligned}$$

(2.12)

almost surely for all $t\ge 0$.

Proof

We prove Theorem 2.3 by rewriting Itô formula (2.4) into Eq. (2.12) under the additional condition that h is bounded, and then we dispense with this condition by approximating h by bounded functions. For notational simplicity we assume $m=1$, for $m>1$ the proof goes in the same way. First, in addition to the conditions (2.2) and (2.3), assume there is a constant K such that $|h|\le K$. By Taylor’s formula for $I^{a}\phi (v)$ and $J^{a}\phi (v)$, introduced in (2.9), for each $v,a\in \mathbb {R}^M$ we have

$$\begin{aligned} |I^a\phi (v)|\le \sup _{|x|\le |a|+|v|}|D\phi (x)||a|, \quad |J^a\phi (v)|\le \sup _{|x|\le |a|+|v|}|D^2\phi (x)||a|^2, \end{aligned}$$

(2.13)

where $|D\phi |^2:=\sum _{i=1}^M|D_i\phi |^2$ and $|D^2\phi |^2:= \sum _{i=1}^M\sum _{j=1}^M|D_iD_j\phi |^2$. Since $(X_t)_{t\ge 0}$ is a cadlag process, $R:=\sup _{t\le T}|X_t|$ is a finite random variable for each fixed T. Thus, we have

$$\begin{aligned} \int _0^T\int _Z|J^{h_t(z)}\phi (X_{t-})|\mu (dz)\,dt \le \sup _{|x|\le R+K}|D^2\phi (x) |\int _0^T\int _Z|h_t(z)|^2\,\mu (dz)\,dt<\infty \end{aligned}$$

(2.14)

and

$$\begin{aligned} \int _0^T\int _Z|J^{h_t(z)}\phi (X_{t-})|^2\mu (dz)\,dt&\le \sup _{|x|\le R+K}|D^2\phi (x)|^2K^2 \int _0^T\int _Z|h_t(z)|^2\,\mu (dz)\,dt\nonumber \\&<\infty \end{aligned}$$

(2.15)

almost surely. Clearly,

$$\begin{aligned} \int _0^T\int _Z|D_i\phi (X_{t-})h^i_t(z)|^2\,\mu (dz)\,dt&\le \sup _{|x|\le R}|D\phi (x)|^2 \int _0^T\int _Z|h_t(z)|^2\,\mu (dz)\,dt\\&\quad <\infty \quad (\mathrm{a.s.}). \end{aligned}$$

Hence, by virtue of (2.15) the stochastic Itô integral

$$\begin{aligned} \int _0^t\int _Z\phi (X_{s-}+h_t(z))-\phi (X_s)\,\tilde{\pi }(dz,ds) =\int _0^t\int _ZI^{h_s(z)}\phi (X_{s-})\tilde{\pi }(dz,ds) \end{aligned}$$

can be decomposed as

$$\begin{aligned}&\int _0^t\int _ZI^{h_s(z)}\phi (X_{s-})\tilde{\pi }(dz,ds) =\int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\,\tilde{\pi }(dz,ds)\\&\quad +\int _0^t\int _ZD_i\phi (X_{s-})h^i_s(z)\,\tilde{\pi }(dz,ds), \end{aligned}$$

and by virtue of (2.14) and (2.15),

$$\begin{aligned}&\int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\,\tilde{\pi }(dz,ds) +\int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\, \mu (dz)\,ds\\&\quad =\int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\,\pi (dz,ds). \end{aligned}$$

Hence,

$$\begin{aligned}&\int _0^t\int _ZI^{h_s(z)}\phi (X_{s-})\,\tilde{\pi }(dz,ds)+ \int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\, \mu (dz)\,ds \\&\quad =\int _0^t\int _ZD_i\phi (X_{s-})h^i_s(z)\,\tilde{\pi }(dz,ds) +\int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\,\pi (dz,ds), \end{aligned}$$

which shows that Theorem 2.3 holds under the additional condition that |h| is bounded. To prove the theorem in full generality, we approximate h by $h^{(n)}=(h^{1(n)},\ldots ,h^{M(n)})$, where $h_t^{in}=-n\vee h_t^{i}\wedge n$ for integers $n\ge 1$, and define

$$\begin{aligned} X^{(n)}_t&:=X_0+\int _0^tf_s\,ds+\int _0^tg_s^{r}\,dw_s^r +\int _0^t\int _Z{\bar{h}}_s(z)\,\pi (dz,ds)\\&\quad +\int _0^t\int _Zh^{(n)}_s(z)\,\tilde{\pi }(dz,ds), \quad t\in [0,T]. \end{aligned}$$

Clearly, for all $(\omega ,t,z)$

$$\begin{aligned} |h^{(n)}|\le \min (|h|, nM)\quad \text {and }\quad h^{(n)}\rightarrow h\quad \text { as }n\rightarrow \infty . \end{aligned}$$

Therefore, Theorem 2.3 for $X^{(n)}$ holds, and

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^T\int _Z|h^{(n)}_t(z)-h_t(z)|^2\,\mu (dz)\,dt=0\,\,(\mathrm {a.s.}), \end{aligned}$$

which implies

$$\begin{aligned} \sup _{t\le T}|X^{(n)}_t-X_t|\rightarrow 0 \quad \text {in probability as}\; n\rightarrow \infty . \end{aligned}$$

Thus, there is a strictly increasing subsequence of positive integers $(n_k)_{k=1}^{\infty }$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup _{t\le T}|X^{(n_k)}_t-X_t|=0\quad (\mathrm {a.s.}), \end{aligned}$$

which implies

$$\begin{aligned} \rho :=\sup _{k\ge 1}\sup _{t\le T}|X^{(n_k)}_t|<\infty \quad (\mathrm {a.s.}). \end{aligned}$$

Hence, it is easy to pass to the limit $k\rightarrow \infty$ in $\phi (X_t^{(n_k)})$ and in the first two integral terms in the equation for $\phi (X_t^{(n_k)})$ in Theorem 2.3. To pass to the limit in the other terms in this equation notice that since $\pi (dz,dt)$ is a counting measure of a point process, from the condition for ${\bar{h}}$ in (2.3) we get

$$\begin{aligned} \xi :={\pi -{\mathrm{ess\,sup}}}\,|{\bar{h}}|<\infty \,\,(\mathrm {a.s.}), \end{aligned}$$

(2.16)

where ${\pi -{\mathrm{ess\,sup}}}$ denotes the essential supremum operator with respect to the measure $\pi (dz,dt)$ over $Z\times [0,T]$. Similarly, from the condition for h we have

$$\begin{aligned} \eta :={\pi -{\mathrm{ess\,sup}}}\,|h|<\infty \,\,(\mathrm {a.s.}). \end{aligned}$$

(2.17)

This can be seen by noting that for the sequence of predictable stopping times

$$\begin{aligned} \tau _j=\inf \left\{ t\in [0,T]:\int _0^t\int _Z|h_s(z)|^2\,\mu (dz)\,ds\ge j \right\} , \quad {j=1,2,\ldots }, \end{aligned}$$

we have

$$\begin{aligned} E\int _0^T\int _Z\mathbf{1}_{t\le \tau _j}|h_t(z)|^2\,\pi (dz,dt) =E\int _0^T\int _Z\mathbf{1}_{t\le \tau _j}|h_t(z)|^2\,\mu (dz)\,dt\le j<\infty , \end{aligned}$$

which gives

$$\begin{aligned}&\int _0^T\int _Z|h_t(z)|^2\,\pi (dz,dt)<\infty \quad \text {almost surely on }\Omega _j=\{\omega \in \Omega :\tau _j\ge T\} \\&\quad \text { for each} j\ge 1. \end{aligned}$$

Since $(\tau _j)_{j=1}^{\infty }$ is an increasing sequence converging to infinity, we have $P(\cup _{j=1}^{\infty }\Omega _j)=1$, i.e.

$$\begin{aligned} \int _0^T\int _Zh^2_t(z)\,\pi (dz,dt)<\infty \,\,(\mathrm {a.s.}), \end{aligned}$$

(2.18)

which implies (2.17). By (2.16) and the first inequality in (2.13), we have

$$\begin{aligned} |I^{{\bar{h}}_t(z)}\phi (X_{t-}^{(n_k)})|+|I^{{\bar{h}}_t(z)}\phi (X_{t-})| \le 2\sup _{|x|\le \rho +\xi }|D\phi (x)||{\bar{h}}_t(z)|<\infty \end{aligned}$$

almost surely for $\pi (dz,dt)$-almost every $(z,t)\in Z\times [0,T]$. Hence, by Lebesgue’s theorem on dominated convergence we get

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _0^T\int _Z |I^{{\bar{h}}_s(z)}\phi (X^{(n_k)}_{s-})-I^{{\bar{h}}_s(z)}\phi (X_{s-})|\,\pi (dz,ds)=0 \quad \mathrm {(a.s.)}, \end{aligned}$$

which implies that, for $k\rightarrow \infty$,

$$\begin{aligned} \int _0^t\int _Z I^{{\bar{h}}_s(z)}\phi (X^{(n_k)}_{s-})\,\pi (dz,ds) \rightarrow \int _0^t\int _Z I^{{\bar{h}}_s(z)}\phi (X_{s-})\,\pi (dz,ds) \end{aligned}$$

almost surely, uniformly in $t\in [0,T]$. Clearly,

$$\begin{aligned} |D_i\phi (X^{(n_k)}_{t-})h^{i(n_k)}_t(z)|^2+|D_i\phi (X_{t-})h^{i}_t(z)|^2 \le 2\sup _{|x|\le \rho }|D\phi (x)|^2|h_t(z)|^2 \end{aligned}$$

almost surely for all $(z,t)\in Z\times [0,T]$. Hence, by Lebesgue’s theorem on dominated convergence,

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _0^T\int _Z |D_i\phi (X^{(n_k)}_{t-})h^{i(n_k)}_t(z)-D_i\phi (X_{t-})h^{i}_t(z)|^2 \,\mu (dz)\,dt=0\quad \mathrm {(a.s.)}, \end{aligned}$$

which implies that, for $k\rightarrow \infty$,

$$\begin{aligned} \int _0^t\int _Z D_i\phi (X^{(n_k)}_{s-})h^{i(n_k)}_s(z)\,\tilde{\pi }(dz,ds) \rightarrow \int _0^t\int _Z D_i\phi (X_{s-})h^{i}_s(z)\,\tilde{\pi }(dz,ds) \end{aligned}$$

in probability, uniformly in $t\in [0,T]$. Finally, note that by using the second inequality in (2.13) together with (2.17) we have

$$\begin{aligned} |J^{h^{(n_k)}_t(z)}\phi (X_{t-}^{(n_k)})|+|J^{h_t(z)}\phi (X_{t-})| \le 2\sup _{|x|\le \rho +\eta }|D^2\phi (x)||h_t(z)|^2 \end{aligned}$$

almost surely for $\pi (dz,dt)$-almost every $(z,t)\in Z\times [0,T]$. Hence, taking into account (2.18), by Lebesgue’s theorem on dominated convergence we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _0^T\int _Z |J^{h^{(n_k)}_t(z)}\phi (X^{(n_k)}_{t-})-J^{h_t(z)}\phi (X_{t-})| \,\pi (dz,dt)=0\quad \mathrm {(a.s.)}, \end{aligned}$$

which implies that, for $k\rightarrow \infty$,

$$\begin{aligned} \int _0^t\int _ZJ^{h^{(n_k)}_s(z)}\phi (X^{(n_k)}_{s-})\,\pi (dz,ds) \rightarrow \int _0^t\int _ZJ^{h_s(z)}\phi (X_{s-})\,\pi (dz,ds) \end{aligned}$$

almost surely, uniformly in $t\in [0,T]$ for every $T>0$, which finishes the proof of the theorem. $\square$

Remark 2.2

One can give a different proof of Theorem 2.3 by showing that for finite measures $\mu ^k$, the Itô formula for general semimartingales, Theorem VIII.27 in [6], applied to $(X_t)_{t\ge 0}$, can be rewritten as Eq. (2.12). Hence, by an approximation procedure one can get the general case of $\sigma$-finite measures $\mu ^k$.

Corollary 2.4

Let conditions (2.2) and (2.3) hold. Then for any $p\ge 2$ the process $|X_t|^p$ is a semimartingale such that

$$\begin{aligned} |X_t|^p&= |\psi |^p+ p\int _0^t|X_s|^{p-2}X^i_sg_s^{ir}\,dw_s^r \nonumber \\&\quad + \tfrac{p}{2}\int _0^t\left( 2|X_s|^{p-2}X^i_sf^i_s+ (p-2)|X_s|^{p-4}|X^i_sg^{i\cdot }_s|_{l_2}^2 +\sum _{i=1}^M|X_s|^{p-2}|g^{i\cdot }_s|_{l_2}^2\right) ds \nonumber \\&\quad + \sum _{k=1}^mp\int _0^t\int _{Z_k}|X_{s-}|^{p-2}X^i_{s-}h^{ik}_s(z)\,\tilde{\pi }^k(dz,ds) \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}(|X_{s-}+\bar{h}^k_s|^p-|X_{s-}|^p)\,\pi ^k(dz,ds) \nonumber \\&\quad +\sum _{k=1}^m\int _0^t\int _{Z_k}\left( |X_{s-}+h^k_s|^p-|X_{s-}|^p-p|X_{s-}|^{p-2}X^i_{s-}h^{ik}_s\right) \,\pi ^k(dz,ds) \qquad \end{aligned}$$

(2.19)

almost surely for all $t\ge 0$, where, and through the paper, the convention $0/0:=0$ is used whenever it occurs.

Proof

Notice that $\phi (x)=|x|^p$ for $p\ge 2$ belongs to $C^2(\mathbb {R}^M)$ with

$$\begin{aligned} D_i|x|^p=p|x|^{p-2}x^i, \quad D_jD_i|x|^p=p(p-2)|x|^{p-4}x^ix^j+p|x|^{p-2}\delta _{ij}, \end{aligned}$$

where $\delta _{ij}=1$ for $i=j$ and $\delta _{ij}=0$ for $i\ne j$. Hence, it is easy to see that Theorem 2.3 applied to $\phi (x)=|x|^p$ gives the corollary. $\square$

The above corollary will be used to obtain an Itô’s formulas for jump processes in $L_p$-spaces presented in the next section.

3 Itô formula in $L_p$ spaces

Itô formulas in infinite-dimensional spaces play important roles in studying stochastic PDEs. Our theorem below is motivated by applications in the theory of stochastic integro-differential equations arising in nonlinear filtering theory of jump diffusions. To present it first we need to introduce some notation, where T is a fixed positive number, and $d\ge 1$ and $M\ge 1$ are fixed integers.

The Borel $\sigma$-algebra of a topological space V is denoted by $\mathcal {B}(V)$. For $p, q\ge 1$ we denote by $L_p=L_p(\mathbb {R}^d,\mathbb {R}^M)$ and $\mathcal {L}_{q}=\mathcal {L}_{q}(Z, \mathbb {R}^M)$ the Banach spaces of $\mathbb {R}^M$-valued Borel-measurable functions of $f=(f^i(x))_{i=1}^M$ and $\mathcal {Z}$-measurable functions $h=(h^{i}(z))_{i=1}^M$ of $x\in \mathbb {R}^d$ and $z\in Z$, respectively, such that

$$\begin{aligned} |f|_{L_p}^p=\int _{\mathbb {R}^d}|f(x)|^p\,dx<\infty \quad \text {and} \quad |h|^{q}_{\mathcal {L}_{q}}=\int _{Z}|h(z)|^{q}\,\mu (dz)<\infty . \end{aligned}$$

The notation $\mathcal {L}_{p,q}$ means the space $\mathcal {L}_{p}\cap \mathcal {L}_{q}$ with the norm

$$\begin{aligned} |v|_{\mathcal {L}_{p,q}}=\max (|v|_{\mathcal {L}_{p}},|v|_{\mathcal {L}_{q}}) \quad \text {for}\quad v\in \mathcal {L}_{p}\cap \mathcal {L}_{q}. \end{aligned}$$

As usual, $W^1_p$ denotes the space of functions $u\in L_p$ such that $D_iu\in L_p$ for every $i=1,2,\ldots ,d$, where $D_iv$ means the generalised derivative of v in $x^i$ for locally integrable functions v on $\mathbb {R}^d$. The norm of $u\in W^1_p$ is defined by

$$\begin{aligned} |u|_{W^1_p}=|u|_{L_p}+\sum _{i=1}^d|D_iu|_{L_p}. \end{aligned}$$

We use the notation $L_p=L_p(\ell _2)$ for $L_p(\mathbb {R}^d,\ell _2)$, the space of Borel-measurable functions $g=(g^{ir})$ on $\mathbb {R}^d$ with values in $\ell _2$ such that

$$\begin{aligned} |g|_{L_p}^p=\int _{\mathbb {R}^d}|g(x)|_{\ell _2}^p\,dx<\infty . \end{aligned}$$

For $p,q\in [0,\infty )$, we denote by $L_{p}=L_p(\mathcal {L}_{p,q})$ and $L_p=L_p(\mathcal {L}_q)$ the Banach spaces of Borel-measurable functions $h=(h^i(x,z))$ and ${\tilde{h}}=({\tilde{h}}^i(x,z))$ of $x\in \mathbb {R}^d$ with values in $\mathcal {L}_{p,q}$ and $\mathcal {L}_q$, respectively, such that

$$\begin{aligned} |h|^p_{L_{p}}=\int _{\mathbb {R}^d}|h(x,\cdot )|^p_{\mathcal {L}_{p,q}}\,dx<\infty \quad \text {and}\quad |{\tilde{h}}|^p_{L_{p}}=\int _{\mathbb {R}^d}|{\tilde{h}}(x,\cdot )|^p_{\mathcal {L}_q}\,dx<\infty . \end{aligned}$$

For $p\ge 2$ and a separable real Banach space V, we denote by $\mathbb {L}_p=\mathbb {L}_p(V)$ the space of predictable V-valued functions $f=(f_t)$ of $(\omega ,t)\in \Omega \times [0,T]$ such that

$$\begin{aligned} |f|_{\mathbb {L}_p}^p=E\int _0^T|f_t|^p_{V}\,dt<\infty . \end{aligned}$$

In the sequel, V will be $L_p(\mathbb {R}^d,\mathbb {R}^M)$, $L_p(\mathbb {R}^d,\ell _2)$ or $L_p(\mathbb {R}^d,\mathcal {L}_{p,2})$. When $V=L_p(\mathbb {R}^d,\mathcal {L}_{p,2})$, then for $\mathbb {L}_p(V)$ the notation $\mathbb {L}_{p,2}$ is also used. For $\varepsilon \in (0,1)$ and locally integrable functions v of $x\in \mathbb {R}^d$, we use the notation $v^{(\varepsilon )}$ for the mollifications of v,

$$\begin{aligned} v^{(\varepsilon )}(x)=\int _{\mathbb {R}^d}v(x-y)k_{\varepsilon }(y)\,dy, \quad x\in \mathbb {R}^d, \end{aligned}$$

(3.1)

where $k_{\varepsilon }(y)=\varepsilon ^{-d}k(y/\varepsilon )$ for $y\in \mathbb {R}^d$ with a fixed function $k\in C_0^{\infty }$ of unit integral. If v is a locally Bochner integrable function on $\mathbb {R}^d$, taking values in a Banach space, then the mollification of v is defined as (3.1) in the sense of the Bochner integral.

Recall that the summation convention with respect to integer-valued indices is used throughout the paper.

Assumption 3.1

Let $\psi ^i$ be an $L_p(\mathbb {R}^d,\mathbb {R})$-valued $\mathcal {F}_0$-measurable random variable, $(u^{i}_{t})_{t\in 0,T}$ be a progressively measurable $L_p$-valued process and let $f^{i\alpha }$, $g^i=(g^{ir})_{r=1}^{\infty }$ and $h^i$ be predictable functions on $\Omega \times [0,T]\times Z$ with values in $L_p(\mathbb {R}^d,\mathbb {R})$, $L_p(\mathbb {R}^d,l_2)$ and $L_p(\mathbb {R}^d,\mathcal {L}_{p,2})$, respectively, for each $i=1,2,\ldots ,M$ and $\alpha =0,1,\ldots ,d$, such that the following conditions are satisfied for each $i=1,2,\ldots ,M$:

• (i) We have $u^i_t\in W^1_p$ for $P\otimes dt$-a.e. $(\omega ,t)\in \Omega \times [0,T]$ such that

  $$\begin{aligned} \int _0^T|u_t^i|^p_{W^1_p}\,dt<\infty \quad \mathrm{(a.s.)}. \end{aligned}$$

  (3.2)

• (ii) Almost surely

  $$\begin{aligned} \mathcal {K}^p_{p}(T):=\sum _{i=1}^M\int _0^T\int _{\mathbb {R}^d}\sum _{\alpha }|f^{i\alpha }_t(x)|^p +|g^i_t(x)|_{l_2}^p+|h^i_t(x)|^p_{\mathcal {L}_{p,2}}\,dx\,dt<\infty . \end{aligned}$$

  (3.3)

• (iii) For every $\varphi \in C_0^{\infty }(\mathbb {R}^d)$, we have

  $$\begin{aligned} (u^i_t,\varphi )&=(\psi ,\varphi ) +\int _0^t(f^{i\alpha }_s,D^{*}_{\alpha }\varphi )\,ds +\int _0^t(g_s^{ir},\varphi )\,dw_s^r\nonumber \\&\quad +\int _0^t\int _Z(h^i_s(z),\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

  (3.4)

  for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$, where $D^{*}_{\alpha }=-D_{\alpha }$ for $\alpha =1,2,\ldots ,d$ and $D_{\alpha }^{*}$ is the identity operator for $\alpha =0$.

In Eq. (3.4), and later on, we use the notation $(v,\phi )$ for the Lebesgue integral over $\mathbb {R}^d$ of the product $v\phi$ for functions v and $\phi$ on $\mathbb {R}^d$ when their product and its integral are well defined. Below u stands for $(u^1,\ldots ,u^M)$.

Theorem 3.1

Let Assumption 3.1 hold with $p\ge 2$. Then there is an $L_p(\mathbb {R}^d,\mathbb {R}^M)$-valued adapted cadlag process ${\bar{u}}=({\bar{u}}^i_t)_{t\in [0,T]}$ such that Eq. (3.4), with ${\bar{u}}$ in place of u, holds for each $\varphi \in C_0^{\infty }(\mathbb {R}^d)$ almost surely for all $t\in [0,T]$. Moreover, $u={\bar{u}}$ for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$, and almost surely

$$\begin{aligned} |{\bar{u}}_t|^p_{L_p}&= |\psi |_{L_p}^p +p\int _0^t\int _{\mathbb {R}^d}|{\bar{u}}_s|^{p-2}{\bar{u}}^i_sg^{ir}_s\,dx\, dw^r_s \nonumber \\&\quad +\frac{p}{2}\int _0^t\int _{\mathbb {R}^d} 2|u_s|^{p-2}{\bar{u}}^i_sf^{i0}_s -2|u_s|^{p-2}D_k u^i_sf_s^{ik}\nonumber \\&\quad -(p-2)|u_s|^{p-4}u^i_sf^{ik}_sD_k|u_s|^2\,dx\,ds \nonumber \\&\quad +\frac{p}{2}\int _0^t\int _{\mathbb {R}^d} (p-2)|u_s|^{p-4} |u^i_sg_s^{ i\cdot }|_{l_2}^2 +|u_s|^{p-2} \sum _{i=1}^M|g^{i\cdot }_s|^2_{l_2}\,dx\,ds \nonumber \\&\quad +p\int _0^t\int _Z\int _{\mathbb {R}^d}|u_{s-}|^{p-2}u^i_{s-}h^i_s \,dx\,\tilde{\pi }(dz,ds) \nonumber \\&\quad +\int _0^t\int _Z\int _{\mathbb {R}^d} (|{\bar{u}}_{s-}+h_s|^p-|{\bar{u}}_{s-}|^p-p|{\bar{u}}_{s-}|^{p-2}{\bar{u}}^i_{s-}h^i_s) \,dx\,\pi (dz,ds) \end{aligned}$$

(3.5)

for all $t\in [0,T]$, where ${\bar{u}}_{s-}$ means the left-hand limit in $L_p$ of ${\bar{u}}$ at s. If $f^i=0$ for $i=1,2,\ldots ,d$, then the above statements hold if Assumption 3.1 is satisfied with (i) replaced in it with the weaker condition that

$$\begin{aligned} \int _0^T|u^i_t|^p_{L_p}\,dt<\infty \quad \mathrm{(a.s.)}. \end{aligned}$$

(3.6)

Notice that for $M=1$ Eq. (3.5) has the simpler form

$$\begin{aligned}&|{\bar{u}}_t|^p_{L_p}= |\psi |_{L_p}^p+p\int _0^t\int _{\mathbb {R}^d}|u_s|^{p-2}u_sg_s^r\,dx\,dw^r_s \nonumber \\&\quad +\frac{p}{2}\int _0^t\int _{\mathbb {R}^d} \big (2|u_s|^{p-2}u_sf^0_s-2(p-1)|u_s|^{p-2}f^i_sD_iu_s\nonumber \\&\quad +(p-1)|u_s|^{p-2}|g_s|^2_{l_2} \big )\,dx\,ds \nonumber \\&\quad +p\int _0^t\int _Z\int _{\mathbb {R}^d} |{\bar{u}}_{s-}|^{p-2}{\bar{u}}_{s-}h_s\,dx\,\tilde{\pi }(dz,ds) \nonumber \\&\qquad +\int _0^t\int _Z\int _{\mathbb {R}^d}\big ( | {\bar{u}}_{s-} +h_s|^p-| {\bar{u}}_{s-}|^p-p| {\bar{u}}_{s-}|^{p-2}{\bar{u}}_{s-}h_s \big )\,dx\,\pi (dz,ds). \end{aligned}$$

(3.7)

Theorem 3.1 generalises Theorem 2.1 from [14], and we use ideas and methods from [14] to prove it. The basic idea in [14] adapted to our situation can be explained as follows: Assume first that $f^{i\alpha }=0$ for $\alpha =1,2,\ldots ,d$, and suppose from (3.4) we could show the existence of a random field ${\bar{u}}={\bar{u}}(t,x)$ and suitable modifications of the integrals of $f^{i}:=f^{i0}_s(x)$, $g=g^{ir}_s(x)$ and $h^i_s(x,z)$ against ds, $dw^r_s$ and $\tilde{\pi }(dz,ds)$, respectively, satisfying appropriate measurability conditions such that the equation

$$\begin{aligned} {\bar{u}}^i_t(x) =\psi ^{i}(x) +\int _0^tf^{i}_s(x)\,ds +\int _0^tg_s^{ir}(x)\,dw_s^r +\int _0^t\int _Zh^i_s(x,z)\,\tilde{\pi }(dz,ds) \end{aligned}$$

(3.8)

holds for every $x\in \mathbb {R}^d$ and $i=1,2,\ldots ,M$. Then applying Itô’s formula (2.19) from Corollary 2.4 to $|{\bar{u}}_t(x)|^p=(\sum _{i}|{\bar{u}}^i_t(x)|^2)^{p/2}$ for every $x\in \mathbb {R}^d$, then integrating over $\mathbb {R}^d$, and finally, using suitable stochastic Fubini theorems, we could obtain (3.5) when $f^{i\alpha }=0$ for $\alpha \ge 1$. When $f^{i \alpha }\ne 0$, we could take

$$\begin{aligned} u^{i(\varepsilon )},\quad \psi ^{i(\varepsilon )},\quad f^{i(\varepsilon )}:=f^{i0(\varepsilon )}+\sum _{k=1}^dD_kf^{ik(\varepsilon )}, \quad g^{ir(\varepsilon )} \quad \text {and}\quad h^{i(\varepsilon )} \end{aligned}$$

instead of $u^i$, $\psi ^i$, $f^{i}$, $g^{ir}$ and $h^{i}$ above, respectively, to apply the theorem in the special case, and let $\varepsilon \rightarrow 0$ in the corresponding Itô formula after integrating by parts in the terms containing $D_kf^{ik(\varepsilon )}$ for $k=1,\ldots ,d$. Notice that we can formally obtain Eq. (3.8) from (3.4) with $f^{i1}=\cdots =f^{id}=0$ and a suitable process ${\bar{u}}$ in place of u, by substituting $\delta _x$, the Dirac delta at x, in place of $\varphi$. Clearly, we cannot substitute $\delta _x$, but we can substitute approximations $k_{\varepsilon }(x-\cdot )$ of it to get

$$\begin{aligned} {\bar{u}}^{i(\varepsilon )}_t(x) =\psi ^{i(\varepsilon )}(x) +\int _0^tf^{i(\varepsilon )}_s(x)\,ds +\int _0^tg_s^{ir(\varepsilon )}(x)\,dw_s^r +\int _0^t\int _Zh^{i(\varepsilon )}_s(x,z)\,\tilde{\pi }(dz,ds) \end{aligned}$$

(3.9)

in place of (3.8). Therefore, the above strategy is modified as follows: One chooses suitable representative of the stochastic integrals in (3.9) so that one could apply Itô’s formula (2.19) to $|{\bar{u}}^{(\varepsilon )}_t(x)|^p$ for each $x\in \mathbb {R}^d$, integrate the obtained formula over $\mathbb {R}^d$, then interchange the order of the integrals, and finally let $\varepsilon \rightarrow 0$ to prove Eq. (3.5) when $f^{ik}=0$ for $i=1,2,\ldots ,M$ and $k=1,2,\ldots ,d$.

To implement the above idea we fix a $p\ge 2$ and introduce a class of functions $\mathcal {U}_p$, the counterpart of the class $\mathcal {U}_p$ given in [14]. Let $\mathcal {U}_p$ denote the set of $\mathbb {R}^M$-valued functions $u=u_t(x)=u_t(\omega ,x)$ on $\Omega \times [0,T]\times \mathbb {R}^d$ such that

• (i) u is $\mathcal {F}\otimes \mathcal {B}([0,T])\otimes \mathcal {B}(\mathbb {R}^d)$-measurable,

• (ii) for each $x\in \mathbb {R}^d$, $u_t(x)$ is $\mathcal {F}_t$-adapted,

• (iii) $u_t(x)$ is cadlag in $t\in [0,T]$ for each $(\omega ,x)$,

• (iv) $u_t(\omega ,\cdot )$ as a function of $(\omega ,t)$ is $L_p$-valued, $\mathcal {F}_t$-adapted and cadlag in t for every $\omega \in \Omega$.

The following lemmas present suitable versions of Lebesgue and Itô integrals with values in $L_p$. The first two of them are obvious corollaries of Lemmas 4.3 and 4.4 in [14].

Lemma 3.2

Let $f\in \mathbb {L}_p(V)$ for $V=L_p(\mathbb {R}^d,\mathbb {R}^M)$. Then there exists a function $m\in \mathcal {U}_p$ such that for each $\varphi \in C_0^\infty$ almost surely

$$\begin{aligned} (m_t,\varphi )=\int _0^t(f_s,\varphi )\,ds \end{aligned}$$

holds for all $t\in [0,T]$. Furthermore, we have

$$\begin{aligned} E\int _{\mathbb {R}^d}\sup _{t\le T}|m_t(x)|^p\,dx\le NT^{p-1}E\int _0^T|f_s|^p_{L_p}\,ds, \end{aligned}$$

with a constant $N=N(p,M)$.

Lemma 3.3

Let g be from $\mathbb {L}_p(V)$ for $V=L_p(\mathbb {R}^d,\ell _2)$. Then there exists a function $a\in \mathcal {U}_p$ such that for each $\varphi \in C_0^\infty$ almost surely

$$\begin{aligned} (a_t,\varphi )=\sum _{r=1}^\infty \int _0^t(g_s^r,\varphi )\,dw_s^r \end{aligned}$$

holds for all $t\in [0,T]$. Furthermore, we have

$$\begin{aligned} E\int _{\mathbb {R}^d}\sup _{t\le T}|a_t(x)|^p\,dx \le NT^{(p-2)/2}E\int _0^T|g_s|^p_{L_p}\,ds \end{aligned}$$

with a constant $N=N(p,M)$.

The proof of the following lemma can be found in [10].

Lemma 3.4

Let $h\in \mathbb {L}_{p,2}$. Then there exists a function $b\in \mathcal {U}_p$ such that for each real-valued $\varphi \in L_q(\mathbb {R}^d)$ with $q=p/(p-1)$, almost surely

$$\begin{aligned} (b_t,\varphi )=\int _0^t\int _Z(h_s,\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

(3.10)

for all $t\in [0,T]$, and

$$\begin{aligned} E\sup _{t\le T}|(b_t,\varphi )| \le 3T^{(p-2)/(2p)}|\varphi |_{L_q}\left( E\int _0^T|h_t|_{L_p(\mathcal {L}_2)}^p\,dt\right) ^{1/p}. \end{aligned}$$

(3.11)

Furthermore,

$$\begin{aligned} E\int _{\mathbb {R}^d}\sup _{t\le T}|b_t(x)|^p\,dx&\le NE\int _0^T|h_t|^p_{L_p(\mathcal {L}_p)}\,dt+NT^{(p-2)/2}E\int _0^T|h_t|^p_{L_p(\mathcal {L}_2)}\,dt\nonumber \\&\le N'|h|^p_{\mathbb {L}_{p,2}} \end{aligned}$$

(3.12)

with constants $N=N(p,M)$ and $N'=N'(p,M,T)$.

We are now in a position to sketch the proof of Theorem 3.1. Technical details can be found in [10].

Proof of Theorem 3.1(Sketch)

By using standard stopping time arguments, we may assume $E|\psi ^i|^p_{L_p}<\infty$ and that

$$\begin{aligned} E\int _0^T|u_t^i|^p_{W^1_p}\,dt<\infty , \quad E\mathcal {K}_p^p(T)<\infty \quad \text {and}\quad E\int _0^T|u_t^i|^p_{L_p}\,dt<\infty \end{aligned}$$

hold in place of (3.2), (3.3) and (3.6), respectively, for every $i=1,2,\ldots ,M$. We prove first the last sentence of the theorem. We have $f^{ik}=0$ for $i=1,2,\ldots ,M$, $k=1,2,\ldots ,d$ and use the notation $f^i:=f^{i0}$. By Lemmas 3.2, 3.3 and 3.4, there exist $a=(a^i)$ and $b=(b^i)$ and $m=(m^i)$ in $\mathcal {U}_p$ such that for each $\varphi \in C_0^\infty$ almost surely

$$\begin{aligned} (a_t^i,\varphi )=\int _0^t(f^i_s,\varphi )ds, \quad (b_t^i,\varphi )=\int _0^t(g^{ir}_s,\varphi )dw_s^r \end{aligned}$$

and

$$\begin{aligned} (m^i_t,\varphi )=\int _0^t\int _Z(h^i_s,\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

for all $t\in [0,T]$ and $i=1,\ldots ,M$. Thus, $a+b+m$ is an $L_p$-valued adapted cadlag process such that for ${\bar{u}}_t:=\psi +a_t+b_t+m_t$ we have $({\bar{u}}_t,\varphi )= (u_t,\varphi )$ for each $\varphi \in C_0^{\infty }$ for $P\otimes dt$ almost every $(\omega ,t)\in \Omega \times [0,T]$. Hence, by taking a countable set $\Phi \subset C_0^{\infty }$ such that $\Phi$ is dense in $L_q$, we get that ${\bar{u}}=u$ for $P\otimes dt$ almost everywhere as $L_p$-valued functions. Moreover, for each $\varphi \in C_0^{\infty }$

$$\begin{aligned} ({\bar{u}}^i_t,\varphi ) =(\psi ,\varphi ) +\int _0^t(f^i_s,\varphi )\,ds +\int _0^t(g_s^{ir},\varphi )\,dw_s^r +\int _0^t\int _Z(h^i_s(z),\varphi )\,\tilde{\pi }(dz,ds)\nonumber \\ \end{aligned}$$

(3.13)

almost surely for all $t\in [0,T]$, $i=1,2,\ldots ,M$, since on both sides we have cadlag processes. By the estimates of Lemmas 3.2, 3.3 and 3.4,

$$\begin{aligned}&E\int _{\mathbb {R}^d}\sup _{t\le T}|u_t(x)|^p\,dx\nonumber \\&\quad \le N\left( E|\psi |^p_{L_p}+ |f|^p_{\mathbb {L}_p} +|g_s|^p_{\mathbb {L}_p} +|h|^p_{\mathbb {L}_{p,2}} \right) <\infty , \end{aligned}$$

(3.14)

where $N=N(p,M, T)$ is a constant. Substituting $k_{\varepsilon }(x-\cdot )$ in place of $\varphi$ in Eq. (3.13), for $\varepsilon >0$ and $x\in \mathbb {R}^d$ we have (3.9) almost surely for all $t\in [0,T]$ for $i=1,2,\ldots ,M$. Hence, by Corollary 2.4 for each $x\in \mathbb {R}^d$ we have almost surely

$$\begin{aligned} |{\bar{u}}^{(\varepsilon )}_t(x)|^p&= |\psi ^{(\varepsilon )}(x)|^p +\int _0^t p|{\bar{u}}^{(\varepsilon )}_{s-}(x)|^{p-2} {\bar{u}}^{i(\varepsilon )}_{s-}(x)g^{ir(\varepsilon )}_s(x)\,dw^r_s \nonumber \\&\quad +\int _0^t p|{\bar{u}}_{s-}^{(\varepsilon )}(x)|^{p-2} {\bar{u}}_{s-}^{(\varepsilon )i}f_s^{i(\varepsilon )}(x) \,ds \nonumber \\&\quad +\tfrac{p}{2}\int _0^t\big ((p-2) |{\bar{u}}_{s-}^{(\varepsilon )}(x)|^{p-4} |{\bar{u}}_{s-}^{i(\varepsilon )}(x)g^{i\cdot (\varepsilon )}_s(x)|^2_{l_2}\nonumber \\&\quad + |{\bar{u}}_{s-}^{(\varepsilon )}(x)|^{p-2} |g_s^{(\varepsilon )}(x)|_{\ell _2}^2 \big )\,ds\nonumber \\&\quad +\int _0^t \int _Z p|{\bar{u}}_{s-}^{(\varepsilon )}(x)|^{p-2} {\bar{u}}_{s-}^{(\varepsilon )i}(x)h_s^{(\varepsilon )i}(x) \,\tilde{\pi }(dz,ds) \nonumber \\&\quad +\int _0^t\int _Z J^{h_s^{(\varepsilon )}(x,z)}|{\bar{u}}_{s-}^{(\varepsilon )}(x)|^p \,\pi (dz,ds), \end{aligned}$$

(3.15)

for all $t\in [0,T]$, where the notation

$$\begin{aligned} J^a|v|^p=|v+a|^p-|v|^p-a^iD_{i}|v|^p=|v+a|^p-|v|^p-pa^i|v|^{p-2}v^i \end{aligned}$$

is used for vectors $a=(a^1,\ldots ,a^M):={\bar{u}}_{s-}^{(\varepsilon )}(x)$ and $(v^1,\ldots ,v^M):=h_s^{(\varepsilon )}(x,z)\in \mathbb {R}^M$. Furthermore, integrating (3.15) over $\mathbb {R}^d$ and using deterministic and stochastic Fubini theorems, see [10], we get

$$\begin{aligned} |{\bar{u}}^{(\varepsilon )}_{t}|_{L_p}^p&= |\psi ^{(\varepsilon )}|_{L_p}^p +\int _0^t\int _{\mathbb {R}^d} p|u^{(\varepsilon )}_s|^{p-2}u^{i(\varepsilon )}_sg^{ir(\varepsilon )}_s\,dx\,dw^r_s \nonumber \\&\quad +\tfrac{p}{2}\int _0^t\int _{\mathbb {R}^d} 2|u_s^{(\varepsilon )}|^{p-2}u_s^{i(\varepsilon )}f_s^{i(\varepsilon )} + (p-2) |u_s^{(\varepsilon )}|^{p-4}|u_s^{i(\varepsilon )}g^{i\cdot (\varepsilon )}_s|_{l_2}^2 \nonumber \\&\quad +|u_s^{(\varepsilon )}|^{p-2}|g_s^{(\varepsilon )}|_{l_2}^2 \,dx\,ds \nonumber \\&\quad +\int _0^t\int _Z\int _{\mathbb {R}^d} p|u_{s-}^{(\varepsilon )}|^{p-2} u_{s-}^{i(\varepsilon )}h_s^{i(\varepsilon )}\,dx\,\tilde{\pi }(dz,ds) \nonumber \\&\quad +\int _0^t\int _Z\int _{\mathbb {R}^d} J^{h^{(\varepsilon )}_s}|u_{s-}^{(\varepsilon )}|^p\,dx \,\pi (dz,ds) \end{aligned}$$

(3.16)

almost surely for all $t\in [0,T]$. Finally, by taking $\varepsilon \rightarrow 0$ in (3.16), we obtain (3.5) with $f^{ik}=0$ for $i=1,2,\ldots ,M$ and $k=1,2,\ldots ,d$.

Let us prove now the other statements of the theorem. By taking $\varphi ^{(\varepsilon )}$ in place of $\varphi$ in Eq. (3.4), we get

$$\begin{aligned} (u_t^{i(\varepsilon )},\varphi )&=(\psi ^{i(\varepsilon )},\varphi ) +\int _0^t(f_s^{(i\varepsilon )},\varphi )\,ds+\int _0^t(g_s^{ir(\varepsilon )},\varphi )\,dw_s^r\nonumber \\&\quad +\int _0^t\int _Z(h_s^{i(\varepsilon )},\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

(3.17)

for $P\otimes dt$ almost every $(\omega ,t)\in \Omega \times [0,T]$ for each $\varphi \in C_0^\infty$, $i=1,2,\ldots ,m$, where

$$\begin{aligned} f_s^{i(\varepsilon )}:=\sum _{k=1}^dD_kf_s^{ik(\varepsilon )}+f_s^{i0(\varepsilon )}, \quad i=1,2,\ldots ,M,\quad k=1,2,\ldots .,d. \end{aligned}$$

Hence, by virtue of what we have proved above we have an $L_p$-valued adapted cadlag process ${\bar{u}}^{\varepsilon }=({\bar{u}}^{i\varepsilon })$ such that for each $\varphi \in C_0^{\infty }$ almost surely (3.17) holds with ${\bar{u}}^{i\varepsilon }$ in place of $u^{i(\varepsilon )}$ for all $t\in [0,T]$. In particular, for each $\varphi \in C_0^{\infty }$ we have $(u^{(\varepsilon )},\varphi )=({\bar{u}}^{\varepsilon },\varphi )$ for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$. Thus, $u^{(\varepsilon )}={\bar{u}}^{\varepsilon }$, as $L_p$-valued functions, for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$, and almost surely (3.16) holds for all $t\in [0,T]$. Moreover, using that by integration by parts

$$\begin{aligned}&\int _{\mathbb {R}^d} |u_{s}^{(\varepsilon )}|^{p-2}u_{s}^{i(\varepsilon )} D_kf_s^{ik(\varepsilon )}\,dx =-\int _{\mathbb {R}^d} |u_s^{(\varepsilon )}|^{p-2}f^{ik(\varepsilon )}_sD_ku_s^{i(\varepsilon )}\,dx \\&\quad -\tfrac{p-2}{2}\int _{\mathbb {R}^d} |u_s^{(\varepsilon )}|^{p-4}D_k|u_s^{(\varepsilon )}|^2f^{ik(\varepsilon )}_su_s^{i(\varepsilon )}\,dx \end{aligned}$$

for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$, we get

$$\begin{aligned} |{\bar{u}}_t^{\varepsilon }|^p_{L_p}&= |\psi ^{(\varepsilon )}|_{L_p}^p +p\int _0^t\int _{\mathbb {R}^d}|{\bar{u}}_s^{\varepsilon }|^{p-2} {\bar{u}}_s^{\varepsilon }g_s^{ir(\varepsilon )}\,dx\,dw^r_s \nonumber \\&\quad +p\int _0^t\int _{\mathbb {R}^d} |{\bar{u}}_s^{\varepsilon }|^{p-2}{\bar{u}}_s^{\varepsilon } f^{0(\varepsilon )}_s- |{\bar{u}}_s^{\varepsilon }|^{p-2} f_s^{ik(\varepsilon )}D_k u_s^{i(\varepsilon )} \,dx\,ds \nonumber \\&\quad -\tfrac{p}{2}\int _0^t\int _{\mathbb {R}^d} (p-2)|u_s^{(\varepsilon )}|^{p-4}D_k|u_s^{(\varepsilon )}|^2f^{ik(\varepsilon )}_su_s^{i(\varepsilon )} \,dx\,ds \nonumber \\&\quad +\int _0^t\int _{\mathbb {R}^d}(p-2) |{\bar{u}}_{s-}^{(\varepsilon )}|^{p-4} |{\bar{u}}_{s-}^{i(\varepsilon )}g^{i\cdot (\varepsilon )}_s|^2_{l_2} + |{\bar{u}}_{s-}^{(\varepsilon )}|^{p-2} |g_s^{(\varepsilon )}|_{\ell _2}^2\,dx\,ds \nonumber \\&\quad +p\int _0^t\int _Z\int _{\mathbb {R}^d} |{\bar{u}}_{s-}^{\varepsilon }|^{p-2}{\bar{u}}_{s-}^{\varepsilon } h_s^{(\varepsilon )}\,dx\,\tilde{\pi }(dz,ds)\nonumber \\&\quad +\int _0^t\int _Z\int _{\mathbb {R}^d} J^{h^{(\varepsilon )}}|{\bar{u}}_{s-}^{\varepsilon }|^p\,dx\,\pi (dz,ds) \end{aligned}$$

(3.18)

almost surely for all $t\in [0,T]$. Hence, by Davis’, Minkowski and Hölder inequalities, using standard estimates we obtain

$$\begin{aligned}&E\sup _{t\le T}|{\bar{u}}_t^{\varepsilon }|^p_{L_p} \le 2\,E|\psi ^{(\varepsilon )}|_{L_p}^p +NE\int _0^T|h_t^{(\varepsilon )}|^p_{L_p(\mathcal {L}_p)}\,dt +NT^{p-1} |f^{0(\varepsilon )}|^p_{\mathbb {L}_p} \nonumber \\ \nonumber \\&\quad + NT^{(p-2)/2} \left( |g^{(\varepsilon )}|^p_{\mathbb {L}_p} +E\int _0^T|h_t^{(\varepsilon )}|^p_{L_p(\mathcal {L}_2)}\,dt +\sum _{\alpha =1}^d|f^{\alpha (\varepsilon )}|_{\mathbb {L}_p}^p +\sum _{\alpha =1}^d|D_{\alpha }u^{(\varepsilon )}|^p_{\mathbb {L}_p} \right) \end{aligned}$$

(3.19)

with a constant $N=N(p,d)$, where $f^{\alpha (\varepsilon )}:=(f^{1\alpha (\varepsilon )},\ldots , f^{M\alpha (\varepsilon )})$, and recall that $|v|_{L_p}$ means the $L_p$-norm of $|(\sum _{i=1}^M|v^i|^2)^{1/2}|$ for $\mathbb {R}^M$-valued functions $v=(v^1,\ldots ,v^M)$ on $\mathbb {R}^d$. Hence,

$$\begin{aligned} E\sup _{t\le T}|{\bar{u}}_t^{\varepsilon }-{\bar{u}}_t^{\varepsilon '}|^p_{L_p} \rightarrow 0 \quad \text {as }\varepsilon ,\,\varepsilon '\rightarrow 0. \end{aligned}$$

Consequently, there is an $L_p$-valued adapted cadlag process ${\bar{u}}=({\bar{u}}_t)_{t\in [0,T]}$ such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}E\sup _{t\le T}|{\bar{u}}_t^{\varepsilon }-{\bar{u}}|^p_{L_p}=0. \end{aligned}$$

Thus, for each $\varphi \in C_0^{\infty }(\mathbb {R}^d)$ we can take $\varepsilon \rightarrow 0$ in

$$\begin{aligned} ({\bar{u}}_t^{i\varepsilon },\varphi )&=(\psi ^{i(\varepsilon )},\varphi )+\int _0^t(f_s^{i(\varepsilon )},\varphi )\,ds +\int _0^t(g_s^{ir(\varepsilon )},\varphi )\,dw_s^r\\&\qquad +\int _0^t\int _Z(h_s^{i(\varepsilon )},\varphi )\,\tilde{\pi }(dz,ds)\\&= (\psi ^{i(\varepsilon )},\varphi )+\int _0^t(f_s^{i0(\varepsilon )},\varphi )\,ds -\int _0^t(f_s^{ik(\varepsilon )},D_k\varphi )\,ds +\int _0^t(g_s^{ir(\varepsilon )},\varphi )\,dw_s^r\\&\quad +\int _0^t\int _Z(h_s^{i(\varepsilon )},\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

and it is easy to see that we get

$$\begin{aligned} ({\bar{u}}^i_t,\varphi )=(\psi ^i,\varphi )+\int _0^t(f_s^{i\alpha },D^{*}_{\alpha }\varphi )\,ds +\int _0^t(g_s^{ir},\varphi )\,dw_s^r+\int _0^t\int _Z(h^i_s,\varphi )\,\tilde{\pi }(dz,ds) \end{aligned}$$

almost surely for all $t\in [0,T]$. Hence, ${\bar{u}}= u$ for $P\otimes dt$-almost every $(\omega ,t)\in \Omega \times [0,T]$. Finally letting $\varepsilon \rightarrow 0$ in (3.18), we obtain (3.7). $\square$