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.
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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
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
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
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
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.
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
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 \(k=1,2,\ldots ,m\)
and
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
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
with
Clearly, \(\phi (X^n_t)=\phi (X^n_0)+A^n_t+B_t^n\) with
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
which gives
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
for all \(t\ge 0\), where
Combining this with (2.6) we get
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
with
Clearly,
for every \(t>0\), which shows that the last integral in (2.4) is infinite. Similarly, one can show that almost surely
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
and
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
Assume first \(\phi \in C_b^2(\mathbb {R}^M)\). Approximate h by \(h^{(n)}=(h^{1(n)},\ldots ,h^{M(n)})\) and define
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,
which implies
in probability uniformly in \(t\in [0,T]\). Consequently, for each \(T>0\) we have
in probability. It is easy to see
in probability uniformly in \(t\in [0,T]\) for \(T>0\). Furthermore, by Taylor’s formula we have
with a constant C independent of n. Hence, by Lebesgue’s theorem on dominated convergence for \(T>0\) we have
and
in probability, which implies
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
uniformly on compact subsets of \(\mathbb {R}^M\) for \(i,j=1,2,\ldots ,M\). Hence, it is easy to see
and
in probability, uniformly in \(t\in [0,T]\) as \(n\rightarrow \infty \). Using the simple identity
with \(\varphi =\phi _n\) and \(a=h_s(z)\), we get
with a constant C independent of n, and since \(\lim _{n\rightarrow \infty }|1-\zeta _n(X_s+h_s(z))|=0\), we have
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
which implies that for \(n\rightarrow \infty \) we have
in probability uniformly in \(t\in [0,T]\) for each \(T>0\). Similarly, we get
for every \(T\ge 0\). Using the identity
with \(\varphi =\phi _n\) and \(a=h_s(z)\), we get
Hence, taking into account \(|(1-\zeta _n(X_s))|\le 1\),
and \(\lim _{n\rightarrow \infty }|(1-\zeta _n(X_s))|=0\), we obtain
with a constant C independent of n, and
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
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 \(t>0\)
and
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
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
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
and
almost surely. Clearly,
Hence, by virtue of (2.15) the stochastic Itô integral
can be decomposed as
and by virtue of (2.14) and (2.15),
Hence,
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
Clearly, for all \((\omega ,t,z)\)
Therefore, Theorem 2.3 for \(X^{(n)}\) holds, and
which implies
Thus, there is a strictly increasing subsequence of positive integers \((n_k)_{k=1}^{\infty }\) such that
which implies
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
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
This can be seen by noting that for the sequence of predictable stopping times
we have
which gives
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.
which implies (2.17). By (2.16) and the first inequality in (2.13), we have
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
which implies that, for \(k\rightarrow \infty \),
almost surely, uniformly in \(t\in [0,T]\). Clearly,
almost surely for all \((z,t)\in Z\times [0,T]\). Hence, by Lebesgue’s theorem on dominated convergence,
which implies that, for \(k\rightarrow \infty \),
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
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
which implies that, for \(k\rightarrow \infty \),
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
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
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
The notation \(\mathcal {L}_{p,q}\) means the space \(\mathcal {L}_{p}\cap \mathcal {L}_{q}\) with the norm
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
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
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
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
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,
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
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
Notice that for \(M=1\) Eq. (3.5) has the simpler form
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
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
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
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
holds for all \(t\in [0,T]\). Furthermore, we have
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
holds for all \(t\in [0,T]\). Furthermore, we have
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
for all \(t\in [0,T]\), and
Furthermore,
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
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
and
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 \(\varphi \in C_0^{\infty }\)
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,
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
for all \(t\in [0,T]\), where the notation
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
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
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
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
for \(P\otimes dt\)-almost every \((\omega ,t)\in \Omega \times [0,T]\), we get
almost surely for all \(t\in [0,T]\). Hence, by Davis’, Minkowski and Hölder inequalities, using standard estimates we obtain
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,
Consequently, there is an \(L_p\)-valued adapted cadlag process \({\bar{u}}=({\bar{u}}_t)_{t\in [0,T]}\) such that
Thus, for each \(\varphi \in C_0^{\infty }(\mathbb {R}^d)\) we can take \(\varepsilon \rightarrow 0\) in
and it is easy to see that we get
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). \(\square \)
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Gyöngy, I., Wu, S. On Itô formulas for jump processes. Queueing Syst 98, 247–273 (2021). https://doi.org/10.1007/s11134-021-09709-8
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DOI: https://doi.org/10.1007/s11134-021-09709-8