On Itô formulas for jump processes

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 Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.


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 R M -valued stochastic processes X = (X t ) t≥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 φ(X t ) by sufficiently smooth functions φ are basic tools in the investigations of stochastic phenomena mod-B István Gyöngy i.gyongy@ed.ac.uk Sizhou Wu Sizhou.Wu@ed.ac.uk 1 School of Mathematics and Maxwell Institute, University of Edinburgh, Scotland, Edinburgh, UK 2 School of Mathematics, University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ, UK elled 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 φ. 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 φ.
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 α and g r of t ∈ [0, T ] and x = (x 1 , . . . , x d ) ∈ R d , and a sequence of martingales (m i ) ∞ i=1 . This stochastic differential is understood in a weak sense, i.e. for each smooth function ϕ with compact support on R d we have the stochastic differential where (u, v) denotes the Lebesgue integral over R d of the product uv for functions u and v of x ∈ R d . In the L 2 -theory of SPDEs f α and g r are L 2 (R d , R)-valued functions of (ω, 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 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 = 2 one should establish a suitable formula for |v t | p L p , which was first achieved in Krylov [14] for p ≥ 2 when (m i ) ∞ i=1 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 2) whereπ(dz, dt) is a Poisson martingale measure with a σ -finite characteristic measure μ on a measurable space (Z , Z) and h is a function on × [0, T ] × Z × 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 ≥ 2. We prove it by adapting ideas and methods from Krylov [14]. In particular, we use the finite-dimensional Itô's formula (2.19) t is an approximation of v t obtained by smoothing it in x. Hence, we integrate both sides of the formula for |v ε t (x)| p over 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 ε → 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, . . . , M, and we need estimates for || 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, . . . , 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 i f 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 i f i . Thus, the Itô formula in these papers cannot be applied to |v t | p L p when the stochastic differential dv 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 ( , F, P) equipped with a right-continuous filtration (F t ) t≥0 such that F 0 contains all P-zero sets of F. The σ -algebra of the predictable subsets of × [0, ∞) is denoted by P. We are given a sequence w = (w 1 t , w 2 t , . . .) t≥0 of F t -adapted independent Wiener processes w r = (w r t ) t≥0 , such that w t −w s is independent of F s for any 0 ≤ s ≤ t. For an integer m ≥ 1, we are given also a sequence of independent Poisson random measures π k (dz, dt) on [0, ∞) × Z k , with intensity measure μ k (dz) dt for k = 1, 2, . . . , m, where μ k is a σ -finite measure on a measurable space (Z k , Z k ) with a countably generated σ -algebra Z k . We assume that the process π k t ( ) := π k ( × (0, t]), t ≥ 0, is F t -adapted and π k t ( ) − π k s ( ) is independent of F s for any 0 ≤ s ≤ t and ∈ Z k such that μ k ( ) < ∞. We use the notationπ k (dz, dt) = π k (dz, dt) − μ k (dz)dt for the compensated Poisson random measure and setπ k t ( ) =π k ( × (0, t]) = π k t ( ) − tμ k ( ) for t ≥ 0 and ∈ Z such that μ k ( ) < ∞. If m = 1, then we write π ,π, Z , Z and μ in place of π 1 ,π 1 , Z 1 , Z 1 and μ 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 ν = (ν 1 , ν 2 , . . .) of vectors ν k ∈ R M with finite norm is denoted by 2 = 2 (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 R M is denoted by C ∞ 0 (R M ). For integers k ≥ 1, the notation C k (R M ) means the space of functions on R M whose derivatives up to order k exist and are continuous, and C k b (R M ) denotes the space of functions on 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.

Itô formulas in finite dimensions
We consider an R M -valued semimartingale X = (X 1 t , . . . , X M t ) t≥0 given by 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 r g 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
holds almost surely for all t ≥ 0.
Proof This theorem with a finite-dimensional Wiener process w = (w 1 , . . . , 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: .. be the jump times of the process N = m k=1 N k . Then ρ k i and τ i are stopping times for every k = 1, 2, . . . , m and i ≥ 1, and for almost every ω ∈ , the set of time points {τ i (ω) : i ≥ 1} contains all points of discontinuities of (X n t (ω)) t≥0 , where the process X n is defined by where we set τ 0 := 0 and X n τ i ∧t− := X n τ i − for t ≥ τ i and X n τ i ∧t− := X n t for t < τ i . By Itô's formula for Itô processes, we have which gives Notice that ρ k i has a density with respect to the Lebesgue measure for i ≥ 1, and ρ k i and ρ l j are independent for k = l. Hence, P(ρ k i = ρ l j ) = 0 for k = l and positive integers i, j. Consequently, for almost every ω ∈ we have 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 Combining this with (2.6) we get Hence, we can finish the proof by letting n → ∞ and using standard facts about convergence of Lebesgue integrals and stochastic integrals with respect to Wiener processes and random measures.
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.
is the measure of jumps of a standard Poisson process andπ(dz, dt) = π(dz, dt)−μ(dz)dt is its compensated measure, where μ = δ 1 is the Dirac measure on Z concentrated at 1. Then obviously conditions (2.2) and (2.3) hold, and for φ(x) = x 4 , the last integrand in (2.4) is 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 toπ(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 φ to ensure that formula (2.4) holds. It is sufficient to assume that the derivatives of φ up to second order are bounded. Such a condition, however, excludes the applicability of Itô's formula to power functions φ(x) = |x| p for p ≥ 2. Notice that for any  = (a 1 , . . . , a M ) ∈ R M and functions φ ∈ C 2 (R M ), we define the functions I a φ and J a φ by (h 1(n) , . . . , h M(n) ) and define in probability uniformly in t ∈ [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 in probability uniformly in t ∈ [0, T ] for each T > 0. Hence, letting n → ∞ in (2.4) with h (n) and X (n) in place of h and X , respectively, we prove the theorem for φ ∈ , and therefore, (2.4) holds with φ n in place of φ. Thus, it remains to take limit as n → ∞ for each term in (2.4) with φ n in place of φ. Clearly, as n → ∞, we have uniformly on compact subsets of R M for i, j = 1, 2, . . . , M. Hence, it is easy to see with a constant C independent of n, and since lim n→∞ |1 − ζ n (X s + h s (z))| = 0, we have lim sup n→∞ |I h s (z) φ n (X s ) − I h s (z) φ(X s )| = 0 for every (ω, s, z).

Hence, by (2.10), taking into account conditions (2.3) and (2.7) on h and I h s (z) φ(X s ),
we can apply Lebesgue's theorem on dominated convergence to obtain which implies that for n → ∞ we have in probability uniformly in t ∈ [0, T ] for each T > 0. Similarly, we get for every T ≥ 0. Using the identity Hence, taking into account |(1 − ζ n (X s ))| ≤ 1, and lim n→∞ |(1 − ζ n (X s ))| = 0, we obtain with a constant C independent of n, and
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 Clearly, for all (ω, t, z) |h (n) | ≤ min(|h|, nM) and h (n) → h as n → ∞.
Therefore, Theorem 2.3 for X (n) holds, and Thus, there is a strictly increasing subsequence of positive integers (n k ) ∞ k=1 such that Hence, it is easy to pass to the limit k → ∞ in φ(X (n k ) t ) and in the first two integral terms in the equation for φ(X (n k ) t ) in Theorem 2.3. To pass to the limit in the other terms in this equation notice that since π(dz, dt) is a counting measure of a point process, from the condition forh in (2.3) we get ξ := π − ess sup |h| < ∞ (a.s.), (2.16) where π − ess sup denotes the essential supremum operator with respect to the measure π(dz, dt) over Z × [0, T ]. Similarly, from the condition for h we have η := π − ess sup |h| < ∞ (a.s.). (2.17) This can be seen by noting that for the sequence of predictable stopping times we have Since (τ j ) ∞ j=1 is an increasing sequence converging to infinity, we have P( which implies (2.17). By (2.16) and the first inequality in (2.13), we have almost surely for π(dz, dt)-almost every (z, t) ∈ Z × [0, T ]. Hence, by Lebesgue's theorem on dominated convergence we get almost surely, uniformly in t ∈ [0, T ]. Clearly, Hence, by Lebesgue's theorem on dominated convergence, in probability, uniformly in t ∈ [0, T ]. Finally, note that by using the second inequality in (2.13) together with (2.17) we have almost surely for π(dz, dt)-almost every (z, t) ∈ Z × [0, T ]. Hence, taking into account (2.18), by Lebesgue's theorem on dominated convergence we obtain almost surely, uniformly in t ∈ [0, T ] for every T > 0, which finishes the proof of the theorem.
almost surely for all t ≥ 0, where, and through the paper, the convention 0/0 := 0 is used whenever it occurs.
Proof Notice that φ(x) = |x| p for p ≥ 2 belongs to C 2 (R M ) with where δ i j = 1 for i = j and δ i j = 0 for i = j. Hence, it is easy to see that Theorem 2.3 applied to φ(x) = |x| p gives the corollary.
The above corollary will be used to obtain an Itô's formulas for jump processes in L p -spaces presented in the next section.

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 ≥ 1 and M ≥ 1 are fixed integers.
The Borel σ -algebra of a topological space V is denoted by B(V ). For p, q ≥ 1 we denote by The notation L p,q means the space L p ∩ L q with the norm As usual, W 1 p denotes the space of functions u ∈ L p such that D i u ∈ L p for every i = 1, 2, . . . , d, where D i v means the generalised derivative of v in x i for locally integrable functions v on R d . The norm of u ∈ W 1 p is defined by We use the notation L p = L p ( 2 ) for L p (R d , 2 ), the space of Borel-measurable functions g = (g ir ) on R d with values in 2 such that For p, q ∈ [0, ∞), we denote by L p = L p (L p,q ) and L p = L p (L q ) the Banach spaces of Borel-measurable functions h = (h i (x, z)) andh = (h i (x, z)) of x ∈ R d with values in L p,q and L q , respectively, such that For p ≥ 2 and a separable real Banach space V , we denote by In the sequel, V will be , then for L p (V ) the notation L p,2 is also used. For ε ∈ (0, 1) and locally integrable functions v of x ∈ R d , we use the notation v (ε) for the mollifica- where k ε (y) = ε −d k(y/ε) for y ∈ R d with a fixed function k ∈ C ∞ 0 of unit integral. If v is a locally Bochner integrable function on 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.
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 iα = 0 for α = 1, 2, . . . , d, and suppose from (3.4) we could show the existence of a random fieldū =ū(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 andπ(dz, ds), respectively, satisfying appropriate measurability conditions such that the equation for every x ∈ R d , then integrating over R d , and finally, using suitable stochastic Fubini theorems, we could obtain (3.5) when f iα = 0 for α ≥ 1. When f iα = 0, we could take instead of u i , ψ i , f i , g ir and h i above, respectively, to apply the theorem in the special case, and let ε → 0 in the corresponding Itô formula after integrating by parts in the terms containing D k f ik(ε) for k = 1, . . . , d. Notice that we can formally obtain Eq. (3.8) from (3.4) with f i1 = · · · = f id = 0 and a suitable processū in place of u, by substituting δ x , the Dirac delta at x, in place of ϕ. Clearly, we cannot substitute δ x , but we can substitute approximations k ε (x − ·) of it to get u i(ε) s (x, z)π(dz, ds) (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 |ū (ε) t (x)| p for each x ∈ R d , integrate the obtained formula over R d , then interchange the order of the integrals, and finally let ε → 0 to prove Eq. (3.5) when f ik = 0 for i = 1, 2, . . . , M and k = 1, 2, . . . , d.
To implement the above idea we fix a p ≥ 2 and introduce a class of functions U p , the counterpart of the class U p given in [14]. Let U p denote the set of R M -valued is cadlag in t ∈ [0, T ] for each (ω, x), (iv) u t (ω, ·) as a function of (ω, t) is L p -valued, F t -adapted and cadlag in t for every ω ∈ .
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].   N ( p, M).
The proof of the following lemma can be found in [10]. We are now in a position to sketch the proof of Theorem 3.1. Technical details can be found in [10].