Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations
Jiahui Zhu Zdzisław Brzeźniak Wei Liu
math.PR math.AP 60H15, 60J75, 35B65, 46B09

	Maximal inequalities]
	 Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations^†
	 [† This work is supported  by NSFC (11501509,11571147,11822106,11831014), NSF of Jiangsu Province
(BK20160004), and the Qing Lan Project and PAPD of Jiangsu Higher Education Institutions.]
J. Zhu,  Z. Brzeźniak and W. Liu]






145mm




	








[
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    today
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Jiahui Zhu^a, Zdzisław Brzeźniak^b, Wei Liu^c,[Corresponding author: weiliu@jsnu.edu.cn]
 a.   School of Science, Zhejiang University of Technology, Hangzhou 310019, China 
  b. Department of Mathematics, University of York,  York YO10 5DD, UK
  c. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
 



We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. 	
 Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.


Keywords.  Burkholder-Davis-Gundy inequality, maximal inequality, exponential estimate, stochastic convolution,  Itô  formula, martingale type r Banach space


AMS Subject Classification.  60H15; 60J75; 35B65; 46B09
 





		
	
	
	
	

§ INTRODUCTION


	Over the past few decades, stochastic partial differential equations (SPDEs) have attracted considerable attention from researchers in a wide variety of fields, including biology, physics, engineering and finance etc. (cf. <cit.> and the references therein).
In the study of SPDEs, the Burkholder-Davis-Gundy (BDG) inequality and maximal inequality play vital roles
	  in proving the existence, uniqueness, and regularity of solutions of SPDEs. There are quite a number of contributions on the study of BDG and maximal inequalities when the state space is a Hilbert space; see <cit.>, <cit.>,  <cit.>, <cit.> and <cit.>.  However, many interesting problems in the theory of SPDEs whose natural settings in function spaces are not Hilbert spaces, but rather Banach spaces (e.g. some Sobolev spaces).  Nevertheless, literature and research studies related to these inequalities on general Banach spaces  are very limited  and this is the motivation of our paper.
	
	The overall goal of this work is to investigate BDG inequalities and maximal regularities of stochastic convolutions driven by Lévy processes in Banach spaces.  We will derive, in Appendix <ref>, two general versions of Itô's  formula for Lévy-type processes in Banach spaces, which are crucial for the proof of our inequalities. 	
	We will work in the martingale type r Banach spaces with 1<r≤ 2.
	This assumption is necessary for establishing a theory of stochastic integration in Banach spaces.
 Typical examples of such spaces are L^p spaces with p∈[r,∞) and Sobolev spaces. 	

 	Now let us state  our problem more explicitly.
	Let (E,|·|_E) be a separable Banach space of martingale type r with 1<r≤ 2 and  (Ω,ℱ,ℙ) be a probability space with the filtration (ℱ_t)_t≥0 satisfying the usual hypotheses, and let  (Z,𝒵) be a measurable space. We first consider the following process
	
    u_t=∫_0^t∫_Zξ(s,z) Ñ(ṣ,ẓ),    t∈[0,T], T>0,

	where
	Ñ is a compensated Poisson random measure on (Z,𝒵) and ξ is an E-valued 𝒫⊗𝒵-measurable process. In this paper we will establish the following BDG and L^p inequalities
		
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r (∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r  for all 1≤ p<∞;
       sup_t∈[0,T] |u_t|_E^p≤ C_p,rmin{ ∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ) ṣ,
       (∫_0^T∫_Z|ξ(s,z)|_E^r ν(ẓ) ṣ)^p/r}  for all 1≤ p≤ r;
       sup_t∈[0,T]|u_t|_E^p≤ C_p,r (∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
       +C_p,r (∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r  for all r≤ p<∞.

	
	Here and in what follows, C_p,r is a generic constant depending only on p and
	r, which may change from line to line. An inequality similar to (<ref>) with p=r^n, n∈ℕ was proved by Hausenblas in <cit.> by using a discrete approximation argument of stochastic integrals. In this paper, by following the Itô formula approach in the spirit of <cit.> and using some ingredients from <cit.>, we give a short and  simple proof of  (<ref>) and extend the inequality to all 1≤ p<∞; see Theorem <ref>. 	
 In a recent work <cit.> by Dirksen, upper-bound inequalities similar to (<ref>) and (<ref>) with E being L^q spaces were obtained via a noncommutative probability approach.  	   In this paper, we propose a different approach and generalize the inequalities to all Banach spaces of martingale type r with 1 < r≤ 2.  Note that all L^q spaces with q∈[r,∞) are of martingale type r.
	 We want to remark that our method is more direct and closer in spirit to the ideas used in Hilbert space and finite dimensional spaces.
	

	Now we assume that (e^tA)_t≥0 is a C_0-semigroup on (E,|·|_E) with a generator A such that e^tA≤ e^tα  for some α≥  0.  Let X_t=∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ). 	 In the second part of the present paper, we will prove that there exists a càdlàg modification X̅ of X such that the following inequalities hold
	
	
  (i) for all r≤ p<∞,
		
    sup_0≤ t≤ T|X̅_t|^p≤ e^α p TC_p,r (∫_0^T∫_Z |ξ(s,z)|^r_E  N(ṣ,ẓ))^p/r,

	 
  (ii) for all 0< p≤ r,
	 
    sup_0≤ t≤ T|X̅_t|^p≤ e^α p TC_p,r (∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r,

		
  (iii) for all r≤ p<∞,
		
    sup_0≤ t≤ T|X̅_t|^p≤ e^α p TC_p,r[(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
    		 +(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r].

	
	
	
	Considering maximal inequalities in infinite dimensional spaces, there exist mainly three different approaches in literature. The first approach is based on the factorization method and stochastic Fubini theorem introduced by Da Prato, Kwapień and Zabczyk in <cit.>. By following this approach, a weaker inequality of  L^p estimates of stochastic convolutions for Wiener processes in Hilbert spaces was obtain in <cit.>. Another approach is to apply isometric dilation theorems on semigroups that admit dilations; see <cit.>, <cit.>, <cit.> and the references therein.
		This unitary dilation method was used in <cit.> to get the maximal inequality for a stochastic convolution driven by a Wiener process and C_0-contraction semigroups in a Hilbert space H. Later  Seidler in <cit.> obtained L^p-estimates for stochastic convolutions of positive contraction semigroups in a 2-smooth Banach space E=L^q(μ), q≥ 2 with a sharp constant C_p=o(√(p)).
		We also refer to Veraar and Weis <cit.>, Dirksen, Maas and van Neerven <cit.> and Dirksen <cit.>.
		
	A classical approach
	to prove the
	maximal inequality for stochastic convolution in finite dimensional spaces is to apply the Itô  formula to a C^2-mapping E ∋ x↦ |x|^p_E, 2≤ p<∞, and then the classical Burkholder inequality; see e.g. <cit.>, <cit.>. When E is a Hilbert space, the function ψ_p: E ∋ x↦ |x|^p_E,  is always  of C^2 class, see Ichikawa <cit.> and Tubaro <cit.> for inequalities of stochastic convolutions in Hilbert spaces. While dealing with more general Banach spaces, we encounter a difficulty that the mapping ψ_p may  not even be twice Fréchet differentiable. 
	In the Gaussian case, this particular problem is addressed in <cit.> where the second named author and Peszat made a certain assumption on E, which they call (H_p). Under a similar assumption on the Banach space E, the first two authors of this paper and Hausenblas derived in <cit.> a version of maximal inequality for Lévy-type noises. More precisely, they assumed  that  for some p∈[2,∞), the function ψ_p  is of C^2 and its first and second Fréchet derivatives are bounded by some constant multiples of |x|_E^p-1 and |x|_E^p-2. By means of the Itô  formula and the Davis inequality, they obtained a maximal inequality for contraction semigroups.
	Note that various spaces do have martingale type 2 property, but fail to satisfy condition (H_p).  For instance, L^2(L^q(0,1)), q>2, are martingale type 2 Banach spaces, but according to Theorem 3.9 in <cit.>, the norm of  L^2(L^q(0,1)) is not twice Fréchet differentiable away from the origin.
In this paper, we follow the Itô formula approach and work in a more general setting of Banach spaces. Compared to <cit.> and <cit.>, in the present paper we weaken the assumptions on the Banach spaces. We only assume that the Banach space is of martingale type r, 1<r≤ 2. Note that this is not an easy question. Due to the lack of the twice differentiability property of the p-th power of the norm in martingale type r Banach spaces, a straightforward extension of the inequalities seems impossible and a new technique is required.
To do this, we derive two general versions of Itô's formula in Banach spaces (see Appendix B) which can be applied to the norm function ψ_p directly.
 By following the Itô formula approach and employing the BDG inequality, e.g. (<ref>), it allows us to estimate each term properly.   Our proofs here are more succinct.


Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingale type 2 Banach space (E,|·|_E)
		and let ξ∈ℳ_T^2(𝒫⊗𝒵,ṭ×ℙ×ν;E).
 We will also show the following exponential tail estimates for stochastic convolutions driven by compensated Poisson random measures: if there exists λ>0 and M_λ>0 such that
	    
    ∫_0^T∫_Z e^λ^1/2|ξ(s,z)|_Eλ|ξ(s,z)|_E^2 ν(ẓ) ṣ≤ M_λ,

	then for every R>0 there exists
	a constant C_λ>0 such that
	
    ℙ( sup_0≤ t≤ T|X_t|_E ≥ R )≤ C_λe^-(1+λ R^2)^1/2.



	Let E be a martingale type 2 Banach space. We will also study stochastic convolution processes of the form
	
    X_t=∫_0^te^(t-s)Ag_s Ẉ_s+∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ), t∈[0,T], T>0.

	Here g:ℝ_+×Ω→γ(H;E) is a progressively measurable process which is stochastic integrable with respect to a cylindrical Wiener process W on H and ξ is defined as above.
	 We will prove that there exists a càdlàg modification X̅ of X such that
	 	
    sup_0≤ t≤ T |X̅_t|_E^p≤ e^α TC_p,r   [(∫_0^T g_s^2_γ(H,E) ṣ)^p/2+(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ).
       .+(∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ) ṣ)^p/2]  for all 2≤ p<∞.

	
	
We emphasize that maximal inequalities developed in this paper  are applicable to many nonlinear SPDEs, including stochastic Euler equations, stochastic reaction-diffusion equations, and stochastic Navier-Stokes equations, etc. As an example we consider in Section <ref> the existence and uniqueness of mild solutions of stochastic two dimensional quasi-geostrophic equations via the application of maximal inequalities. We establish the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equations under much weaker assumptions in terms of L^p theory.



The rest of the paper is organized as follows. In the next section we will study BDG inequalities for stochastic integrals. Maximal inequalities and exponential estimates for stochastic convolutions will be treated in Section <ref>. In Section 4 we show the existence and uniqueness of solutions to stochastic quasi-geostrophic equations as the application of our main results. Finally, in Appendix A we give a brief review of some results of stochastic integral w.r.t. Poisson random measures, while in Appendix B we prove two versions of Itô's formula for Banach space valued Lévy processes.





	 

§ BDG INEQUALITIES FOR STOCHASTIC INTEGRALS DRIVEN BY LÉVY  PROCESSES


	      Throughout the whole section we assume that (E,|·|_E) is a real separable Banach space of martingale type r, 1<r≤ 2, see Appendix A for the definition.
	      Let  (Ω,ℱ,𝔽,ℙ), where 𝔽 = (ℱ_t )_t≥ 0, be a filtered probability space satisfying the usual hypothesis.
	
	      Let (Z,𝒵) be a measurable space and ν be a σ-finite measure on it.  Assume that  π is a σ-finite and stationary Poisson point process with the characteristic measure ν such that the counting measure
  N associated with π is a time homogeneous Poisson random measure; see Appendix A for the existence of such a process and also <cit.>. We denote by 𝒫 the predictable σ-field on [0,T]×Ω, i.e. the σ-field generated by all left continuous and ℱ_t-adapted
		 real-valued processes on [0,T]×Ω.	


	
	   For the convenience of the reader we repeat some notations and standard facts about stochastic integration w.r.t. the compensated Poisson random measure which we will be frequently used in the paper. The detailed discussions can be found in Appendix A.
	
	   We use the symbol ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E)
		 to denote the space of all 𝒫⊗𝒵-measurable functions
		ξ:[0,T]×Ω× Z→ E such that ξ∈ L^r(Ω;L^r([0,T]× Z;E)).
			
		 Take ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E). According to <cit.>, the stochastic integral process
		 
    I_t(ξ)=∫_0^t∫_Z ξ(s,·,z) Ñ(ṣ,ẓ),   t∈[0,T],

		  is a càdlàg r-integrable E-valued martingale and it satisfies the following: inequality
	         	
    𝔼|∫_0^T∫_Z ξ(s,z) Ñ(ṣ,ẓ)|_E^r
    		≤ C_r(E) 𝔼∫_0^T∫_Z|ξ(s,z)|_E^r ν(ẓ) ṣ.

		If
			ξ is a 𝒫⊗𝒵-measurable function in ℳ_T^1(𝒫⊗𝒵,ṭ×ℙ×ν;E),
			then one can show (see Appendix A for the proof) that
			
    ∫_0^T∫_Zξ(s,z) N(ṣ,ẓ)=∫_0^T∫_Zξ(s,z) ν(ẓ) ṣ.

		
			In particular, if
			ξ∈ℳ_T^1(𝒫⊗𝒵,ṭ×ℙ×ν;E)∩ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E), then the stochastic integral I_t(ξ) can be written as a sum of two Bochner integrals
			
    ∫_0^t∫_Zξ(s,·,z) Ñ(ṣ,ẓ)=
    			     ∑_s∈(0,t]∩𝒟(π)ξ(s,·,π(s))-∫_0^t∫_Zξ(s,·,z) ν(ẓ) ṣ,  -a.s.

			Here 𝒟(π) is the domain of the Poisson point process π.
			

	  Set
		
    u_t:=∫_0^t∫_Zξ(s,z) Ñ(ṣ,ẓ),   t∈[0,T].

	In this section, we aim to establish some types of inequalities and L^p estimates for (u_t)_0≤ t≤ T.
	
		
		  Let f:[0,T]×Ω× Z→ℝ be a real-valued function in ℳ_T^1(𝒫⊗𝒵,ṭ×ℙ×ν;ℝ).  			Recall that ∫_0^t∫_Z |f(s,z)|Ñ(ṣ,ẓ), t∈[0,T], is an ℝ-valued and 1-summable martingale; see <cit.>. Moreover, it can be written as a sum of two Lebesgue integrals in the following way:
			
    ∫_0^t∫_Z |f(s,z)| Ñ(ṣ,ẓ)=∫_0^t∫_Z |f(s,z)| N(ṣ,ẓ)-∫_0^t∫_Z |f(s,z)| ν(ẓ) ṣ,  ℙ-a.s.

	So we can deduce that
			
    ( ∫_0^T∫_Z|f(s,z)| N(ṣ,ẓ)|ℱ_t)
    			        ≥( ∫_0^T∫_Z|f(s,z)| ν(ẓ) ṣ-∫_0^t∫_Z|f(s,z)|ν(ẓ) ṣ|ℱ_t).

			  A simple application of Proposition 25.21 in <cit.> with X=∫_0^T∫_Z|f(s,z)| N(ṣ,ẓ) and 
 A_t=∫_0^t∫_Z|f(s,z)| ν(ẓ) ṣ, t∈ [0,T], yields for all 1≤ p<∞,
		   
    (∫_0^T∫_Z|f(s,z)| ν(ẓ) ṣ)^p≤ p^p (∫_0^T∫_Z|f(s,z)| N(ṣ,ẓ))^p.

		
	
	    In Proposition <ref> and Theorem <ref> below, we will formulate and prove some inequalities for stochastic integrals with respect to compensated Poisson random measures.  We shall follow a common strategy for proving a BDG inequality for real-valued local martingales used in <cit.> to obtain inequality (<ref>). The idea of this proof can be traced back to <cit.>, where a version of the BDG inequality was derived for discrete-time martingales with p=1.
	 Similar estimates, analogous to (<ref>), (<ref>), and (<ref>) in finite dimension space, were first studied by Novikov in <cit.>. The main ingredient of his proof is a version of the change-of-variable formula (Itô formula) for the transformation of a stochastic integral which requires the function only to be continuously differentiable.
	  In a Hilbert space setting, r=2, inequalities (<ref>) and (<ref>) together with the inequality (<ref>) in Theorem <ref> can be read as
	
    sup_t∈[0,T] |u_t|_E^p≤ C  ⟨ u⟩_T^p/2,   0<p<2,
       sup_t∈[0,T] |u_t|_E^p≤ C  [ u]_T^p/2,   1≤ p<∞.

	Here ⟨ u⟩_t=∫_0^t∫_Z|ξ(s,z)|^2 ν(ẓ)ṣ and [u]_t=∫_0^t∫_Z|ξ(s,z)|^2 N(ṣ,ẓ) are, respectively, the Meyer  and the quadratic variation processes of u; see <cit.> for the definitions. Inequality (<ref>) is known as the BDG inequality. Inequalities (<ref>) with p=2 and (<ref>) with 2≤ p<∞ were studied by Métivier <cit.> for Hilbert space valued right-continuous local martingales. Ichikawa in <cit.> established a stopped version of inequality (<ref>) for right-continuous martingales in Hilbert spaces. A BDG inequality for a stochastic integral with respect to a cylindrical Brownian motion in Orlicz-type spaces was obtained recently by Xie and Zhang in <cit.>.
	
	Let us point out that inequality (<ref>) below has already  been stated and proved in <cit.>, but we include it here for  the readers' convenience.
	
	Throughout the paper, we use C_a_1,a_2,⋯ to denote a generic positive constant whose value may change from
line to line, but depends only on the designated variables a_1,a_2,⋯.

		
		  Let ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E) and (u_t)_t∈[0,T] be defined by (<ref>).  Then the following inequalities hold provided the expressions on the right hand sides are finite:
		   
		      
  * for 0<p≤ r,
		      	
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r (∫_0^T∫_Z|ξ(s,z)|_E^r ν(ẓ) ṣ)^p/r;

		      
  * for 1≤ p≤ r,
		   
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r (∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r;

		   
  * for 1≤ p≤ r,
		   
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r ∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ) ṣ;

		      
  * for 1≤ p≤ r,
	          
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r ∫_0^T∫_Z|ξ(s,z)|_E^p N(ṣ,ẓ).

		   
		

		
			(1) See Corollary C.2 in <cit.>.
	
	 (2). Let us fix r ≥ 1. Note that u_t=ξ(t,π(t))1_𝒟(π)(π(t)) and also observe that
			
    ∑_s≤ t|ξ(s,π(s))|_E1_{|ξ(s,π(s))|_E>2sup_v<s|ξ(v,π(v))|_E}   ≤ 2∑_s≤ t(sup_v≤ s|ξ(v,π(v))|_E-sup_v<s|ξ(v,π(v))|_E) 
       ≤ 2sup_s≤ t |ξ(s,π(s))|_E≤ 2 (∑_s≤ t|ξ(s,π(s))|_E^r)^1/r
       =2(∫_0^t∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^1/r.

	Thus we can define a process A_t, t∈[0,T], by
	    
    A_t   :=∑_s≤ tξ(s,π(s))1_𝒟(π)(π(s))1_{| u_s|_E>2sup_v<s| u_v|_E},  t∈[0,T].

	Since the process (u_t) is right-continuous with left limits, the function X_t:=sup_s<t| u_s|_E
	is left-continuous. So this together with the adaptedness implies that X_t is 𝒫-measurable. By assumption, the function ξ is 𝒫⊗𝒵-measurable. Hence we infer that 	{(s,ω,z):|ξ(s,ω,z)|_E>2sup_s< t| u_s|_E}∈𝒫⊗𝒵. Similarly, one can find that {(s,ω,z):|ξ(s,ω,z)|_E≤2sup_s< t| u_s|_E}∈𝒫⊗𝒵.

	  Notice that A_t=∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} N(ṣ,ẓ).
	
	 By the predictability of the function ξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E}, we have
	
    ∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} ν(ẓ) ṣ=∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} N(ṣ,ẓ)<∞.

		 Hence by Equality (<ref>), we infer that the process  Â_t:=∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} ν(ẓ) ṣ, t∈ [0,T] is the compensator of the process A. Moreover the process
		      
    D_t:=A_t-Â_t= ∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} Ñ(ṣ,ẓ),    t∈ [0,T]

	is an L_1-integrable martingale. Hence,
			by Remark <ref>  and an argument analogous to (<ref>), we infer, for every 1≤ p<∞,
			
    sup_t≤ T|D_t|_E^p
    				  				≤    2^p-1(A_T)^p
    				+2^p-1(Â_T)^p
    ≤    2^p-1(1+p^p)(∫_0^T∫_Z|ξ(s,z)|_E1_{|ξ(s,z)|_E>2sup_v<s| u_v|_E} N(ṣ,ẓ))^p 
    ≤    C_p (∫_0^t∫_Z|ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r.
	
			Set
			
    H_t:=u_t-D_t
    				=∫_0^t∫_Zξ(s,z)1_{|ξ(s,z)|_E≤2sup_v<s| u_v|_E} Ñ(ṣ,ẓ),   t∈[0,T].

			Clearly, the process H=(H_t)_t∈[0,T],  is an r-integrable E-valued martingale. 	
			Let us fix an auxiliary number λ>0 and introduce the following random variable
			
    τ_λ:=inf{t:(∫_0^t∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤ 2sup_v<s| u_v|_E} N(ṣ,ẓ))^1/r∨sup_s≤ t| u_s|_E>λ}∧ T.

		   By the right-continuity of ∫_0^t∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤ 2sup_v<s| u_v|_E} N(ṣ,ẓ) and sup_s≤ t| u_s|_E, we infer that τ_λ is a stopping time; see Example I.5.1 in <cit.>.
			Hence, 			by using first the Chebyshev and Doob inequalities and then (<ref>) we infer
			
    ℙ{sup_t≤τ_λ|H_t|_E>λ}   ≤C_r/λ^r (∫_0^τ_λ∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤2sup_v<s| u_v|_E} N(ṣ,ẓ))
       ≤C_r/λ^r (∫_0^τ_λ-∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤2sup_v<s| u_v|_E} N(ṣ,ẓ)+2^rsup_v<τ_λ|ξ(v,π(v))|_E^r)  
       ≤ C_rλ^r/λ^r.

			  Here C_r is a constant depending only on r but whose value may change from line to line.
		                  Hence it follows that
		
    ℙ{sup_t≤τ_λ|H_t|_E>λ}≤ C_r1/λ^r ((∫_0^T∫_Z|ξ(s,z)|_E^r  N(ṣ,ẓ))∧λ^r).

		Observe that
		
    ℙ{sup_t≤ T|H_t|_E>λ}   ≤ℙ{sup_t≤ T|H_t|_E>λ,τ_λ=T}+ℙ{τ_λ<T}
       ≤ℙ{sup_t≤τ_λ|H_t|_E>λ}+ℙ{sup_t≤ T|ξ(t,π(t))|_E>λ}
       +ℙ{(∫_0^T∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤2sup_v<s| u_v|_E} N(ṣ,ẓ))^1/r>λ}.

		Let us now fix p:  1≤ p<r. 		By using the following standard equalities from <cit.>,
		 
    X^p   =∫_0^∞pλ^p-1ℙ{X>λ} λ̣,
     X^p   =r-p/r∫_0^∞(X^r∧λ^r) pλ^p-r-1 λ̣,
	
		as well as  inequalities (<ref>), (<ref>), and (<ref>), we get
		
    sup_s≤ T|H_t|_E^p
    			  			                          ≤   ∫_0^∞pλ^p-1ℙ{sup_t≤τ_λ|H_t|_E>λ} λ̣+ ∫_0^∞pλ^p-1ℙ{sup_t≤ T|ξ(t,π(t))|_E>λ} λ̣
       +∫_0^∞pλ^p-1ℙ{∫_0^T∫_Z|ξ(s,z)|_E^r1_{|ξ(s,z)|_E≤2sup_v<s| u_v|_E} N(ṣ,ẓ))^1/r>λ} λ̣
    ≤    C_r ∫_0^∞pλ^p-r-1((∫_0^T∫_Z|ξ(s,z)|_E^r  N(ṣ,ẓ))∧λ^r) λ̣
       +(sup_t≤ T|ξ(s,π(s))|_E)^p
    			+( ∫_0^T∫_Z|ξ(s,z)|_E^r
    						 N(ṣ,ẓ))^p/r
    ≤    C_p,r ( ∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r.

		This together with (<ref>) allows us to infer
		
    sup_t≤ T|u_t|_E^p
    			       ≤ C_p,r ( ∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r.

		
	  (3).  Let us fix numbers r≥ 1 and p∈ [1,r]. Interpreting ∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ) as ∑_s∈[0,T]|ξ(s,π(s))|^r, we easily obtain
	  	
    (∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r   =(∑_s∈[0,T]|ξ(s,π(s))|_E^r1_𝒟(π)(π(s)))^p/r
       ≤∑_s∈[0,T]|ξ(s,π(s))|_E^p1_𝒟(π)(π(s))
       =∫_0^T∫_Z|ξ(s,z)|_E^p N(ṣ,ẓ)
       =∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ) ṣ.

		Combining the above inequality with (<ref>) we get the result.
		

	   (4).   This is implicit in the proof of part (3).
		

	  Before proceeding further, let us recall that according to <cit.>(see also <cit.>), every Banach space (E,|·|_E) of martingale type r, 1<r≤ 2, admits an equivalent norm, for simplicity denoted also by |·|_E, such that (E,|·|_E) is an r-smooth Banach space and,  for every p ∈ (r,∞) the first derivative of the p-th power of the norm ψ_p(·):=|·|_E^p,  is locally (r-1)-Hölder continuous on E, i.e.
	       
    ψ_p^'(x)-ψ_p^'(y)_L(E)≤ C_p,r(|x|_E+|y|_E)^p-r|x-y|_E^r-1,    x,y∈ E.

		The proof of the above inequality can be found in <cit.>.
		
	 Suppose that ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E) and (u_t)_t∈[0,T] is defined by (<ref>).  Then for r≤ p<∞, Don't we need to mention the modification here?
		
    sup_t∈[0,T] |u_t|_E^p≤ C_p,r (∫_0^T∫_Z |ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r.


	

	
	
Let us fix r and p:  1<r≤ 2, r≤ p<∞. For the simplicity of notation we  put ψ:=ψ_p.
			  An application of the Itô  formula (<ref>) to the function ψ and the process (u_t)_t∈[0,T] yields for t∈[0,T], ℙ-a.s.
		
    ψ(u_t)
    		   =I_1(t)+I_2(t),

		where
		
    I_1(t)=∫_0^t∫_Zψ^'(u_s-)(ξ(s,z)) Ñ(ṣ,ẓ),

		
    I_2(t)=∫_0^t∫_Z[ψ(u_s-+ξ(s,z))-ψ(u_s-)-ψ^'(u_s-)(ξ(s,z))] N(ṣ,ẓ).

		Let ε>0. Applying inequality (<ref>) with p=1 gives 		
		
    sup_0≤ t≤ T|I_1(t)|
    		   ≤ C (∫_0^T∫_Z|ψ^'(u_s-)(ξ(s,z))|^r N(ṣ,ẓ))^1/r
       ≤ C_p sup_0≤ t≤ T|u_t|_E^p-1(∫_0^T∫_Z|ξ(s,z)|^r_E N(ṣ,ẓ))^1/r
       ≤ε C_p  sup_0≤ t≤ T|u_t|_E^p+C_1/ε,p (∫_0^T∫_Z|ξ(s,z)|^r_E N(ṣ,ẓ))^p/r,

		where in the last step we used Hölder's inequality and Young's inequality.


		On the other hand, applying the mean value theorem, we find that for every s∈(0,T]
		
    |ψ(u_s-+ξ(s,π(s)))-ψ(u_s-)-ψ^'(u_s-)(ξ(s,π(s)))|
       ≤|ψ^'(u_s-+θξ(s,π(s)))-ψ^'(u_s-)|_L(E,ℝ)|ξ(s,π(s))|_E.

			Observe that for all 0≤ s≤ T,
			
    |u_s-|_E≤sup_0≤ s≤
    			T|u_s-|_E≤sup_0≤ s≤
    			T|u_s|_E.

		  Also, since u_s-+ξ(s,π(s))=u_s, we get
			
    |u_s-+ξ(s,π(s))|_E≤sup_0≤ s≤
    			T|u_s|_E.

		  	By using the
				fact that |x+θ y|_E≤max{|x|_E,|x+y|_E} for 0<θ<1 and all x,y∈
		   	E, we obtain
			
    |u_s-+θξ(s,π(s))|_E≤max{|u_s-|_E,|u_s-+ξ(s,π(s))|_E}≤sup_0≤ s≤ T|u_s|_E.

	      Hence by (<ref>), we infer
	    
    |ψ^'(u_s-+θξ(s,π(s)))-ψ^'(u_s-)|_L(E,ℝ)   ≤ C_p,r(|u_s-|_E+|u_s-+θξ(s,π(s))|_E)^p-r|ξ(s,π(x))|_E^r-1
       ≤ C_p,rsup_s∈[0,T]|u_s|^p-r_E|ξ(s,π(s))|^r-1_E.

		It then follows from (<ref>) and (<ref>)  that for every s∈(0,T],
		
    |ψ(u_s-+ξ(s,π(s)))-ψ(u_s-)-ψ^'(u_s-)(ξ(s,π(s)))|
    		≤ C_p,rsup_s∈[0,T]|u_s|^p-r_E|ξ(s,π(s))|^r_E.

	Hence by Hölder's  and Young's inequalities we have
	
    sup_t∈[0,T]|I_2(t)|   ≤∫_0^T∫_Z|ψ(u_s-+ξ(s,z))-ψ(u(s-)-ψ^'(u_s-)(ξ(s,z))| N(ṣ,ẓ)
       ≤ C_p,r sup_t∈[0,T]|u_t|^p-r_E∫_0^T∫_Z |ξ(s,π(s))|^r_E   N(ṣ,ẓ)           
       ≤ C_p,r (ε sup_t∈[0,T]|u_t|_E^p)^p-r/p((1/ε)^p-r/r (∫_0^T∫_Z|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r)^r/p
       ≤ C_p,r ε sup_t∈[0,T]|u_t|^p_E+C_p,r(1/ε)^p-r/r (∫_0^T∫_Z|ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r.
	

				Combining (<ref>) and (<ref>) and choosing a suitable small value of ε, we get
	 				
    sup_t∈[0,T∧σ_n]|u_t|_E^p≤ C_p,r (∫_0^T∫_Z|ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r.

		

	 	The following L^p estimate (<ref>) of (u_t)_t∈[0,T] for all r≤ p<∞, as a consequence of Theorem <ref>, is an important and useful tool in studying solutions of SPDEs, especially the regularity of solutions.
	
	  For all r≤ p<∞, we have
		
    sup_t∈[0,T]| u_t |_E^p≤ C_p,r [(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)+(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r].

	



	 Let us assume that the right-hand side of (<ref>) is finite, as otherwise the inequality follows trivially.
To prove inequality (<ref>), we start with the case where r<p≤ r^2. According to the inequality (<ref>), by  taking into account the following equality
	      
    ∫_0^t∫_Z|ξ(s,z)|^r_E N(ṣ,ẓ)=∫_0^t∫_Z|ξ(s,z)|_E^r Ñ(ṣ,ẓ)+∫_0^t∫_Z|ξ(s,z)|_E^r ν(ẓ) ṣ,
 we obtain
	       
    sup_t∈[0,T]| ∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ) |_E^p   ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E N(ṣ,ẓ))^p/r
       ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|_E^r Ñ(ṣ,ẓ))^p/r+    C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r.

	       Notice that 1<p/r≤ r, hence by applying the inequality (<ref>) to the first term on the right-hand side of the above inequality, we infer that
	       
    sup_t∈[0,T]| ∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ) |_E^p≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)+C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r.

This completes the proof of  inequality (<ref>) in  the case when r<p≤ r^2.


	       Let now assume that r^2<p≤ r^3.  Applying inequality (<ref>) twice and then  inequality (<ref>) yields  the following estimate:
	       
    sup_t∈[0,T]| ∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ) |_E^p   ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E N(ṣ,ẓ))^p/r
       ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|_E^r Ñ(ṣ,ẓ))^p/r+C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r
       ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^r^2_E N(ṣ,ẓ))^p/r^2+C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r
       ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^r^2_E Ñ(ṣ,ẓ))^p/r^2+C_p,r(∫_0^t∫_Z |ξ(s,z)|^r^2_E ν(ẓ) ṣ)^p/r^2
       +C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r
       ≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)+C_p,r(∫_0^t∫_Z |ξ(s,z)|_E^r^2 ν(ẓ) ṣ)^p/r^2
       +C_p,r(∫_0^t∫_Z |ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r.

  Suppose that  r^n-1<p≤ r^n for some natural number n  ≥ 3. Then  by induction we obtain the following estimate:
	                   
    sup_t∈[0,T]| ∫_0^t∫_Z ξ(s,z)  Ñ(ṣ,ẓ) |^p≤ C_p,r(∫_0^t∫_Z |ξ(s,z)|_E^p ν(ẓ) ṣ)+C_p,r∑_j=1^n-1(∫_0^T∫_Z|ξ(s,z)|^r^j_Eν(ẓ) ṣ)^p/r^j.

	                   Note that for r≤ m≤ p, by making use of Hölder's inequality and Young's inequality, we have
	                   
    (∫_0^T∫_Z|ξ(s,z)|^m_E ν(ẓ)ṛ)^1/m   ≤(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ)ṣ)^p-m/p-r·1/m(∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ)ṣ)^m-r/p-r·1/m
       ≤r(p-m)/m(p-r)(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ)ṣ)^1/r+p(m-r)/m(p-r)(∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ)ṣ)^1/p
       ≤(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ)ṣ)^1/r+(∫_0^T∫_Z|ξ(s,z)|^p_E ν(ẓ)ṣ)^1/p.

	                         Hence, by combing the last two inequalities  we infer that
	                                      
    sup_t∈[0,T]| ∫_0^t∫_Z ξ(s,z) Ñ(ds,dz) |_E^p≤ C_p,r(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)+C_p,r(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r.

The proof is complete.
	
	
	As a byproduct of  the proof of the above corollary, one can deduce  that for r≤ p<∞, the following inequality holds:
		
    ( ∫_0^T∫_Z |ξ(s,z)|^r_E  N(ṣ,ẓ))^p/r≤ C_p,r(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)+C_p,r(∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r.

	    For a comparison, let us recall Remark <ref> which says for p≥ r, there holds the following inequality:
		
    (∫_0^T∫_Z|ξ(s,z)|^r_E ν(ẓ) ṣ)^p/r≤(p/r)^p/r(∫_0^T∫_Z|ξ(s,z)|^r_E N(ṣ,ẓ))^p/r.

	  	



With the help of two versions of the Itô formula given  in Appendix B and Corollary <ref> we can now prove the following BDG inequality for Lévy-type processes in martingale type 2 Banach spaces (r=2). Let (W_t)_t≥0 be a cylindrical Brownian motion in a Hilbert space H and let γ(H;E) be the space of all γ-radonifying operators from H to E. Suppose that (g_t)_t≥0 is a progressively measurable process with values in γ(H;E) such that
	
    (∫_0^Tg_s^2_γ(H;E)ṣ)^p/2<∞.

	We may denote by M^p([0,T];γ(H;E)) the space of such functions (g_t)_t≥0. Let's recall some basic properties of stochastic integrals which we shall use in the proof; see <cit.>.
	
	 

  * For all g ∈ M^2([0,T];γ(H,E)) and 0≤ u<t≤ T,
 
    (| ∫_u^tg_s Ẉ_s|_E ^2|_u)≤
    C(∫_u^tξ_s ^2_γ(H,E) ṣ |_u).


  * (Burkholder's inequality)
For all 0<p<∞ there exists a constant C, depending only on p and E, such
that for all g∈ M^p([0,T];γ(H,E)) and t∈ [0,T],
 
    sup_s∈[0,t]| ∫_0^sg_u Ẉ_u|_E ^p≤
    C(∫_0^t g_s ^2_γ(H,E) ṣ)^p/2.



 Suppose that r=2, that is, (E,|·|_E) is a martingale type 2 Banach space. Let p≥ 2 and let (a_t)_t∈[0,T] be an E-valued progressively measurable process such that

    (∫_0^T|a_t|_E ṭ)^p<∞

and let (g_t)_t∈[0,T] be a process in M^2([0,T];γ(H;E)) and ξ be a process in ℳ_T^2(𝒫⊗𝒵,ṭ×ℙ×ν;E).
 Let X be a process given by

    X_t=∫_0^t a_s ṣ+∫_0^tg_s Ẉ_s+∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ).

Then there exists a constant C_p, depending only on p and E, such that

    sup_0≤ t≤ T|X_t|_E^p≤    C_p (∫_0^T|a_s|_E ṣ)^p+C_p (∫_0^Tg_s^2_γ(H;E)ṣ)^p/2
       +C_p (∫_0^T∫_Z|ξ(s,z)|^2_E N(ṣ,ẓ))^p/2  for 2≤ p<∞,

and

    sup_0≤ t≤ T|X_t|_E^p≤    C_p (∫_0^T|a_s|_E ṣ)^p+C_p (∫_0^Tg_s^2_γ(H;E)ṣ)^p/2+C_p (∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
       +C_p (∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ) ṣ)^p/2  for 2≤ p<∞.

In particular, if  we make the additional assumption that

    N(t,Z)<∞ for t∈(0,T]     and  ψ^'(X_s)(a_s)≤ 0   for all (s,ω)∈[0,T]×Ω,

then we have

    sup_0≤ t≤ T|X_t|_E^p≤    C_p (∫_0^Tg_s^2_γ(H;E) ṣ)^p/2+C_p (∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
        +C_p (∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ) ṣ)^p/2  for 2≤ p<∞.





Since the Poisson point process π is σ-finite, there exists a sequence of sets {D_n}_n∈ℕ such that D_n⊂ D_n+1, ∪_n∈ℕD_n=Z and N(T,D_n)<∞ for n∈ℕ. So there is contained only a finite number of jumps of π in D_n and at most a countably many number of jumps in Z. We shall denote by {τ_m}_m=1^∞ the corresponding sequence of jump times of  π until time T in Z.  In this way we order the set {s ∈𝒟(π)∩ (0,T]: π(s)∈ Z } according to magnitude by 0<τ_1<τ_2<⋯<τ_m<⋯≤ T.  See more details in Appendix B.
Define a sequence {ξ^n}_n∈ of functions by
       
    ξ^n(s,ω,z):=ξ(s,ω,z)1_D_n(z), (s,ω,z)∈ℝ_+×Ω× Z,  n∈,

          and a sequence {X^n}_n∈ of process X^n:=(X^n_t)_t≥0 by
             
    X^n_t=∫_0^ta_sṣ+∫_0^tg_s Ẉ_s+∫_0^t∫_Z ξ^n(s,z) Ñ(ṣ,ẓ), t≥0,  n∈ℕ.

Similar to the proof of Theorem <ref>, we have
    
    ψ(X_t^n)-ψ(X_0)
                =∑_m=1^∞[ψ(X^n_t∧τ_m)-ψ(X^n_t∧τ_m-) ]+∑_m=1^∞[ψ(X^n_t∧τ_m-)-ψ(X^n_t∧τ_m-1) ]
       =: I_1+I_2.

   Here as usual we set τ_0=0.
    A similar argument as in the proof of Theorem <ref> gives
    
    I_1   =∫_0^t∫_Z[ψ(X^n_s-+ξ^n(s,z,ω)) -ψ(X^n_s-)    ]Ñ(ṣ,ẓ)
        +∫_0^t∫_Z[ψ(X^n_s-+ξ^n(s,z,ω)) -ψ(X^n_s-)    ]ν(ẓ)ṣ.

                   To approximate I_2, let Π_j={0=t_0^j<t_1^j<⋯<t_l(j)^j=T}, j∈ℕ be any sequence of partitions of the interval [0,T] whose meshes Π_j:=max_0≤ i≤ l(j)-1 |t_i+1^j-t_i^j| tend to 0 as j→∞.   Set for u∈[0,T],
		                 
    Y^n,m_t
    			  =X^n_τ_m-1+∫_0^t1_[τ_m-1,T][a(s)-∫_Zξ^n(s,z) ν(ẓ)]ṣ+∫_0^t
    			 1_[τ_m-1,T] g_s Ẉ_s.

		                 Since the process X^n is continuous in the random time interval (t∧τ_m-1,t∧τ_m), we have X_t^n=Y_t^n,m for t∈[τ_m-1,τ_m), m=1,2,….
Since the function ψ(x)=|x|_E^p, p≥ 2, is Fréchet differentiable and its first derivative is Lipschitz continuous, we can apply Theorem 3.1 in <cit.> to the process Y^n,m and get

    ψ(Y^n,m_t∧τ_m)-ψ(Y^n,m_τ_m-1)=   ∫_τ_m-1^t∧τ_mψ^'(Y^n,m_s)(a_s) ṣ-∫_τ_m-1^t∧τ_m∫_Zψ^'(Y^n,m_s)(ξ^n(s,z)) ν(ẓ)ṣ
       +∫_τ_m-1^t∧τ_mψ^'(Y^n,m_s)(g_s) Ẉ_s
    						+lim_j→∞∑_i=0^l(j)-1 R(Y^n,m_t∧τ_m∧ t_i^j,Y^n,m_t∧τ_m∧ t_i+1^j).

Here R(x,y)=∫_0^1(ψ'(x+θ(y-x))(y-x)-ψ'(x)(y-x)) θ̣.
Recall that  Y^n,m_t=X^n_τ_m-1, for t∈[0,τ_m-1]; Y_t^n,m=X_t^n for t∈[τ_m-1,τ_m) and Y_t∧τ_m^n,m=Y_τ_m^n,m=X^n_τ_m-=X^n_t∧τ_m-,  for t≥τ_m and also ψ(Y^n,m_t∧τ_m)-ψ(Y^n,m_τ_m-1)=ψ(X^n_t∧τ_m-)-ψ(X^n_t∧τ_m-1).  Now we have
			
    I_2=∑_mψ(X^n_t∧τ_m-)-ψ(X^n_t∧τ_m-1)=   ∫_0^tψ^'(X^n_s)(a_s) ṣ-∫_0^t∫_Zψ^'(X^n_s)(ξ^n(s,z)) ν(ẓ)ṣ
       +∫_0^tψ^'(X^n_s)(g_s) Ẉ_s
    					+∑_mlim_j→∞∑_i=0^l(j)-1 R(Y^n,m_t∧τ_m∧ t_i^j,Y^n,m_t∧τ_m∧ t_i+1^j).

	Using (<ref>) and (<ref>) in (<ref>) gives
	  
    ψ(X_t^n)-ψ(X_0)=   ∫_0^tψ^'(X^n_s)(a_s) ṣ+∫_0^tψ^'(X^n_s)(g_s) Ẉ_s
             +∫_0^t∫_Z ψ^'(X^n_s-)(ξ^n(s,z,ω)) Ñ(ṣ,ẓ)
        +∫_0^t∫_Z[ψ(X^n_s-+ξ^n(s,z,ω)) -ψ(X^n_s-) -ψ^'(X^n_s-) ξ^n(s,z)   ] N(ṣ,ẓ)
       +R(Y),

	where R(Y)=∑_mlim_j→∞∑_i=0^l(j)-1 R(Y^n,m_t∧τ_m∧ t_i^j, Y^n,m_t∧τ_m∧ t_i+1^j).
	Now  taking first the supreme then the expectation to both sides of the equality (<ref>) gives that
	                      
    sup_t∈[0,T]ψ(X_t)≤   sup_t∈[0,T]|∫_0^tψ^'(X_s^n)(a_s) ṣ|+sup_t∈[0,T]|∫_0^tψ^'(X_s)(g_s) Ẉ_s|
       +sup_t∈[0,T]|∫_0^t∫_Z ψ^'(X^n_s-)(ξ^n(s,z,ω)) Ñ(ṣ,ẓ)|
       
    		                    +sup_t∈[0,T]|∫_0^t∫_Z[ψ(X^n_s-+ξ^n(s,z,ω)) -ψ(X^n_s-) -ψ^'(X^n_s-) ξ^n(s,z)   ]N(ṣ,ẓ)|
       +sup_0≤ t≤ T R(Y)
       :=J_1+J_2+J_3+J_4+J_5.

		                  It's worth mentioning here that if we impose the additional conditions  (<ref>), i.e. N(t,Z)<∞ and ψ^'(X_s^n)(a_s)≤ 0, then X^n=X and the first term on the right hand side of  (<ref>) satisfies ∫_0^tψ^'(X^n_s)(a_s)≤ 0. In this case we do not need to consider J_1.	
		
		              Now let ε>0.  By Young's inequality  we have
		               		                  
    J_1≤εsup_0≤ t≤ T|X_s^n|_E^p+C_p,1/ε^p(∫_0^T|a_s|_E ṣ)^p.

		
		            	
		            For the term J_2, according to <cit.>, the process t↦ψ^'(X^n_t)( g_t) is progressively measurable and satisfies (∫_0^Tψ^'(X^n_t)( g_t)^2_Hṭ)^1/2<∞. Then we can apply Burkholder's inequality (<ref>), Hölder's inequality, and Young's inequality to obtain, for any ε>0,
			    
    J_2   ≤ C (∫_0^Tψ^'(X^n_s)( g_s)^2_H  ṣ)^1/2
       ≤ C_p (∫_0^T |X^n_s|_E^2(p-1)g_s^2_γ(H;E) ṣ)^1/2
       ≤ C_p sup_t∈[0,T]|X_t^n|_E^p-1(∫_0^T g_s^2_γ(H;E) ṣ)^1/2
       ≤ C_pε sup_t∈[0,T]|X^n_t|_E^p+C_p,1/ε(∫_0^Tg_s_γ(H;E)^2 ṣ)^p/2.

			    
			    	 
	By adopting the same procedure as in the proof of Theorem <ref> (with  r=2), we get
	
    J_3   ≤ C_p ε sup_t∈[0,T]|X_t^n|^p_E+C_p,1/ε(∫_0^T∫_Z|ξ(s,z)|_E^2  N(ṣ,ẓ))^p/2,
    
    	J_4   ≤ C_p ε sup_t∈[0,T]|X_t^n|^p_E+C_p,1/ε(∫_0^T∫_Z|ξ(s,z)|_E^2  N(ṣ,ẓ))^p/2.

	The estimate of J_5 is similar to that in <cit.>. By applying Lemma 3.5 in <cit.>, we have
	 
    J_5   ≤sup_0≤ t≤ T∑_mlim inf_j→∞∑_i=0^l(j)-1 |R(Y^n,m_t∧τ_m∧ t_i^j,Y^n,m_t∧τ_m∧ t_i+1^j)|
       ≤∑_mlim inf_j→∞∑_i=0^l(j)-1 |R(Y^n,m_τ_m∧ t_i^j,Y^n,m_τ_m∧ t_i+1^j)|.

	Since R(x,y)=∫_0^1(ψ'(x+θ (y-x))(y-x)-ψ'(x)(y-x)) θ̣, by (<ref>) we infer
	
    |R(x,y)|≤ C_p∫_0^1(|x+θ(y-x)|_E+|x|_E)^p-2|x-y|_E θ̣≤ C_p|x|_E^p-2|x-y|_E^2+C_p|x-y|^p_E.

	It follows that
	
    |R(Y^n,m_τ_m∧ t_i^j,Y^n,m_τ_m∧ t_i+1^j)|≤ C_p|Y^n,m_τ_m∧ t_i^j|_E^p-2|Y^n,m_τ_m∧ t_i+1^j-Y^n,m_τ_m∧ t_i^j|_E^2+C_p|Y^n,m_τ_m∧ t_i+1^j-Y^n,m_τ_m∧ t_i^j|^p_E.

	
	For the first term, by (<ref>), we obtain

    ∑_i=0^l(j)-1| Y^n,m_τ_m∧ t_i^j|_E ^p-2 | Y^n,m_τ_m∧ t_i+1^j-Y^n,m_τ_m∧ t_i^j|_E ^2 
       ≤
     2∑_i=0^l(j)-1 | Y^n,m_τ_m∧ t_i^j|_E ^p-2|  ∫_  t_i^j^ t_i+1^j1_[τ_m-1,τ_m][a_s-∫_Zξ^n(s,z) ν(ẓ)]ṣ|_E ^2
    
           +2∑_i=0^l(j)-1| Y^n,m_τ_m∧ t_i^j|_E ^p-2|  ∫_ t_i^j^ t_i+1^j1_[τ_m-1,τ_m]g_s
     Ẉ_s|_E ^2 =:K_1^j+K_2^j.

   For the term K_1^j, it's easy to see
     
    lim_j→∞K_1^j=0,ℙ-a.s.

To estimate K_2^j, by using the Itô isometry property (<ref>) and Young's inequality with >0, we infer

    ∑_m lim inf_j K_2^j    ≤∑_mlim inf_j   K_2^j 
       =∑_mlim inf_j ∑_i=0^l(j)-1| Y^n,m_τ_m∧ t_i^j|_E ^p-2|  ∫_ t_i^j^ t_i+1^j1_[τ_m-1,τ_m]g_s
     Ẉ_s|_E ^2
       =∑_m lim inf_j∑_i=0^l(j)-1(| Y^n,m_τ_m∧ t_i^j|_E ^p-2(| ∫_t_i^j∧ t^t_i+1^j1_[τ_m-1,τ_m]g_s
     Ẉ_s|_E^2|_t_i^j))
       ≤∑_m
    Clim inf_j∑_i=0^l(j)-1(| Y^n,m_τ_m∧ t_i^j|_E ^p-2(∫_t_i^j^t_i+1^j1_[τ_m-1,τ_m] g_s ^2_γ(H,E) ṣ|_t_i^j))


    ≤∑_m
    Clim inf_j∑_i=0^l(j)-1(| Y^n,m_τ_m∧ t_i^j|_E ^p-2(∫_t_i^j^t_i+1^j1_[τ_m-1,τ_m] g_s ^2_γ(H,E) ṣ))
       ≤
    C∑_m lim inf_j(sup_s∈[0,T]| Y^n,m_s∧τ_m|_E ^p-2∑_i=0^l(j)-1∫_
    t_i^j^t_i+1^j 1_[τ_m-1,τ_m]g_s ^2_γ(H,E) ṣ)
       ≤
    C∑_m(sup_s∈[0,T]| X^n_s∧τ_m|_E ^p-2∫_
    0^T 1_[τ_m-1,τ_m]g_s ^2_γ(H,E) ṣ)
       ≤
    C(sup_s∈[0,T]| X^n_s|_E ^p-2∫_
    0^ T g_s ^2_γ(H,E) ṣ)
       ≤
    C_pε (sup_s∈[0,T]| X^n_s|_E ^p)
    + C_p,1/ε (∫_0^T g_s ^2_γ(H,E) ṣ)^p/2,

where we also used the fact that sup_s∈[0,T]|Y^n,m_s∧τ_m|^p-2≤sup_s∈[0,T]|X_s^n|_E^p-2.


Next we estimate the second term in (<ref>). Observe that

    ∑_i=0^l(j)-1|Y^n,m_τ_m∧ t_i+1^j-Y^n,m_τ_m∧ t_i^j|^p_E
       ≤  C∑_i=0^l(j)-1|  ∫_  t_i^j^ t_i+1^j1_[τ_m-1,τ_m][a_s-∫_Zξ^n(s,z) ν(ẓ)]ṣ|_E ^p
    
           +C∑_i=0^l(j)-1|  ∫_ t_i^j^ t_i+1^j1_[τ_m-1,τ_m]g_s
     Ẉ_s|_E ^p=: K_3^j+K_4^j.

Clearly,

    lim_j→∞ K_3^j   ≤
    Clim_j→∞sup_0≤ i≤ l(j)-1|  ∫_  t_i^j^ t_i+1^j1_[τ_m-1,τ_m][a_s-∫_Zξ^n(s,z)ν(ẓ)] ṣ|_E^p-1·∫_0^ T|a_s-∫_Zξ^n(s,z) ν(ẓ)|_Eṣ
        =0.

  Moreover, by Burkholder's inequality (<ref>),
  
    ∑_m lim inf_j K_4^j ≤∑_mlim inf_j   K_4^j
       =∑_mlim inf_j ∑_i=0^l(j)-1|  ∫_ t_i^j^ t_i+1^j1_[τ_m-1,τ_m]g_s
     Ẉ_s|_E ^p
       ≤∑_m
    Clim inf_j∑_i=0^l(j)-1(∫_t_i^j^t_i+1^j1_[τ_m-1,τ_m] g_s ^2_γ(H,E) ṣ)^p/2
       ≤
    C ∑_m lim inf_j(∑_i=0^l(j)-1∫_
    t_i^j^t_i+1^j 1_[τ_m-1,τ_m]g_s ^2_γ(H,E) ṣ)^p/2
       ≤
    C∑_m(∫_
    0^T 1_[τ_m-1,τ_m]g_s ^2_γ(H,E) ṣ)^p/2
       ≤
     C (∫_0^T g_s ^2_γ(H,E) ṣ)^p/2.

Collecting all the estimates, for any >0
we obtain
	 
    J_5		
    		≤ε C_p sup_s∈[0,T]|X^n_s|_E^p+C_p,1/ε (∫_0^Tg_s_γ(H,E)^2 ṣ)^p/2.
	
				Finally, we can collect together all terms and choose a suitable value of ε to get
	
	
    sup_t∈[0,T]|X^n_t|_E^p≤   C_p(∫_0^T|a_s|_E ṣ)^p+ C_p(∫_0^Tg_s^2_γ(H;E) ṣ)^p/2
       +C_p(∫_0^T∫_Z|ξ(s,z)|_E^2  N(ṣ,ẓ))^p/2.

	By applying (<ref>) with r=2, we have
	
    sup_t∈[0,T]|X_t-X^n_t|_E^p			 ≤   C_p(∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E^p ν(ẓ)ṣ))
       + C_p(∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E^2 ν(ẓ)ṣ))^p/2→0, as n→∞,

	and inequality (<ref>) follows.
	Inequality (<ref>)  follows immediately from (<ref>) and (<ref>).




	

	

§ MAXIMAL INEQUALITIES AND EXPONENTIAL ESTIMATES FOR STOCHASTIC CONVOLUTIONS W.R.T. LÉVY PROCESSES

		

 §.§ Maximal L^p estimates for stochastic convolutions w.r.t. Poisson random measures 

		 Suppose that (E,|·|_E) is a martingale type r Banach space, 1<r≤ 2. Here, as before, |·|_E is the equivalent r-smooth norm. Let (e^tA)_t≥0 be a C_0-semigroup on E, with the generator denoted by A,  of contraction type with respect to the r-smooth norm |·|_E such that e^tA≤ e^tα  for some α≥ 0. We start by considering the following stochastic convolution driven by purely discontinuous noises:
	
    X_t=∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ).

	
		 Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingale type r Banach space (E,|·|_E), 1<r≤ 2,
		and let ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E). Then there exists a càdlàg modification X̅ of X such that for r≤ p<∞,
		
    sup_0≤ t≤ T|X̅_t|_E^p≤ e^α p TC_p,r (∫_0^T∫_Z |ξ(s,z)|^r_E  N(ṣ,ẓ))^p/r.


	
	 
	
		Step 1  First fix p≥ r and suppose that ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;𝒟(A)) and α=0, i.e. e^tA_L(E)≤1.
		It is known (see <cit.>) that process
		X_t, t∈[0,T] is a unique strong solution to the following problem
		
    dX_t   =AX_tdt+∫_Zξ(t,z) Ñ(ṭ,ẓ),  t∈[0,T],
    
    		   X_0   =0.

		Apparently X_t is an E-valued càdlàg process. Let τ be a stopping time with values in [0.T]. We have
		
    X_t∧τ=∫_0^t1_[0,τ](s)AX_sṣ+∫_0^t∫_Z1_[0,τ](s)ξ(s,z) Ñ(ṣ,ẓ).

	   Hence, by applying the Itô  formula (<ref>) to ψ(·)=|·|_E^p and then using the fact that ψ^'(x)(Ax)≤ 0 for all x∈ D(A) (see e.g. Lemma 4.7 in <cit.>), we obtain for t∈[0,T], ℙ-a.s.
		
    ψ(X_t∧τ)   ≤∫_0^t∫_Z1_[0,τ](s)ψ(X_s-+ξ(s,z))-ψ(X_s-) Ñ(ṣ,ẓ)
       +∫_0^t∫_Z1_[0,τ](s)[ψ(X_s-+ξ(s,z))-ψ(X_s-)-ψ^'(X_s-)(ξ(s,z))] ν(ẓ)ṣ
       =     ∫_0^t∫_Z1_[0,τ](s)ψ^'(X_s-)(ξ(s,z)) Ñ(ṣ,ẓ)
       +∫_0^t∫_Z1_[0,τ](s)[ψ(X_s-+ξ(s,z))-ψ(X_s-)-ψ^'(X_s-)(ξ(s,z))] N(ṣ,ẓ)
       =:I_1(t)+I_2(t).

	 We again follow the same line of argument as used in deducing (<ref>) and (<ref>) to get
	 	
    sup_0≤ t≤ TI_1(t)   ≤ C_p εsup_t∈[0,T]|X_t∧τ|^p_E+C_p,1/ε(∫_0^T∫_Z1_[0,τ]|ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r,
    sup_0≤ t≤ TI_2(t)   ≤ C_p εsup_t∈[0,T]|X_t∧τ|^p_E+C_p,1/ε(∫_0^T∫_Z1_[0,τ]|ξ(s,z)|_E^r  N(ṣ,ẓ))^p/r.

		By choosing a suitable number ε such that ε C_p,r=1/4, we get
	       	
    sup_0≤ t≤ T|X_t∧τ|^p_E≤
    		C_p,r(∫_0^t∫_Z1_[0,τ](s)|ξ(s,z)|_E^r N(ṣ,ẓ))^p/r.

			 Step 2
	Suppose that α=0. Take 	ξ∈ℳ_T^p(𝒫⊗𝒵,ṭ×ℙ×ν;E). Put ξ^n(t,ω,z)=nR(n,A)ξ(t,ω,z) on [0,T]×Ω× Z, where R(n,A)=(nI-A)^-1. Since A is the infinitesimal generator of the contraction C_0-semigroup e^tA, t≥0, by the Hille-Yosida theorem, we have ξ^n→ξ pointwisely on [0,T]×Ω× Z and |ξ^n|_E≤ |ξ|_E. Define,  for each n∈ℕ, a process X^n by
		
    X^n_t=∫_0^te^(t-s)Aξ^n(s,z) Ñ(ṣ,ẓ),  t∈ [0,T].

		Then for each n∈ℕ, X^n is an E-valued càdlàg process.
		By the discussion in step 1, we infer
		 
    lim_n,m→∞sup_0≤ t≤ T|X^n_t-X^m_t|_E^p=0,

		from which we may deduce by using the usual argument that X^n is almost surely uniformly convergent on [0,T] to some  càdlàg process X̅ satisfying lim_n→∞sup_t∈[0,T]|X̅_t-X^n_t|_E^p=0 and for any stopping time τ in [0,T]
	       	
    sup_0≤ t≤ T|X̅_t∧τ|_E^p≤
    		C_p,r(∫_0^T∫_Z1_[0,τ](s)|ξ(s,z)|^r_E N(ṣ,ẓ))^p/r.

	    Meanwhile, since
	            	
    |X̅_t-X_t|_E^r
    			      ≤ 2^r|X̅_t-X^n_t|_E^p+2^r|∫_0^t∫_Z(e^(t-s)Aξ^n(s,z)-e^(t-s)Aξ(s,z)) Ñ(ṣ,ẓ)|^r
       ≤ 2^r|X̅_t-X^n_t|^r_E+C_r∫_0^t∫_Z|ξ^n(s,z)-ξ(s,z)|^r_E ν(ẓ) ṣ,

	we infer that that X^n_t converges to X_t in L_p(Ω) for every t∈[0,T]. Hence X̅ is a modification of X. The required result then follows.   

		Step 3 Let α>0. Define
	
    X̃_t=e^-α tX_t=∫_0^t∫_Z e^(A-α I)(t-s)(e^-sαξ(s,z)) Ñ(ṣ,ẓ).

	Note that e^(A-α I)t is a C_0-semigroup of contractions. Applying (<ref>), we have
	
    sup_0≤ t≤ T|X_t|^p_E≤ e^α p Tsup_0≤ t≤ T|X̃_t|^p_E≤ e^α p TC_p,r(∫_0^T∫_Z |ξ(s,z)|^r_E  N(ṣ,ẓ))^p/r.

	
	
		   Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingale type r Banach space (E,|·|_E), 1<r≤ 2
		and let ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E). Then there exists a càdlàg modification X̅ such that
		
    sup_t∈[0,T]|X̅_t|_E^p≤   e^α pT C_p,r(∫_0^T∫_Z |ξ(s,z)|^r_E  ν(ẓ) ṣ)^p/r  for0< p≤ r.

	
	
	By applying (<ref>) with p=r, we have
	
    sup_0≤ t≤ T|∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ)|_E^r   ≤ e^α p TC_r(∫_0^T∫_Z1_[0,τ] |ξ(s,z)|^r_E  N(ṣ,ẓ))
       =e^α p TC_r(∫_0^T1_[0,τ]∫_Z |ξ(s,z)|^r_E ν(ẓ)ṣ).

	 Let us set
	 
    Z_t=|∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ)|_E^r and A_t=∫_0^t∫_Z|ξ(s,z)|_E^r ν(ẓ)ṣ,   t∈[0,T] .

	 We find that for any stopping time τ in [0,T],
	
    Z_τ≤ C_r A_τ.

	Moreover, Z is a càdlàg process or more precisely it has a càdlàg version and the process A is increasing. Put k=p/r. For p∈(0,r), notice that k∈(0,1). By means of Proposition IV4.7 from <cit.>, we deduce
	
    sup_0≤ t≤ TZ_t^k≤2-k/1-k A_T^k.

		Hence we infer for 0< p<r,
	
    sup_0≤ t≤ T|∫_0^t∫_Z e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ)|_E^p ≤ e^α pTC_p,r(∫_0^T∫_Z|ξ(s,z)|_E^r ν(ẓ)ṣ)^p/r .

This proves inequality (<ref>).
		
	In the same way as we deduce inequality (<ref>) from Theorem <ref>, we can get the following L^p inequality for stochastic convolutions from the preceding theorem immediately.
	
	  Let E be a martingale type r Banach space, 1<r≤ 2.Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingle type r Banach space (E,|·|_E), 1<r≤ 2
		and let ξ∈ℳ_T^r(𝒫⊗𝒵,ṭ×ℙ×ν;E). Then there exists a càdlàg modification X̅ such that for r≤ p<∞,
		
    sup_t∈[0,T]|X̅_t|_E^p≤    e^α pTC_p,r[(∫_0^T∫_Z |ξ(s,z)|^p_E  ν(ẓ) ṣ)+(∫_0^T∫_Z |ξ(s,z)|^r_E  ν(ẓ) ṣ)^p/r].

	


	
		 Note that if (e^tA)_t≥0 is an analytic semigroup with generators A such that -A has a bounded H^∞-calculus,  we can always find an equivalent norm on E for which E is 2-smooth and (e^tA)_t≥0 is a contraction C_0-semigroup; see <cit.>. For a related earlier result see <cit.>.
	Both our Theorems 3.1 and 3.3 from <cit.> are applicable to C_0-semigroups of positive contractions on L^q(S) spaces, where S is a measure space and q ∈ (1,∞); see Example 3.1 in <cit.>. However, more precise understanding of the relationship  between our results and those from <cit.> would require further analysis.   Our results have roots in the PhD thesis of the first named author which was known to the authors of  <cit.>.  	

		
		   To do this properly, by <cit.>, we may define an equivalent norm x_E=T_x_γ(ℝ_+,E), where the operator T_x is defined by T_x(t)=(-A)^1/2e^tAx, t∈ℝ_+. Since there exists an isometry from γ(ℝ_+,E) to L^2(Ω,E), it is enough to show that L^2(Ω,E) is 2-smooth with respect to the norm ·_L^2(Ω,E). Observe that for all u∈ L^2(Ω,E)
			
    (y^2_L^2(Ω,E))'(u)=∫_Ω|(y(ω)|_E^2)'(u(ω))P(ω)

			and
			 
    |(y^2_L^2(Ω,E))'(u)-(x^2_L^2(Ω,E))'(u)|_ℝ   =|∫_Ω(|(y(ω)|_E^2)'(u(ω))-(|(x(ω)|_E^2)'(u(ω))P(ω)|_ℝ
       ≤∫_Ω(|(y(ω)|_E^2)'-(|(x(ω)|_E^2)'_L(E,ℝ)|u(ω)|P(ω)
       ≤ L∫_Ω |y(ω)-x(ω)|_E|u(ω)|_E P(ω)
       ≤ Ly-x_L^2(Ω,E)u_L^2(Ω,E),

			where the constant L comes from the Lipschitz continuity of the first derivative of |·|_E^2, since E is 2-smooth.
		 Therefore, the first derivative of ·^2_L^2(Ω,E) is Lipschitz continuous. This implies the space L^2(Ω,E) with the norm ·_L^2(Ω,E) is 2-smooth. Hence the space (E,·_E) is a 2-smoothness Banach space. Also, we see that
		
    e^tAx_E=T_e^tAx_γ(ℝ_+,E)=T_x_γ([s,∞),E)≤T_x_γ(ℝ_+,E)=x_E,

		which shows that (e^tA)_t≥0 is a contraction semigroup on (E,·_E).
	


	
	
	
	

 §.§ Exponential tail estimates

	With more effort the method we described above to derive (<ref>) and (<ref>) can be applied to obtain the following exponential tail estimates for stochastic convolutions driven by compensated Poisson random measures.
	Assume that E is a martingale type 2 Banach space. Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingale type 2 Banach space (E,|·|_E)
		and let ξ∈ℳ_T^2(𝒫⊗𝒵,ṭ×ℙ×ν;E).
	 If there exists λ>0 and M_λ>0 such that
	    
    ∫_0^T∫_Z e^λ^1/2|ξ(s,z)|_Eλ |ξ(s,z)|_E^2 ν(ẓ) ṣ+∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ)ṣ+∫_0^Tg_s^2_γ(H,E)ṣ≤ M_λ,  ℙ-a.s.,

	then for every R>0,
	
    ℙ( sup_0≤ t≤ T|X_t|_E ≥ R )≤ C_λe^-(1+λ R^2)^1/2

	with C_λ=e^1+3CM_λ.
	

	
		Define f_λ=(1+λ|x|_E^2)^1/2, x∈ E. It is easy to see that f^'_λ(x)=λ(1+λ |x|_E^2)^-1/2|x|_Ef_x, where f_x is the derivative of |·|_E at the point x≠ 0. One can show that for every x,y∈ E,
		
    |f^'_λ(x)-f^'_λ(y)|_L(E;ℝ)≤ Cλ |x-y|_E,

		where the constant C depends on the constant appearing in the martingale type 2 Banach space property.
		As in the proof of Theorem 3.1, we may assume that ξ∈ M_T^2(𝒫⊗𝒵,ṭ×ℙ×ν;𝒟(A)) and consider the process
		
    X_t=∫_0^t AX_sṣ+∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ).

		Since f^'_λ(x)(Ax)≤ 0, for all x∈ E, applying the Itô  formula (<ref>) yields
		
    f_λ(X_t)=   f_λ(X_0)+∫_0^tf_λ'(X_s)(Au_s) ṣ+∫_0^t∫_Z( f_λ(X_s-+ξ(s,z))-f_λ(X_s-)) Ñ(ṣ,ẓ)
       +∫_0^t∫_Z (f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-f^'_λ(X_s-)(ξ(s,z))) ν(ẓ) ṣ
    ≤    1+ ∫_0^t∫_Z( f_λ(X_s-+ξ(s,z))-f_λ(X_s-)) Ñ(ṣ,ẓ) 
       +∫_0^t∫_Z (f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-f^'_λ(X_s-)(ξ(s,z))) ν(ẓ) ṣ.

	   Let us set
		   	
    Y_t   =∫_0^t∫_Z( f_λ(X_s-+ξ(s,z))-f_λ(X_s-)) Ñ(ṣ,ẓ)
       -∫_0^t∫_Z( e^f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-1-(f_λ(X_s-+ξ(s,z))-f_λ(X_s-))) ν(ẓ) ṣ.

	 Again by applying the Itô  formula (<ref>) to Y_t, we get
		
    e^Y_t=   1
    -   ∫_0^t∫_Z e^Y_s-(e^f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-1-(f_λ(X_s-+ξ(s,z))-f_λ(X_s-))) ν(ẓ) ṣ
       +∫_0^t∫_Z e^Y_s-+f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-e^Y_s- Ñ(ṣ,ẓ)
       + ∫_0^t∫_Z (e^Y_s-+f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-e^Y_s--e^Y_s-(f_λ(X_s-+ξ(s,z))-f_λ(X_s-))) ν(ẓ) ṣ
    
    			=   1+∫_0^t∫_Z (e^Y_s-+f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-e^Y_s-) Ñ(ṣ,ẓ).

		Therefore, Z_t=e^Y_t is a nonnegative local martingale. 
		
		By using the mean value theorem twice, the fact |f_λ'(x)|_L(E;ℝ)≤λ^1/2, and also (<ref>), we have
		
    |e^f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-1-f_λ'(X_s-)(ξ(s,z))|_E
    
    		    =    e^-f_λ(X_s-)[ e^f_λ(X_s-+θξ(s,z))f^'_λ(X_s-+θξ(s,z))(ξ(s,z))-e^f_λ(X_s-)f_λ'(X_s-)(ξ(s,z))]
    ≤    e^f_λ(X_s-+θξ(s,z))-f_λ(X_s-)|f_λ'(X_s-+θξ(s,z))-f^'_λ(X_s-) |_E|ξ(s,z)|_E
       +e^-f_λ(X_s-)[(e^f_λ(X_s-+θξ(s,z))-e^f_λ(X_s-))(f^'_λ(X_s-)(ξ(s,z))]
    
    			 =    e^f_λ(X_s-+θξ(s,z))-f_λ(X_s-)|f_λ'(X_s-+θξ(s,z))-f^'_λ(X_s-) |_E|ξ(s,z)|_E
       +e^f_λ(X_s-+θ_1ξ(s,z))-f_λ(X_s-)[f_λ'(X_s-+θ_1ξ(s,z))(θξ(s,z))f_λ'(X_s-)(ξ(s,z))] 
    ≤    2Cλ e^λ^1/2|ξ(s,z)||ξ(s,z)|_E^2+Cλ e^λ^1/2|ξ(s,z)||ξ(s,z)|_E^2,

		where 0<θ,θ_1<1. Hence
		
    f_λ(X_t)   ≤1+log Z_t
    			  +∫_0^t∫_Z( e^f_λ(u_s-+ξ(s,z))-f_λ(u_s-)-1-(f_λ(X_s-+ξ(s,z))-f_λ(X_s-)))ν(ẓ) ṣ
       +∫_0^t∫_Z (f_λ(X_s-+ξ(s,z))-f_λ(X_s-)-f^'_λ(X_s-)(ξ(s,z))) ν(ẓ) ṣ
       ≤   1+log Z_t+3Cλ∫_0^T∫_Z e^λ^1/2|ξ(s,z)||ξ(s,z)|^2ν(ẓ) ṣ
    
    			   ≤ 1+log Z_t+3C M_λ.

	It follows that
	
    ℙ( sup_0≤ t≤ T|X_t|≥ R)   = ℙ( sup_0≤ t≤ Tf_λ(X_t)≥ (1+λ R^2)^1/2)
       ≤ℙ(sup_0≤ t≤ Tlog Z_t≥ (1+λ R^2)^1/2-1-3CM_λ) 
       ≤ℙ(sup_0≤ t≤ T Z_t≥ e^(1+λ R^2)^1/2-1-3C M_λ)
       ≤ e^1+3C M_λ-(1+λ R^2)^1/2sup_0≤ t≤ TZ_t 
       ≤ e^1+3C M_λ-(1+λ R^2)^1/2
       ≤ C_λe^-(1+λ R^2)^1/2,

	where C_λ=e^1+3C M_λ.
	
	Set β(λ)=1+3Cλ M-(1+λ R^2)^1/2. If R^2≤ 6CM_λ, (<ref>) always holds since
	
    ℙ( sup_0≤ t≤ T|X_t|_E≥ R)≤ 1< 3e^-1/2≤ 3e^-R^2/12 CM_λ.

When R^2>6C M, let us choose λ_0>0 such that 1+λ_0 R^2=(R^2/6CM_λ)^2.
	Since (<ref>) holds for any λ>0, we conclude
	
    ℙ( sup_0≤ t≤ T|X_t|_E≥ R)≤ e^1+3Cλ_0M_λ-(1+λ_0 R^2)^1/2< e^1-R^2/12CM_λ< 3e^-R^2/12CM_λ.

 
	










	 

 §.§ Maximal L^p estimates for stochastic convolutions w.r.t. Lévy processes

	 Now let us consider the issue of maximal L^p estimates for the stochastic convolutions driven by a more general Lévy-type process. Here we assume that r=2, i.e. E is a martingale type 2 Banach space.  Let (e^tA)_t≥0 be a C_0-semigroup on E, with the generator denoted by A,  of contraction type with respect to the equivalent 2-smooth norm |·|_E such that e^tA≤ e^tα  for some α≥ 0.

	 We will establish a type of maximal inequality for the following Lévy-type stochastic convolution
		               
    X_t=∫_0^te^(t-s)Ag_s Ẉ_s+∫_0^t∫_Z e^(t-s)Aξ(s,z)Ñ(ṣ,ẓ),     t≥ 0.



	
	Let (e^tA)_t≥0 be a C_0-contraction semigroup on a martingale type 2 Banach space (E,|·|_E)
		and let (g_t)_t∈[0,T] be a process in M([0,T];γ(H;E)) and ξ∈ℳ_T^2(𝒫⊗𝒵,ṭ×ℙ×ν;E).
	   There exists a càdlàg modification X̅ such that for 2≤ p<∞ and some constant C_p depending on p such that
	
    sup_0≤ t≤ T |X̅_t|_E^p≤ e^α TC_p   [(∫_0^T g_s^2_γ(H,E)ṣ)^p/2+(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
       +(∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ) ṣ)^p/2].

	

	
	 Note that (<ref>) is trivially satisfied if the right-hand side is infinite. So let us assume 	
    (∫_0^T g_s^2_γ(H,E)ṣ)^p/2+(∫_0^T∫_Z |ξ(s,z)|^p_E ν(ẓ) ṣ)
    		 +(∫_0^T∫_Z|ξ(s,z)|^2_E ν(ẓ) ṣ)^p/2<∞.

	Just as in the proofs of Theorem <ref>, we may assume that e^tA≤ 1 and show that inequality (<ref>) holds for  g∈ M^2([0,T],γ(H;𝒟(A))) and ξ∈ℳ^2_T(𝒫⊗𝒵,dt×ℙ×ν;𝒟(A)). Since the Poisson point process π is σ-finite, there exists a sequence of sets {D_n}_n∈ℕ such that ∪_n∈ℕD_n=Z and N(t,D_n)<∞ for every 0<t<∞ and n∈ℕ.
	
	       	  Let us define a sequence {X^n}_n∈ of process X^n:=(X^n_t)_t≥0 by
	
    X^n_t=∫_0^te^(t-s)Ag_s Ẉ_s+∫_0^t∫_Z1_D_n(z) e^(t-s)Aξ(s,z) Ñ(ṣ,ẓ),  n∈ℕ.
 	
	Then (see <cit.> and <cit.>) X^n is a strong solution to the following equation:
	
    X̣_t^n=AX_t^n ṭ+g_t Ẉ_t+∫_D_nξ(t,z) Ñ(ṭ,ẓ).

	Hence we may apply Theorem <ref> and use the fact that ψ^'(x)(Ax)≤ 0 for all x∈ D(A) to infer	
    sup_t∈[0,T]|X_t^n|^p_E≤ C_p(∫_0^T|g_s|^2 ṣ)^p/2+C_p(∫_0^T∫_Z|ξ(s,z)|_E^2 ν(ẓ)ṣ)^p/2
    				+C_p∫_0^T∫_Z|ξ(s,z)|^p ν(ẓ)ṣ.

Meanwhile, by inequality (<ref>) and (<ref>) we observe
			
    sup_t∈[0,T]|X_t^n-X_t|^p_E   =sup_t∈[0,T]|∫_0^T∫_Z e^(t-s)A(ξ(s,z)1_D_n-ξ(s,z)) Ñ(ṣ,ẓ)|_E^p
       ≤ C_p(∫_0^T∫_Z|ξ(s,z)1_D_n-ξ(s,z)|^2_E ν(ẓ)ṣ)^p/2
       +C_p(∫_0^T∫_Z|ξ(s,z)1_D_n-ξ(s,z)|^p_E ν(ẓ)ṣ)
       →0,  as n→∞.

			Note that here the constant C_p is independent of n.
				Therefore, an argument similar to that in the proof of Theorem <ref> shows that there exists a càdlàg modification X̅ of X such that
				
    sup_t∈[0,T]|X̅_t|^p_E≤ C_p(∫_0^T|g_s|^2 ṣ)^p/2+C_p(∫_0^T∫_Z|ξ(s,z)|_E^2 ν(ẓ)ṣ)^p/2
    				+C_p∫_0^T∫_Z|ξ(s,z)|^p ν(ẓ)ṣ.
   	
	




§ APPLICATION TO STOCHASTIC 2D QUASI-GEOSTROPHIC EQUATIONS

We consider the following stochastic quasi-geostrophic equation in ^2,
 
    θ̣(t)+[(v(t)·∇ )θ(t)-Δθ(t)]ṭ= ∫_Zξ(t,z)Ñ(ṭ,ẓ),   t>0,
       θ(0)=θ_0.

  Here θ:^+×^2→ denotes the temperature, v:^2→^2 is the 2D velocity field and θ_0∈ L^2(ℝ^2,ℝ). We refer to <cit.> for the background and more references on this model. Let ψ :^2→ be the stream function which satisfies
  
    (-Δ)^1/2ψ=-θ.

  The velocity is expressed in terms of the stream function by
  
    v=curl ψ.

Clearly, the velocity v can be represented in terms of the temperature by
  
    v=ℛθ=(-ℛ_2θ,ℛ_1θ),

  where ℛ_jθ=ℱ^-1[ξ_j/|ξ|ℱθ], j=1,2 is the j-th Riesz transform.

  
  Note that since the operator -(-Δ)^α/2 is also a generator of a C_0 contraction semigroup on L^p(^2) for p∈[1,∞) (see e.g. <cit.>), our results could be generalized to a fractionally dissipative quasi-geostrophic equation for some α∈ (0,2) in a similar way. It will be further investigated in future works.
  
  
  Let C^∞_c(^2;^d) be  the space of all infinitely differential functions ϕ:^2→^d with compact support contained in ^2.
For m ∈ℕ and p ∈ [1,∞],  denote by W^m,p(^2) the Banach space of all elements of L^p(^2;) whose weak partial derivatives up to order m belong to L^p(^2;) as well. The space W^1,2(^2) will be denoted by H^1(^2).
 It is well known that the set C^∞_c(^2;) is dense in the space W^m,p(^2) if p∈ [1,∞).   Let 𝒱 denotes the space of all v∈ C_c^∞(^2;^2) such that div v=0.
We will denote by L^p_sol(^2;^2) the closure of 𝒱 in L^p(^2;^2).

  

  
  Now let b:C_c^∞(^2;^2)× C_c^∞(^2;)× C_c^∞(^2;) → be a trilinear form defined by
    
    b(v,ϕ,η)   :
                =∫_^2 (v·∇) ϕ η x̣
                =∑_i=1^2∫_^2v_iD^iϕ η x̣,

  whenever
 v ∈ C_c^∞(^2;^2) and ϕ,η∈ C_c^∞(^2;) such that the integral on
the right-hand side exists.

 If div v=0, then we have
  
    ∫_^2 (v·∇) ϕ η x̣=-∫_^2 (v·∇) η ϕ x̣.

  This gives that
  
    b(v,ϕ,η)=-b(v,η,ϕ).

  So we infer
   
    b(v,ϕ,ϕ)=0.

 Using the Hölder inequality and equality (<ref>) we can deduce  the following estimate:

    | b(v,ϕ, η)|= | b(v,η,ϕ)|≤| v|_L^4|ϕ|_L^4|∇η|
    _L^2.

Hence b can be extended to a trilinear continuous form on L^4_sol(^2;^2)× L^4(^2;)× H^1(^2).
Define a bilinear map B by B(v,ϕ):=b(v,ϕ,·). It follows from (<ref>) that

    B:L^4_sol(^2;^2)× L^4(^2;)→ H^-1(^2)

and it satisfies that

    B(v,ϕ)_H^-1(^2)≤| v|_L^4|ϕ|_L^4,   v∈ L^4_sol(^2;^2), ϕ∈ L^4(^2;).

Here H^-1(^2) is defined as the dual space of H^1(^2).
Also we have for v∈ L^4_sol(^2;^2), ϕ∈ L^4(^2;), and w∈ H^1(^2),

    ⟨ B(v,ϕ),w⟩ =b(v,ϕ,w).

Hence by (<ref>),

    ⟨ B(v,ϕ),ϕ⟩ =b(v,ϕ,ϕ)=0,    v∈ L^4_sol(^2;^2), ϕ∈ L^4(^2;).

  Observe that ℛ is a linear continuous operator such that for all 1<p<∞, ℛ:L^p(^2;)→ L^p_sol(^2;^2) (see <cit.>). Clearly, for all θ∈ L^p(^2;), applying the Fourier transform gives that
      
    ℱ[div ℛθ](ξ)   =ℱ[D_1(-ℛ_2θ)+D_2(ℛ_1θ)](ξ)
       =-iξ_1ℱ(ℛ_2θ)+iξ_2ℱ(ℛ_1θ)
       =-iξ_1ξ_2/|ξ|θ̂+iξ_2ξ_1/|ξ|θ̂
       =i[-ξ_1ξ_2+ξ_2ξ_1]1/|ξ|θ̂=0.

Thus we have div ℛθ=0.
  



  Let us define the operator A:D(A)⊂ L^4(^2;)→ L^4(^2;) by
  
    D(A)=W^2,4(^2)∩ W^1,4(^2),
       Au=-Δ u,   u∈ D(A).

  Then (<ref>) can be rewritten as
  
    θ̣(t)   =-[Aθ(t)+B(ℛθ(t),θ(t))]ṭ+ ∫_Zξ(t,z)Ñ(ṭ,ẓ),
    θ(0)   =θ_0.

  
   Note that Equation (<ref>) is closely related to the Navier-Stokes equation
    
    ụ(t)   =-[A_1u(t)+B(u(t),u(t))]ṭ+ ∫_Zξ(t,z)Ñ(ṭ,ẓ),
    
            u(0)   =u_0

  in the following sense: both equations have the same linear part Δθ and Δ u and the nonlinearities satisfy the cancellation property ⟨ B(ℛu,v),v⟩=0 and ⟨ B(u,v),v⟩=0 respectively. For the Navier-Stokes equation,  ℛ is replaced by I. The main difference between these two equations is that for quasi-geostrophic equation the operator -A generates a contraction semigroup on L^p for 1≤ p<∞, however, for the Navier-Stokes equation, the negative Stokes operator -A_1 generates a contraction semigroup only for p=2.
    



  
There exists a constant C such that for all T>0 and all θ_1,θ_2∈ L^4(0,T; L^4(^2;)), the following inequality holds:
  
    |B(ℛθ_1,θ_1)-B(ℛθ_2,θ_2)|_L^2(0,T;H^-1(^2))≤ C(|θ_1|_L^4(0,T;L^4(^2;))+|θ_2|_L^4(0,T;L^4(^2;)) )|θ_1-θ_2|_L^4(0,T;L^4(^2;)).

  

  
  By (<ref>) and the fact that Riesz transforms are bounded on L^p for any p∈(1,∞) we obtain
     
    |B(ℛθ,ϕ)|_H^-1(^2)≤|ℛθ|_L^4(^2;^2)|ϕ|_L^4(^2;)≤ C |θ|_L^4(^2;)|ϕ|_L^4(^2;),  θ,ϕ∈ L^4(^2;).


  Then we infer
       
    ∫_0^T |B(ℛθ(t),θ(t))|^2_H^-1(^2)ṭ≤ C∫_0^T|θ(t)|_L^4^4 ṭ.

      
    If θ_1,θ_2∈ L^4(0,T;L^4(^2;)) and w∈ H^1(^2) we have
    
    ⟨  B(ℛθ_1,θ_1)-B(ℛθ_2,θ_2),w⟩   =b(ℛθ_1,θ_1,w)-b(ℛθ_2,θ_2,w)
       =b(ℛθ_1,θ_1-θ_2,w)+b(ℛθ_1-ℛθ_2,θ_2,w)
       =⟨ B(ℛθ_1,θ_1-θ_2),w⟩+⟨ B(ℛ(θ_1-θ_2),θ_2),w⟩.

    It follows that
     
    |B(ℛθ_1,θ_1)-B(ℛθ_2,θ_2)|_H^-1(^2)≤ C(|θ_1|_L^4+|θ_2|_L^4)|θ_1-θ_2|_L^4.

    Hence we have for θ_1,θ_2∈ L^4(0,T; L^4(^2;)),
    
    ∫_0^T   |B(ℛθ_1(t),θ_1(t))-B(ℛθ_2(t),θ_2(t))|_H^-1(^2)^2 ṭ
       ≤ 4C(∫_0^T|θ_1(t)-θ_2(t)|^4_L^4ṭ)^1/2(∫_0^T|θ_1(t)|^4_L^4+|θ_2(t)|^4_L^4ṭ)^1/2.

 



 Define a process Z by
  
    Z(t)=∫_0^te^-(t-s)Aξ(s,z) Ñ(ṣ,ẓ),    t≥ 0.

  Equivalently, Z is the unique solution to the following stochastic Langevin equation (see <cit.>):
  
    Ẓ=-AZṭ+∫_Zξ(t,z)Ñ(ṭ,ẓ),
       Z(0)=0.

Let us fix a number s∈ (0,1/2).
 
 Assume that ξ:[0,T]×Ω× Z→ W^-s,4(^2) is a predictable process which satisfies
 
    ∫_0^T∫_Z|ξ(s,z)|^2_W^-s,4(^2)ν(ẓ)ṣ<∞.

 
For the definition of the spaces W^s,p(^2) see e.g. <cit.>. We aim to look for a solution of (<ref>) with the following form
  
    θ(t)=e^-t Aθ_0+Y(t)+Z(t).

   For this purpose we shall first prove the following result.
   Under Assumption <ref>, we have for T>0, Z∈ L^4(0,T;L^4(^2;)) ℙ-a.s., that is,
   
    ∫_0^T |Z(t)|^4_L^4 ṭ<∞,    ℙ-a.s.

   

  
By the Gagliardo-Nirenberg interpolation inequality (see e.g. <cit.>) we have for 0<s<∞
    |Z|_L^4^4≤ C|Z|_W^s,4(^2)^2|Z|_W^-s,4(^2)^2.

It follows that

    ∫_0^T |Z(t)|^4_L^4 d t   ≤ C∫_0^T |Z(t)|^2_W^s,4(^2)|Z(t)|^2_W^-s,4(^2)ṭ
       ≤ Csup_0≤ t≤ T|Z(t)|^2_W^-s,4(^2)∫_0^T|Z(t)|^2_W^s,4(^2)ṭ.

Since the space E=W^-s,4(^2) is of martingale type 2 (see <cit.>) and -A generates a contraction semigroup in E, we may apply inequality (<ref>) to get

    sup_0≤ t≤ T|Z(t)|^2_W^-s,4(^2)≤ C∫_0^T|ξ(t,z)|^2_W^-s,4(^2)ν(ẓ)ṭ<∞,

which also gives a W^-s,4(^2;)-valued càdlàg modification of Z(t).

On the other hand because W^s,4(^2;) is also of martingale type 2, by using inequality (1.7) from <cit.> we have

    ∫_0^T|Z(t)|^2_W^s,4(^2)ṭ   =∫_0^T|A^s/2Z(t)|^2_L^4(^2;)ṭ
       =∫_0^T|Z(t)|^2_D(A^s/2)ṭ
       ≤∫_0^T∫_Z |ξ(t,z)|^2_D(A^s/2-1/2)ν(ẓ)ṭ
       =∫_0^T∫_Z |ξ(t,z)|^2_W^s-1,4(^2)ν(ẓ)ṭ
       ≤∫_0^T∫_Z|ξ(t,z)|^2_W^-s,4ν(ẓ)ṭ<∞,

provided -s≥ s-1, i.e. s≤1/2.
 

  
 By the classical Gagliardo-Nirenberg inequality <cit.>, we infer that

    |Z|^4_L^4(0,T;L^4)≤ |Z|^2_L^∞(0,T;L^2)|Z|^2_L^2(0,T;H^1(^2)).

 So one sufficient condition for (<ref>) is
  
    ∫_0^T |Z(t)|^2_H^1(^2)ṭ<∞

  and
   
    sup_0≤ t≤ T |Z(t)|^2_L^2ṭ<∞.

  

 Notice that (<ref>) can be rewritten as
  
    Ỵ+AYṭ+B(ℛ(Y+Z),Y+Z)ṭ=0,
        Y(0)=Y_0.

  
  
  
Assume that Z(·)∈ L^4(0,T;L^4(^2;)). Then for every Y_0 ∈ L^2(^2;), there exists a unique solution Y  to the equation (<ref>)
  and it satisfies


  
  
    Y(·)∈ L^4(0,T,L^4(^2;)),


  
 
    Y(·)∈ L^∞(0,T;L^2(^2;))∩ L^2(0,T;H^1(^2)).


Moreover, we have

    sup_t∈[0,T]
          |Y(t)|^2_L^2≤ e^C_1 ∫_0^T|Z(t)|^4_L^4ṭ|Y_0|^2_L^2+C_2∫_0^Te^C_1∫_s^T|Z(r)|^4_L^4ṛ|Z(s)|^4_L^4ṣ

and

    ∫_0^T |∇ Y(t)|^2_L^2ṭ≤ |Y_0|^2_L^2+C_1sup_t∈[0,T] |Y(t)|^2_L^2∫_0^T|Z(t)|^4_L^4ṭ+ C_2∫_0^T |Z(t)|^2_L^4ṭ.






Let us put

    ρ:=|Y_0|^2_L^2  e^C_1 ∫_0^T|Z(t)|^4_L^4ṭ+C_2∫_0^Te^C_1∫_t^T|Z(s)|^4_L^4ṣ|Z(t)|^4_L^4ṭ.

Note that   | Y_0 |_L^2^2≤ρ.
Because of Lemma <ref>,  applying similar arguments as in <cit.> and then the Banach fixed point theorem yields the existence of some T_1∈ (0,T] and
 unique local solution in L^4(0,T_1;L^4(^2;)) to (<ref>). Below, we will show that  this solution could be extended to the whole interval [0,T].

For this aim, by applying Lemma III.1.2 in <cit.> we have for t ∈ [0,T_1],

    1/2/ṭ|Y(t)|^2_L^2   =⟨ Y(t), /ṭY(t)⟩_L^2
       =-⟨ Y(t), AY(t)⟩ +⟨ B(ℛ(Y(t)+Z(t)),Y(t)+Z(t)),Y(t) ⟩
       =-∫_^2∇ Y(t)  ^2x̣+⟨ B(ℛ(Y(t)+Z(t)),Y(t)+Z(t)),Y(t)⟩
       =-|∇ Y(t)|^2_L^2-b(ℛ(Y(t)),Y(t),Z(t))-b(ℛ(Z(t)),Y(t),Z(t)).

Then by (<ref>) and the boundedness of the operator ℛ on L^4 we obtain that

    |b(ℛ(Y(t)),Y(t),Z(t))|≤ C|Y(t)|_L^4|∇ Y(t)|_L^2|Z(t)|_L^4,
       |b(ℛ(Z(t)),Y(t),Z(t))|≤ C|Z(t)|_L^4|∇ Y(t)|_L^2|Z(t)|_L^4.

Due to the Ladyzhenskaya inequality (see Lemma III 3.3 in <cit.>)

    |Y(t)|_L^4≤ 2^1/4|∇ Y(t)|_L^2^1/2|Y(t)|_L^2^1/2,

 we have

    |b(ℛ(Y(t)),Y(t),Z(t))|   ≤
            C2^1/4|∇ Y(t)|_L^2^3/2|Y(t)|^1/2_L^2|Z(t)|_L^4.

By Young's inequality we get

    |b(ℛ(Y(t)),Y(t),Z(t))|≤1/4|∇ Y(t)|_L^2^2+27/2C^4|Y(t)|^2_L^2|Z(t)|^4_L^4.

Again, Young's inequality gives that

    |b(ℛ(Z(t)),Y(t),Z(t))|   ≤ C|Z(t)|^2_L^4|∇ Y(t)|_L^2
       ≤1/4 |∇ Y(t)|_L^2^2+C^2|Z(t)|^4_L^4.

It follows that

    1/2/ṭ|Y(t)|^2_L^2+1/2|∇ Y(t)|_L^2^2
        ≤27/4C^4|Y(t)|^2_L^2|Z(t)|^4_L^4+C^2|Z(t)|^2_L^4
       =1/2C_1 |Y(t)|^2_L^2|Z(t)|^4_L^4+ 1/2C_2 |Z(t)|^2_L^4,

where C_1=27/2C^4 and C_2=2C^2.

 Now by Gronwall's lemma we have for t∈ [0,T_1],

    |Y(t)|^2_L^2≤ e^C_1 ∫_0^T|Z(s)|^4_L^4ṣ|Y_0|^2_L^2+C_2∫_0^Te^C_1∫_s^T|Z(r)|^4_L^4ṛ|Z(s)|^4_L^4ṣ,

 where the constants C_1,C_2 are independent of Y_0 and Z.


 Integrating (<ref>) we obtain

    ∫_0^T_1|∇ Y(t)|^2_L^2ṭ≤ |Y_0|^2_L^2+C_1sup_t∈[0,T_1] |Y(t)|^2_L^2∫_0^T|Z(t)|^4_L^4ṭ+ C_2∫_0^T |Z(t)|^2_L^4ṭ.

Since by assumption Z(·)∈ L^4(0,T;L^4(^2;)), it follows from the inequality (<ref>) that  |Y(t)|^2_L^2≤ρ, for all t ∈ [0,T_1]. In other words, the local solutions cannot blow up in finite time. Thus, by a simple and standard contradiction argument we infer that T_1=T. Inequalities (<ref>) and (<ref>) follow from the above two inequalities. The proof is complete.




 A W^-s,4(^2)-valued adapted càdlàg process θ(t), t∈[0,T], is a solution to (<ref>) if it satisfies

    ∫_0^T|θ(t)|^4_L^4ṭ<∞ a.s.

and for t∈[0,T], the following equality holds a.s.,

    θ(t)=θ_0-∫_0^t e^-(t-s)AB(ℛθ(s),θ(s))ṣ+∫_0^te^-(t-s)Aξ(s,z) Ñ(ṣ,ẓ).



 Suppose Assumption <ref> holds. Then for any θ_0∈ W^-s,4(ℝ^2), there exists a unique solution θ(t), t∈[0,T], to (<ref>).


Notice that if Z satisfies (<ref>) and Y satisfies (<ref>), then θ defined by (<ref>) with Y_0=0 satisfies

    θ (·)∈ L^4([0,T],L^4(^2;))  a.s.

Now let us show the uniqueness. Assume that θ, θ̃ are two solutions of (<ref>).
We define

    Ỹ(t)=θ̃(t)-Z(t),

 hence both Y and Ỹ solve (<ref>).  Moreover, since Y,Ỹ∈ L^4(0,T;L^4(^2;)) a.s., then by the uniqueness of (<ref>) we have Y=Ỹ a.s.





One might also study (<ref>) by following the method from previous works <cit.>. However, our approach here is different. For example, comparing with <cit.>, where u∈ L^∞(0,T;L^2(D))∩ L^2(0,T;H^1_0(D)), in this paper we get the solution of  θ∈ L^4(0,T;L^4(^2)) under much weaker assumptions on the jump process than that in <cit.>, in which even the Ornstein-Uhlenbeck process Z is assumed to satisfy Z∈  L^2(0,T;H^1_0(D)). A generalization of the current result to multiplicative noise will be investigated in future work.




Stochastic Navier-Stokes equations with Poisson noise in L^p-spaces are investigated in <cit.>. The key tool employed in the proof of existence and uniqueness of a local mild solution theory in <cit.> is a type of maximal inequality that was recently developed by the first two authors of this paper and Hausenblas in <cit.>. It's worth mentioning that <cit.> deals with maximal inequalities with respect to stronger norms. More precisely, the p-th power of the norm is assumed to be of C^2. Our results here avoid such conditions and our assumptions here are much easier to verify. The advantage of the maximal inequalities here becomes more transparent when one deals with closed subspace of Banach spaces.







	





			
 

§ STOCHASTIC INTEGRATION W.R.T. POISSON RANDOM
MEASURES

In this section we give a brief review of  some basic terminology and known results on the stochastic integral w.r.t. Poisson random measures.

Suppose that (Z,𝒵) is a measurable space and ν is a nonnegative σ-finite measure on it.  It is known that there exists a stationary Poisson point process π=(π(t))_t≥ 0 on (Z,𝒵) with the intensity measure ν; see <cit.>. Let N be the counting measure associated with π which is defined by
  
    N(U,ω):=∑_s∈𝒟(π(ω))I_U(s,π(s,ω)),  U∈ℬ((0,∞))⊗𝒵,  ω∈Ω.

 In particular, we have
 
    N((0,t]× A,ω)=∑_s∈(0,t]∩𝒟(π(ω))I_A(π(s,ω)),  A∈𝒵,   0<t<∞.



According to <cit.>, N is a Poisson random measure with  the intensity measure ν. That is for every U∈ℬ(0,∞)⊗𝒵 with N(U)<∞, the random variable N(U) is Poisson distributed and for any pairwise disjoint sets U_1,…,U_n∈ℬ(0,∞)⊗𝒵,
 the random variables N(U_1),… N(U_n) are independent. Here as usual we shall employ the notation

    Ñ=N-⊗ν

to denote the compensated Poisson random measure of N.

We will define the stochastic integral of 𝒫⊗𝒵-measurable functions under the martingale type r Banach space setting.  Let us recall the definition of martingale type r Banach spaces. 	
	      
A Banach space E with norm |·|_E is of martingale type r, for r∈[1,2] if there exists a constant C_r(E)>0 such that for any E-valued discrete martingale {M_k}_k=1^n the following inequality holds

    |M_n|_E^r≤ C_r(E)∑_k=0^n|M_k-M_k-1|_E^r,

with M_-1=0 as usual.

           Note that every real separable Banach space is of martingale type 1 and every Hilbert space is of martingale type 2. If a real separable Banach space is of martingale type p for some 1 < p ≤2 then it is of martingale type r for all r∈ [1, p].
            All L^p spaces for p≥ 1 and Sobolev spaces W^k,p for p≥ 1 and k>0 are martingale type p∧ 2 Banach spaces. For more details of this subject we refer the reader to <cit.>.





For T∈ (0,∞), let ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E) denote the
linear space consisting of (equivalence classes of) all 𝒫⊗𝒵-measurable functions
ξ:[0,T]×Ω× Z→ E such that


    ∫_0^T∫_Z𝔼 |ξ(t,z)|^r_E ν(ẓ) ṭ<∞.


Note that ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E)
is the usual L^r space of E-valued functions on [0,T]×Ω× Z with respect to the σ-field 𝒫⊗𝒵 and the measure
⊗ℙ⊗ν.

Let ℳ^r_step([0,T]×Ω× Z,𝒫⊗𝒵;E) be the space of all functions f:[0,T]×Ω× Z→ E for which there exists a partition 0=t_0< t_1<…<t_n=T and a finite family A_j-1^k, j=1,…,n, k=1,…,m, of sets from
𝒵 with ν(A_j-1^k)<∞  such that
   
    f(t,ω,z)=∑_k=1^m∑_j=1^nξ^k_j-1(ω)1_(t_j-1,t_j](t)1_A^k_j-1(z),   (t,ω,z)∈[0,T]×Ω× Z.

 Here ξ^k_j-1 is an E-valued ℱ_t_j-1-measurable random variable, for every j=1,…,n and k=1,…,m, and for each j=1,…,n, the sets A_j-1^k, k=1,…,m, are pairwise disjoint.

For each ξ∈ℳ^r_step([0,T]×Ω× Z,𝒫⊗𝒵;E), we set
  
    I̅(ξ):=∑_k=1^m∑_j=1^nξ^k_j-1(ω)Ñ((t_j-1,t_j]× A_j-1^k).




<cit.> Assume that E is a martingale type r Banach space, r∈(1,2]. For every ξ∈ℳ^r_step([0,T]×Ω× Z,𝒫⊗𝒵;E), I̅(ξ)∈  L^r(Ω,ℱ,E), I(ξ)=0, and

    |I̅(ξ)|^r≤ C ∫_0^T∫_Z|ξ(t,z)|^r_E ν(ẓ)ṭ,

where the constant C depends on C_r(E) from (<ref>).


<cit.> Assume that E is a martingale type r Banach space, r∈(1,2]. For every ξ∈ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E), there exists a unique bounded linear operator

    I: ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E)→ L^r(Ω,ℱ,E)

such that

    |I(ξ)|^r≤ C ∫_0^T∫_Z|ξ(t,z)|^r_E ν(ẓ)ṭ.

In particular, if ξ∈ℳ^r_step([0,T]×Ω× Z,𝒫⊗𝒵;E), then I(ξ)=I̅(ξ).

 Moreover, the process ∫_0^t∫_Z ξ(s,z)Ñ(ṣ, ẓ):=I(1_[0,t]ξ), t∈[0,T], is an E-valued r-integrable càdlàg martingale with mean 0 and it satisfies

    |∫_0^t∫_Zξ(s,z) Ñ(ṣ,ẓ)|^r_E≤∫_0^t∫_Z|ξ(s,z)|_E^r ν(ẓ)ṣ,   t∈[0,T].




Analogously,  if τ is a stopping time with ℙ{τ≤ T}=1,  we shall define ∫_0^τ∫_Zf(s,z)Ñ(ds,dz):=I(1_(0,τ]ξ) and by Theorem <ref> it satisfies

    |∫_0^τ∫_Zξ(s,z) Ñ(ṣ,ẓ)|^r_E≤∫_0^τ∫_Z|ξ(s,z)|_E^r ν(ẓ)ṣ.


It's worth mentioning that when the function is predictable, the stochastic integral studied in <cit.> for progressively measurable functions coincides with the stochastic integral we defined here for predictable functions. But for predictable but not progressively measurable functions, the stochastic integral in <cit.> differs from the one considered in <cit.> where the stochastic integral is defined as a limit of a sequence of Bochner integrals.




  Assume that E is a Banach space. Let ℒ be a linear space of ℬ([0,T])⊗ℱ_T⊗𝒵-measurable E-valued (resp. ℝ_+-valued) functions defined on [0,T] ×Ω× Z.
		If 𝒦 is a linear subspace of ℒ satisfying the following two conditions.
		
			
  * 𝒦 contains all functions of the form
			          
    ∑_i=1^nx_i1_F_i1_(t_i,t_i+1]1_B_i+x_01_F_01_{0}1_B_0,

				where 0=t_0<t_1<⋯<t_n=T, x_i∈ E, (resp. x_i≥0), B_i∈𝒵 with ν(B_i)<∞, and F_i∈ℱ_t_i for  i=0,…,n;
				 
  * 
				if {f_n} is a sequence (resp. monotone increasing sequence) in 𝒦, f∈ℒ and
				   |f_n-f|_E decreases to 0 on [0,T]×Ω× Z, then f∈𝒦,
		
		then 𝒦 contains all 𝒫⊗𝒵-measurable E-valued (resp. ℝ_+-valued) functions in ℒ.
	

	
	   Note that for every E-valued
		𝒫⊗𝒵-measurable function f in ℒ, we can always find a sequence of simple functions {f^n} of the form ∑_j=1^m x_j1_A_j, x_j∈ E, A_j∈𝒫⊗𝒵 such that |f^n-f|_E decreases to 0, as n→∞.  In particular, if f is a 𝒫⊗𝒵-measurable and positive function, then  f is a monotone increasing limit of  𝒫⊗𝒵-measurable and positive simple functions. Hence by condition <ref> and linearity of 𝒦 we only need to show that for every x∈ E and A∈𝒫⊗𝒵, x1_A∈𝒦.

		It is known (e.g. <cit.>) that the predictable σ-field is also generated by the following family of sets
		
    ℛ= {(s,t]× F: 0≤ s≤ t≤ T, F∈ℱ_s}∪{{0}× F, F∈ℱ_0}.

		Let 𝒢:={B∈ Z: ν(B)<∞}. Since (Z,𝒵,ν) is a σ-finite measure space, 𝒵 contains an exhausting sequence (D_j)_j∈ℕ of sets such that ν(D_j)<∞ for all j∈ℕ. So it is easy to see that σ(𝒢)=𝒵 and 𝒢 contains the exhausting sequence (D_j)_j∈ℕ. Hence we have 𝒫⊗𝒵=σ(ℛ)⊗σ(𝒢)=σ(ℛ×𝒢). In other words, 𝒫⊗𝒵 is generated by
		    
    ℛ̂= {(s,t]× F× B: 0≤ s≤ t≤ T, F∈ℱ_s, B∈𝒢}∪{{0}× F× B, F∈ℱ_0, B∈𝒢}.

		Let us set
		
    𝒫'={A: A∈𝒫⊗𝒵 s.t. x1_A∈𝒦, ∀ x∈ E}

		Then it is straightforward to see that  𝒫' is a Dynkin system. Let 𝒜 be the collection of all finite unions of sets in ℛ̂.
		Clearly, 𝒜 is closed under finite intersections. Indeed, take A=∪_i=1^k A_i and B=∪_j=1^l from 𝒜, where A_i, B_j∈ℛ̂. Then A∩ B=∪_i≤ k,j≤ l(A_i∩ B_j)∈𝒜 which shows that 𝒜 admits finite intersections, i.e. 𝒜 is a π system. Also by (1), we find 𝒜⊂𝒫'. Hence we may apply Dynkin's lemma to get 𝒫⊗𝒵=σ(𝒜)⊂𝒫'. Therefore, we obtain 𝒫'=𝒫⊗𝒵 which shows that for every x∈ E and A∈𝒫⊗𝒵, x1_A∈𝒦.
	
	 Assume that E is a Banach space. If
		ξ:  [0,T] ×Ω× Z → E is a 𝒫⊗𝒵-measurable function such that either 
    ∫_0^T∫_Z|ξ(s,z)|_E N(ṣ,ẓ)<∞
 or ∫_0^T∫_Z|ξ(s,z)|_E ν(ẓ)ṣ<∞, then we have
		
    ∫_0^T∫_Zξ(s,z) N(ṣ,ẓ)=∫_0^T∫_Zξ(s,z) ν(ẓ)ṣ.

	
	 We first show that equality (<ref>) holds for any positive 𝒫⊗𝒵-measurable function f.
		 Let ℒ_1:={f:[0,T]×Ω× Z→ℝ_+ |  f is ℬ([0,T])⊗ℱ_T⊗𝒵/ℬ(ℝ_+)-measurable}.   	 Define
		
    𝒦_1={f:[0,T]×Ω× Z→ℝ_+ | f is in ℒ_1 and satisfies (<ref>)}.

		Take a function f of the form
		  
    ∑_i=1^na_i1_F_i1_(t_i,t_i+1]1_B_i+a_01_F_01_{0}1_B_0,  t_i<t_i+1, a_i≥0, B_i∈𝒢, F_i∈ℱ_t_i, for  i=0,⋯,n.

		 Then by the independence of 1_F_i and N((t_i,t_i+1]× B_i) and the stationary property of the Poisson random measure N, we have
		
    ∫_0^T∫_Zf(s,z)N(ṣ,ẓ)   =∑_i=1^ma_i1_F_iN((t_i,t_i+1]× B_i)
    			                                        =∑_i=1^ma_i1_F_i N((t_i,t_i+1]× B_i)
       =∑_i=1^ma_i1_F_iν(B_i)(t_i+1-t_i)
    			=∫_0^T∫_Zf(s,z) ν(ẓ)ṣ.

Suppose that {f_n}⊂𝒦_1 is a monotone increasing sequence of positive functions and f_n converges to f. Then by the monotone convergence theorem

    ∫_0^T∫_Zf(s,z)N(ṣ,ẓ)    =sup_n  ∫_0^T∫_Zf^n(s,z) N(ṣ,ẓ) 
        = sup_n∫_0^T∫_Zf^n(s,z) ν(ẓ)ṣ= ∫_0^T∫_Zf(s,z) ν(ẓ)ṣ.

	It follows from Lemma <ref> that 𝒦_1 contains all 𝒫⊗𝒵-measurable positive functions.
	
		                    	Now let us take an E-valued and 𝒫⊗𝒵-measurable function ξ.
		From above discussion, we know
		      
    ∫_0^T∫_Z|ξ(s,z)|_E ν(ẓ)ṣ=∫_0^T∫_Z|ξ(s,z)|_E N(ṣ,ẓ),

	   from which deduce  that whenever one side of the equality makes sense, so does the other. Hence if we suppose either
		    ∫_0^T∫_Z|ξ(s,z)|_Eν(ẓ)ṣ<∞ or ∫_0^T∫_Z|ξ(s,z)|_EN(ẓ,ṣ)<∞,	
			then both integrals ∫_0^T∫_Zξ(s,z)ν(ẓ)ṣ and ∫_0^T∫_Zξ(s,z)N(ẓ,ṣ) are well defined as Lebesgue-Bochner integrals, ℙ-a.s.

			   Define
			
    ℒ_2   ={ξ:[0,T]×Ω× Z→ E,| ξ is ℬ([0,T])⊗ℱ_T⊗𝒵-measurable and ∫_0^T∫_Z|ξ(s,z)|_E ν(ẓ)ṣ<∞},     
    𝒦_2   ={ξ:[0,T]×Ω× Z→ E | ξ∈ℒ_2 and (<ref>) holds}.

			We can repeat the argument before to show that every function ξ of the form (<ref>) is in ℒ_2 and  satisfies (<ref>).
		   Now suppose that {ξ^n}_n∈ℕ⊂𝒦_2, ξ∈ℒ_2, and |ξ^n-ξ|_E decreases to 0 as n→∞. Then according to the monotone convergence theorem, we obtain
			
    lim_n→∞∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E N(ṣ,ẓ)=0   and lim_n→∞∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E ν(ẓ)ṣ=0

  Since ξ^n satisfies (<ref>), we infer

    ∫_0^T∫_Zξ(s,z) N(ṣ,ẓ)   =lim_n→∞∫_0^T∫_Zξ^n(s,z) N(ṣ,ẓ)
       = lim_n→∞∫_0^T∫_Zξ^n(s,z) ν(ẓ)ṣ= ∫_0^T∫_Zξ(s,z) ν(ẓ)ṣ.
  		
			Again by Lemma <ref>, 𝒦_2 contains all 𝒫⊗𝒵-measurable functions in ℒ_2.
	
	
	
	 Let E be a Banach space. Suppose that ξ:[0,T]×Ω× Z→ E is a ℬ([0,T])⊗ℱ_T⊗𝒵-measurable function such that ∫_0^T∫_Z|ξ(s,z)|_EN(ṣ,ẓ)<∞. Then
		
    ∫_0^T∫_Zξ(s,·,z)  N(ṣ,ẓ)(ω)=∑_s∈(0,T]∩𝒟(π)ξ(s,ω,π(s,ω))

	 
  
	   Since ξ is a ℬ([0,T])⊗ℱ_T⊗𝒵-measurable function and ∫_0^T∫_Z|ξ(s,z)|_EN(ṣ,ẓ)<∞, there exists a set Ω̂⊂Ω with probability 1 such that for every ω∈Ω̂, ξ(·,ω,·) is ℬ([0,T])⊗𝒵-measurable and ∫_0^T∫_Z|ξ(s,·,z)|_EN(ṣ,ẓ)(ω)<∞. This implies that ξ(·,ω,·) is Bochner integrable with respect to N ℙ-a.s. Given ω∈Ω, we can always find a sequence of functions {ξ^n} of the form
	
    ∑_i=1^nx_i1_A_i,  A_i∈ℬ([0,T])⊗𝒵

		such that |ξ^n(t,ω,z)-ξ(t,z)|_E decreases to 0 as n→∞ for every (t,z)∈[0,T]× Z. So it's enough to verify (<ref>) for the function ξ of the form (<ref>). For this, observe that
		
    ∫_0^t∫_Zξ(s,·,z) N(ṣ,ẓ)(ω)   = ∑_i=1^nx_iN(A_i∩(0,t]× Z)(ω)= ∑_i=1^nx_i∑_s∈(0,t]∩𝒟(π)1_A_i(π(s,ω))
       = ∑_s∈(0,t]∩𝒟(π)∑_i=1^nx_i1_A_i(π(s,ω))=∑_s∈(0,t]∩𝒟(π)ξ(s,ω,π(ω)).

	
	
	
 Let E be a martingale type r Banach space.
 If
	ξ∈ℳ^1_T(𝒫⊗𝒵,ṭ×ℙ×ν;E)∩ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E), then  for each t≥0, ℙ-a.s.
	
    ∫_0^t∫_Zξ(s,·,z) Ñ(ds,dz)   =∫_0^t∫_Z ξ(s,·,z)  N(ds,dz)-∫_0^t∫_Zξ(s,·,z) ν(dz)ds
        = ∑_s∈(0,t]∩𝒟(π)ξ(s,·,π(s))-∫_0^t∫_Zξ(s,·,z) ν(dz)ds.

	Here the integral ∫_0^t∫_Zξ(s,·,z) Ñ(ds,dz) on the left side is understood as the stochastic integral taking values in E.
	

	
	 The proof could be done in the same manner as earlier in the proof of Proposition <ref>. First the equality (<ref>) can be easily verified for functions of simple structures (<ref>).
	Next, an approximating step introduced in Lemma <ref> allows us to extend the equality to a general 𝔉-predictable process in ℳ^1_T(𝒫⊗𝒵,ṭ×ℙ×ν;E). To do this, suppose that {ξ_n}_n∈ℕ is a sequence of functions in ℳ^1_T(𝒫⊗𝒵,ṭ×ℙ×ν;E)∩ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E) such that for every n∈ℕ, ξ^n satisfies (<ref>) and |ξ^n-ξ|_E decreases to 0, as n→∞ on [0,T]×Ω× Z. Hence we have
	
    lim_n→∞∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E ν(ẓ)ṣ=0,
       lim_n→∞∫_0^T∫_Z|ξ^n(s,z)-ξ(s,z)|_E N(ṣ,ẓ)=0.

	By (<ref>), we may find a subsequence ξ^n_k such that for every t∈[0,T],
	
    ∫_0^t∫_Zξ(s,·,z) Ñ(ṣ,ẓ)    =lim_k→∞∫_0^t∫_Zξ^n_k(s,·,z) Ñ(ṣ,ẓ)
       = lim_k→∞[∫_0^t∫_Zξ^n_k(s,·,z) N(ṣ,ẓ)-∫_0^t∫_Zξ^n_k(s,·,z) ν(dz)ds]
       = ∫_0^t∫_Zξ(s,·,z) N(ṣ,ẓ)-∫_0^t∫_Zξ(s,·,z) ν(dz)ds,   ℙ-a.s.

	
	
	
	Proposition <ref> is still valid under the weaker constrain on E and ξ. Namely, we only assume that E is a Banach space and ξ∈ℳ^1_T(𝒫⊗𝒵,ṭ×ℙ×ν;E). Then E is of martingale type 1, and we can still define the stochastic integral ∫_0^t∫_Zξ(s,·,z)Ñ(ds,dz) which is an L^1-integrable martingale and satisfies
	
    |∫_0^t∫_Zξ(s,·,z) Ñ(ṣ,ẓ)|_E≤ C∫_0^t∫_Z|ξ(s,·,z)|_E ν(ẓ)ṣ.

	In this case, the stochastic integrable coincides with the Bochner integral almost everywhere, that is (<ref>) holds.
	It is worth noting that (<ref>) is only satisfied when the function ξ is predictable (specifically 𝒫⊗𝒵-measurable).
	




§ ITÔ'S  FORMULA





Let E be a martingale of type r (1<r≤ 2) Banach space.
In this section, we will study  Itô's formula for the processes of the following type:

    X_t=X_0+∫_0^t a(s) ṣ+∫_0^tg(s) Ẉ_s+∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ)+∫_0^t∫_Z η(s,z) N(ṣ,ẓ), t≥0.

Here a :_+ ×Ω→ E is an E-valued progressively measurable process on the space (_+×Ω,ℬ(_+)⊗ℱ) such that for all t≥0, ∫_0^ta(s,ω)ds<∞, ℙ-a.s., N is a Poisson random measure associated with a Poisson point process π,
 ξ∈ℳ^p_loc(𝒫̂,ν;E), η∈ℳ_loc(𝒫̂,N;E), and g=(g(t))_t∈[0,T] is a process in M([0,T];γ(H;E)). Assume that |ξ(t,ω,z)|_E|η(t,ω,z)|_E=0 for all (t,ω,z)∈_+×Ω× Z. Without loss of generality, we may assume that the process X is right-continuous with left-limits.
 
 
 In order to make the paper self-contained, we formulate the following auxiliary lemma.
(see <cit.>)
  Let E and F be separable Banach spaces and ϕ:E→ F be a Fréchet differentiable function whose  Fréchet  derivative ϕ^': E→ L(E;F) is locally α-Hölder continuous.  In other words, for all r>0, there exists H=H(r)<∞ such that
  
    sup_|x|_E,|y|_E≤ r,x≠ yϕ'(x)-ϕ'(y)_L(E;F)/|x-y|^α_E≤ H(r).

  Then we have that  for every x,y∈ E,
  
    ϕ(y)=ϕ(x)+ϕ^'(x)(y-x)+R_ϕ(x,y),

  where R_ϕ(x,y)=∫_0^1(ϕ^'(x+θ (y-x))(y-x)-ϕ^'(x)(y-x)) dθ.
   Moreover,  for every r>0, there exists C=C(r)>0 such that
   
    |R_ϕ(x,y)|_F≤ C|y-x|_E^α+1 |x|_E,|y|_E≤ r.



(Itô's  formula 1)  Assume that E is a martingale type 2 Banach space. Let X be a process given by (<ref>). Let ϕ:E→ E be twice Fréchet differentiable and its first and second derivative are all continuous. Then for every t>0, we have ℙ-a.s.

    ϕ(X_t)   =ϕ(X_0)+∫_0^t ϕ^'(X_s) (a(s)) ṣ+∫_0^tϕ^'(X_s)(g(s)) Ẉ_s+1/2∫_0^t tr ϕ^''(X_s)(g(s),g(s)) ṣ
       +  ∫_0^t∫_Z [ϕ( X_s-+η(s,z))-ϕ(X_s-)]  N(ṣ,ẓ)
       +∫_0^t∫_Z [ϕ( X_s-+ξ(s,z))-ϕ(X_s-)] Ñ(ṣ,ẓ)
       +∫_0^t∫_Z [ϕ( X_s-+ξ(s,z))-ϕ(X_s-)-ϕ^'(X_s-)(ξ(s,z))] ν(ẓ)ṣ.




(Itô's  formula 2)
Assume that E is a martingale type r Banach space, r∈(1,2].
Let ϕ:E→ E be a function of class C^1 such that the first derivative ϕ^':E→ L(E;E) is locally (r-1)-Hölder continuous. Let X be a non-Gaussian Lévy process given by

    X_t=X_0+∫_0^t a(s) ṣ+∫_0^t∫_Z ξ(s,z) Ñ(ṣ,ẓ)+∫_0^t∫_Z η(s,z) N(ṣ,ẓ), t≥0.

Then for every t>0, we have ℙ-a.s.

    ϕ(X_t)   =ϕ(X_0)+∫_0^t ϕ^'(X_s) (a(s)) ṣ+
           ∫_0^t∫_Z [ϕ( X_s-+η(s,z))-ϕ(X_s-)] N(ṣ,ẓ)
       +∫_0^t∫_Z [ϕ( X_s-+ξ(s,z))-ϕ(X_s-)] Ñ(ṣ,ẓ)
       +∫_0^t∫_Z [ϕ( X_s-+ξ(s,z))-ϕ(X_s-)-ϕ^'(X_s-)(ξ(s,z))] ν(ẓ)ṣ.








Without loss of generality, we may assume that the process X is bounded, namely, there exists r>0 such that

    sup_0≤ s≤ t|X_s|_E≤ r.

 Then we can relax the boundedness assumption (<ref>) by the usual localization argument.
        Since the Poisson point process π is σ-finite, there exists a sequence of sets {D_n}_n∈ℕ such that D_n⊂ D_n+1, ∪_n∈ℕD_n=Z, and N(t,D_n)<∞ for every 0<t<∞ and n∈ℕ. Define a sequence {ξ^n}_n∈ of functions by
       
    ξ^n(s,ω,z):=ξ(s,ω,z)1_D_n(z), (s,ω,z)∈ℝ_+×Ω× Z,  n∈.

 Since |ξ^n|_E≤ |ξ|_E and by the assumption, ξ∈ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E), we infer that ξ^n∈ℳ^r_T(𝒫⊗𝒵,ṭ×ℙ×ν;E). By the definition of stochastic integrals, we have
 
    ∫_0^t∫_Zξ^n(s,z) Ñ(ṣ,ẓ)   = ∫_0^t∫_Z1_D_nξ(s,z) Ñ(ṣ,ẓ)=∫_0^t∫_D_nξ(s,z) Ñ(ṣ,ẓ).

 Now applying Proposition <ref> yields that
        
    ∫_0^t∫_Zξ^n(s,z) Ñ(ṣ,ẓ)=
            ∑_s∈(0,t]∩𝒟(π)ξ(s,π(s))1_D_n(π(s))-∫_0^t∫_D_nξ(s,z) ν(ẓ)ṣ.

        Similarly, we define a sequence {g^n}_n∈ of functions by
               
    η^n(s,ω,z)=η(s,ω,z)1_D_n(z),   (s,ω,z)∈ℝ_+×Ω× Z, n∈.

   It follows that

    ∫_0^t∫_Zη^n(s,z)N(ṣ,ẓ)=∑_s∈(0,t]∩𝒟(π)η(s,π(s))1_D_n(π(s)).

     Since N(t,D_n)<∞, t≥0,
        we see that for almost every ω∈Ω, the set {s≤ t:π(s,ω)∈ D_n∩𝒟(π)} contains only finitely many points in each time interval (0,t], for t>0.
 Hence we may denote these points according to their magnitude by 0=τ_0(ω)<τ_1(ω)<τ_2(ω)<⋯<τ_m(ω)<⋯.  In other words, we put
 
    τ_0   =0;
    τ_m   =inf{s∈(0,t]∩𝒟(π): π(s)∈ D_n; s>τ_m-1}, m≥1.

         The random times τ_1,τ_2,… form a random configuration of points in (0,t] with π(τ_i)∈ D_n and for each m, the random time τ_m is a stopping time. Indeed, for every u>0, we find that
         
    {τ_m≤ u}={N(u,D_n)≥ m}∈ℱ_u.

             Let us define a sequence {X^n}_n∈ of process X^n:=(X^n_t)_t≥0 by
             
    X^n_t=X_0^t+∫_0^ta(s)ṣ+∫_0^tg(s)Ẉ_s+∫_0^t∫_Z ξ^n(s,z)Ñ(ṣ,ẓ)+∫_0^t∫_Z η^n(s,z)N(ṣ,ẓ), t≥0,  n∈ℕ.

It follows from (<ref>) and (<ref>) that for every n∈ and all t≥0, ℙ-a.s.
                
    X^n_t
                      
                         =X_0+∫_0^ta(s)ṣ+∑_s∈(0,t]∩𝒟(π)ξ(s,π(s))1_D_n(π(s))-∫_0^t∫_D_nξ(s,z)ν(ẓ)ṣ
       + ∑_s∈(0,t]∩𝒟(π)η(s,π(s))1_D_n(π(s))
       = X_0+∫_0^t a(s)ṣ+∫_0^tg(s)Ẉ_s-∫_0^t∫_Zξ^n(s,z)ν(ẓ)ṣ+∑_mξ^n(τ_m,π(τ_m),·)1_{τ_m≤ t}
       +∑_m η^n(τ_m,π(τ_m),·)1_{τ_m≤ t}.

    Note that
    
    ϕ(X_t^n)-ϕ(X_0)   =∑_m[ϕ(X^n_t∧τ_m)-ϕ(X^n_t∧τ_m-1) ]
       =∑_m[ϕ(X^n_t∧τ_m)-ϕ(X^n_t∧τ_m-) ]+∑_m[ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1) ]
       =: I_1+I_2.

 Here X^n_t∧τ_m=(X^n)_t^τ_m, t≥0 is the process X^n stopped at time τ_m, and X^n_t∧τ_m-=(X̅^n)_t^τ_m, t≥0 is the process X^n stopped strictly before time τ_m.  Namely,
               
    X^n_t∧τ_m(ω)=(X^n)_t^τ_m(ω)={[      X^n_t(ω)  if t≤τ_m(ω),; X^n_τ_m(ω)(ω)  if t≥τ_m(ω). ].

                        and
               
    X^n_t∧τ_m-(ω)=   (X̅^n)_t^τ_m(ω)={[       X^n_t(ω)   if t< τ_m(ω); X^n_τ_m(ω)-(ω)   if t≥τ_m(ω). ].

 Note that the jumps of X^n occur only at times {τ_m}. So X^n_t∧τ_m≠ X^n_t∧τ_m- if and only if 
    ξ^n(τ_m,π(τ_m),·)1_{τ_m≤ t}+ η^n(τ_m,π(τ_m),·)1_{τ_m≤ t}≠ 0.
  Since by assumption |f|_E·|g|_E=0, we infer that X^n_t∧τ_m≠ X^n_t∧τ_m- if and only if
             
    ξ^n(τ_m,π(τ_m),·)1_{τ_m≤ t}≠ 0 and η^n(τ_m,π(τ_m),·)1_{τ_m≤ t}=0

             or
                          
    ξ^n(τ_m,π(τ_m),·)1_{τ_m≤ t}= 0 and η^n(τ_m,π(τ_m),·)1_{τ_m≤ t}≠ 0

             Hence
             
    X^n_t∧τ_m   = X^n_t∧τ_m-+ξ^n(τ_m,π(τ_m))1_{τ_m≤ t}+ η^n(τ_m,π(τ_m))1_{τ_m≤ t}
         =
                       {[     X^n_t∧τ_m-+ ξ^n(τ_m,π(τ_m))1_{τ_m≤ t} if ξ^n(τ_m,π(τ_m))≠ 0, η^n(τ_m,π(τ_m))=0,;     X^n_t∧τ_m-+ η^n(τ_m,π(τ_m))1_{τ_m≤ t} if ξ^n(τ_m,π(τ_m))= 0, η^n(τ_m,π(τ_m))≠0. ].

It follows that
             
    I_1   =∑_m[ϕ(X^n_t∧τ_m)-ϕ(X^n_t∧τ_m-) ]
        =∑_m[ϕ(X^n_t∧τ_m)-ϕ(X^n_t∧τ_m-) ]1_{ξ(τ_m,π(τ_m))≠0}∩{η(τ_m,π(τ_m))=0}1_{τ_m≤ t}
       + ∑_m[ϕ(X^n_t∧τ_m)-ϕ(X^n_t∧τ_m-) ]1_{ξ(τ_m,π(τ_m))=0}∩{η(τ_m,π(τ_m))≠0}1_{τ_m≤ t}
        =∑_m[ϕ(X^n_t∧τ_m-+ ξ^n(τ_m,π(τ_m)))-ϕ(X^n_t∧τ_m-) ]1_{ξ(τ_m,π(τ_m))≠0}∩{η(τ_m,π(τ_m))=0}1_{τ_m≤ t}
       + ∑_m[ϕ(X^n_t∧τ_m-+ η^n(τ_m,π(τ_m)))-ϕ(X^n_t∧τ_m-) ]1_{ξ(τ_m,π(τ_m))=0}∩{η(τ_m,π(τ_m))≠0}1_{τ_m≤ t}
       =∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)    ]N(ds,dz)
       +∫_0^t∫_Z[ϕ(X^n_s-+η^n(s,z,ω)) -ϕ(X^n_s-)    ]N(ds,dz)
       =∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)    ]Ñ(ds,dz)
       +∫_0^t∫_Z[ϕ(X^n_s-+η^n(s,z,ω)) -ϕ(X^n_s-)    ]N(ds,dz)
        +∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)    ]ν(dz)ds


    Notice that there are no jumps of X in the random time interval (t∧τ_m-1,t∧τ_m), in other words X contains only the continuous components. Set for u∈[0,T],
		                 
    Y^n,m_t
    			  =X^n_τ_m-1+∫_0^t1_[τ_m-1,T][a(s)-∫_Zξ^n(s,z) ν(ẓ)]ṣ+∫_0^t
    			 1_[τ_m-1,T] g(s) Ẉ_s.

		                 Then we have Y^n,m_t=X^n_τ_m-1 for t∈[0,τ_m-1], Y_t^n,m=X_t^n for t∈[τ_m-1,τ_m), and Y_τ_m^n,m=Y_τ_m-^n,m=X_τ_m-.		                 By applying the Itô formula of Gaussian processes in <cit.> to the process Y^n,m_u, we obtain
		                                 
    ϕ(Y^n,m_t∧τ_m)=   ϕ(X^n_τ_m-1)-∫_τ_m-1^t∧τ_mϕ^'(Y^n,m_s)(a(s))ṣ-∫_τ_m-1^t∧τ_mϕ^'(Y^n,m_s)(ξ^n(s,z))ν(ẓ)ṣ
       +∫_τ_m-1^t∧τ_mϕ^'(Y^n,m_s)(g(s))Ẉ_s
        +1/2∫_τ_m-1^t∧τ_mtr ϕ^''(Y^n,m_s)(g(s),g(s))ṣ,  for t∈[τ_m-1,τ_m]  ℙ-a.s.

     Moreover, when t<τ_m-1, we have 
    ϕ(Y^n,m_t∧τ_m)-ϕ(X^n_τ_m-1)=0=ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1);
  when τ_m-1≤ t<τ_m, 
    ϕ(Y^n,m_t∧τ_m)-ϕ(X^n_τ_m-1)=ϕ(Y^n,m_t)-ϕ(X^n_τ_m-1)=ϕ(X^n_t)-ϕ(X^n_τ_m-1)=ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1);

     when t≥τ_m, we have
     
    ϕ(Y^n,m_t∧τ_m)-ϕ(X^n_τ_m-1)=ϕ(Y_τ_m^n,m)-ϕ(X^n_τ_m-1)=ϕ(X^n_τ_m-)-ϕ(X^n_τ_m-1)=ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1).

     Hence,
		                      
    ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1)=   ∫_τ_m-1^t∧τ_mϕ^'(X^n_s)(a(s))ṣ-∫_τ_m-1^t∧τ_mϕ^'(X^n_s)(ξ^n(s,z))ν(ẓ)ṣ
       +∫_τ_m-1^t∧τ_mϕ^'(X^n_s)(g(s))Ẉ_s
        +1/2∫_τ_m-1^t∧τ_mtr ϕ^''(X^n_s)(g(s),g(s))ṣ.

     Therefore, we have
     
    I_2= ∑_m[ϕ(X^n_t∧τ_m-)-ϕ(X^n_t∧τ_m-1) ]=   ∫_0^tϕ^'(X^n_s)(a(s)) ṣ-∫_0^tϕ^'(X^n_s)(ξ^n(s,z)) ν(ẓ)ṣ
       +∫_0^tϕ^'(X^n_s)(g(s)) Ẉ_s
        +1/2∫_0^t tr ϕ^''(X^n_s)(g(s),g(s)) ṣ.




 
                         Combining I_1 and I_2 yields that
                  
    ϕ(X_t^n)-ϕ(X_0)
                          =∫_0^t ϕ^'(X^n_s)(a(s)) ds+∫_0^tϕ^'(X^n_s)(g(s)) Ẉ_s
        +1/2∫_0^t tr ϕ^''(X^n_s)(g(s),g(s)) ṣ
       +∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)    ]Ñ(ds,dz)
       +∫_0^t∫_Z[ϕ(X^n_s-+η^n(s,z,ω)) -ϕ(X^n_s-)    ]N(ds,dz)
        +∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)- ϕ^'(X^n_s-)(ξ^n(s,z))    ]ν(dz)ds.

                 This shows that Itô formula (<ref>) holds for the process X^n.
    Now let us consider the general case. On the basis of the inequality (<ref>)  and the Lebesgue dominated convergence theorem, we infer
    
    lim_n→∞| ∫_0^t∫_Z ξ^n(s,z)Ñ(ds,dz)-∫_0^t∫_Z ξ(s,z)Ñ(ds,dz)   |_E^2
       ≤ Clim_n→∞∫_0^t∫_Z |ξ^n(s,z)-ξ(s,z)|_E^2 ν(dz)ds=0.

   This allows  us to find a subsequence such that ∫_0^t∫_Z ξ^n(s,z)Ñ(ds,dz) uniformly converges to ∫_0^t∫_Z ξ(s,z)Ñ(ds,dz) on any finite interval [0,T] ℙ-a.s. Similarly, we can prove that ∫_0^t∫_Z η^n(s,ω,z) N(ds,dz) uniformly converges to ∫_0^t∫_Z η(s,ω,z) N(ds,dz) on [0,T] ℙ-a.s. as well. Hence we infer that X^n_s converges uniformly to X_s as n→∞, ℙ-a.s. on [0,T]. Also, X^n_s- converges uniformly to X_s- ℙ-a.s. on [0,T] as n→∞. Hence by the continuity of ϕ, ϕ^', and ϕ^'', we infer that
   
    ∫_0^tϕ^'(X_s^n)(a(s)) ṣ→∫_0^tϕ^'(X_s)(a(s)) ṣ,ℙ-a.s.,
       ∫_0^tϕ^'(X_s^n)(g(s)) Ẉ_s→∫_0^tϕ^'(X_s)(g(s)) Ẉ_sin L_2,
       ∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)    ]Ñ(ds,dz)
       ⟶∫_0^t∫_Z[ϕ(X_s-+ξ(s,z,ω)) -ϕ(X_s-)    ]Ñ(ds,dz)in L_2,
       ∫_0^t∫_Z[ϕ(X^n_s-+η^n(s,z,ω)) -ϕ(X^n_s-)    ]N(ds,dz)
       ⟶∫_0^t∫_Z[ϕ(X_s-+η(s,z,ω)) -ϕ(X_s-)    ]N(ds,dz),ℙ-a.s.
       ∫_0^t∫_Z[ϕ(X^n_s-+ξ^n(s,z,ω)) -ϕ(X^n_s-)- ϕ^'(X^n_s-)(ξ^n(s,z))    ]ν(dz)ds,
    
       ⟶∫_0^t∫_Z[ϕ(X_s-+ξ(s,z,ω)) -ϕ(X_s-)- ϕ^'(X_s-)(ξ(s,z))    ]ν(dz)ds,ℙ-a.s.

   and
   
    1/2∫_0^t tr ϕ^''(X^n_s)(g(s),g(s)) ṣ→1/2∫_0^t tr ϕ^''(X_s)(g(s),g(s)) ṣℙ-a.s.

  Hence the Itô formula follows by passing the limit in the equality (<ref>) along a subsequence.
   




The proofs are almost identical to the one given in the proof of Theorem <ref>, the major change being the
proof of the term I_2. Since ϕ^' is (r-1)-Hölder continuous, by applying Lemma <ref>,  routine computations give rise to

    I_2=∑_m[ϕ(X^n_t∧τ_m-)-ψ(X^n_t∧τ_m-1) ]
    		  		=∫_0^tϕ^'(X_s)(a_s) ṣ-∫_0^t∫_Zξ^n(s,z) ν(ẓ)ṣ.





§ ACKNOWLEDGMENT

 The authors would like to thank the referees for their very careful reading of the manuscript and especially for their very valuable suggestions and constructive comments.




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