The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes.

Running maximumdensity functionsMalliavin calculusstochastic differential equationsLévy processes60H1060G5260H07KAKENHI23K12507The second author was supported by JSPS KAKENHI Grant Number 23K12507. Introduction

We consider a solution of the following one-dimensional SDE
dXt=b(Xt)dt+σ1dWt+σ2dLt,X0=x∈R,
for t≥0, where σ1 and σ2 are constants, b:R→R is differentiable and its derivative is bounded, W={Wt}t∈[0,T] is a standard Brownian motion and L={Lt}t∈[0,T] is a Lévy process with the Lévy triplet (0,0,ν). The infinitesimal generator A of L is defined by
Af(x):=∫R∖{0}{f(x+y)−f(x)−1{|y|<1}yf′(x)}ν(dy)
for any f∈Cb2(R) and x∈R. See, e.g., equation (3.18) in [1]. The Lévy measure ν satisfies assumptions (2.1) and (2.2). We assume that W and L are independent. In considering SDE (1.1), we introduce the following SDE for each n∈N:
dXt(n)=b(Xt(n))dt+σ1dWt+σ2dLt(n),X0(n)=x∈R,
for t∈[0,T], where L(n)={Lt(n)}t∈[0,T] is a truncated pure-jump process L with jump sizes larger than n. Let X∗:={Xt∗}t∈[0,T] and X(∗,n):={Xt(∗,n)}t∈[0,T], defined as
Xt∗=supt∈[0,t]Xs,Xt(∗,n)=sups∈[0,t]Xs(n)for eacht∈[0,T]andn∈N,
be the running maximums of the solutions X and X(n) to SDE (1.1) and (1.2), respectively. It is well known that when b is Lipschitz continuous, SDEs (1.1) and (1.2) have a unique solution (e.g., see [12]). The purpose of this paper is to show that the distribution of Xt∗ is absolutely continuous with respect to the Lebesgue measure on R for all t>0. The running maximum process has received widespread attention in recent years as an interesting objectboth practically and theoretically (cf. [6, 3]). The following results for the special cases of the law of X∗ are known. The density function of the maximum of Brownian motion (i.e. x=b=σ2=0 and σ1=1) is well known. See, e.g., [7]. The law of the maximum of Lévy motion (i.e. x=b=σ1=0 and σ2=1) is also well known. See, for example, [4, 10].

The following prior studies are based on the simultaneous dealing of Brownian motion and truncation Lévy processes. If L is a compound Poisson process, Coutin et al. [5] consider a joint density of (Xt∗,Xt). Song and Zhang [15] study the existence of distributional density of Xt and the weak continuity in the first variable of the distributional density under full Hörmander’s conditions. This proof is given by showing the statement for Xt(1). Song and Xie [14] show the existence of density functions for the running maximum Xt(∗,1) of a Lévy–Itô diffusion. They claimed that if b is Lipschitz continuous in Lemma 4.3 of [14], they can prove the existence of the density function of Xt(∗,1). However, we cannot follow them because the product of weakly convergent sequences does not necessarily converge to the product of their limits. These [15, 14] are proved similarly if jump size n is a finite value. However, to the best of our knowledge, the results of the nontruncated Lévy process are not known. This is since Bismut’s approach to the Malliavin calculus for jump processes in [2] can simply calculate the concrete form only for finite jumps by using Proposition 2.11 of [15]. In this paper, we show the existence of a density function for X∗ using the proof method of [14] and the fact that the Malliavin calculus for L can be defined by the limit of that of L(n).

The structure of this paper is as follows. In Section 2, we introduce the notations employed throughout this paper and present our main theorem. Section 3 revisits Bismut’s approach to the Malliavin calculus with jumps. In Section 4, we discuss the results of Song and Xie [14] and extend their results. Section 5 is dedicated to applying the outcomes derived in the preceding section to our stochastic differential equations. Our primary contribution, Theorem 2.1, is proven in Section 6. Lastly, Section A offers several lemmas essential for the proof of the our main results.

Notations and a result

Let L={Lt}0≤t≤T and L(n)={Lt(n)}t∈[0,T] be a pure-jump process and the one truncated by [−n,n]∖{0}, respectively. The jump size of L and L(n) at time t is defined by ΔLt=Lt−Lt− and ΔLt(n):=Lt(n)−Lt−(n) for any t>0 and ΔL0:=0 and ΔL0(n):=0. The Poisson random measures associated with L and L(n) on B([0,T])×B(R∖{0}) are denoted by N(t,F)=∑0≤s≤t1F(ΔLs) and N(n)(t,G)=∑0≤s≤t1F(ΔLs(n)) for t∈[0,T] and F∈B(R∖{0}), respectively. The Lévy measures of L and L(n) on B(R∖{0}) are defined as ν(dz)=c(z)dz and c(z)1{|z|≤n}(z)dz, where the positive function c satisfies the following requirements: there exist some constants β>1 and C>0 such that ∫R∖{0}(1∧|z|2)ν(dz)<∞,limn→∞∫R∖{0}|z|p1{|z|>n}(z)ν(dz)=0for anyp∈(1,β),supn∈N|∫1≤|z|≤nzν(dz)|<∞and∫0+ν(dz)=∞,|c′(z)c(z)|≤C(1∨1|z|)for anyz≠0. The compensated Poisson random measures of L and L(n) are defined as N˜ and N˜(n), respectively.

If L is a symmetric α-stable process with α∈(1,2), then for any z≠0c(z)=cα|z|1+α,wherecα=π−1Γ(α+1)sin(απ2),
so that a Lévy measure ν satisfies assumptions (2.1) and (2.2) for any p∈(1,α).

Our results are described below.

Assume thatb:R→Ris once differentiable and its derivative is bounded, and that a Lévy measure ν of L satisfies (2.1) and (2.2). Let{Xt}t∈[0,T]be the solution to equation (1.1). Ifσ12+σ22≠0, then for anyT>0the law ofXT∗is absolutely continuous with respect to the Lebesgue measure.

We will prove this result in Section 6. To prepare for that proof, we introduce the Malliavin calculus.

Bismut’s approach to the Malliavin calculus with jumps

This section provides a brief overview of Bismut’s approach in the context of Malliavin calculus for jump processes (cf. [2, 14, 15], etc.). Consider an open set Γ⊂Rd containing the origin. We define
Γ0:=Γ∖{0},ϱ(z):=1∨d(z,Γ0c)−1,
where d(z,Γ0c) is the distance of z to the complement of Γ0. Let Ω denote the canonical space consisting of all pairs ω=(w,μ), where

w:[0,1]→Rd is a continuous function satisfying w(0)=0;

μ represents an integer-valued measure on [0,1]×Γ0 such that μ(A)<+∞ for any compact subset A⊂[0,1]×Γ0.

Let us define the canonical process on Ω by setting for ω=(w,μ):
Wt(ω):=w(t),N(ω;dt,dz):=μ(ω;dt,dz):=μ(dt,dz).
We consider the smallest right-continuous filtration (Ft)t∈[0,1] on Ω ensuring that both W and N are optional processes. Throughout our discussion, we set F:=F1. The space (Ω,F) is equipped with a unique probability measure P satisfying the following conditions:

W is a standard d-dimensional Brownian motion;

N is a Poisson random measure with intensity dtν(dz), where ν(dz)=κ(z)dz with
κ∈C1(Γ0;(0,∞)),∫Γ0(1∧|z|2)κ(z)dz<+∞,|∇logκ(z)|≤Cϱ(z),
where ϱ(z) is defined by equation (3.1);

W and N are independent.

We denote the compensated Poisson random measure N by
N˜(dt,dz):=N(dt,dz)−ν(dz)dt.
Let p≥1, i∈{1,2}, and let k be an integer. We introduce the following spaces for subsequent discussions.

We denote Lp(Ω) as the space of all F-measurable random variables for which the norm represented by
‖F‖p:=E[|F|p]1p
is finite.

Let Lpi be the space of all predictable processes ξ:Ω×[0,1]×Γ0→Rk with finite norm
‖ξ‖Lpi:=E[(∫01∫Γ0|ξ(s,z)|iν(dz)ds)pi]1p+E[∫01∫Γ0|ξ(s,z)|pν(dz)ds]1p<∞.

We introduce Hp as the set of all measurable adapted processes h:Ω×[0,1]→Rd that possess a finite norm defined by
‖h‖Hp:=E[(∫01|h(s)|2ds)p2]1p<∞.

Consider the space Vp of all predictable processes v:Ω×[0,1]×Γ0→Rd that satisfy the finite norm condition
‖v‖Vp:=‖∇zv‖Lp1+‖vϱ‖Lp1<∞,
where ϱ(z) is defined by equation (3.1). For later discussions, we will use the notations
H∞−:=⋂p≥1Hp,V∞−:=⋂p≥1Vp.

The space H0 encompasses all bounded, measurable, and adapted processes h:Ω×[0,1]→Rd.

The space V0 is constituted of all predictable processes v:Ω×[0,1]×Γ0→Rd satisfying the following conditions:

both v and ∇zv are bounded;

there exists a compact subset U⊂Γ0 such that
v(t,z)=0,∀z∈U.

Let Cp∞(Rm) denote the set of smooth functions on Rm for which all derivatives exhibit at most polynomial growth. Define the collection of Wiener–Poisson functionals on Ω given by
F(ω)=f(W(h1),…,W(hm1),N(g1),…,N(gm2)),
where f belongs to Cp∞(Rm1+m2), h1,…,hm1 are elements of H0, and g1,…,gm2 are in V0, with all of them being nonrandom and real-valued. Additionally, for each j in the range 1≤j≤m1 and each k in the range 1≤k≤m2, we define
W(hj):=∫01⟨hj(s),dWs⟩RdandN(gk):=∫01∫Γ0gk(s,z)N(ds,dz).
Given any p>1 and Θ=(h,v)∈Hp×Vp, we denote
DΘF:=∑i=1m1(∂if)(·)∫01⟨h(s),hi⟩Rdds+∑j=1m2(∂j+m1f)(·)∫01∫Γ0⟨v(s,z),∇zgj⟩RdN(ds,dz),
where “(·)” represents the collection W(h1),…,W(hm1),N(g1),…,N(gm2).

Given p>1 and Θ=(h,v)∈Hp×Vp, we introduce the first-order Sobolev space WΘ1,p as the completion of FCp in Lp(Ω) with respect to the norm
‖F‖Θ;1,p:=‖F‖Lp+‖DΘF‖Lp.

It is well known that the Banach space WΘ1,p possesses weak compactness, a crucial property for the proof of Theorem 2.1 (see Lemma 2.3 in [14]). We next present the results obtained by applying the Malliavin calculus developed above to the running maximum processes.

Regularity of the running maximum processes

In this section we discuss the results of Song and Xie [14] and their extensions. Let X(n)={Xs(n)}s≥0 be a right continuous real-valued process. For any fixed T>0 and n∈N, in the following we shall write
XT(∗,n):=sups∈[0,T]Xs(n),XT∗:=sups∈[0,T]Xs.

LetX(n)={Xs(n)}s≥0andX={Xs}s≥0be a right continuous process for eachn∈N. Suppose that for somep>1andΘ=(h,v)∈H∞−×V∞−:

supn∈NE[|Xs(∗,n)|p]<∞, and for anys∈[0,T],Xs(n)∈WΘ1,p, andsupn∈NE[sups∈[0,T]|DΘXs(n)|p]<∞;

the process{DΘXs(n)}s∈[0,T]possesses a right continuous version for eachn∈N;

limn→∞E[sups∈[0,T]|Xs(n)−Xs|p]=0.

ThenXT∗∈Wθ1,pand the sequence{DΘXs(n)}s∈[0,T]converges toDΘX={DΘXs}s∈[0,T]in the weak topology ofLp(Ω×[0,T]). Moreover, if thisDΘXhas a right continuous version andP(DΘXt≠0on{t∈(0,T]:Xt=XT∗})=1,then the law ofXT∗is absolutely continuous with respect to the Lebesgue measure.

It can be seen that XT(∗,n)∈Wθ1,p follows from Proposition 3.1 in [14] for each n∈N. From Lemma 2.3 in [14], we obtain XT∗∈Wθ1,p and
limn→∞DΘX·(n)=DΘX·weakly inLp(Ω×[0,T]).
In exactly the same way as in Theorem 3.2 in [14], the following equality follows if DΘX has a right continuous path almost surely:
1=P({∃t∈[0,T]such thatDΘXt≠DΘXT∗andXt=XT∗}c)=P(DΘXt=DΘXT∗on{t∈[0,T]:Xt=Xt∗})≤P(DΘXt=DΘXT∗on{t∈(0,T]:Xt=Xt∗})=1.
Subsequently, we prove that XT∗ has a density function if (4.1) holds. In addition, by the closability of DΘ (see Theorem 2.6 in [14]), we obtain
P(1A(DΘXT∗)DΘXT∗=0)=1,
for any A∈B(R) with Leb(A)=0. With these facts and (4.1) we obtain the following equation:
1=P({1A(DΘXT∗)DΘXT∗=0}∩{DΘXt=DΘXT∗on{t∈(0,T]:Xt=Xt∗}}∩{DΘXt≠0on{t∈(0,T]:Xt=Xt∗}})=P({1A(DΘXT∗)DΘXt=0on{t∈(0,T]:Xt=Xt∗}}∩{DΘXt=DΘXT∗on{t∈(0,T]:Xt=Xt∗}}∩{DΘXt≠0on{t∈(0,T]:Xt=Xt∗}})=P({1A(DΘXT∗)=0}∩{DΘXt=DΘXT∗on{t∈(0,T]:Xt=Xt∗}}∩{DΘXt≠0on{t∈(0,T]:Xt=Xt∗}})≤P(1A(DΘXT∗)=0)=1.
Therefore, this lemma is completed. □

Now we know the relationship between the Malliavin calculus of the running maximum processes and the existence of the density function. Next, we note the results of applying of the Malliavin calculus to the SDE (1.1).

Applying of Malliavin calculus to SDEs

In this section, to find an equation satisfied by DΘX for X in equation (1.1), we check an equation satisfied by DΘX(n) for X(n) in equation (1.2). The following lemma is shown in the same way as for Lemma 4.3 in [14].

Assume thatb:R→Ris once differentiable and its derivative is bounded. Then for anyΘ=(h,v)∈H∞−×V∞−andt∈[0,T],Xt(n)∈WΘ1,2andDΘXt(n)=∫0tb′(Xs(n))DΘXs(n)ds+σ1∫0th(s)ds+σ2∫0t∫0<|z|≤nv(s,z)N(ds,dz).

The following lemma defines DΘX and confirms that it satisfies (5.1) below.

Assume the same assumptions as in Lemma5.1. Then for somep∈(1,β), for somen∈N, for anyq∈(1,β)and for anyΘ=(h,v)∈H∞−×V∞−, wherelimn→∞∫0T∫|z|>n|v(s,z)|qβdsν(dz)=0and∫0T∫|z|>n|v(s,z)|qdsν(dz)<∞,Xt∗∈WΘ1,pfor anyt∈[0,T], andDΘXt=∫0tb′(Xs)DΘXsds+σ1∫0th(s)ds+σ2∫0t∫|z|>0v(s,z)N(ds,dz).Then thisDΘXis the limit of weakLp(Ω×[0,T])convergence of the sequenceDΘX(n), and this sequence is stronglyLp(Ω×[0,T])convergent in practice.

It can be seen immediately from Lemma 5.1 that Assumptions 1 and 2 of Lemma 4.1 are satisfied. Note that the Lp integrability of DΘX(n) in Assumption 1 shall be checked later. See Lemma A.5 for the fact that Assumption 3 is satisfied. By using Lemma 2.3 in [14], Lemma A.5 and the closability of DΘ (cf. Lemma 2.7 in [15]), we have
limn→∞DΘX·(n)=DΘX·weakly inLp(Ω×[0,T]).
We verify that this DΘX={DΘXt}t∈[0,T] satisfies equation (5.1). We set {Yt}t∈[0,T] as a solution of
Yt=∫0tb′(Xs)Ysds+Ct,whereCt=σ1∫0th(s)ds+σ2∫0t∫|z|>0v(s,z)N(ds,dz),Ct(n)=σ1∫0th(s)ds+σ2∫0t∫0<|z|≤nv(s,z)N(ds,dz).
We prove
limn→∞E[supt∈[0,T]|DΘXt(n)−Yt|p]=0.
By using an inequality |a+b|p≤2p−1(|a|p+|b|p) for any a,b∈R, we obtain
E[supt∈[0,T]|DΘXt(n)−Yt|p]≤2p−1E[supt∈[0,T]|∫0t{b′(Xs(n))DΘXs(n)−b′(Xs)Ys}ds|p]+2p−1E[supt∈[0,T]|Ct(n)−Ct|p]≤22(p−1)∫0TE[|b′(Xs(n))−b′(Xs)|p|DΘXs(n)|p]ds+22(p−1)‖b′‖∞p∫0TE[|DΘXs(n)−Ys|p]ds+2p−1E[supt∈[0,T]|Ct(n)−Ct|p].
The last and last second inequalities in the last chain follow from Jensen’s inequality and Fubini’s theorem. By using Gronwall’s inequality, we have
E[supt∈[0,T]|DΘXt(n)−Yt|p]≤22(p−1)exp(22(p−1)‖b′‖∞p)∫0TE[|b′(Xs(n))−b′(Xs)|p|DΘXs(n)|p]ds+22(p−1)exp(22(p−1)‖b′‖∞p)E[supt∈[0,T]|Ct(n)−Ct|p].
We show limn→∞∫0TE[|b′(Xs(n))−b′(Xs)|p|DΘXs(n)|p]ds=0,limn→∞E[supt∈[0,T]|Ct(n)−Ct|p]=0. See Lemma A.6 for proof of (5.4). Here we show equation (5.3). Notice that p∈(1,β), there exists q>1 such that pq<β because of the denseness of rational numbers. By using the Hölder inequality, we have
∫0TE[|b′(Xs(n))−b′(Xs)|p|DΘXs(n)|p]ds≤∫0TE[|b′(Xs(n))−b′(Xs)|pqq−1]q−1qE[|DΘXs(n)|pq]1qds≤TE[supt∈[0,T]|b′(Xt(n))−b′(Xt)|pqq−1]q−1qE[supt∈[0,T]|DΘXt(n)|pq]1q.
Due to an inequality |a+b|p≤2p−1(|a|p+|b|p) for any a,b∈R and p≥1 and Jensen’s inequality, we have
E[supt∈[0,T]|DΘXt(n)|pq]≤2pq−1E[supt∈[0,T]|∫0tb′(Xs(n))DΘXs(n)ds|pq]+22(pq−1)E[supt∈[0,T]|∫0th(s)ds|pq]+22(pq−1)E[supt∈[0,T]|∫0t∫0<|z|≤nv(s,z)N(ds,dz)|pq]≤2pq−1‖b′‖∞pq∫0TE[supu∈[0,s]|DΘXu(n)|pq]ds+22(pq−1)(E[∫0T|h(s)|pqds]+E[supt∈[0,T]|∫0t∫0<|z|≤nv(s,z)N(ds,dz)|pq]).
Gronwall’s inequality implies
E[supt∈[0,T]|DΘXt(n)|pq]≤22(pq−1)(E[∫0T|h(s)|pqds]+E[supt∈[0,T]|∫0t∫0<|z|≤nv(s,z)N(ds,dz)|pq])e2pq−1T‖b′‖∞pq.
The boundedness of the mean of sup with respect to time can be proved as in Lemma A.5 (ii). Due to assumptions on h and v, we have
supn∈NE[supt∈[0,T]|DΘXt(n)|pq]<∞.
This allows us to confirm the Lp(Ω×[0,T]) integrability of DΘX(n) for assumption 1 in Lemma 4.1, and that DΘX can be defined as the limit of weak Lp(Ω×[0,T]) convergence of DΘX(n). By Lemma A.5, boundedness of b′ and continuous mapping theorem, we have
limn→∞E[supt∈[0,T]|b′(Xt(n))−b′(Xt)|pqq−1]q−1q=0,
so that we obtain (5.3). Thus, by (5.2) and completeness of Lp(Ω×[0,T]), (5.1) follows. □

From this proof, we see that X(n) and X satisfy the assumptions of Lemma 4.1. The proof of Theorem 2.1 is now ready to be presented.

Proof of Theorem <xref rid="j_vmsta245_stat_002">2.1</xref>

Applying the Itô formula to e−∫0tb′(Xs)dsDΘXt (e.g., see [12], Corollary (Integration by Parts), P. 84), we obtain
e−∫0tb′(Xs)dsDΘXt=∫0+te−∫0sb′(Xu)du∘dDΘXs−+∫0+tDΘXs−∘de−∫0sb′(Xu)du=∫0te−∫0sb′(Xu)dudDΘXs−+∫0tDΘXs−de−∫0sb′(Xu)du=∫0te−∫0sb′(Xu)du(b′(Xs)DΘXs+σ1h(s))ds+∫0te−∫0sb′(Xu)duσ2∫|z|>0v(s,z)N(ds,dz)+∫0tDΘXs−(−b′(Xs)∫0te−∫0sb′(Xu)du)ds.
For any t>0, we set
h(t):=σ1e−∫0tb′(Xs)ds,v(t,z):=σ2e−∫0tb′(Xs)dsη(z),η(z)=|z|2,|z|≤14,0,|z|>12,smooth,otherwise.
Since the function v is bounded, it satisfies the assumptions of Lemma 5.2. Hence, of course, X satisfies the assumptions of Lemma 4.1. Substituting these, we have
DΘXt=e∫0tb′(Xs)ds(σ12∫0te−2∫0sb′(Xu)duds+σ22∫0t∫|z|>0e−2∫0sb′(Xu)duη(z)N(ds,dz))≥e∫0t(b′(Xs)−2‖b‖Lip)ds(σ12t+σ22∫0t∫|z|>0η(z)N(ds,dz)).
Noticing the condition σ12+σ22≠0 and the fact
P(∫0t∫|z|>0η(z)N(ds,dz)>0,∀t>0)=1,
we have
P(DΘXt>0,∀t∈(0,T])=1.
See Section A.1 for a proof of equation (6.2). So we have
1=P(DΘXt>0,∀t∈(0,T])≤P(DΘXt≠0on{t∈(0,T]:Xt=XT∗})=1.
Therefore, we conclude by Lemma 4.1 that the law of XT∗ is absolutely continuous with respect to the Lebesgue measure. □

Conflict of interest

The authors have no relevant financial or nonfinancial interests to disclose.

Appendices

We give some lemmas to show Theorem 2.1. In Subsection A.1, we prove equation (6.2). In Subsection A.3, we confirm equation (6.2) and in Subsection A.2 we provide some results useful for Subsection A.3.

A proof of equation (<xref rid="j_vmsta245_eq_047">6.2</xref>)

In this section, we prove equation (6.2). We set t>0, η as (6.1) and εk=12k for any k∈N. Since
limk→∞∫0t∫|z|>εkη(z)N(ds,dz)=∫0t∫|z|>0η(z)N(ds,dz)inL2(Ω),limk→∞∫0t∫|z|>εkη(z)N(ds,dz)=∫0t∫|z|>0η(z)N(ds,dz)in distribution.
Noting that the support of η, for any t>0, we have
P(∫0t∫|z|>0η(z)N(ds,dz)=0)=limk→∞P(∫0t∫|z|>εkη(z)N(ds,dz)=0)=limk→∞P(∫|z|>εkη(z)N(t,dz)=0)≤limk→∞P(N(t,(εk,12]∪(−12,−εk])=0).
Here, since for any A∈B(R)∖{0}, {N(t,A)}t≥0 is a Poisson process with intensity ν(A) (e.g., see [1], Th. 2.3.5), we have
limk→∞P(N(t,(εk,12]∪(−12,−εk])=0)=limk→∞exp(−tν((εk,12]∪(−12,−εk]))=0.
The last equation follows from assumption (2.2). We set for each t>0,
It=∫0t∫|z|>0η(z)N(ds,dz);
from countable additivity we have
P(⋃t∈(0,∞)∩Q{It=0})≤∑t∈(0,∞)∩QP({It=0})=0(see, e.g., [16], 1.9(b)).
Thus we obtain
P(⋂t∈(0,∞)∩Q{It>0})=1.
Here, since η≥0, we obtain
P(⋂0≤s≤u{Is≤Iu})=1.
By denseness of rational numbers, we obtain the following:
P(∀t>0,It>0)≥P(⋂t∈(0,∞)∩Q{It>0}∩⋂0≤s≤u{Is≤Iu})=1.

Preparation for proof of convergence of <inline-formula id="j_vmsta245_ineq_196"><alternatives><mml:math>
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To prove Theorem 2.1, we apply a variation of the method introduced by Komatsu ([8], proof of Theorem 1) in order to prove the convergence of X(n). This technique has been used in [11].

Forε>0,δ>1andr∈(0,1], we can choose a smooth functionψδ,εwhich satisfies the conditionsψδ,ε(x)=between0and2(xlogδ)−1,εδ−1<x<ε,0,otherwise,and∫εδ−1εψδ,ε(y)dy=1. We defineur(x)=|x|randur,δ,ε=ur∗ψδ,ε. Then,ur,δ,ε∈C2and for anyx∈R,|x|r≤εr+ur,δ,ε(x),ur,δ,ε(x)≤|x|r+εr.

We introduce a quasimartingale and its properties. Let T∈[0,∞] and Z be a càdlàg adapted process defined on [0,T]. A finite subdivision of [0,T] is defined by Δt=(t0,t1,…,tn+1) such that 0=t0<t1<⋯<tn+1=T.

The mean variation of X is defined by
VT(X):=supΔtE[∑i=0n|E[Xti−Xti+1|Fti]|].

A càdlàg adapted process Z is a quasimartingale on [0,T] if for each t∈[0,T], E[|Zt|]<∞ and VT(Z)<∞.

Kurtz [9] proved the following lemma by using Rao’s theorem ([12], Section III, Theorem 17). ([<xref ref-type="bibr" rid="j_vmsta245_ref_009">9</xref>], Lemma 5.3).

Let Z be a càdlàg adapted process defined on[0,T]. Suppose that for eacht∈[0,T],E[|Zt|]<∞andVt(Z)<∞. Then, for eachh>0,hP(supt∈[0,T]|Zt|>h)≤VT(Z)+E[|ZT|].

In this section, we prove several lemmas in order to give complete proof of Lemma 5.2. To that purpose, we show the following two statements.

We assume the same assumptions as in Theorem2.1. Thenlimn→∞E[supt∈[0,T]|Xt(n)−Xt|p]=0for anyp∈(1,β).This is because the following two conditions are valid.

supt∈[0,T]|Xt(n)−Xt|p→0in probability asn→∞.

The class of random variables{sups∈[0,t]|Xs(n)−Xs|p}t∈[0,T]is uniformly integrable.

(i) For clarity, we write r=pβ. By using the triangle inequality and Jensen’s inequality, we have
|Xt(n)−Xt|≤∫0t|b(Xs(n))−b(Xs)|ds+σ2|Lt(n)−Lt|.
By the definition of supremum, we have
supt∈[0,T]|Xt(n)−Xt|≤Tsupt∈[0,T]|b(Xt(n))−b(Xt)|+σ2supt∈[0,T]|Lt(n)−Lt|.
Since b is Lipschitz continuous, we have
supt∈[0,T]|Xt(n)−Xt|≤CTsupt∈[0,T]|Xt(n)−Xt|ds+σ2supt∈[0,T]|Lt(n)−Lt|.
By using Gronwall’s inequality and Jensen’s inequality, we have
supt∈[0,T]|Xt(n)−Xt|≤Cσ2exp(CT)supt∈[0,T]|Lt(n)−Lt|.
Here, by the above inequality and Lemma A.4, for any h>0,
P(supt∈[0,T]|Xt(n)−Xt|p>h)≤P(supt∈[0,T]|Lt(n)−Lt|r>(hCσ2exp(CT))1β)≤P(supt∈[0,T](εp+ur,δ,ε(Lt(n)−Lt))>(hCσ2exp(CT))1β)≤(Cσ2exp(CT)h)1β(VT(εp+ur,δ,ε(L(n)−L))+E[|εp+ur,δ,ε(LT(n)−LT)|]).
Here, by the definition of the mean variation and (A.2), we have
VT(εp+ur,δ,ε(L(n)−L))=VT(ur,δ,ε(L(n)−L)),E[|εp+ur,δ,ε(Lt(n)−Lt)|]=E[εp+ur,δ,ε(LT(n)−LT)].
By using the Lévy–Itô decomposition ([1], Theorem 2.4.16), we have
Lt(n)−Lt=∫0t∫|z|>1z1{0<|z|≤n}N(ds,dz)+∫0t∫0<|z|≤1z1{0<|z|≤n}N˜(ds,dz)−∫0t∫|z|>1zN(ds,dz)−∫0t∫0<|z|≤1zN˜(ds,dz),=−∫0t∫|z|>1z1{|z|>n}N(ds,dz)−∫0t∫0<|z|≤1z1{|z|>n}N˜(ds,dz),=−∫0t∫|z|>1z1{|z|>n}N(ds,dz).
The last equality follows by n≥1. Using the Itô formula ([1], Theorem 4.4.7), N(dt,dz)=N˜(dt,dz)+ν(dz)dt and the function ur,δ,ε defined in Lemma A.1, we have
ur,δ,ε(Lt(n)−Lt)=∫0t∫|z|≥1{ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)}N(ds,dz)=∫0t∫|z|≥1{ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)}N˜(ds,dz)+∫0t∫|z|≥1{ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)}dsν(dz)=:Mtδ,ε+Itδ,ε.
Here, by (A.1), for any x,y∈R,
−ur,δ,ε(y)≤εr−|y|r,ur,δ,ε(x)−ur,δ,ε(y)≤2εr+|x|r−|y|r≤2εr+||x|r−|y|r|≤2εr+|x−y|r.
So we have
ur,δ,ε(LT(n)−LT)≤∫0T∫|z|≥1{ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)}N˜(ds,dz)+∫0T∫|z|≥1{2εr+|z|r1{|z|>n}}dsν(dz).
Also,
∫0T∫|z|≥1{2εr+|z|r1{|z|>n}}dsν(dz)≤2CTεr+T∫R∖{0}|z|r1{|z|>n}ν(dz).
We can evaluate
VT(ur,δ,ε(L(n)−L))=supΔtE[∑i=0n|E[ur,δ,ε(Lti(n)−Lti)−ur,δ,ε(Lti+1(n)−Lti+1)∣Fti]|]=supΔtE[∑i=0n|E[Mtiδ,ε+Itiδ,ε−Mti+1δ,ε−Iti+1δ,ε∣Fti]|]=supΔtE[∑i=0n|E[Itiδ,ε−Iti+1δ,ε∣Fti]|].
The last equality is valid, since (Mtδ,ε)t∈[0,T] is a martingale. By using Jensen’s inequality, we have
VT(ur,δ,ε(L(n)−L))≤supΔtE[∑i=0nE[|Itiδ,ε−Iti+1δ,ε|∣Fti]]=supΔt∑i=0nE[|∫titi+1∫|z|≥1{ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)}dsν(dz)|]≤supΔt∑i=0nE[∫titi+1∫|z|≥1|ur,δ,ε(Ls−(n)−Ls−−z1{|z|>n})−ur,δ,ε(Ls−(n)−Ls−)|dsν(dz)]≤supΔt∑i=0nE[∫titi+1∫|z|≥1(2εr+|z|r1{|z|>n})dsν(dz)]≤∫0T∫|z|≥1(2εr+|z|r1{|z|>n})dsν(dz).
Therefore, we have
P(supt∈[0,T]|Xt(n)−Xt|p>h)≤(Cσ2exp(CT)h)1β{(4CT+1)εr+2T∫R∖{0}|z|r1{|z|>n}ν(dz)}.
Since the above inequality holds for any ε>0, we obtain
P(supt∈[0,T]|Xt(n)−Xt|p>h)≤2T(Cσ2exp(CT)h)1β∫R∖{0}|z|r1{|z|>n}ν(dz).
By the assumption, for any h>0, we have
limn→∞P(supt∈[0,T]|Xt(n)−Xt|p>h)=0.

(ii) Next, we show that the process
{supt∈[0,T]|Xt(n)−Xt|p}T≥0
is uniformly integrable. To show this, by using inequality (A.3) it suffices to show for some q>1,
E[(supt∈[0,T]|Lt(n)−Lt|p∨1)q]<∞.
By assumption of (2.1) and the denseness of rational numbers, we can set q>1 such that pq<β so that for each n∈N,
∫|z|>n|z|pqν(dz)<∞.
Here, we set g(x)=|x|pq∨1; then g is a nonnegative increasing submultiplicative function and limx→∞g(x)=∞. Using Theorem 25.18 in [13], we will show that
E[g(|Lt(n)−Lt|)]<∞for somet>0.
For some t>0,
E[g(|Lt(n)−Lt|)]=E[1{|Lt(n)−Lt|≤1}]+E[|Lt(n)−Lt|pq1{|Lt(n)−Lt|>1}]≤1+E[|Lt(n)−Lt|pq]<∞.
The last inequality does not depend on n∈N because of (A.4) and Example 25.10 in [13]. Since we have shown (i) and (ii) from the above, the proof is completed. □

We recall
Ct=σ1∫0th(s)ds+σ2∫0t∫|z|>0v(s,z)N(ds,dz),Ct(n)=σ1∫0th(s)ds+σ2∫0t∫0<|z|≤nv(s,z)N(ds,dz).
To complete the proof of Lemma 5.2, we show the following.

In the setup of Lemma5.2, the following convergence holds:limn→∞E[supt∈[0,T]|Ct(n)−Ct|p]=0.

It can be proved in the same way as in Lemma A.5. In fact, by using the function ur,δ,ε defined in Lemma A.1 with r=pβ, we have
ur,δ,ε(Ct(n)−Ct)=∫0t∫|z|≥1{ur,δ,ε(Ct(n)−Ct+v(s,z)1{|z|>n})−ur,δ,ε(Ct(n)−Ct)}N(ds,dz).
Note that {Ct(n)−Ct}t≥0 is a compound Poisson process and assumptions of Lemma 5.2 are fulfilled. We can confirm the Lp(Ω×[0,T]) convergence by showing convergence in probability and uniform integrability in the same way as in Lemma A.5. □

Acknowledgement

We would like to express our heartfelt gratitude to the editor Prof. Yuliya Mishura and the anonymous referees, whose suggestions and attentive reading resulted in considerable alterations from the original version, thereby significantly enhancing the article’s readability. Additionally, the authors extend our deepest gratitude to Prof. Arturo Kohatsu-Higa, Prof. Atsushi Takeuchi and Dr. Gô Yûki for their invaluable discussions and insightful comments, which have significantly enriched this paper.

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