VMSTA Modern Stochastics: Theory and Applications 2351-60542351-60462351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA245 10.15559/24-VMSTA245 Research Article Existence of density function for the running maximum of SDEs driven by nontruncated pure-jump Lévy processes https://orcid.org/0000-0003-0379-5833 NakagawaTakuyatakuya.nakagawa73@gmail.coma∗ https://orcid.org/0000-0001-9979-1882 SuzukiRyoichirsuzukimath@gmail.comb Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan Corresponding author. 2024 2312024113303321196202317112023712024 © 2024 The Author(s). Published by VTeX2024 Open access article under the CC BY license.

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 maximum density functions Malliavin calculus stochastic differential equations Lévy processes 60H10 60G52 60H07 KAKENHI23K12507The second author was supported by JSPS KAKENHI Grant Number 23K12507. Introduction

We consider a solution of the following one-dimensional SDE d X t = b ( X t ) d t + σ 1 d W t + σ 2 d L t , X 0 = x ∈ R , for t ≥ 0, where σ 1 and σ 2 are constants, b : R → R is differentiable and its derivative is bounded, W = { W t } t ∈ [ 0 , T ] is a standard Brownian motion and L = { L t } t ∈ [ 0 , T ] is a Lévy process with the Lévy triplet ( 0 , 0 , ν ). The infinitesimal generator A of L is defined by A f ( x ) : = ∫ R ∖ { 0 } { f ( x + y ) − f ( x ) − 1 { | y | < 1 } y f ′ ( x ) } ν ( d y ) for any f ∈ C b 2 ( 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: d X t ( n ) = b ( X t ( n ) ) d t + σ 1 d W t + σ 2 d L t ( n ) , X 0 ( n ) = x ∈ R , for t ∈ [ 0 , T ], where L ( n ) = { L t ( n ) } t ∈ [ 0 , T ] is a truncated pure-jump process L with jump sizes larger than n. Let X ∗ : = { X t ∗ } t ∈ [ 0 , T ] and X ( ∗ , n ) : = { X t ( ∗ , n ) } t ∈ [ 0 , T ] , defined as X t ∗ = sup t ∈ [ 0 , t ] X s , X t ( ∗ , n ) = sup s ∈ [ 0 , t ] X s ( n ) for each t ∈ [ 0 , T ] and n ∈ 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 X t ∗ 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 = σ 2 = 0 and σ 1 = 1) is well known. See, e.g., [7]. The law of the maximum of Lévy motion (i.e. x = b = σ 1 = 0 and σ 2 = 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 ( X t ∗ , X t ). Song and Zhang [15] study the existence of distributional density of X t 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 X t ( 1 ) . Song and Xie [14] show the existence of density functions for the running maximum X t ( ∗ , 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 X t ( ∗ , 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 = { L t } 0 ≤ t ≤ T and L ( n ) = { L t ( 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 Δ L t = L t − L t − and Δ L t ( n ) : = L t ( n ) − L t − ( n ) for any t > 0 and Δ L 0 : = 0 and Δ L 0 ( 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 ≤ t 1 F ( Δ L s ) and N ( n ) ( t , G ) = ∑ 0 ≤ s ≤ t 1 F ( Δ L s ( 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 ( d z ) = c ( z ) d z and c ( z ) 1 { | z | ≤ n } ( z ) d z, 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 ( d z ) < ∞ , lim n → ∞ ∫ R ∖ { 0 } | z | p 1 { | z | > n } ( z ) ν ( d z ) = 0 for any p ∈ ( 1 , β ) , sup n ∈ N | ∫ 1 ≤ | z | ≤ n z ν ( d z ) | < ∞ and ∫ 0 + ν ( d z ) = ∞ , | c ′ ( z ) c ( z ) | ≤ C ( 1 ∨ 1 | z | ) for any z ≠ 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 ≠ 0 c ( z ) = c α | z | 1 + α , where c α = π − 1 Γ ( α + 1 ) sin ( α π 2 ) , so that a Lévy measure ν satisfies assumptions (2.1) and (2.2) for any p ∈ ( 1 , α ).

Our results are described below.

Assume that b : R → R is once differentiable and its derivative is bounded, and that a Lévy measure ν of L satisfies (2.1) and (2.2). Let { X t } t ∈ [ 0 , T ] be the solution to equation (1.1). If σ 1 2 + σ 2 2 ≠ 0, then for any T > 0 the law of X T ∗ 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 ⊂ R d containing the origin. We define Γ 0 : = Γ ∖ { 0 } , ϱ ( z ) : = 1 ∨ d ( z , Γ 0 c ) − 1 , where d ( z , Γ 0 c ) is the distance of z to the complement of Γ 0 . Let Ω denote the canonical space consisting of all pairs ω = ( w , μ ), where

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

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

 Let us define the canonical process on Ω by setting for ω = ( w , μ ): W t ( ω ) : = w ( t ) , N ( ω ; d t , d z ) : = μ ( ω ; d t , d z ) : = μ ( d t , d z ) . We consider the smallest right-continuous filtration ( F t ) t ∈ [ 0 , 1 ] on Ω ensuring that both W and N are optional processes. Throughout our discussion, we set F : = F 1 . 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 d t ν ( d z ), where ν ( d z ) = κ ( z ) d z with κ ∈ C 1 ( Γ 0 ; ( 0 , ∞ ) ) , ∫ Γ 0 ( 1 ∧ | z | 2 ) κ ( z ) d z < + ∞ , | ∇ log κ ( z ) | ≤ C ϱ ( 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 ˜ ( d t , d z ) : = N ( d t , d z ) − ν ( d z ) d t . Let p ≥ 1, i ∈ { 1 , 2 }, and let k be an integer. We introduce the following spaces for subsequent discussions.

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

Let L p i be the space of all predictable processes ξ : Ω × [ 0 , 1 ] × Γ 0 → R k with finite norm ‖ ξ ‖ L p i : = E [ ( ∫ 0 1 ∫ Γ 0 | ξ ( s , z ) | i ν ( d z ) d s ) p i ] 1 p + E [ ∫ 0 1 ∫ Γ 0 | ξ ( s , z ) | p ν ( d z ) d s ] 1 p < ∞ .

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

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

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

The space V 0 is constituted of all predictable processes v : Ω × [ 0 , 1 ] × Γ 0 → R d satisfying the following conditions:

both v and ∇ z v are bounded;

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

Let C p ∞ ( R m ) denote the set of smooth functions on R m for which all derivatives exhibit at most polynomial growth. Define the collection of Wiener–Poisson functionals on Ω given by F ( ω ) = f ( W ( h 1 ) , … , W ( h m 1 ) , N ( g 1 ) , … , N ( g m 2 ) ) , where f belongs to C p ∞ ( R m 1 + m 2 ), h 1 , … , h m 1 are elements of H 0 , and g 1 , … , g m 2 are in V 0 , with all of them being nonrandom and real-valued. Additionally, for each j in the range 1 ≤ j ≤ m 1 and each k in the range 1 ≤ k ≤ m 2 , we define W ( h j ) : = ∫ 0 1 ⟨ h j ( s ) , d W s ⟩ R d and N ( g k ) : = ∫ 0 1 ∫ Γ 0 g k ( s , z ) N ( d s , d z ) . Given any p > 1 and Θ = ( h , v ) ∈ H p × V p , we denote D Θ F : = ∑ i = 1 m 1 ( ∂ i f ) ( · ) ∫ 0 1 ⟨ h ( s ) , h i ⟩ R d d s + ∑ j = 1 m 2 ( ∂ j + m 1 f ) ( · ) ∫ 0 1 ∫ Γ 0 ⟨ v ( s , z ) , ∇ z g j ⟩ R d N ( d s , d z ) , where “ ( · )” represents the collection W ( h 1 ) , … , W ( h m 1 ) , N ( g 1 ) , … , N ( g m 2 ).

Given p > 1 and Θ = ( h , v ) ∈ H p × V p , we introduce the first-order Sobolev space W Θ 1 , p as the completion of F C p in L p ( Ω ) with respect to the norm ‖ F ‖ Θ ; 1 , p : = ‖ F ‖ L p + ‖ D Θ F ‖ L p .

It is well known that the Banach space W Θ 1 , 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 ) = { X s ( n ) } s ≥ 0 be a right continuous real-valued process. For any fixed T > 0 and n ∈ N, in the following we shall write X T ( ∗ , n ) : = sup s ∈ [ 0 , T ] X s ( n ) , X T ∗ : = sup s ∈ [ 0 , T ] X s .

Let X ( n ) = { X s ( n ) } s ≥ 0 and X = { X s } s ≥ 0 be a right continuous process for each n ∈ N. Suppose that for some p > 1 and Θ = ( h , v ) ∈ H ∞ − × V ∞ − :

sup n ∈ N E [ | X s ( ∗ , n ) | p ] < ∞, and for any s ∈ [ 0 , T ], X s ( n ) ∈ W Θ 1 , p , and sup n ∈ N E [ sup s ∈ [ 0 , T ] | D Θ X s ( n ) | p ] < ∞ ;

the process { D Θ X s ( n ) } s ∈ [ 0 , T ] possesses a right continuous version for each n ∈ N;

lim n → ∞ E [ sup s ∈ [ 0 , T ] | X s ( n ) − X s | p ] = 0 .

 Then X T ∗ ∈ W θ 1 , p and the sequence { D Θ X s ( n ) } s ∈ [ 0 , T ] converges to D Θ X = { D Θ X s } s ∈ [ 0 , T ] in the weak topology of L p ( Ω × [ 0 , T ] ). Moreover, if this D Θ X has a right continuous version and P ( D Θ X t ≠ 0 on { t ∈ ( 0 , T ] : X t = X T ∗ } ) = 1 , then the law of X T ∗ is absolutely continuous with respect to the Lebesgue measure.

It can be seen that X T ( ∗ , n ) ∈ W θ 1 , p follows from Proposition 3.1 in [14] for each n ∈ N. From Lemma 2.3 in [14], we obtain X T ∗ ∈ W θ 1 , p and lim n → ∞ D Θ X · ( n ) = D Θ X · weakly in L p ( Ω × [ 0 , T ] ) . In exactly the same way as in Theorem 3.2 in [14], the following equality follows if D Θ X has a right continuous path almost surely: 1 = P ( { ∃ t ∈ [ 0 , T ] such that D Θ X t ≠ D Θ X T ∗ and X t = X T ∗ } c ) = P ( D Θ X t = D Θ X T ∗ on { t ∈ [ 0 , T ] : X t = X t ∗ } ) ≤ P ( D Θ X t = D Θ X T ∗ on { t ∈ ( 0 , T ] : X t = X t ∗ } ) = 1 . Subsequently, we prove that X T ∗ has a density function if (4.1) holds. In addition, by the closability of D Θ (see Theorem 2.6 in [14]), we obtain P ( 1 A ( D Θ X T ∗ ) D Θ X T ∗ = 0 ) = 1 , for any A ∈ B ( R ) with Leb ( A ) = 0. With these facts and (4.1) we obtain the following equation: 1 = P ( { 1 A ( D Θ X T ∗ ) D Θ X T ∗ = 0 } ∩ { D Θ X t = D Θ X T ∗ on { t ∈ ( 0 , T ] : X t = X t ∗ } } ∩ { D Θ X t ≠ 0 on { t ∈ ( 0 , T ] : X t = X t ∗ } } ) = P ( { 1 A ( D Θ X T ∗ ) D Θ X t = 0 on { t ∈ ( 0 , T ] : X t = X t ∗ } } ∩ { D Θ X t = D Θ X T ∗ on { t ∈ ( 0 , T ] : X t = X t ∗ } } ∩ { D Θ X t ≠ 0 on { t ∈ ( 0 , T ] : X t = X t ∗ } } ) = P ( { 1 A ( D Θ X T ∗ ) = 0 } ∩ { D Θ X t = D Θ X T ∗ on { t ∈ ( 0 , T ] : X t = X t ∗ } } ∩ { D Θ X t ≠ 0 on { t ∈ ( 0 , T ] : X t = X t ∗ } } ) ≤ P ( 1 A ( D Θ X T ∗ ) = 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 D Θ X for X in equation (1.1), we check an equation satisfied by D Θ X ( n ) for X ( n ) in equation (1.2). The following lemma is shown in the same way as for Lemma 4.3 in [14].

([<xref ref-type="bibr" rid="j_vmsta245_ref_014">14</xref>], Lemma 4.3).

Assume that b : R → R is once differentiable and its derivative is bounded. Then for any Θ = ( h , v ) ∈ H ∞ − × V ∞ − and t ∈ [ 0 , T ], X t ( n ) ∈ W Θ 1 , 2 and D Θ X t ( n ) = ∫ 0 t b ′ ( X s ( n ) ) D Θ X s ( n ) d s + σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) .

The following lemma defines D Θ X and confirms that it satisfies (5.1) below.

Assume the same assumptions as in Lemma 5.1. Then for some p ∈ ( 1 , β ), for some n ∈ N, for any q ∈ ( 1 , β ) and for any Θ = ( h , v ) ∈ H ∞ − × V ∞ − , where lim n → ∞ ∫ 0 T ∫ | z | > n | v ( s , z ) | q β d s ν ( d z ) = 0 and ∫ 0 T ∫ | z | > n | v ( s , z ) | q d s ν ( d z ) < ∞ , X t ∗ ∈ W Θ 1 , p for any t ∈ [ 0 , T ], and D Θ X t = ∫ 0 t b ′ ( X s ) D Θ X s d s + σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ | z | > 0 v ( s , z ) N ( d s , d z ) . Then this D Θ X is the limit of weak L p ( Ω × [ 0 , T ] ) convergence of the sequence D Θ X ( n ) , and this sequence is strongly L p ( 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 L p integrability of D Θ X ( 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 D Θ (cf. Lemma 2.7 in [15]), we have lim n → ∞ D Θ X · ( n ) = D Θ X · weakly in L p ( Ω × [ 0 , T ] ) . We verify that this D Θ X = { D Θ X t } t ∈ [ 0 , T ] satisfies equation (5.1). We set { Y t } t ∈ [ 0 , T ] as a solution of Y t = ∫ 0 t b ′ ( X s ) Y s d s + C t , where C t = σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ | z | > 0 v ( s , z ) N ( d s , d z ) , C t ( n ) = σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) . We prove lim n → ∞ E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) − Y t | p ] = 0 . By using an inequality | a + b | p ≤ 2 p − 1 ( | a | p + | b | p ) for any a , b ∈ R, we obtain E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) − Y t | p ] ≤ 2 p − 1 E [ sup t ∈ [ 0 , T ] | ∫ 0 t { b ′ ( X s ( n ) ) D Θ X s ( n ) − b ′ ( X s ) Y s } d s | p ] + 2 p − 1 E [ sup t ∈ [ 0 , T ] | C t ( n ) − C t | p ] ≤ 2 2 ( p − 1 ) ∫ 0 T E [ | b ′ ( X s ( n ) ) − b ′ ( X s ) | p | D Θ X s ( n ) | p ] d s + 2 2 ( p − 1 ) ‖ b ′ ‖ ∞ p ∫ 0 T E [ | D Θ X s ( n ) − Y s | p ] d s + 2 p − 1 E [ sup t ∈ [ 0 , T ] | C t ( n ) − C t | 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 [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) − Y t | p ] ≤ 2 2 ( p − 1 ) exp ( 2 2 ( p − 1 ) ‖ b ′ ‖ ∞ p ) ∫ 0 T E [ | b ′ ( X s ( n ) ) − b ′ ( X s ) | p | D Θ X s ( n ) | p ] d s + 2 2 ( p − 1 ) exp ( 2 2 ( p − 1 ) ‖ b ′ ‖ ∞ p ) E [ sup t ∈ [ 0 , T ] | C t ( n ) − C t | p ] . We show lim n → ∞ ∫ 0 T E [ | b ′ ( X s ( n ) ) − b ′ ( X s ) | p | D Θ X s ( n ) | p ] d s = 0 , lim n → ∞ E [ sup t ∈ [ 0 , T ] | C t ( n ) − C t | 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 p q < β because of the denseness of rational numbers. By using the Hölder inequality, we have ∫ 0 T E [ | b ′ ( X s ( n ) ) − b ′ ( X s ) | p | D Θ X s ( n ) | p ] d s ≤ ∫ 0 T E [ | b ′ ( X s ( n ) ) − b ′ ( X s ) | p q q − 1 ] q − 1 q E [ | D Θ X s ( n ) | p q ] 1 q d s ≤ T E [ sup t ∈ [ 0 , T ] | b ′ ( X t ( n ) ) − b ′ ( X t ) | p q q − 1 ] q − 1 q E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) | p q ] 1 q . Due to an inequality | a + b | p ≤ 2 p − 1 ( | a | p + | b | p ) for any a , b ∈ R and p ≥ 1 and Jensen’s inequality, we have E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) | p q ] ≤ 2 p q − 1 E [ sup t ∈ [ 0 , T ] | ∫ 0 t b ′ ( X s ( n ) ) D Θ X s ( n ) d s | p q ] + 2 2 ( p q − 1 ) E [ sup t ∈ [ 0 , T ] | ∫ 0 t h ( s ) d s | p q ] + 2 2 ( p q − 1 ) E [ sup t ∈ [ 0 , T ] | ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) | p q ] ≤ 2 p q − 1 ‖ b ′ ‖ ∞ p q ∫ 0 T E [ sup u ∈ [ 0 , s ] | D Θ X u ( n ) | p q ] d s + 2 2 ( p q − 1 ) ( E [ ∫ 0 T | h ( s ) | p q d s ] + E [ sup t ∈ [ 0 , T ] | ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) | p q ] ) . Gronwall’s inequality implies E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) | p q ] ≤ 2 2 ( p q − 1 ) ( E [ ∫ 0 T | h ( s ) | p q d s ] + E [ sup t ∈ [ 0 , T ] | ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) | p q ] ) e 2 p q − 1 T ‖ b ′ ‖ ∞ p q . 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 sup n ∈ N E [ sup t ∈ [ 0 , T ] | D Θ X t ( n ) | p q ] < ∞ . This allows us to confirm the L p ( Ω × [ 0 , T ] ) integrability of D Θ X ( n ) for assumption 1 in Lemma 4.1, and that D Θ X can be defined as the limit of weak L p ( Ω × [ 0 , T ] ) convergence of D Θ X ( n ) . By Lemma A.5, boundedness of b ′ and continuous mapping theorem, we have lim n → ∞ E [ sup t ∈ [ 0 , T ] | b ′ ( X t ( n ) ) − b ′ ( X t ) | p q q − 1 ] q − 1 q = 0 , so that we obtain (5.3). Thus, by (5.2) and completeness of L p ( 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 − ∫ 0 t b ′ ( X s ) d s D Θ X t (e.g., see [12], Corollary (Integration by Parts), P. 84), we obtain e − ∫ 0 t b ′ ( X s ) d s D Θ X t = ∫ 0 + t e − ∫ 0 s b ′ ( X u ) d u ∘ d D Θ X s − + ∫ 0 + t D Θ X s − ∘ d e − ∫ 0 s b ′ ( X u ) d u = ∫ 0 t e − ∫ 0 s b ′ ( X u ) d u d D Θ X s − + ∫ 0 t D Θ X s − d e − ∫ 0 s b ′ ( X u ) d u = ∫ 0 t e − ∫ 0 s b ′ ( X u ) d u ( b ′ ( X s ) D Θ X s + σ 1 h ( s ) ) d s + ∫ 0 t e − ∫ 0 s b ′ ( X u ) d u σ 2 ∫ | z | > 0 v ( s , z ) N ( d s , d z ) + ∫ 0 t D Θ X s − ( − b ′ ( X s ) ∫ 0 t e − ∫ 0 s b ′ ( X u ) d u ) d s . For any t > 0, we set h ( t ) : = σ 1 e − ∫ 0 t b ′ ( X s ) d s , v ( t , z ) : = σ 2 e − ∫ 0 t b ′ ( X s ) d s η ( z ) , η ( z ) = | z | 2 , | z | ≤ 1 4 , 0 , | z | > 1 2 , 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 D Θ X t = e ∫ 0 t b ′ ( X s ) d s ( σ 1 2 ∫ 0 t e − 2 ∫ 0 s b ′ ( X u ) d u d s + σ 2 2 ∫ 0 t ∫ | z | > 0 e − 2 ∫ 0 s b ′ ( X u ) d u η ( z ) N ( d s , d z ) ) ≥ e ∫ 0 t ( b ′ ( X s ) − 2 ‖ b ‖ Lip ) d s ( σ 1 2 t + σ 2 2 ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) ) . Noticing the condition σ 1 2 + σ 2 2 ≠ 0 and the fact P ( ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) > 0 , ∀ t > 0 ) = 1 , we have P ( D Θ X t > 0 , ∀ t ∈ ( 0 , T ] ) = 1 . See Section A.1 for a proof of equation (6.2). So we have 1 = P ( D Θ X t > 0 , ∀ t ∈ ( 0 , T ] ) ≤ P ( D Θ X t ≠ 0 on { t ∈ ( 0 , T ] : X t = X T ∗ } ) = 1 . Therefore, we conclude by Lemma 4.1 that the law of X T ∗ 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 ε k = 1 2 k for any k ∈ N. Since lim k → ∞ ∫ 0 t ∫ | z | > ε k η ( z ) N ( d s , d z ) = ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) in L 2 ( Ω ) , lim k → ∞ ∫ 0 t ∫ | z | > ε k η ( z ) N ( d s , d z ) = ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) in distribution. Noting that the support of η, for any t > 0, we have P ( ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) = 0 ) = lim k → ∞ P ( ∫ 0 t ∫ | z | > ε k η ( z ) N ( d s , d z ) = 0 ) = lim k → ∞ P ( ∫ | z | > ε k η ( z ) N ( t , d z ) = 0 ) ≤ lim k → ∞ P ( N ( t , ( ε k , 1 2 ] ∪ ( − 1 2 , − ε k ] ) = 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 lim k → ∞ P ( N ( t , ( ε k , 1 2 ] ∪ ( − 1 2 , − ε k ] ) = 0 ) = lim k → ∞ exp ( − t ν ( ( ε k , 1 2 ] ∪ ( − 1 2 , − ε k ] ) ) = 0 . The last equation follows from assumption (2.2). We set for each t > 0, I t = ∫ 0 t ∫ | z | > 0 η ( z ) N ( d s , d z ) ; from countable additivity we have P ( ⋃ t ∈ ( 0 , ∞ ) ∩ Q { I t = 0 } ) ≤ ∑ t ∈ ( 0 , ∞ ) ∩ Q P ( { I t = 0 } ) = 0 ( see, e.g., [16], 1.9(b) ) . Thus we obtain P ( ⋂ t ∈ ( 0 , ∞ ) ∩ Q { I t > 0 } ) = 1 . Here, since η ≥ 0, we obtain P ( ⋂ 0 ≤ s ≤ u { I s ≤ I u } ) = 1 . By denseness of rational numbers, we obtain the following: P ( ∀ t > 0 , I t > 0 ) ≥ P ( ⋂ t ∈ ( 0 , ∞ ) ∩ Q { I t > 0 } ∩ ⋂ 0 ≤ s ≤ u { I s ≤ I u } ) = 1 .

Preparation for proof of convergence of <inline-formula id="j_vmsta245_ineq_196"><alternatives><mml:math> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic">X</mml:mi> </mml:mrow> <mml:mrow> <mml:mo mathvariant="normal" fence="true" stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">n</mml:mi> <mml:mo mathvariant="normal" fence="true" stretchy="false">)</mml:mo> </mml:mrow> </mml:msup></mml:math><tex-math><![CDATA[${X^{(n)}}$]]></tex-math></alternatives></inline-formula>

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].

For ε > 0, δ > 1 and r ∈ ( 0 , 1 ], we can choose a smooth function ψ δ , ε which satisfies the conditions ψ δ , ε ( x ) = between 0 and 2 ( x log δ ) − 1 , ε δ − 1 < x < ε , 0 , otherwise , and ∫ ε δ − 1 ε ψ δ , ε ( y ) d y = 1. We define u r ( x ) = | x | r and u r , δ , ε = u r ∗ ψ δ , ε . Then, u r , δ , ε ∈ C 2 and for any x ∈ R, | x | r ≤ ε r + u r , δ , ε ( x ) , u r , δ , ε ( x ) ≤ | x | r + ε r .

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 Δ t = ( t 0 , t 1 , … , t n + 1 ) such that 0 = t 0 < t 1 < ⋯ < t n + 1 = T.

The mean variation of X is defined by V T ( X ) : = sup Δ t E [ ∑ i = 0 n | E [ X t i − X t i + 1 | F t i ] | ] .

A càdlàg adapted process Z is a quasimartingale on [ 0 , T ] if for each t ∈ [ 0 , T ], E [ | Z t | ] < ∞ and V T ( 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 each t ∈ [ 0 , T ], E [ | Z t | ] < ∞ and V t ( Z ) < ∞. Then, for each h > 0, h P ( sup t ∈ [ 0 , T ] | Z t | > h ) ≤ V T ( Z ) + E [ | Z T | ] .

Proof of <inline-formula id="j_vmsta245_ineq_221"><alternatives><mml:math> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">p</mml:mi> </mml:mrow> </mml:msup></mml:math><tex-math><![CDATA[${L^{p}}$]]></tex-math></alternatives></inline-formula>-convergence of <inline-formula id="j_vmsta245_ineq_222"><alternatives><mml:math> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic">X</mml:mi> </mml:mrow> <mml:mrow> <mml:mo mathvariant="normal" fence="true" stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">n</mml:mi> <mml:mo mathvariant="normal" fence="true" stretchy="false">)</mml:mo> </mml:mrow> </mml:msup></mml:math><tex-math><![CDATA[${X^{(n)}}$]]></tex-math></alternatives></inline-formula>

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 Theorem 2.1. Then lim n → ∞ E [ sup t ∈ [ 0 , T ] | X t ( n ) − X t | p ] = 0 for any p ∈ ( 1 , β ) . This is because the following two conditions are valid.

sup t ∈ [ 0 , T ] | X t ( n ) − X t | p → 0 in probability as n → ∞.

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

(i) For clarity, we write r = p β . By using the triangle inequality and Jensen’s inequality, we have | X t ( n ) − X t | ≤ ∫ 0 t | b ( X s ( n ) ) − b ( X s ) | d s + σ 2 | L t ( n ) − L t | . By the definition of supremum, we have sup t ∈ [ 0 , T ] | X t ( n ) − X t | ≤ T sup t ∈ [ 0 , T ] | b ( X t ( n ) ) − b ( X t ) | + σ 2 sup t ∈ [ 0 , T ] | L t ( n ) − L t | . Since b is Lipschitz continuous, we have sup t ∈ [ 0 , T ] | X t ( n ) − X t | ≤ C T sup t ∈ [ 0 , T ] | X t ( n ) − X t | d s + σ 2 sup t ∈ [ 0 , T ] | L t ( n ) − L t | . By using Gronwall’s inequality and Jensen’s inequality, we have sup t ∈ [ 0 , T ] | X t ( n ) − X t | ≤ C σ 2 exp ( C T ) sup t ∈ [ 0 , T ] | L t ( n ) − L t | . Here, by the above inequality and Lemma A.4, for any h > 0, P ( sup t ∈ [ 0 , T ] | X t ( n ) − X t | p > h ) ≤ P ( sup t ∈ [ 0 , T ] | L t ( n ) − L t | r > ( h C σ 2 exp ( C T ) ) 1 β ) ≤ P ( sup t ∈ [ 0 , T ] ( ε p + u r , δ , ε ( L t ( n ) − L t ) ) > ( h C σ 2 exp ( C T ) ) 1 β ) ≤ ( C σ 2 exp ( C T ) h ) 1 β ( V T ( ε p + u r , δ , ε ( L ( n ) − L ) ) + E [ | ε p + u r , δ , ε ( L T ( n ) − L T ) | ] ) . Here, by the definition of the mean variation and (A.2), we have V T ( ε p + u r , δ , ε ( L ( n ) − L ) ) = V T ( u r , δ , ε ( L ( n ) − L ) ) , E [ | ε p + u r , δ , ε ( L t ( n ) − L t ) | ] = E [ ε p + u r , δ , ε ( L T ( n ) − L T ) ] . By using the Lévy–Itô decomposition ([1], Theorem 2.4.16), we have L t ( n ) − L t = ∫ 0 t ∫ | z | > 1 z 1 { 0 < | z | ≤ n } N ( d s , d z ) + ∫ 0 t ∫ 0 < | z | ≤ 1 z 1 { 0 < | z | ≤ n } N ˜ ( d s , d z ) − ∫ 0 t ∫ | z | > 1 z N ( d s , d z ) − ∫ 0 t ∫ 0 < | z | ≤ 1 z N ˜ ( d s , d z ) , = − ∫ 0 t ∫ | z | > 1 z 1 { | z | > n } N ( d s , d z ) − ∫ 0 t ∫ 0 < | z | ≤ 1 z 1 { | z | > n } N ˜ ( d s , d z ) , = − ∫ 0 t ∫ | z | > 1 z 1 { | z | > n } N ( d s , d z ) . The last equality follows by n ≥ 1. Using the Itô formula ([1], Theorem 4.4.7), N ( d t , d z ) = N ˜ ( d t , d z ) + ν ( d z ) d t and the function u r , δ , ε defined in Lemma A.1, we have u r , δ , ε ( L t ( n ) − L t ) = ∫ 0 t ∫ | z | ≥ 1 { u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) } N ( d s , d z ) = ∫ 0 t ∫ | z | ≥ 1 { u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) } N ˜ ( d s , d z ) + ∫ 0 t ∫ | z | ≥ 1 { u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) } d s ν ( d z ) = : M t δ , ε + I t δ , ε . Here, by (A.1), for any x , y ∈ R, − u r , δ , ε ( y ) ≤ ε r − | y | r , u r , δ , ε ( x ) − u r , δ , ε ( y ) ≤ 2 ε r + | x | r − | y | r ≤ 2 ε r + | | x | r − | y | r | ≤ 2 ε r + | x − y | r . So we have u r , δ , ε ( L T ( n ) − L T ) ≤ ∫ 0 T ∫ | z | ≥ 1 { u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) } N ˜ ( d s , d z ) + ∫ 0 T ∫ | z | ≥ 1 { 2 ε r + | z | r 1 { | z | > n } } d s ν ( d z ) . Also, ∫ 0 T ∫ | z | ≥ 1 { 2 ε r + | z | r 1 { | z | > n } } d s ν ( d z ) ≤ 2 C T ε r + T ∫ R ∖ { 0 } | z | r 1 { | z | > n } ν ( d z ) . We can evaluate V T ( u r , δ , ε ( L ( n ) − L ) ) = sup Δ t E [ ∑ i = 0 n | E [ u r , δ , ε ( L t i ( n ) − L t i ) − u r , δ , ε ( L t i + 1 ( n ) − L t i + 1 ) ∣ F t i ] | ] = sup Δ t E [ ∑ i = 0 n | E [ M t i δ , ε + I t i δ , ε − M t i + 1 δ , ε − I t i + 1 δ , ε ∣ F t i ] | ] = sup Δ t E [ ∑ i = 0 n | E [ I t i δ , ε − I t i + 1 δ , ε ∣ F t i ] | ] . The last equality is valid, since ( M t δ , ε ) t ∈ [ 0 , T ] is a martingale. By using Jensen’s inequality, we have V T ( u r , δ , ε ( L ( n ) − L ) ) ≤ sup Δ t E [ ∑ i = 0 n E [ | I t i δ , ε − I t i + 1 δ , ε | ∣ F t i ] ] = sup Δ t ∑ i = 0 n E [ | ∫ t i t i + 1 ∫ | z | ≥ 1 { u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) } d s ν ( d z ) | ] ≤ sup Δ t ∑ i = 0 n E [ ∫ t i t i + 1 ∫ | z | ≥ 1 | u r , δ , ε ( L s − ( n ) − L s − − z 1 { | z | > n } ) − u r , δ , ε ( L s − ( n ) − L s − ) | d s ν ( d z ) ] ≤ sup Δ t ∑ i = 0 n E [ ∫ t i t i + 1 ∫ | z | ≥ 1 ( 2 ε r + | z | r 1 { | z | > n } ) d s ν ( d z ) ] ≤ ∫ 0 T ∫ | z | ≥ 1 ( 2 ε r + | z | r 1 { | z | > n } ) d s ν ( d z ) . Therefore, we have P ( sup t ∈ [ 0 , T ] | X t ( n ) − X t | p > h ) ≤ ( C σ 2 exp ( C T ) h ) 1 β { ( 4 C T + 1 ) ε r + 2 T ∫ R ∖ { 0 } | z | r 1 { | z | > n } ν ( d z ) } . Since the above inequality holds for any ε > 0, we obtain P ( sup t ∈ [ 0 , T ] | X t ( n ) − X t | p > h ) ≤ 2 T ( C σ 2 exp ( C T ) h ) 1 β ∫ R ∖ { 0 } | z | r 1 { | z | > n } ν ( d z ) . By the assumption, for any h > 0, we have lim n → ∞ P ( sup t ∈ [ 0 , T ] | X t ( n ) − X t | p > h ) = 0 .

(ii) Next, we show that the process { sup t ∈ [ 0 , T ] | X t ( n ) − X t | p } T ≥ 0 is uniformly integrable. To show this, by using inequality (A.3) it suffices to show for some q > 1, E [ ( sup t ∈ [ 0 , T ] | L t ( n ) − L t | p ∨ 1 ) q ] < ∞ . By assumption of (2.1) and the denseness of rational numbers, we can set q > 1 such that p q < β so that for each n ∈ N, ∫ | z | > n | z | p q ν ( d z ) < ∞ . Here, we set g ( x ) = | x | p q ∨ 1; then g is a nonnegative increasing submultiplicative function and lim x → ∞ g ( x ) = ∞. Using Theorem 25.18 in [13], we will show that E [ g ( | L t ( n ) − L t | ) ] < ∞ for some t > 0 . For some t > 0, E [ g ( | L t ( n ) − L t | ) ] = E [ 1 { | L t ( n ) − L t | ≤ 1 } ] + E [ | L t ( n ) − L t | p q 1 { | L t ( n ) − L t | > 1 } ] ≤ 1 + E [ | L t ( n ) − L t | p q ] < ∞ . 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 C t = σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ | z | > 0 v ( s , z ) N ( d s , d z ) , C t ( n ) = σ 1 ∫ 0 t h ( s ) d s + σ 2 ∫ 0 t ∫ 0 < | z | ≤ n v ( s , z ) N ( d s , d z ) . To complete the proof of Lemma 5.2, we show the following.

In the setup of Lemma 5.2, the following convergence holds: lim n → ∞ E [ sup t ∈ [ 0 , T ] | C t ( n ) − C t | p ] = 0 .

It can be proved in the same way as in Lemma A.5. In fact, by using the function u r , δ , ε defined in Lemma A.1 with r = p β , we have u r , δ , ε ( C t ( n ) − C t ) = ∫ 0 t ∫ | z | ≥ 1 { u r , δ , ε ( C t ( n ) − C t + v ( s , z ) 1 { | z | > n } ) − u r , δ , ε ( C t ( n ) − C t ) } N ( d s , d z ) . Note that { C t ( n ) − C t } t ≥ 0 is a compound Poisson process and assumptions of Lemma 5.2 are fulfilled. We can confirm the L p ( 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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