1 Introduction
We consider a solution of the following one-dimensional SDE
for $t\ge 0$, where ${\sigma _{1}}$ and ${\sigma _{2}}$ are constants, $b:\mathbb{R}\to \mathbb{R}$ is differentiable and its derivative is bounded, $W={\{{W_{t}}\}_{t\in [0,T]}}$ is a standard Brownian motion and $L={\{{L_{t}}\}_{t\in [0,T]}}$ is a Lévy process with the Lévy triplet $(0,0,\nu )$. The infinitesimal generator A of L is defined by
for $t\in [0,T]$, where ${L^{(n)}}={\{{L_{t}^{(n)}}\}_{t\in [0,T]}}$ is a truncated pure-jump process L with jump sizes larger than n. Let ${X^{\ast }}:={\{{X_{t}^{\ast }}\}_{t\in [0,T]}}$ and ${X^{(\ast ,n)}}:={\{{X_{t}^{(\ast ,n)}}\}_{t\in [0,T]}}$, defined as
(1.1)
\[ \mathrm{d}{X_{t}}=b({X_{t}})\mathrm{d}t+{\sigma _{1}}\mathrm{d}{W_{t}}+{\sigma _{2}}\mathrm{d}{L_{t}},\hspace{2.5pt}{X_{0}}=x\in \mathbb{R},\]
\[ Af(x):={\int _{\mathbb{R}\setminus \{0\}}}\big\{f(x+y)-f(x)-{1_{\{|y|\lt 1\}}}y{f^{\prime }}(x)\big\}\nu (\mathrm{d}y)\]
for any $f\in {C_{b}^{2}}(\mathbb{R})$ and $\hspace{2.5pt}x\in \mathbb{R}$. 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 $n\in \mathbb{N}$:
(1.2)
\[ \mathrm{d}{X_{t}^{(n)}}=b\big({X_{t}^{(n)}}\big)\mathrm{d}t+{\sigma _{1}}\mathrm{d}{W_{t}}+{\sigma _{2}}\mathrm{d}{L_{t}^{(n)}},\hspace{2.5pt}{X_{0}^{(n)}}=x\in \mathbb{R},\]
\[ {X_{t}^{\ast }}=\underset{t\in [0,t]}{\sup }{X_{s}},\hspace{2.5pt}{X_{t}^{(\ast ,n)}}=\underset{s\in [0,t]}{\sup }{X_{s}^{(n)}}\hspace{2.5pt}\text{for each}\hspace{2.5pt}t\in [0,T]\hspace{2.5pt}\text{and}\hspace{2.5pt}n\in \mathbb{N},\]
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 ${X_{t}^{\ast }}$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$ for all $t\gt 0$. 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 ${X^{\ast }}$ are known. The density function of the maximum of Brownian motion (i.e. $x=b={\sigma _{2}}=0$ and ${\sigma _{1}}=1$) is well known. See, e.g., [7]. The law of the maximum of Lévy motion (i.e. $x=b={\sigma _{1}}=0$ and ${\sigma _{2}}=1$) 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 $({X_{t}^{\ast }},{X_{t}})$. 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 ${X_{t}^{(\ast ,1)}}$ 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 ${X_{t}^{(\ast ,1)}}$. 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 ${X^{\ast }}$ 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.
2 Notations and a result
Let $L={\{{L_{t}}\}_{0\le t\le T}}$ and ${L^{(n)}}={\{{L_{t}^{(n)}}\}_{t\in [0,T]}}$ be a pure-jump process and the one truncated by $[-n,n]\setminus \{0\}$, respectively. The jump size of L and ${L^{(n)}}$ at time t is defined by $\Delta {L_{t}}={L_{t}}-{L_{t-}}$ and $\Delta {L_{t}^{(n)}}:={L_{t}^{(n)}}-{L_{t-}^{(n)}}$ for any $t\gt 0$ and $\Delta {L_{0}}:=0$ and $\Delta {L_{0}^{(n)}}:=0$. The Poisson random measures associated with L and ${L^{(n)}}$ on $\mathcal{B}([0,T])\times \mathcal{B}(\mathbb{R}\setminus \{0\})$ are denoted by $N(t,F)={\textstyle\sum _{0\le s\le t}}{1_{F}}(\Delta {L_{s}})$ and ${N^{(n)}}(t,G)={\textstyle\sum _{0\le s\le t}}{1_{F}}(\Delta {L_{s}^{(n)}})$ for $t\in [0,T]$ and $F\in \mathcal{B}(\mathbb{R}\setminus \{0\})$, respectively. The Lévy measures of L and ${L^{(n)}}$ on $\mathcal{B}(\mathbb{R}\setminus \{0\})$ are defined as $\nu (\mathrm{d}z)=c(z)\mathrm{d}z$ and $c(z){1_{\{|z|\le n\}}}(z)\mathrm{d}z$, where the positive function c satisfies the following requirements: there exist some constants $\beta \gt 1$ and $C\gt 0$ such that
The compensated Poisson random measures of L and ${L^{(n)}}$ are defined as $\widetilde{N}$ and ${\widetilde{N}^{(n)}}$, respectively.
(2.1)
\[\begin{aligned}{}& {\int _{\mathbb{R}\setminus \{0\}}}\hspace{-0.1667em}\big(1\wedge |z{|^{2}}\big)\nu (\mathrm{d}z)\hspace{-0.1667em}\lt \hspace{-0.1667em}\infty ,\hspace{1em}\underset{n\to \infty }{\lim }{\int _{\mathbb{R}\setminus \{0\}}}|z{|^{p}}{1_{\{|z|\gt n\}}}(z)\nu (\mathrm{d}z)\hspace{-0.1667em}=\hspace{-0.1667em}0\hspace{2.5pt}\text{for any}\hspace{2.5pt}p\hspace{-0.1667em}\in \hspace{-0.1667em}(1,\beta ),\end{aligned}\](2.2)
\[\begin{aligned}{}& \underset{n\in \mathbb{N}}{\sup }\bigg|{\int _{1\le |z|\le n}}z\nu (\mathrm{d}z)\bigg|\lt \infty \hspace{5pt}\text{and}\hspace{5pt}{\int _{0+}}\nu (\mathrm{d}z)=\infty ,\hspace{1em}\bigg|\frac{{c^{\prime }}(z)}{c(z)}\bigg|\le C\bigg(1\vee \frac{1}{|z|}\bigg)\\ {} & \hspace{1em}\text{for any}\hspace{2.5pt}z\ne 0.\end{aligned}\]Example 2.1.
If L is a symmetric α-stable process with $\alpha \in (1,2)$, then for any $z\ne 0$
\[ c(z)=\frac{{c_{\alpha }}}{|z{|^{1+\alpha }}},\hspace{2.5pt}\text{where}\hspace{2.5pt}{c_{\alpha }}={\pi ^{-1}}\Gamma (\alpha +1)\sin \bigg(\frac{\alpha \pi }{2}\bigg),\]
so that a Lévy measure ν satisfies assumptions (2.1) and (2.2) for any $p\in (1,\alpha )$.Our results are described below.
Theorem 2.1.
Assume that $b:\mathbb{R}\to \mathbb{R}$ is once differentiable and its derivative is bounded, and that a Lévy measure ν of L satisfies (2.1) and (2.2). Let ${\{{X_{t}}\}_{t\in [0,T]}}$ be the solution to equation (1.1). If ${\sigma _{1}^{2}}+{\sigma _{2}^{2}}\ne 0$, then for any $T\gt 0$ the law of ${X_{T}^{\ast }}$ 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.
3 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 $\Gamma \subset {\mathbb{R}^{d}}$ containing the origin. We define
where $\mathbf{d}(z,{\Gamma _{0}^{c}})$ is the distance of z to the complement of ${\Gamma _{0}}$. Let Ω denote the canonical space consisting of all pairs $\omega =(w,\mu )$, where
Let us define the canonical process on Ω by setting for $\omega =(w,\mu )$:
We denote the compensated Poisson random measure N by
Let ${C_{p}^{\infty }}({\mathbb{R}^{m}})$ denote the set of smooth functions on ${\mathbb{R}^{m}}$ for which all derivatives exhibit at most polynomial growth. Define the collection of Wiener–Poisson functionals on Ω given by
where f belongs to ${C_{p}^{\infty }}({\mathbb{R}^{{m_{1}}+{m_{2}}}})$, ${h_{1}},\dots ,{h_{{m_{1}}}}$ are elements of ${\mathbb{H}_{0}}$, and ${g_{1}},\dots ,{g_{{m_{2}}}}$ are in ${\mathbb{V}_{0}}$, with all of them being nonrandom and real-valued. Additionally, for each j in the range $1\le j\le {m_{1}}$ and each k in the range $1\le k\le {m_{2}}$, we define
(3.1)
\[ {\Gamma _{0}}:=\Gamma \setminus \{0\},\hspace{1em}\varrho (z):=1\vee \mathbf{d}{\big(z,{\Gamma _{0}^{c}}\big)^{-1}},\]
\[ {W_{t}}(\omega ):=w(t),\hspace{1em}N(\omega ;\mathrm{d}t,\mathrm{d}z):=\mu (\omega ;\mathrm{d}t,\mathrm{d}z):=\mu (\mathrm{d}t,\mathrm{d}z).\]
We consider the smallest right-continuous filtration ${({\mathcal{F}_{t}})_{t\in [0,1]}}$ on Ω ensuring that both W and N are optional processes. Throughout our discussion, we set $\mathcal{F}:={\mathcal{F}_{1}}$. The space $(\Omega ,\mathcal{F})$ is equipped with a unique probability measure $\mathbb{P}$ satisfying the following conditions:
-
• W is a standard d-dimensional Brownian motion;
-
• N is a Poisson random measure with intensity $\mathrm{d}t\nu (\mathrm{d}z)$, where $\nu (\mathrm{d}z)=\kappa (z)\mathrm{d}z$ withwhere $\varrho (z)$ is defined by equation (3.1);
(3.2)
\[ \kappa \in {C^{1}}\big({\Gamma _{0}};(0,\infty )\big),\hspace{2.5pt}{\int _{{\Gamma _{0}}}}\big(1\wedge |z{|^{2}}\big)\kappa (z)\mathrm{d}z\lt +\infty ,\hspace{2.5pt}|\nabla \log \kappa (z)|\le C\varrho (z),\] -
• W and N are independent.
\[ \widetilde{N}(\mathrm{d}t,\mathrm{d}z):=N(\mathrm{d}t,\mathrm{d}z)-\nu (\mathrm{d}z)\mathrm{d}t.\]
Let $p\ge 1$, $i\in \{1,2\}$, and let k be an integer. We introduce the following spaces for subsequent discussions.
-
• Let ${\mathbb{L}_{p}^{i}}$ be the space of all predictable processes $\xi :\Omega \times [0,1]\times {\Gamma _{0}}\to {\mathbb{R}^{k}}$ with finite norm\[\begin{aligned}{}\| \xi {\| _{{\mathbb{L}_{p}^{i}}}}& :=\mathbb{E}{\Bigg[{\Bigg({\int _{0}^{1}}{\int _{{\Gamma _{0}}}}|\xi (s,z){|^{i}}\nu (\mathrm{d}z)\mathrm{d}s\Bigg)^{\frac{p}{i}}}\Bigg]^{\frac{1}{p}}}\\ {} & \hspace{1em}+\mathbb{E}{\Bigg[{\int _{0}^{1}}{\int _{{\Gamma _{0}}}}|\xi (s,z){|^{p}}\nu (\mathrm{d}z)\mathrm{d}s\Bigg]^{\frac{1}{p}}}\lt \infty .\end{aligned}\]
-
• We introduce ${\mathbb{H}_{p}}$ as the set of all measurable adapted processes $h:\Omega \times [0,1]\to {\mathbb{R}^{d}}$ that possess a finite norm defined by
-
• Consider the space ${\mathbb{V}_{p}}$ of all predictable processes $\mathbf{v}:\Omega \times [0,1]\times {\Gamma _{0}}\to {\mathbb{R}^{d}}$ that satisfy the finite norm condition\[ \| \mathbf{v}{\| _{{\mathbb{V}_{p}}}}:=\| {\nabla _{z}}\mathbf{v}{\| _{{\mathbb{L}_{p}^{1}}}}+\| \mathbf{v}\varrho {\| _{{\mathbb{L}_{p}^{1}}}}\lt \infty ,\]where $\varrho (z)$ is defined by equation (3.1). For later discussions, we will use the notations
-
• The space ${\mathbb{H}_{0}}$ encompasses all bounded, measurable, and adapted processes $h:\Omega \times [0,1]\to {\mathbb{R}^{d}}$.
-
• The space ${\mathbb{V}_{0}}$ is constituted of all predictable processes $\mathbf{v}:\Omega \times [0,1]\times {\Gamma _{0}}\to {\mathbb{R}^{d}}$ satisfying the following conditions:
\[ W({h_{j}}):={\int _{0}^{1}}{\langle {h_{j}}(s),\mathrm{d}{W_{s}}\rangle _{{\mathbb{R}^{d}}}}\hspace{2.5pt}\text{and}\hspace{2.5pt}N({g_{k}}):={\int _{0}^{1}}{\int _{{\Gamma _{0}}}}{g_{k}}(s,z)N(\mathrm{d}s,\mathrm{d}z).\]
Given any $p\gt 1$ and $\Theta =(h,\mathbf{v})\in {\mathbb{H}_{p}}\times {\mathbb{V}_{p}}$, we denote
\[\begin{aligned}{}{D_{\Theta }}F& :={\sum \limits_{i=1}^{{m_{1}}}}({\partial _{i}}f)(\cdot ){\int _{0}^{1}}{\langle h(s),{h_{i}}\rangle _{{\mathbb{R}^{d}}}}\mathrm{d}s\\ {} & \hspace{1em}+{\sum \limits_{j=1}^{{m_{2}}}}({\partial _{j+{m_{1}}}}f)(\cdot ){\int _{0}^{1}}{\int _{{\Gamma _{0}}}}{\langle \mathbf{v}(s,z),{\nabla _{z}}{g_{j}}\rangle _{{\mathbb{R}^{d}}}}N(\mathrm{d}s,\mathrm{d}z),\end{aligned}\]
where “$(\cdot )$” represents the collection $W({h_{1}}),\dots ,W({h_{{m_{1}}}}),N({g_{1}}),\dots ,N({g_{{m_{2}}}})$.4 Regularity of the running maximum processes
In this section we discuss the results of Song and Xie [14] and their extensions. Let ${X^{(n)}}={\{{X_{s}^{(n)}}\}_{s\ge 0}}$ be a right continuous real-valued process. For any fixed $T\gt 0$ and $n\in \mathbb{N}$, in the following we shall write
Lemma 4.1.
Let ${X^{(n)}}={\{{X_{s}^{(n)}}\}_{s\ge 0}}$ and $X={\{{X_{s}}\}_{s\ge 0}}$ be a right continuous process for each $n\in \mathbb{N}$. Suppose that for some $p\gt 1$ and $\Theta =(h,\mathbf{v})\in {\mathbb{H}_{\infty -}}\times {\mathbb{V}_{\infty -}}$:
Then ${X_{T}^{\ast }}\in {\mathbb{W}_{\theta }^{1,p}}$ and the sequence ${\{{D_{\Theta }}{X_{s}^{(n)}}\}_{s\in [0,T]}}$ converges to ${D_{\Theta }}X={\{{D_{\Theta }}{X_{s}}\}_{s\in [0,T]}}$ in the weak topology of ${L^{p}}(\Omega \times [0,T])$. Moreover, if this ${D_{\Theta }}X$ has a right continuous version and
then the law of ${X_{T}^{\ast }}$ is absolutely continuous with respect to the Lebesgue measure.
-
1. ${\sup _{n\in \mathbb{N}}}\mathbb{E}[|{X_{s}^{(\ast ,n)}}{|^{p}}]\lt \infty $, and for any $s\in [0,T]$, ${X_{s}^{(n)}}\in {\mathbb{W}_{\Theta }^{1,p}}$, and
-
2. the process ${\{{D_{\Theta }}{X_{s}^{(n)}}\}_{s\in [0,T]}}$ possesses a right continuous version for each $n\in \mathbb{N}$;
-
3. ${\lim \nolimits_{n\to \infty }}\mathbb{E}[{\sup _{s\in [0,T]}}|{X_{s}^{(n)}}-{X_{s}}{|^{p}}]=0.$
(4.1)
\[ \mathbb{P}\big({D_{\Theta }}{X_{t}}\ne 0\hspace{2.5pt}\textit{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{T}^{\ast }}\big\}\big)=1,\]Proof.
It can be seen that ${X_{T}^{(\ast ,n)}}\in {\mathbb{W}_{\theta }^{1,p}}$ follows from Proposition 3.1 in [14] for each $n\in \mathbb{N}$. From Lemma 2.3 in [14], we obtain ${X_{T}^{\ast }}\in {\mathbb{W}_{\theta }^{1,p}}$ and
\[ \underset{n\to \infty }{\lim }{D_{\Theta }}{X_{\cdot }^{(n)}}={D_{\Theta }}{X_{\cdot }}\hspace{5pt}\text{weakly in}\hspace{2.5pt}{L^{p}}\big(\Omega \times [0,T]\big).\]
In exactly the same way as in Theorem 3.2 in [14], the following equality follows if ${D_{\Theta }}X$ has a right continuous path almost surely:
\[\begin{aligned}{}1& =\mathbb{P}\big({\big\{\exists t\in [0,T]\hspace{2.5pt}\text{such that}\hspace{2.5pt}{D_{\Theta }}{X_{t}}\ne {D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{and}\hspace{2.5pt}{X_{t}}={X_{T}^{\ast }}\big\}^{c}}\big)\\ {} & =\mathbb{P}\big({D_{\Theta }}{X_{t}}={D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in [0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big)\\ {} & \le \mathbb{P}\big({D_{\Theta }}{X_{t}}={D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big)\\ {} & =1.\end{aligned}\]
Subsequently, we prove that ${X_{T}^{\ast }}$ has a density function if (4.1) holds. In addition, by the closability of ${D_{\Theta }}$ (see Theorem 2.6 in [14]), we obtain
\[ \mathbb{P}\big({1_{A}}\big({D_{\Theta }}{X_{T}^{\ast }}\big){D_{\Theta }}{X_{T}^{\ast }}=0\big)=1,\]
for any $A\in \mathcal{B}(\mathbb{R})$ with $\text{Leb}(A)=0$. With these facts and (4.1) we obtain the following equation:
\[\begin{aligned}{}1& =\mathbb{P}\big(\big\{{1_{A}}\big({D_{\Theta }}{X_{T}^{\ast }}\big){D_{\Theta }}{X_{T}^{\ast }}=0\big\}\cap \big\{{D_{\Theta }}{X_{t}}={D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\\ {} & \hspace{1em}\hspace{1em}\cap \big\{{D_{\Theta }}{X_{t}}\ne 0\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\big)\\ {} & =\mathbb{P}\big(\big\{{1_{A}}\big({D_{\Theta }}{X_{T}^{\ast }}\big){D_{\Theta }}{X_{t}}=0\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\\ {} & \hspace{1em}\hspace{1em}\cap \big\{{D_{\Theta }}{X_{t}}={D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\\ {} & \hspace{1em}\hspace{1em}\cap \big\{{D_{\Theta }}{X_{t}}\ne 0\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\big)\\ {} & =\mathbb{P}\big(\big\{{1_{A}}\big({D_{\Theta }}{X_{T}^{\ast }}\big)=0\big\}\cap \big\{{D_{\Theta }}{X_{t}}={D_{\Theta }}{X_{T}^{\ast }}\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\\ {} & \hspace{1em}\hspace{1em}\cap \big\{{D_{\Theta }}{X_{t}}\ne 0\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{t}^{\ast }}\big\}\big\}\big)\\ {} & \le \mathbb{P}\big({1_{A}}\big({D_{\Theta }}{X_{T}^{\ast }}\big)=0\big)\\ {} & =1.\end{aligned}\]
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).
5 Applying of Malliavin calculus to SDEs
In this section, to find an equation satisfied by ${D_{\Theta }}X$ for X in equation (1.1), we check an equation satisfied by ${D_{\Theta }}{X^{(n)}}$ for ${X^{(n)}}$ in equation (1.2). The following lemma is shown in the same way as for Lemma 4.3 in [14].
Lemma 5.1 ([14], Lemma 4.3).
Assume that $b:\mathbb{R}\to \mathbb{R}$ is once differentiable and its derivative is bounded. Then for any $\Theta =(h,\mathbf{v})\in {\mathbb{H}_{\infty -}}\times {\mathbb{V}_{\infty -}}$ and $t\in [0,T]$, ${X_{t}^{(n)}}\in {\mathbb{W}_{\Theta }^{1,2}}$ and
The following lemma defines ${D_{\Theta }}X$ and confirms that it satisfies (5.1) below.
Lemma 5.2.
Assume the same assumptions as in Lemma 5.1. Then for some $p\in (1,\beta )$, for some $n\in \mathbb{N}$, for any $q\in (1,\beta )$ and for any $\Theta =(h,\mathbf{v})\in {\mathbb{H}_{\infty -}}\times {\mathbb{V}_{\infty -}}$, where
Then this ${D_{\Theta }}X$ is the limit of weak ${L^{p}}(\Omega \times [0,T])$ convergence of the sequence ${D_{\Theta }}{X^{(n)}}$, and this sequence is strongly ${L^{p}}(\Omega \times [0,T])$ convergent in practice.
\[ \underset{n\to \infty }{\lim }{\int _{0}^{T}}{\int _{|z|\gt n}}{\big|\mathbf{v}(s,z)\big|^{\frac{q}{\beta }}}\mathrm{d}s\nu (\mathrm{d}z)=0\hspace{5pt}\textit{and}\hspace{5pt}{\int _{0}^{T}}{\int _{|z|\gt n}}{\big|\mathbf{v}(s,z)\big|^{q}}\mathrm{d}s\nu (\mathrm{d}z)\lt \infty ,\]
${X_{t}^{\ast }}\in {\mathbb{W}_{\Theta }^{1,p}}$ for any $t\in [0,T]$, and
(5.1)
\[ {D_{\Theta }}{X_{t}}={\int _{0}^{t}}{b^{\prime }}({X_{s}}){D_{\Theta }}{X_{s}}\mathrm{d}s+{\sigma _{1}}{\int _{0}^{t}}h(s)\mathrm{d}s+{\sigma _{2}}{\int _{0}^{t}}{\int _{|z|\gt 0}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z).\]Proof.
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 ${D_{\Theta }}{X^{(n)}}$ 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 ${D_{\Theta }}$ (cf. Lemma 2.7 in [15]), we have
By using an inequality $|a+b{|^{p}}\le {2^{p-1}}(|a{|^{p}}+|b{|^{p}})$ for any $a,b\in \mathbb{R}$, we obtain
See Lemma A.6 for proof of (5.4). Here we show equation (5.3). Notice that $p\in (1,\beta )$, there exists $q\gt 1$ such that $pq\lt \beta $ because of the denseness of rational numbers. By using the Hölder inequality, we have
\[ \underset{n\to \infty }{\lim }{D_{\Theta }}{X_{\cdot }^{(n)}}={D_{\Theta }}{X_{\cdot }}\hspace{5pt}\text{weakly in}\hspace{5pt}{L^{p}}\big(\Omega \times [0,T]\big).\]
We verify that this ${D_{\Theta }}X={\{{D_{\Theta }}{X_{t}}\}_{t\in [0,T]}}$ satisfies equation (5.1). We set ${\{{Y_{t}}\}_{t\in [0,T]}}$ as a solution of
\[\begin{aligned}{}{Y_{t}}& ={\int _{0}^{t}}{b^{\prime }}({X_{s}}){Y_{s}}\mathrm{d}s+{C_{t}},\hspace{5pt}\text{where}\\ {} {C_{t}}& ={\sigma _{1}}{\int _{0}^{t}}h(s)\mathrm{d}s+{\sigma _{2}}{\int _{0}^{t}}{\int _{|z|\gt 0}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z),\\ {} {C_{t}^{(n)}}& ={\sigma _{1}}{\int _{0}^{t}}h(s)\mathrm{d}s+{\sigma _{2}}{\int _{0}^{t}}{\int _{0\lt |z|\le n}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z).\end{aligned}\]
We prove
(5.2)
\[ \underset{n\to \infty }{\lim }\mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}-{Y_{t}}\big|^{p}}\Big]=0.\]
\[\begin{aligned}{}& \mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}-{Y_{t}}\big|^{p}}\Big]\\ {} & \le {2^{p-1}}\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}\big\{{b^{\prime }}\big({X_{s}^{(n)}}\big){D_{\Theta }}{X_{s}^{(n)}}-{b^{\prime }}({X_{s}}){Y_{s}}\big\}\mathrm{d}s\Bigg|^{p}}\Bigg]\\ {} & \hspace{1em}+{2^{p-1}}\mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{C_{t}^{(n)}}-{C_{t}}\big|^{p}}\Big]\\ {} & \le {2^{2(p-1)}}{\int _{0}^{T}}\mathbb{E}\big[{\big|{b^{\prime }}\big({X_{s}^{(n)}}\big)-{b^{\prime }}({X_{s}})\big|^{p}}{\big|{D_{\Theta }}{X_{s}^{(n)}}\big|^{p}}\big]\mathrm{d}s\\ {} & \hspace{1em}+{2^{2(p-1)}}{\big\| {b^{\prime }}\big\| _{\infty }^{p}}{\int _{0}^{T}}\mathbb{E}\big[{\big|{D_{\Theta }}{X_{s}^{(n)}}-{Y_{s}}\big|^{p}}\big]\mathrm{d}s+{2^{p-1}}\mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{C_{t}^{(n)}}-{C_{t}}\big|^{p}}\Big].\end{aligned}\]
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
\[\begin{aligned}{}& \mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}-{Y_{t}}\big|^{p}}\Big]\\ {} & \le {2^{2(p-1)}}\exp \big({2^{2(p-1)}}\| {b^{\prime }}{\| _{\infty }^{p}}\big){\int _{0}^{T}}\mathbb{E}\big[{\big|{b^{\prime }}\big({X_{s}^{(n)}}\big)-{b^{\prime }}({X_{s}})\big|^{p}}{\big|{D_{\Theta }}{X_{s}^{(n)}}\big|^{p}}\big]\mathrm{d}s\\ {} & \hspace{1em}+{2^{2(p-1)}}\exp \big({2^{2(p-1)}}\| {b^{\prime }}{\| _{\infty }^{p}}\big)\mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{C_{t}^{(n)}}-{C_{t}}\big|^{p}}\Big].\end{aligned}\]
We show
\[\begin{aligned}{}& {\int _{0}^{T}}\mathbb{E}\big[{\big|{b^{\prime }}\big({X_{s}^{(n)}}\big)-{b^{\prime }}({X_{s}})\big|^{p}}{\big|{D_{\Theta }}{X_{s}^{(n)}}\big|^{p}}\big]\mathrm{d}s\\ {} & \le {\int _{0}^{T}}\mathbb{E}{\big[{\big|{b^{\prime }}\big({X_{s}^{(n)}}\big)-{b^{\prime }}({X_{s}})\big|^{\frac{pq}{q-1}}}\big]^{\frac{q-1}{q}}}\mathbb{E}{\big[{\big|{D_{\Theta }}{X_{s}^{(n)}}\big|^{pq}}\big]^{\frac{1}{q}}}\mathrm{d}s\\ {} & \le T\mathbb{E}{\Big[\underset{t\in [0,T]}{\sup }{\big|{b^{\prime }}\big({X_{t}^{(n)}}\big)-{b^{\prime }}({X_{t}})\big|^{\frac{pq}{q-1}}}\Big]^{\frac{q-1}{q}}}\mathbb{E}{\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}\big|^{pq}}\Big]^{\frac{1}{q}}}.\end{aligned}\]
Due to an inequality $|a+b{|^{p}}\le {2^{p-1}}(|a{|^{p}}+|b{|^{p}})$ for any $a,b\in \mathbb{R}$ and $p\ge 1$ and Jensen’s inequality, we have
\[\begin{aligned}{}& \mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}\big|^{pq}}\Big]\\ {} & \le {2^{pq-1}}\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}{b^{\prime }}\big({X_{s}^{(n)}}\big){D_{\Theta }}{X_{s}^{(n)}}\mathrm{d}s\Bigg|^{pq}}\Bigg]+{2^{2(pq-1)}}\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}h(s)\mathrm{d}s\Bigg|^{pq}}\Bigg]\\ {} & \hspace{1em}+{2^{2(pq-1)}}\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}{\int _{0\lt |z|\le n}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z)\Bigg|^{pq}}\Bigg]\\ {} & \le {2^{pq-1}}\| {b^{\prime }}{\| _{\infty }^{pq}}{\int _{0}^{T}}\mathbb{E}\Big[\underset{u\in [0,s]}{\sup }{\big|{D_{\Theta }}{X_{u}^{(n)}}\big|^{pq}}\Big]\mathrm{d}s\\ {} & \hspace{1em}+{2^{2(pq-1)}}\Bigg(\mathbb{E}\Bigg[{\int _{0}^{T}}{\big|h(s)\big|^{pq}}\mathrm{d}s\Bigg]+\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}{\int _{0\lt |z|\le n}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z)\Bigg|^{pq}}\Bigg]\Bigg).\end{aligned}\]
Gronwall’s inequality implies
\[\begin{aligned}{}& \mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}\big|^{pq}}\Big]\\ {} & \le {2^{2(pq-1)}}\Bigg(\mathbb{E}\Bigg[{\int _{0}^{T}}{\big|h(s)\big|^{pq}}\mathrm{d}s\Bigg]\\ {} & \hspace{1em}+\mathbb{E}\Bigg[\underset{t\in [0,T]}{\sup }{\Bigg|{\int _{0}^{t}}{\int _{0\lt |z|\le n}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z)\Bigg|^{pq}}\Bigg]\Bigg){e^{{2^{pq-1}}T\| {b^{\prime }}{\| _{\infty }^{pq}}}}.\end{aligned}\]
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
\[ \underset{n\in \mathbb{N}}{\sup }\mathbb{E}\Big[\underset{t\in [0,T]}{\sup }{\big|{D_{\Theta }}{X_{t}^{(n)}}\big|^{pq}}\Big]\lt \infty .\]
This allows us to confirm the ${L^{p}}(\Omega \times [0,T])$ integrability of ${D_{\Theta }}{X^{(n)}}$ for assumption 1 in Lemma 4.1, and that ${D_{\Theta }}X$ can be defined as the limit of weak ${L^{p}}(\Omega \times [0,T])$ convergence of ${D_{\Theta }}{X^{(n)}}$. By Lemma A.5, boundedness of ${b^{\prime }}$ and continuous mapping theorem, we have
\[ \underset{n\to \infty }{\lim }\mathbb{E}{\Big[\underset{t\in [0,T]}{\sup }{\big|{b^{\prime }}\big({X_{t}^{(n)}}\big)-{b^{\prime }}({X_{t}})\big|^{\frac{pq}{q-1}}}\Big]^{\frac{q-1}{q}}}=0,\]
so that we obtain (5.3). Thus, by (5.2) and completeness of ${L^{p}}(\Omega \times [0,T])$, (5.1) follows. □6 Proof of Theorem 2.1
Proof.
Applying the Itô formula to ${e^{-{\textstyle\int _{0}^{t}}{b^{\prime }}({X_{s}})\mathrm{d}s}}{D_{\Theta }}{X_{t}}$ (e.g., see [12], Corollary (Integration by Parts), P. 84), we obtain
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
we have
See Section A.1 for a proof of equation (6.2). So we have
\[\begin{aligned}{}{e^{-{\textstyle\textstyle\int _{0}^{t}}{b^{\prime }}({X_{s}})\mathrm{d}s}}{D_{\Theta }}{X_{t}}& ={\int _{0+}^{t}}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\circ \mathrm{d}{D_{\Theta }}{X_{s-}}+{\int _{0+}^{t}}{D_{\Theta }}{X_{s-}}\circ \mathrm{d}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\\ {} & ={\int _{0}^{t}}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\mathrm{d}{D_{\Theta }}{X_{s-}}+{\int _{0}^{t}}{D_{\Theta }}{X_{s-}}\mathrm{d}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\\ {} & ={\int _{0}^{t}}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\big({b^{\prime }}({X_{s}}){D_{\Theta }}{X_{s}}+{\sigma _{1}}h(s)\big)\mathrm{d}s\\ {} & \hspace{2em}+{\int _{0}^{t}}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}{\sigma _{2}}{\int _{|z|\gt 0}}\mathbf{v}(s,z)N(\mathrm{d}s,\mathrm{d}z)\\ {} & \hspace{2em}+{\int _{0}^{t}}{D_{\Theta }}{X_{s-}}\Bigg(-{b^{\prime }}({X_{s}}){\int _{0}^{t}}{e^{-{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\Bigg)\mathrm{d}s.\end{aligned}\]
For any $t\gt 0$, we set
(6.1)
\[\begin{aligned}{}& h(t):={\sigma _{1}}{e^{-{\textstyle\textstyle\int _{0}^{t}}{b^{\prime }}({X_{s}})\mathrm{d}s}},\hspace{1em}\mathbf{v}(t,z):={\sigma _{2}}{e^{-{\textstyle\textstyle\int _{0}^{t}}{b^{\prime }}({X_{s}})\mathrm{d}s}}\eta (z),\\ {} & \eta (z)=\left\{\begin{array}{l@{\hskip10.0pt}l}|z{|^{2}},\hspace{1em}& |z|\le \frac{1}{4},\\ {} 0,\hspace{1em}& |z|\gt \frac{1}{2},\\ {} \text{smooth},\hspace{1em}& \text{otherwise}.\end{array}\right.\end{aligned}\]
\[\begin{aligned}{}& {D_{\Theta }}{X_{t}}\\ {} & ={e^{{\textstyle\textstyle\int _{0}^{t}}{b^{\prime }}({X_{s}})\mathrm{d}s}}\Bigg({\sigma _{1}^{2}}{\int _{0}^{t}}{e^{-2{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\mathrm{d}s+{\sigma _{2}^{2}}{\int _{0}^{t}}{\int _{|z|\gt 0}}{e^{-2{\textstyle\textstyle\int _{0}^{s}}{b^{\prime }}({X_{u}})\mathrm{d}u}}\eta (z)N(\mathrm{d}s,\mathrm{d}z)\Bigg)\\ {} & \ge {e^{{\textstyle\textstyle\int _{0}^{t}}({b^{\prime }}({X_{s}})-2\| b{\| _{\text{Lip}}})\mathrm{d}s}}\Bigg({\sigma _{1}^{2}}t+{\sigma _{2}^{2}}{\int _{0}^{t}}{\int _{|z|\gt 0}}\eta (z)N(\mathrm{d}s,\mathrm{d}z)\Bigg).\end{aligned}\]
Noticing the condition ${\sigma _{1}^{2}}+{\sigma _{2}^{2}}\ne 0$ and the fact
(6.2)
\[ \mathbb{P}\Bigg({\int _{0}^{t}}{\int _{|z|\gt 0}}\eta (z)N(\mathrm{d}s,\mathrm{d}z)\gt 0,\hspace{2.5pt}\forall t\gt 0\Bigg)=1,\]
\[ 1=\mathbb{P}\big({D_{\Theta }}{X_{t}}\gt 0,\hspace{2.5pt}\forall t\in (0,T]\big)\le \mathbb{P}\big({D_{\Theta }}{X_{t}}\ne 0\hspace{2.5pt}\text{on}\hspace{2.5pt}\big\{t\in (0,T]:{X_{t}}={X_{T}^{\ast }}\big\}\big)=1.\]
Therefore, we conclude by Lemma 4.1 that the law of ${X_{T}^{\ast }}$ is absolutely continuous with respect to the Lebesgue measure. □
