Modern Stochastics: Theory and Applications

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 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 $n\in \mathbb{N}$: 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 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.

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.

Our results are described below.

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

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 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: We denote the compensated Poisson random measure N by Let $p\ge 1$, $i\in \{1,2\}$, and let k be an integer. We introduce the following spaces for subsequent discussions. 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 Given any $p\gt 1$ and $\Theta =(h,\mathbf{v})\in {\mathbb{H}_{p}}\times {\mathbb{V}_{p}}$, we denote where “$(\cdot )$” represents the collection $W({h_{1}}),\dots ,W({h_{{m_{1}}}}),N({g_{1}}),\dots ,N({g_{{m_{2}}}})$.

It is well known that the Banach space ${\mathbb{W}_{\Theta }^{1,p}}$ 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.

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

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).

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].

The following lemma defines ${D_{\Theta }}X$ and confirms that it satisfies (5.1) below.

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.

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