Exponential Stability of Solutions to Stochastic Differential Equations Driven by G-Lévy Process

Abstract

In this paper, BDG-type inequality for G-stochastic calculus with respect to G-Lévy process is obtained, and solutions of the stochastic differential equations driven by the G-Lévy process under the non-Lipschitz condition are constructed. Furthermore, the mean square exponential stability and quasi-sure exponential stability of the solution using the G-Lyapunov function method is established.

Appendix

Proof of Lemma 2.3

First, let us consider the case:

$$\begin{aligned} K(r,z)(w)=\sum _{k=1}^{n-1}\sum _{l=1}^{m}F_{k,l}(w) {\mathbb {I}}_{]t_{k},t_{k+1}]}(r)\psi _{l}(z)\in H_{G}^{S}([0,T]\times R_{0}^{d}). \end{aligned}$$

For this case, the proof is similar to [9, Theorem 27]. However, for the sake of completeness, we present the prove as follows. Based on the definition of the Itô integral and Theorem 2.4, we have

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K(r,z)L(dr,dz)\right) ^{2}\right] =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P^{\theta }}\left[ \sup _{0\le t\le T}\left( \sum _{0\le r\le t}K(r,\triangle X_{r})\right) ^{2}\right] \nonumber \\&\quad =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P^{\theta }}\nonumber \\&\quad \left[ \sup _{0\le t\le T}\left( \sum _{0\le r\le t}\sum _{k=1}^{n-1}\sum _{l=1}^{m}\phi _{k,l}\left( X_{t_{1}\wedge t},\ldots ,X_{t_{k}\wedge t}-X_{t_{k-1}\wedge t}\right) {\mathbb {I}}_{]t_{k}\wedge t,t_{k+1}\wedge t]}(r)\psi _{l}(\triangle X_{r})\right) ^{2}\right] \nonumber \\&\quad =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\nonumber \\&\quad \left[ \sup _{0\le t\le T} \left( \sum _{0\le r\le t}\sum _{k=1}^{n-1}\sum _{l=1}^{m}\phi _{k,l}\left( B_{t_{1}\wedge t}^{0,\theta },\ldots ,B_{t_{k}\wedge t}^{t_{k-1}\wedge t,\theta }\right) {\mathbb {I}}_{]t_{k}\wedge t,t_{k+1}\wedge t]}(r)\psi _{l}\left( \triangle B_{r}^{0,\theta }\right) \right) ^{2}\right] \nonumber \\&\quad =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\left[ \sup _{0\le t\le T}\left( \sum _{0\le r\le t}\sum _{k=1}^{n-1}\sum _{l=1}^{m}F_{k,l}^{\theta }{\mathbb {I}}_{]t_{k}\wedge t,t_{k+1}\wedge t]}(r)\psi _{l}(\theta ^{d}(r,\triangle N_{r}))\right) ^{2}\right] , \end{aligned}$$

(5.1)

where $F_{k,l}^{\theta }:=\phi _{k,l}(B_{t_{1}\wedge t}^{0,\theta },\ldots ,B_{t_{k}\wedge t}^{t_{k-1}\wedge t,\theta })$ and $N_{t}$ is the Poisson process in Theorem 2.4. We define a predictable process $K^{\theta }(r,z)$ as

$$\begin{aligned} K^{\theta }(r,z):=\sum _{k=1}^{n-1}\sum _{l=1}^{m}F_{k,l}^{\theta }{\mathbb {I}}_{]t_{k}\wedge t,t_{k+1}\wedge t]}(r)\psi _{l}(\theta ^{d}(r,z)). \end{aligned}$$

We can then rewrite (5.1) as

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K(r,z)L(dr,dz)\right) ^{2}\right] \nonumber \\&\quad =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z)N(dr,dz)\right) ^{2}\right] \nonumber \\&\quad =\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z){\tilde{N}}(dr,dz) +\int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z)\mu (dz)dr\right) ^{2}\right] , \end{aligned}$$

(5.2)

$\square$

where N(dr, dz) and ${\tilde{N}}(dr,dz)$ are respectively the Poisson random measure and compensated Poisson measure associated with the Lévy process with the Lévy triplet $(0,0,\mu ).$ Using the standard BDG inequality and Hölder inequality, we obtain

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K(r,z)L(dr,dz)\right) ^{2}\right] \nonumber \\&\quad \le 2\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\left\{ \sup _{0\le t\le T}\left[ \left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z){\tilde{N}}(dr,dz)\right) ^{2}\right. \right. \nonumber \\&\qquad \left. \left. +\,\left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z)\mu (dz)dr\right) ^{2}\right] \right\} \nonumber \\&\quad \le 2\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}}\left\{ \sup _{0\le t\le T}\left[ \left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z){\tilde{N}}(dr,dz)\right) ^{2}\right. \right. \nonumber \\&\qquad \left. \left. +\, t\int _{0}^{t}\int _{R_{0}^{d}}(K^{\theta }(r,z))^{2}\mu (dz)dr\right] \right\} \nonumber \\&\quad \le 2\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\left\{ E^{P_{0}}\sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K^{\theta }(r,z){\tilde{N}}(dr,dz)\right) ^{2}\right. \nonumber \\&\qquad \left. +\,E^{P_{0}}T\int _{0}^{T}\int _{R_{0}^{d}}\left( K^{\theta }(r,z)\right) ^{2}\mu (dz)dr\right\} \nonumber \\&\quad \le 2\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\left\{ {\tilde{C}}_{T}E^{P_{0}} \int _{0}^{T}\int _{R_{0}^{d}}(K^{\theta }(r,z))^{2}\mu (dz)dr\right. \nonumber \\&\qquad \left. +E^{P_{0}}T\int _{0}^{T}\int _{R_{0}^{d}}(K^{\theta }(r,z))^{2}\mu (dz)dr\right\} \nonumber \\&\quad =C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\int _{0}^{T} \int _{R_{0}^{d}}E^{P_{0}}\left( \sum _{k=1}^{n-1}\sum _{l=1}^{m}F_{k,l}^{\theta } {\mathbb {I}}_{]t_{k},t_{k+1}]}(r)\psi _{l}(\theta ^{d}(r,z))\right) ^{2}\mu (dz)dr, \end{aligned}$$

(5.3)

where ${\tilde{C}}_{T}$ is a constant that depends on T and $C_{T}=2(T+{\tilde{C}}_{T}).$ It should be noted that the intervals $]t_{k},t_{k+1}]$ are mutually disjoint, and hence,

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K(r,z)L(dr,dz)\right) ^{2}\right] \nonumber \\&\quad \le C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\sum _{k=1}^{n-1}\sum _{l=1}^{m} \int _{t_{k}}^{t_{k+1}}E^{P_{0}}\left[ \left( F_{k,l}^{\theta }\right) ^{2}\int _{R_{0}^{d}} \psi _{l}^{2}\left( \theta ^{d}(r,z)\right) \mu (dz)\right] dr \nonumber \\&\quad = C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\sum _{k=1}^{n-1}\int _{t_{k}}^{t_{k+1}} E^{P_{0}}\left[ \sum _{l=1}^{m}\phi _{k,l}^{2}\left( B_{t_{1}}^{0,\theta },\ldots , B_{t_{k}}^{t_{k-1},\theta }\right) \int _{R_{0}^{d}}\psi _{l}^{2}(\theta ^{d}(r,z))\mu (dz)\right] dr .\end{aligned}$$

(5.4)

Based on the assumptions on the process $\theta ^{d},$ we know that for a.a. w and a.e. r function $z\rightarrow \theta ^{d}(r,z)(w)$ is equal to $g_{v}$ for $v\in {\mathcal {V}}.$ Hence, we can transform (5.4) to obtain

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( \int _{0}^{t}\int _{R_{0}^{d}}K(r,z)L(dr,dz)\right) ^{2}\right] \nonumber \\&\quad \le C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\sum _{k=1}^{n-1}\int _{t_{k}}^{t_{k+1}} E^{P_{0}}\left[ \sum _{l=1}^{m}\phi _{k,l}^{2}\left( B_{t_{1}}^{0,\theta }, \ldots ,B_{t_{k}}^{t_{k-1},\theta }\right) \int _{R_{0}^{d}}\psi _{l}^{2}(z)v(dz)\right] dr \nonumber \\&\quad \le C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}} \left[ \sum _{k=1}^{n-1}\int _{t_{k}}^{t_{k+1}}\sup _{v\in {\mathcal {V}}}\sum _{l=1}^{m} \phi _{k,l}^{2}\left( B_{t_{1}}^{0,\theta },\ldots ,B_{t_{k}}^{t_{k-1},\theta }\right) \int _{R_{0}^{d}}\psi _{l}^{2}(z)v(dz)dr\right] \nonumber \\&\quad \le C_{T}\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}E^{P_{0}} \left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}\sum _{k=1}^{n-1}\sum _{l=1}^{m} \phi _{k,l}^{2}\left( B_{t_{1}}^{0,\theta },\ldots ,B_{t_{k}}^{t_{k-1},\theta }\right) {\mathbb {I}}_{]t_{k},t_{k+1}]}(r)\psi _{l}^{2}(z)v(dz)dr\right] \nonumber \\&\quad = C_{T}\hat{{\mathbb {E}}}\left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}} K^{2}(r,z)v(dz)dr\right] . \end{aligned}$$

(5.5)

For general $K(r,z)\in H_{G}^{2}([0,T]\times R_{0}^{d}),$ we choose $\{K^{n},n\ge 1\} \subset H_{G}^{S}([0,T]\times R_{0}^{d}),$ such that

$$\begin{aligned} \Vert K-K^{n}\Vert _{H_{G}^{2}([0,T] \times R_{0}^{d})}\rightarrow 0 \quad as \quad n\rightarrow \infty . \end{aligned}$$

We set $Y_{t}^{n}=\int _{0}^{t}\int _{R_{0}^{d}}K^{n}(r,z)L(dr,dz).$ Then, as $n,m\rightarrow \infty ,$

$$\begin{aligned} \hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( Y_{t}^{n}-Y_{t}^{m}\right) ^{2}\right] \le C_{T}\Vert K^{n}-K^{m}\Vert ^{2}_{H_{G}^{2}([0,T] \times R_{0}^{d})}\rightarrow 0, \end{aligned}$$

and thus, there exists a subsequence $\{Y_{t}^{n_{k}},k\ge 1\},$ such that for any $k\ge 1,$

$$\begin{aligned} \left( \hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right) ^{2}\right] \right) ^{\frac{1}{2}}\le \frac{1}{2^{k}}. \end{aligned}$$

Then

$$\begin{aligned}&\left( \hat{{\mathbb {E}}}\left[ \sum _{k=1}^{\infty }\sup _{0\le t\le T}\left( Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right) \right] ^{2}\right) ^{\frac{1}{2}}=\sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\left( E^{P^{\theta }}\left( \sum _{k=1}^{\infty }\sup _{0\le t\le T}\left( Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right) \right) ^{2}\right) ^{\frac{1}{2}}\nonumber \\&\quad \le \sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\sum _{k=1}^{\infty }\left( E^{P^{\theta }}\left( \sup _{0\le t\le T}\left( Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right) \right) ^{2}\right) ^{\frac{1}{2}}\le \sum _{k=1}^{\infty }\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left( Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right) ^{2}\right] \right) ^{\frac{1}{2}}\nonumber \\&\quad \le 1, \end{aligned}$$

(5.6)

which implies that

$$\begin{aligned} \sum _{k=1}^{\infty }\sup _{0\le t\le T}\left| Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}\right| < \infty , \; q.s. \end{aligned}$$

If we set ${\tilde{Y}}_{t}=Y_{t}^{n_{1}}+\sum _{k=1}^{\infty }(Y_{t}^{n_{k+1}}-Y_{t}^{n_{k}}),$ then ${\tilde{Y}}_{t}$ is q.s. defined on $\Omega$ for all $t\in [0,T]$ and for q.s. w,  $t\rightarrow {\tilde{Y}}_{t}(w)$ is càdlàg. Moreover, $(\hat{{\mathbb {E}}}[\sup _{0\le t\le T}{\tilde{Y}}_{t}^{2}])^{\frac{1}{2}}<\infty ,$ and

$$\begin{aligned}&\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left| Y_{t}^{n_{k}}-{\tilde{Y}}_{t}\right| ^{2}\right] \right) ^{\frac{1}{2}} \le \left( \hat{{\mathbb {E}}}\left( \sum _{l=k}^{\infty }\sup _{0\le t\le T}\left| Y_{t}^{n_{l+1}}-Y_{t}^{n_{l}}\right| \right) ^{2}\right) ^{\frac{1}{2}}\nonumber \\&\quad \le \sup _{\theta \in {\mathcal {A}}_{0,T}^{{\mathcal {U}}}}\left( E^{P^{\theta }}\left( \sum _{l=k}^{\infty }\sup _{0\le t\le T}\left| Y_{t}^{n_{l+1}}-Y_{t}^{n_{l}}\right| \right) ^{2}\right) ^{\frac{1}{2}}\nonumber \\&\quad \le \sum _{l=k}^{\infty }\left( \hat{{\mathbb {E}}}\sup _{0\le t\le T}\left| Y_{t}^{n_{l+1}}-Y_{t}^{n_{l}}\right| ^{2}\right) ^{\frac{1}{2}}\rightarrow 0\quad as \; k\rightarrow \infty . \end{aligned}$$

(5.7)

Moreover, by using $\Vert K-K^{n_{k}}\Vert _{H_{G}^{2}([0,T] \times R_{0}^{d})}\rightarrow 0,$ we have

$$\begin{aligned}&\left| \left( \hat{{\mathbb {E}}}\left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}K^{2}(r,z)v(dz)dr\right] \right) ^{\frac{1}{2}} -\left( \hat{{\mathbb {E}}}\left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}\left| K^{n_{k}}(r,z)\right| ^{2}v(dz)dr\right] \right) ^{\frac{1}{2}}\right| \nonumber \\&\quad \le \left\{ \hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}\sup _{v\in {\mathcal {V}}} \int _{R_{0}^{d}}K^{2}(r,z)v(dz)dr\right) ^{\frac{1}{2}}-\left( \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}|K^{n_{k}}(r,z)|^{2} v(dz)dr\right) ^{\frac{1}{2}}\right] ^{2}\right\} ^{\frac{1}{2}}\nonumber \\&\quad \le \left( \hat{{\mathbb {E}}}\left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}\left| K(r,z)-K^{n_{k}}(r,z)\right| ^{2}v(dz)dr\right] \right) ^{\frac{1}{2}} \rightarrow 0 \end{aligned}$$

(5.8)

as $k\rightarrow \infty .$ Then, on combining (5.5)–(5.8), we obtain

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left| {\tilde{Y}}_{t}\right| ^{2}\right] \le \hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left| {\tilde{Y}}_{t}-Y_{t}^{n_{k}}\right| ^{2}\right] +\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}\left| Y_{t}^{n_{k}}\right| ^{2}\right] \nonumber \\&\quad \le C_{T}\hat{{\mathbb {E}}}\left[ \int _{0}^{T}\sup _{v\in {\mathcal {V}}}\int _{R_{0}^{d}}K^{2}(r,z)v(dz)dr\right] . \end{aligned}$$

(5.9)

Finally, as for any $t\in [0,T],$ $\hat{{\mathbb {E}}}[|Y_{t}^{n_{k}}-Y_{t}|^{2}]\rightarrow 0,$ we have $\hat{{\mathbb {E}}}[|Y_{t}-{\tilde{Y}}_{t}|^{2}]= 0,$ and thus, ${\tilde{Y}}$ is a càdlàg modification of Y.