# Existence of solution for stochastic differential equations driven by *G*-Lévy process with discontinuous coefficients

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## Abstract

The existence theory for the vector-valued stochastic differential equations driven by *G*-Brownian motion and pure jump *G*-Lévy process (*G*-SDEs) of the type \(dY_{t}=f(t,Y_{t})\, dt+g_{j,k}(t,Y_{t})\, d\langle B^{j},B^{k}\rangle _{t}+\sigma_{i}(t,Y_{t}) \, dB^{i}_{t}+\int _{R_{0}^{d}}K(t,Y_{t},z)L(dt,dz)\), \(t\in[0,T]\), with first two and last discontinuous coefficients, is established. It is shown that the *G*-SDEs have more than one solution if the coefficients *f*, *g*, *K* are discontinuous functions. The upper and lower solution method is used.

## Keywords

stochastic differential equations*G*-Lévy process upper and lower solution discontinuous coefficients

## MSC

60H05 60H10 60H20## 1 Introduction

In recent years much effort has been made to develop the theory of sublinear expectations connected with the volatility uncertainty and the so-called *G*-Brownian motion. *G*-Brownian motion was introduced by Shige Peng in [1, 2] as a way to incorporate the unknown volatility into financial models. Its theory is tightly associated with the uncertainty problems involving an undominated family of probability measures. Soon other connections have been discovered, not only in the field of financial mathematics, but also in the theory of path-dependent partial differential equations or backward stochastic differential equations. Thus *G*-Brownian motion and connected *G*-expectation are attractive mathematical objects.

Returning, however, to the original problem of volatility uncertainty in the financial models, one feels that *G*-Brownian motion is not sufficient to model the financial world, as both *G*- and the standard Brownian motion share the same property, which makes them often unsuitable for modeling, namely, the continuity of paths. Therefore, it is not surprising that Hu and Peng [3] introduced the process with jumps, which they called *G*-Lévy process. Then Ren [4] introduced the representation of the sublinear expectation as an upper-expectation. In [5], the author concentrated on establishing the integration theory for *G*-Lévy process with finite activity, introduced the integral w.r.t. the jump measure associated with the pure jump *G*-Lévy process and gave the Itô formula for general *G*-Itô Lévy process.

*G*-Lévy process, Paczka [5] established the existence and uniqueness of solutions for the following stochastic differential equation driven by

*G*-Brownian motion and pure jump

*G*-Lévy process with Lipschitz continuous coefficients:

*G*-Brownian motion \((B_{t})_{t\geq0}\), \(L(t,z)\) is pure jump

*G*-Lévy process. For each \(x\in R^{n}\), the coefficients \(f(t,x),g_{j,k}(t,x),\sigma _{i}(t,x)\) are in the space \(\hat{M}_{G}^{2}(0,T; R^{n})\), \(K(t,x,z)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n} )\) (which will be introduced in Section 2). A process \(Y_{t}\) belonging to \(\hat {M}_{G}^{2}(0,T;R^{n})\) and satisfying

*G*-SDE (1.1) is said to be its solution.

Motivated by the importance of discontinuous functions, Faizullah and Piao [6] established the existence of solutions for the stochastic differential equations driven by *G*-Brownian motion with a discontinuous drift coefficient. Then Faizullah [7] developed the existence theory when the coefficient *f* or the coefficients *f* and *g* simultaneously are discontinuous functions. Motivated by the aforementioned works, in this paper, we consider equation (1.1) and assume that \(f(t,x)\), \(g(t,x) \) and \(K(t,x,z)\) are discontinuous for all \(x\in R^{n}\).

The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, the existence of solutions for *G*-SDE (1.1) with simultaneous discontinuous coefficients *f*, *g* and *K* is developed.

## 2 Preliminaries

In this section, we introduce some notations and preliminary results in *G*-framework which are needed in the sequence. More details can be found in [5, 8, 9, 10, 11, 12, 13].

### Definition 2.1

- (i)
Monotonicity: \(\mathbb{E}[X]\geq\mathbb{E}[Y]\) if \(X\geq Y\);

- (ii)
Constant preserving: \(\mathbb{E}[C]=C\) for \(C\in R\);

- (iii)
Sub-additivity: \(\mathbb{E}[X+Y]\leq\mathbb {E}[X]+\mathbb{E}[Y]\);

- (iv)
Positive homogeneity: \(\mathbb{E}[\lambda X]=\lambda \mathbb {E}[X]\) for \(\lambda\geq0\).

The triple \((\Omega,\mathcal{H},\mathbb{E})\) is called a sublinear expectation space. \(X\in\mathcal{H}\) is called a random variable in \((\Omega,\mathcal{H},\mathbb{E})\). We often call \(Y=(Y_{1},\ldots,Y_{d}), Y_{i}\in\mathcal{H}\) a *d*-dimensional random vector in \((\Omega,\mathcal{H},\mathbb{E})\).

### Definition 2.2

*n*-dimensional random vector \(Y=(Y_{1},\ldots,Y_{n})\) is said to be independent from an

*m*-dimensional random vector \(X=(X_{1},\ldots,X_{m})\) if for each \(\varphi\in C_{b,\mathrm{lip}}(R^{m+ n})\),

### Definition 2.3

*n*-dimensional random vectors defined on sublinear expectation spaces \((\Omega_{1},\mathcal{H}_{1},\mathbb {E}_{1})\) and \((\Omega_{2},\mathcal{H}_{2},\mathbb{E}_{2})\), respectively. They are called identically distributed, denoted by \(X_{1}\overset{d}{=}X_{2}\), if

*X̄*is said to be an independent copy of

*X*if

*X̄*is identically distributed with

*X*and independent of

*X*.

### Definition 2.4

*G*-Lévy process

*d*-dimensional càdlàg process on a sublinear expectation space \((\Omega,\mathcal{H},\mathbb{E})\). We say that

*X*is a Lévy process if:

- (i)
\(X_{0}=0\),

- (ii)
for each \(s,t\geq0\), the increment \(X_{t+s}-X_{s}\) is independent of \((X_{t_{1}},\ldots, X_{t_{n}})\) for every \(n\in N\) and every partition \(0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{n}\leq s\),

- (iii)
the distribution of the increment \(X_{t+s}-X_{s}\), \(s, t\geq0\) is stationary, i.e., does not depend on

*s*.

*X*is a

*G*-Lévy process if it satisfies additionally the following conditions:

- (iv)
there is a 2

*d*-dimensional Lévy process \((X_{t}^{c},X_{t}^{d})_{t\geq0}\) such that for each \(t\geq0 \), \(X_{t}=X_{t}^{c}+X_{t}^{d}\), - (v)processes \(X_{t}^{c}\) and \(X_{t}^{d}\) satisfy the following conditions:$$ \lim_{t\downarrow0}\mathbb{E}\bigl[ \bigl\vert X_{t}^{c} \bigr\vert ^{3}\bigr]t^{-1}=0; \qquad\mathbb{E}\bigl[ \bigl\vert X_{t}^{d} \bigr\vert \bigr]< Ct \quad\mbox{for all } t\geq0. $$

Peng and Hu noticed in their paper that each *G*-Lévy process might be characterized by a non-local operator \(G_{X}\).

### Theorem 2.1

[3]

*Let*

*X*

*be a*

*G*-

*Lévy process in*\(R^{d}\).

*For every*\(f\in C_{b}^{3}(R^{d})\)

*such that*\(f(0)=0\),

*we put*

*The above limit exists*.

*Moreover*, \(G_{X}\)

*has the following Levy*-

*Khintchine representation*:

*where*\(R_{0}^{d}:=R^{d}\backslash\{0\}\), \(\mathcal{U}\)

*is a subset*\(\mathcal{U}\subset\mathcal{V}\times R^{d}\times R^{d\times d}\)

*and*\(\mathcal{V}\)

*is a set of all Borel measures on*\((R_{0}^{d}, \mathcal {B}(R_{0}^{d}))\).

*We know additionally that*\(\mathcal{U}\)

*has the property*

### Theorem 2.2

[3]

*Let*

*X*

*be a*

*d*-

*dimensional*

*G*-

*Lévy process*.

*For each*\(\phi\in C_{b,\mathrm{lip}}(R^{d})\),

*define*\(u(t,x):=\mathbb{E}[\phi (x+X_{t})]\).

*Then*

*u*

*is the unique viscosity solution of the following integro*-

*PDE*:

*with the initial condition*\(u(0,x)=\phi(x)\).

### Theorem 2.3

*Let* \(\mathcal{U}\) *satisfy* (2.1). *Consider the canonical space* \(\Omega :=\mathbb{D}_{0}(R^{+},R^{d})\) *of all càdlàg functions taking values in* \(R^{d}\) *equipped with the Skorohod topology*. *Then there exists a sublinear expectation* \(\hat{\mathbb{E}}\) *on* \(\mathbb {D}_{0}(R^{+},R^{d})\) *such that the canonical process* \((X_{t})_{t\geq 0}\) *is a* *G*-*Lévy process satisfying Levy*-*Khintchine representation with the same set* \(\mathcal{U}\).

The proof might be found in [3]. We will give, however, the construction of \(\hat{\mathbb{E}}\) as it is important to understand it.

*u*is a unique viscosity solution of integro-PDE (2.2) with the initial condition \(u(0,x)=\phi(x)\). For general

*σ*-algebra of Ω. It was proved in [4] that there exists a weakly compact probability measure family \(\mathcal{P}\) defined on \((\Omega, \mathcal{B}(\Omega ))\) such that

*P*.

### Definition 2.5

*c*associated with \(\hat{\mathbb{E}}\) by putting

### Remark 2.1

The condition (v) in Definition 2.4 implies that \(X^{c}\) is a *d*-dimensional generalized *G*-Brownian motion and the pure jump part \(X^{d}\) is of finite variation (see [3]). Moreover, \(X^{c}\) is just the *d*-dimensional *G*-Brownian motion \(B_{t}\) when \(p=0\) in (2.1). In this paper, we always let \(p=0\), i.e., the *G*-Lévy process *X* consists of *G*-Brownian motion \(B_{t}\) and the pure jump part.

For each \(\eta\in M_{G}^{p}(0,T)\), \(p\geq2\), the *G*-Itô integral \(\{\int_{0}^{t}\eta_{s}\, dB^{i}_{s}\}_{t\in[0,T]}\) is well defined. For each \(\eta_{s}^{j,k}\in M_{G}^{p}(0,T)\), \(p\geq1\), the integral \(\{ \int_{0}^{t}\eta_{s}^{j,k}\, d\langle B^{j},B^{k}\rangle_{s}\}_{t\in [0,T]}\) is well defined. \(i,j,k=1,\ldots,d\). See Peng [12] and Li et al. [13].

### Lemma 2.1

[7]

*Let*\(\eta_{t}^{j,k},\zeta_{t}^{j,k}\in M_{G}^{1}(0,T)\).

*If*\(\eta _{t}^{j,k}\leq\zeta_{t}^{j,k}\)

*for*\(t\in[0,T]\),

*then*

*G*-Lévy process

*X*has finite activity, i.e.,

*X*at point

*u*, \(\Delta X_{u}=X_{u}-X_{u-}\), then we can define a random measure \(L(\cdot, \cdot)\) associated with the

*G*-Lévy process

*X*by putting

### Definition 2.6

*L*is defined as

### Lemma 2.2

*For every* \(K\in H_{G}^{S}([0,T]\times R_{0}^{d})\), *we have that* \(\int _{0}^{T}\int_{R_{0}^{d}}K(u,z)L(du,dz)\) *is an element of* \(L_{G}^{2}(\Omega_{T})\).

Let \(H_{G}^{p}([0,T]\times R_{0}^{d})\) denote the topological completion of \(H_{G}^{S}([0,T]\times R_{0}^{d})\) under the norm \(\| \cdot\|_{H_{G}^{p}([0,T]\times R_{0}^{d})}\), \(p=1,2\). Then Itô integral can be continuously extended to the whole space \(H_{G}^{p}([0,T]\times R_{0}^{d})\), \(p=1,2\). Moreover, by Lemma 2.2 we know that the extended operator takes value in \(L_{G}^{2}(\Omega _{T})\), \(p=1,2\).

### Lemma 2.3

*Let*\(K^{1}(t,z),K^{2}(t,z)\in H_{G}^{2}([0,T]\times R_{0}^{d})\).

*If*\(K^{1}(t,z)\leq K^{2}(t,z)\)

*for*\(t\in[0,T]\),

*then*

### Proof

*G*-SDEs, let us introduce the new norm on the integrands: for a process

*η*, define

*G*-SDE driven by

*d*-dimensional

*G*-Brownian motion

*B*and the pure jump

*G*-Lévy process

*L*(in this paper we always use Einstein’s convention):

Let \(\hat{M}_{G}^{2}(0,T;R^{n})\) denote the space of \(R^{n}\)-valued process and for each element belong to \(\hat{M}_{G}^{2}(0,T)\). We can define the space \(\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\) in a similar way.

### Theorem 2.4

[5]

*Suppose that* \(f(t,x)\), \(g_{j,k}(t,x)\), \(\sigma_{i}(t,x)\), \(K(t,x)\) *are Lipschitz continuous w*.*r*.*t*. *x* *uniformly*. *For each* \(x\in R^{n}\), \(f(\cdot,x),g_{j,k}(\cdot,x), \sigma_{i}(\cdot,x) \in\hat {M}_{G}^{2}(0,T;R^{n})\), \(K(\cdot,x,\cdot)\in\hat {H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\). *Then* *G*-*SDE* (2.3) *with the initial condition* \(Y_{0}\in R^{n}\) *has a unique solution* \(Y_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\).

## 3 Existence of solution for *G*-SDEs with discontinuous coefficients

### Definition 3.1

*G*-SDE (2.3) on the interval \([0,T]\).

### Definition 3.2

*G*-SDE (2.3) on the interval \([0,T]\).

*G*-SDE

*x*. Define two functions \(p,q:[0,T]\times R^{n}\times\Omega\rightarrow R^{n}\) by

*G*-SDE:

*f̃*, \(\tilde{g}_{j,k}\), \(\tilde{\sigma }_{i}\),

*K̃*are Lipschitz continuous in

*x*, thus the conditions of Theorem 2.4 are satisfied. Then

*G*-SDE (3.5) has a unique solution \(Y_{t}\in\hat{M}_{G}^{2}(0,T;R^{n})\).

### Lemma 3.1

*Suppose that* \(U_{t}\) *and* \(L_{t}\) *are the respective upper and lower solutions of* *G*-*SDE* (3.3) *satisfying* \(L_{t}\leq U_{t}\) *for* \(t\in [0,T]\). *Then* \(U_{t}\) *and* \(L_{t}\) *are the upper and lower solutions of* *G*-*SDE* (3.5), *respectively*.

### Proof

*G*-SDE (3.5). One can show that \(L_{t}\) is a lower solution of

*G*-SDE (3.5) in a similar way as above. □

*G*-SDE (3.3), then they are the respective upper and lower solutions for

*G*-SDE (3.5). Suppose that \(Y_{t}\) is the solution of

*G*-SDE (3.5) such that

*G*-SDE (3.3). Thus, if we can show that any solution \(Y_{t}\) of problem (3.5) does satisfy inequality (3.8), then \(Y_{t}\) is also the solution of

*G*-SDE (3.3).

### Theorem 3.1

*Suppose that*

- (i)
\(f(\cdot,w),g_{j,k}(\cdot,w)\in\hat{M}_{G}^{2}(0,T;R^{n})\), \(K(\cdot,w,\cdot)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\)

*for*\(w\in\Omega\), \(\sigma_{i}(\cdot,x)\in\hat {M}_{G}^{2}(0,T;R^{n})\)*for each*\(x\in R^{n}\)*and*\(\sigma(t,x)\)*is Lipschitz continuous in**x*; - (ii)
*the respective upper and lower solutions*\(U_{t}\)*and*\(L_{t}\)*of**G*-*SDE*(3.3)*satisfy*\(L_{t}< U_{t}\)*for*\(t\in[0,T]\); - (iii)
\(Y_{0}\in R^{n}\)

*is a given initial value with*\(\hat {\mathbb {E}}[|Y_{0}|^{2}]<\infty\)*and*\(L_{0}< Y_{0}< U_{0}\).

*Then there exists a unique solution*\(Y_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\)

*of*

*G*-

*SDE*(3.3)

*such that*\(L_{t}< Y_{t}< U_{t}\)

*for*\(t\in[0,T]\),

*q*.

*s*.

### Proof

*G*-SDE (3.5) does satisfy inequality (3.8). Assume that there exists an arbitrary interval \((t_{1},t_{2})\subset[0,T]\) such that \(Y_{t_{1}}=L_{t_{1}}\) and \(Y_{t}< L_{t}\) for \(t\in(t_{1},t_{2})\), then we have

*G*-SDE:

*x*, only \(\sigma_{i}(t,x)\) is Lipschitz continuous in

*x*.

### Theorem 3.2

*Suppose that*

- (i)
*for each*\(x\in R^{n}\), \(f(\cdot,x), g_{j,k}(\cdot ,x),\sigma _{i}(\cdot,x)\in\hat{M}_{G}^{2}(0,T; R^{n})\), \(K(\cdot,x,\cdot)\in \hat{H}_{G}^{2}([0,T]\times R_{0}^{d}; R^{n})\); - (ii)
\(\sigma_{i}(t,x)\)

*is Lipschitz continuous in**x*, \(f(t,x)\), \(g_{j,k}(t,x)\)*and*\(K(t,x,z)\)*are increasing in**x*; - (iii)
\(U_{t}\)

*and*\(L_{t}\)*are the respective upper and lower solutions of**G*-*SDE*(3.12).*Moreover*, \(K(\cdot,U_{t},\cdot), K(\cdot ,L_{t},\cdot)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d}; R^{n})\)*and*\(L_{t}\leq U_{t}\)*for*\(t\in[0,T]\).

*Then there exists at least one solution*\(Y_{t}\in\hat{M}_{G}^{2}(0,T; R^{n})\)

*of*

*G*-

*SDE*(3.12)

*such that*\(L_{t}\leq Y_{t}\leq U_{t}\)

*for*\(t\in[0,T]\),

*q*.

*s*.

### Proof

Denote the order interval \([L,U]\) in \(\hat{M}_{G}^{2}(0,T; R^{n})\) by \(\mathcal{H}\), that is, \(\mathcal{H}=\{Y:Y \in\hat{M}_{G}^{2}(0,T; R^{n}) \mbox{ and } L_{t}\leq Y_{t}\leq U_{t}\}\) for \(t\in[0,T]\), which is closed and bounded. By using the monotone convergence theorem in [5], one can prove the convergence of a monotone sequence that belongs to \(\mathcal{H}\) in \(\hat{M}_{G}^{2}(0,T; R^{n})\). Thus \(\mathcal{H}\) is a regularly ordered metric space with the norm of \(\hat{M}_{G}^{2}(0,T; R^{n})\).

*x*, it is easy to see that for any process \(V\in\mathcal{H}\), \(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions for the

*G*-SDE

*G*-SDE (3.13) has a unique solution \(Y_{t}\in\hat{M}_{G}^{2}(0,T; R^{n})\) such that \(L_{t}\leq Y_{t}\leq U_{t}\) for \(t\in[0,T]\), q.s.

*Y*is the unique solution of

*G*-SDE (3.13). For all \(t\in[0,T]\), let \(V^{1}_{t}, V^{2}_{t} \in\mathcal{H}\) and \(V^{1}_{t}\leq V^{2}_{t}\) and define \(Y^{1}_{t}=F(V^{1}_{t})\), \(Y^{2}_{t}=F(V^{2}_{t})\). Since

*f*,

*g*,

*K*are increasing functions, then

*G*-SDE

*G*-SDE (3.15) \(Y^{2}_{t}\) satisfies \(Y^{1}_{t}\leq Y^{2}_{t}\leq U_{t}\). Hence

*F*is an increasing mapping and, by Theorem 3.3, it has a fixed point \(Y^{*}=F(Y^{*})\in\mathcal{H}\) such that \(L_{t}\leq Y^{*}_{t} \leq U_{t}\), q.s. and

### Example 3.1

For the following definition and theorem, see [14].

### Definition 3.3

An ordered metric space *M* is called regularly (resp. fully regularly) ordered if each monotone and order (resp. metrically ) bounded ordinary sequence of *M* converges.

### Theorem 3.3

*If* \([a,b]\) *is a non*-*empty order interval in a regularly ordered metric space*, *then each increasing mapping* \(F: [a,b]\rightarrow[a,b]\) *has the least and the greatest fixed points*.

## Notes

### Acknowledgements

The work is supported in part by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant No. 11526103), National Natural Science Foundation of China (Grant No. 11601203).

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