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

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 dYt=f(t,Yt)dt+gj,k(t,Yt)d〈Bj,Bk〉t+σi(t,Yt)dBti+∫R0dK(t,Yt,z)L(dt,dz)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$dY_{t}=f(t,Y_{t})\, dt+g_{j,k}(t,Y_{t})\, d\langle B^{j},B^{k}\rangle _{t}+\sigma_{i}(t,Y_{t}) \, dB^{i}_{t}+\int _{R_{0}^{d}}K(t,Y_{t},z)L(dt,dz)$\end{document}, t∈[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in[0,T]$\end{document}, 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.


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 [, ] 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 [] introduced the process with jumps, which they called G-Lévy process. Then Ren [] introduced the representation of the sublinear expectation as an upper-expectation. In [], 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.
Under the integration theory for G-Lévy process, Paczka [] 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: R n ) (which will be introduced in Section ). A process Y t belonging toM  G (, T; R n ) and satisfying G-SDE (.) is said to be its solution.
Motivated by the importance of discontinuous functions, Faizullah and Piao [] established the existence of solutions for the stochastic differential equations driven by G-Brownian motion with a discontinuous drift coefficient. Then Faizullah [] 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 (.) and assume that f (t, x), g(t, x) and K(t, x, z) are discontinuous for all x ∈ R n .
The rest of this paper is organized as follows. In Section , we introduce some preliminaries. In Section , the existence of solutions for G-SDE (.) with simultaneous discontinuous coefficients f , g and K is developed.

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 [, -].
Definition . Let be a given set, and let H be a linear space of real-valued functions defined on . Moreover, if X i ∈ H, i = , , . . . , d, then ϕ(X  , . . . , is the space of all bounded real-valued Lipschitz continuous functions. A sublinear expectation E is a functional E : H → R satisfying the following properties: for all X, Y ∈ H, we have: Definition . In a sublinear expectation space ( , H, E), an n-dimensional random vector Y = (Y  , . . . , Y n ) is said to be independent from an m-dimensional random vector Definition . Let X  , X  be two n-dimensional random vectors defined on sublinear expectation spaces (  , H  , E  ) and (  , H  , E  ), respectively. They are called identically distributed, denoted by X is said to be an independent copy of X ifX is identically distributed with X and independent of X.
Definition . (G-Lévy process) Let X = (X t ) t≥ be a d-dimensional càdlàg process on a sublinear expectation space ( , H, E). We say that X is a Lévy process if: , does not depend on s. Moreover, we say that a Lévy process X is a G-Lévy process if it satisfies additionally the following conditions: t and X d t satisfy the following conditions: Peng and Hu noticed in their paper that each G-Lévy process might be characterized by a non-local operator G X .
The above limit exists. Moreover, G X has the following Levy-Khintchine representation: . We know additionally that U has the property . Then u is the unique viscosity solution of the following integro-PDE: The proof might be found in []. We will give, however, the construction ofÊ as it is important to understand it.
We denote T := {w ·∧T : w ∈ }. Put where X t (w) = w t is the canonical process on the space D  (R + , R d ) and L  ( ) is the space of all random variables, which are measurable to the filtration generated by the canonical process. We also set Firstly, consider the random variable ξ = φ(X t+s -X s ), φ ∈ C b,lip (R d ). We definê where u is a unique viscosity solution of integro-PDE (.) with the initial condition we setÊ[ξ ] := φ n , where φ n is obtained via the following iterated procedure: Lastly, we extend the definition ofÊ on the completion of Lip( T ) (respectively Lip( )) under the norm · p p =Ê[| · | p ], p ≥ . We denote such a completion by L p G ( T ) (or resp. L p G ( )). Let B( ) be the Borel σ -algebra of . It was proved in [] that there exists a weakly compact probability measure family P defined on ( , B( )) such that where E P is the linear expectation with respect to P. We will say that a set A ∈ B( ) is polar if c(A) = . We say that a property holds quasi-surely (q.s.) if it holds outside a polar set.

Remark . The condition (v) in Definition . implies that X c is a d-dimensional generalized G-Brownian motion and the pure jump part X d is of finite variation (see []).
Moreover, X c is just the d-dimensional G-Brownian motion B t when p =  in (.). In this paper, we always let p = , i.e., the G-Lévy process X consists of G-Brownian motion B t and the pure jump part.
Assume that the G-Lévy process X has finite activity, i.e., Let X u-denote the left limit of X at point u, X u = X u -X u-, then we can define a random measure L(·, ·) associated with the G-Lévy process X by putting

L(]s, t], A)
for any  < s < t < ∞ and A ∈ B(R d  ). The random measure is well defined and may be used to define the pathwise integral. Let We introduce the norm on this space , by Definition ., the following holds: , the inequality still holds under a regular argument.
To consider the solution of G-SDEs, let us introduce the new norm on the integrands: for a process η, define The completion of the space under this norm will be denoted asM thus appropriate integrals will be always well defined.
Similarly, we need to adjust the space of integrands for the jump measure.
We consider the following 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): Then G-SDE (.) with the initial condition Y  ∈ R n has a unique solution Y t ∈M  G (, T; R n ).

Existence of solution for G-SDEs with discontinuous coefficients
Definition . If, for any  ≤ s ≤ t, the process U t ∈M  G (, T; R n ) satisfies the following inequality: Definition . If, for any  ≤ s ≤ t, the process L t ∈M  G (, T; R n ) satisfies the following inequality: q.s., then it is said to be a lower solution of G-SDE (.) on the interval [, T].
Suppose that U t and L t are the respective upper and lower solutions of the G-SDE where f (·, w), g j,k (·, w) ∈M  G (, T; R n ), K(·, w, ·) ∈Ĥ  G ([, T] × R d  ; R n ) for w ∈ , σ i (·, x) ∈ M  G (, T; R n ) for each x ∈ R n and σ i (t, x) is Lipschitz continuous in x. Define two functions p, q : [, T] × R n × → R n by p(t, x, w) = max L t (w), min U t (w), x , q(t, x, w) = p(t, x, w)x, (.) and consider the following G-SDE: with a given constant initial condition Y  ∈ R n , wherẽ f (t, x, w) = f (t, w) + q(t, x, w), g j,k (t, x, w) = g j,k (t, w) + q(t, x, w), σ i (t, x, w) = σ i t, p(t, x, w) , K(t, x, z) = K(t, w, z) + q(t, x, w). (iii) Y  ∈ R n is a given initial value withÊ[|Y  |  ] < ∞ and L  < Y  < U  . Then there exists a unique solution Y t ∈M  G (, T; R n ) of G-SDE (.) such that L t < Y t < U t for t ∈ [, T], q.s.