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

R E S E A R C H

Open Access

Existence of solution for stochastic

differential equations driven by

G

-Lévy

process with discontinuous coefficients

Bingjun Wang

_{and Mingxia Yuan}

*_{Correspondence:}

[email protected]

1_{School of Mathematical Science,}

Nanjing Normal University, Nanjing, 210023, P.R. China

2_{Jinling Institute of Technology,}

Nanjing, 211169, P.R. China Full list of author information is available at the end of the article

Abstract

The existence theory for the vector-valued stochastic differential equations driven by G-Brownian motion and pure jumpG-Lévy process (G-SDEs) of the type

dYt=f(t,Yt)dt+gj,k(t,Yt)dBj,Bkt+

σ

Rd₀K(t,Yt,z)L(dt,dz),t∈[0,T], with first two and last discontinuous coefficients, is established. It is shown that theG-SDEs have more than one solution if the coefficientsf,g,Kare discontinuous functions. The upper and lower solution method is used.

MSC: 60H05; 60H10; 60H20

Keywords: stochastic differential equations;G-Lévy process; upper and lower solution; discontinuous coefficients

1 Introduction

In recent years much effort has been made to develop the theory of sublinear expecta-tions connected with the volatility uncertainty and the so-calledG-Brownian motion. G-Brownian motion was introduced by Shige Peng in [, ] as a way to incorporate the un-known volatility into financial models. Its theory is tightly associated with the uncertainty problems involving an undominated family of probability measures. Soon other connec-tions 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. ThusG-Brownian motion and connectedG-expectation are attractive mathe-matical objects.

Returning, however, to the original problem of volatility uncertainty in the financial models, one feels thatG-Brownian motion is not sufficient to model the financial world, as bothG- 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 surpris-ing that Hu and Peng [] introduced the process with jumps, which they calledG-Lévy process. Then Ren [] introduced the representation of the sublinear expectation as an upper-expectation. In [], the author concentrated on establishing the integration theory forG-Lévy process with finite activity, introduced the integral w.r.t. the jump measure as-sociated with the pure jumpG-Lévy process and gave the Itô formula for generalG-Itô Lévy process.

Under the integration theory forG-Lévy process, Paczka [] established the existence and uniqueness of solutions for the following stochastic differential equation driven by G-Brownian motion and pure jumpG-Lévy process with Lipschitz continuous coefficients:

Yt=Y+ t



f(s,Ys)ds+ t



gj,k(s,Ys)dBj,Bk_s

+ t



σi(s,Ys)dBis+ t



Rd

K(s,Ys,z)L(ds,dz), (.)

where Y∈Rn_{, (B}j_,Bk

t)t≥ is the mutual variation process of the G-Brownian mo-tion (Bt)t≥, L(t,z) is pure jump G-Lévy process. For each x ∈ Rn_{, the coefficients} f(t,x),gj,k(t,x),σi(t,x) are in the spaceMˆG(,T;Rn),K(t,x,z)∈ ˆHG([,T]×Rd;Rn) (which will be introduced in Section ). A processYt belonging toMˆG(,T;Rn) and satisfying G-SDE (.) is said to be its solution.

Motivated by the importance of discontinuous functions, Faizullah and Piao [] es-tablished 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 coefficientf or the coefficientsf andgsimultaneously are dis-continuous functions. Motivated by the aforementioned works, in this paper, we consider equation (.) and assume thatf(t,x),g(t,x) andK(t,x,z) are discontinuous for allx∈Rn_. The rest of this paper is organized as follows. In Section , we introduce some prelimi-naries. In Section , the existence of solutions forG-SDE (.) with simultaneous discon-tinuous coefficientsf,gandKis 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 [, –].

Definition . Letbe a given set, and letHbe a linear space of real-valued functions de-fined on. Moreover, ifXi∈H,i= , , . . . ,d, thenϕ(X, . . . ,Xd)∈Hfor allϕ∈Cb,lip(Rd_), whereCb,lip(Rd_{) is the space of all bounded real-valued Lipschitz continuous functions.} A sublinear expectationEis a functionalE:H→Rsatisfying the following properties: for allX,Y∈H, we have:

(i) Monotonicity:E[X]≥E[Y]ifX≥Y; (ii) Constant preserving:E[C] =CforC∈R; (iii) Sub-additivity:E[X+Y]≤E[X] +E[Y]; (iv) Positive homogeneity:E[λX] =λE[X]forλ≥.

The triple (,H,E) is called a sublinear expectation space.X∈His called a random variable in (,H,E). We often callY= (Y, . . . ,Yd),Yi∈Had-dimensional random vector in (,H,E).

Definition . In a sublinear expectation space (,H,E), ann-dimensional random vec-tor Y = (Y, . . . ,Yn) is said to be independent from an m-dimensional random vector X= (X, . . . ,Xm) if for eachϕ∈Cb,lip(Rm+n_),

Definition . LetX, X be twon-dimensional random vectors defined on sublinear expectation spaces (,H,E) and (,H,E), respectively. They are called identically distributed, denoted byX=dX, if

E

ϕ(X)=E

ϕ(X), ∀ϕ∈Cb,lipRn .

¯

Xis said to be an independent copy ofXifX¯ is identically distributed withXand inde-pendent ofX.

Definition .(G-Lévy process) LetX= (Xt)t≥be ad-dimensional càdlàg process on a sublinear expectation space (,H,E). We say thatXis a Lévy process if:

(i) X= ,

(ii) for eachs,t≥, the incrementXt+s–Xsis independent of(Xt, . . . ,Xtn)for every

n∈Nand every partition≤t≤t≤ · · · ≤tn≤s,

(iii) the distribution of the incrementXt+s–Xs,s,t≥is stationary, i.e., does not depend ons.

Moreover, we say that a Lévy processXis aG-Lévy process if it satisfies additionally the following conditions:

(iv) there is ad-dimensional Lévy process(X_tc,X_td)t≥such that for eacht≥,

Xt=Xtc+Xtd,

(v) processesXc

t andXtdsatisfy the following conditions:

lim

t↓EX c t



t–= ; EXd_t<Ct for allt≥.

Peng and Hu noticed in their paper that eachG-Lévy process might be characterized by a non-local operatorGX.

Theorem .([]) Let X be a G-Lévy process in Rd_{.For every f∈C}

b(Rd)such that f() = , we put

GX

f(·):=lim δ↓E

f(Xδ)

δ–.

The above limit exists.Moreover,GXhas the following Levy-Khintchine representation:

GXf(·)= sup

(v,p,Q)∈U

Rd_

f(z)v(dz) +Df(),p+ tr

Df()QQT,

where Rd

:=Rd\{},Uis a subsetU⊂V×Rd×Rd×dandVis a set of all Borel measures on(Rd

,B(Rd)).We know additionally thatUhas the property

sup

(v,p,Q)∈U

Rd_

|z|v(dz) +|p|+trQQT<∞. (.)

integro-PDE:

 =∂tu(t,x) –GXu(t,x+·) –u(t,x) =∂tu(t,x) – sup

(v,p,Q)∈U

Rd



u(t,x+z) –u(t,x)v(dz)

+Du(t,x),p+ tr

Du(t,x)QQT, (.)

with the initial condition u(,x) =φ(x).

Theorem . LetUsatisfy(.).Consider the canonical space:=D(R+,Rd)of all càdlàg functions taking values in Rd equipped with the Skorohod topology. Then there exists a sublinear expectationEˆ onD(R+,Rd)such that the canonical process(Xt)t≥is a G-Lévy process satisfying Levy-Khintchine representation with the same setU.

The proof might be found in []. We will give, however, the construction ofEˆ as it is important to understand it.

We denoteT:={w·∧T:w∈}. Put

Lip(T) :=

ξ∈L(T) :ξ=φ(Xt,Xt–Xt, . . . ,Xtn–Xtn–),

φ∈Cb,lip

Rd×n , ≤t<· · ·<tn<T

,

whereXt(w) =wtis the canonical process on the spaceD(R+_,Rd_{) andL}_{() is the space} of all random variables, which are measurable to the filtration generated by the canonical process. We also set

Lip() := ∞

T=

Lip(T).

Firstly, consider the random variableξ=φ(Xt+s–Xs),φ∈Cb,lip(Rd). We define

ˆ

E[ξ] :=u(s, ),

where u is a unique viscosity solution of integro-PDE (.) with the initial condition u(,x) =φ(x). For general

ξ=φ(Xt,Xt–Xt, . . . ,Xtn–Xtn–), φ∈Cb,lip

Rd×n ,

we setEˆ[ξ] :=φn, whereφnis obtained via the following iterated procedure: φ(x, . . . ,xn–) =Eˆφ(x, . . . ,xn–,Xtn–Xtn–)

,

φ(x, . . . ,xn–) =Eˆφ(x, . . . ,xn–,Xtn––Xtn–)

,

.. .

φn–(x) =Eˆφn–(x,Xt–Xt)

,

φn=Eˆ

φn–(Xt)

Lastly, we extend the definition ofEˆ on the completion ofLip(T) (respectivelyLip()) under the norm · pp=Eˆ[| · |p],p≥. We denote such a completion byLpG(T) (or resp. Lp_G()).

LetB() be the Borelσ-algebra of. It was proved in [] that there exists a weakly compact probability measure familyPdefined on (,B()) such that

ˆ

E[X] =sup

p∈PEP[X], ∀X∈L

 G(),

whereEPis the linear expectation with respect toP.

Definition . We define the capacitycassociated withEˆ by putting

c(A) :=sup

p∈PP(A), A∈B().

We will say that a setA∈B() is polar ifc(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 thatXc_{is ad-dimensional}

gen-eralized G-Brownian motion and the pure jump part Xd _{is of finite variation (see []).} Moreover,Xc_{is just thed-dimensionalG-Brownian motionBtwhenp=  in (.). In this} paper, we always letp= , i.e., theG-Lévy processXconsists ofG-Brownian motionBt and the pure jump part.

LetM_G,p(,T) be the collection of processes of the following form: for a given partition

{t, . . . ,tN}=πTof [,T],

ηt= N–

i–

ξi(w)I[ti,ti+)(t),

whereξi∈LpG(ti),i= , , . . . ,N– ,p≥. For eachp≥, denote byMpG(,T) the com-pletion ofM,p_G (,T) under the normη_Mp

G:= (

ˆ

E[_T|ηt|pdt])



p_.

For eachη∈M_Gp(,T),p≥, theG-Itô integral{_tηsdBis}t∈[,T]is well defined. For each ηj,ks ∈MGp(,T),p≥, the integral{

t η

j,k

s dBj,Bks}t∈[,T]is well defined.i,j,k= , . . . ,d. See Peng [] and Li et al. [].

Lemma .([]) Letη_tj,k,ζ_tj,k∈M_G(,T).Ifηj,k_t ≤ζ_tj,kfor t∈[,T],then T



ηj,k_t dBj,Bk_t≤ T



ζ_tj,kdBj,Bk_t.

Assume that theG-Lévy processXhas finite activity, i.e.,

λ:=sup

v∈Vv

Rd

LetXu–denote the left limit ofXat pointu,Xu=Xu–Xu–, then we can define a random measureL(·,·) associated with theG-Lévy processXby putting

L(]s,t],A) = s<u≤t

IA(Xu), q.s.

for any  <s<t<∞andA∈B(Rd_). The random measure is well defined and may be used to define the pathwise integral.

LetHS

introduce the norm on this space

To consider the solution ofG-SDEs, let us introduce the new norm on the integrands: for a processη, define

ηp_ˆ Mp_G(,T):=

T



ˆ

E|ηt|p

dt, p≥.

The completion of the space under this norm will be denoted asMˆp_G(,T). Note that

ˆ

E

T



|ηt|pdt

≤

T



ˆ

E|ηt|p

dt,

thus appropriate integrals will be always well defined.

Similarly, we need to adjust the space of integrands for the jump measure. LetHˆ_G([, T]×Rd_) denote the completion of allH_GS([,T]×Rd_) under the norm

K_Hˆ

G([,T]×Rd):=

T



ˆ

E

sup

v∈V

Rd_

K(u,z)v(dz)

du.

We consider the followingG-SDE driven byd-dimensionalG-Brownian motionBand the pure jumpG-Lévy processL(in this paper we always use Einstein’s convention):

dYt=f(t,Yt)dt+gj,k(t,Yt)d

Bj,Bk_t+σi(t,Yt)dBit

+

Rd

K(t,Yt,z)L(dt,dz). (.)

LetMˆ

G(,T;Rn) denote the space ofRn-valued process and for each element belong to

ˆ

M

G(,T). We can define the spaceHˆG([,T]×Rd;Rn) in a similar way.

Theorem . ([]) Suppose that f(t,x), gj,k(t,x), σi(t,x), K(t,x) are Lipschitz

continu-ous w.r.t. x uniformly. For each x∈Rn, f(·,x),gj,k(·,x),σi(·,x)∈ ˆM_G(,T;Rn), K(·,x,·)∈

ˆ

H_G([,T]×Rd_;Rn_{).Then G-SDE(.)with the initial condition Y∈R}n_{has a unique} so-lution Yt∈ ˆM

G(,T;Rn).

3 Existence of solution forG-SDEs with discontinuous coefficients

Definition . If, for any ≤s≤t, the processUt∈ ˆMG(,T;Rn) satisfies the following inequality:

Ut≥Us+ t

s

f(u,Uu)du+ t

s

gj,k(u,Uu)dBj,Bk_u

+ t

s

σi(u,Uu)dBiu+ t

s

Rd

K(u,Uu,z)L(du,dz), (.)

Definition . If, for any ≤s≤t, the processLt∈ ˆM

G(,T;Rn) satisfies the following inequality:

Lt≤Ls+ t

s

f(u,Lu)du+ t

s

gj,k(u,Lu)d

Bj,Bk_u

+ t

s

σi(u,Lu)dBi_u+ t

s

Rd_

K(u,Lu,z)L(du,dz), (.)

q.s., then it is said to be a lower solution ofG-SDE (.) on the interval [,T].

Suppose thatUtandLt are the respective upper and lower solutions of theG-SDE

dYt=f(t,w)dt+gj,k(t,w)dBj,Bk_t+σi(t,Yt)dBi_t

+

Rd



K(t,w,z)L(dt,dz), t∈[,T],w∈, (.)

wheref(·,w),gj,k(·,w)∈ ˆM

G(,T;Rn),K(·,w,·)∈ ˆHG([,T]×Rd;Rn) forw∈,σi(·,x)∈

ˆ

M

G(,T;Rn) for eachx∈Rnandσi(t,x) is Lipschitz continuous inx. Define two functions p,q: [,T]×Rn_×→Rn_by

p(t,x,w) =maxLt(w),min

Ut(w),x

,

q(t,x,w) =p(t,x,w) –x,

(.)

and consider the followingG-SDE:

dYt=f˜(t,Yt)dt+gj,k(t,˜ Yt)dBj,Bk_t+σ˜i(t,Yt)dBi_t +

Rd_

˜

K(t,Yt,z)L(dt,dz), t∈[,T] (.)

with a given constant initial conditionY∈Rn, where

˜

f(t,x,w) =f(t,w) +q(t,x,w),

˜

gj,k(t,x,w) =gj,k(t,w) +q(t,x,w),

˜

σi(t,x,w) =σi

t,p(t,x,w) ,

˜

K(t,x,z) =K(t,w,z) +q(t,x,w).

(.)

It is clear thatf˜,gj,k,˜ σ˜i,K˜ are Lipschitz continuous inx, thus the conditions of Theorem . are satisfied. ThenG-SDE (.) has a unique solutionYt∈ ˆM_G(,T;Rn_).

Lemma . Suppose that Utand Ltare the respective upper and lower solutions of G-SDE

Proof For ≤s≤t, we have ofG-SDE (.). One can show thatLt is a lower solution ofG-SDE (.) in a similar way

as above.

By Lemma . we know that ifUtandLt are upper and lower solutions ofG-SDE (.), then they are the respective upper and lower solutions forG-SDE (.). Suppose thatYtis the solution ofG-SDE (.) such that

Lt<Yt<Ut, t∈[,T], q.s. (.)

which implies thatYtis a solution ofG-SDE (.). Thus, if we can show that any solution Ytof problem (.) does satisfy inequality (.), thenYtis also the solution ofG-SDE (.).

Theorem . Suppose that

(i) f(·,w),gj,k(·,w)∈ ˆM_G(,T;Rn),K(·,w,·)∈ ˆH_G([,T]×Rd_;Rn)forw∈,

σi(·,x)∈ ˆM

G(,T;Rn)for eachx∈Rnandσ(t,x)is Lipschitz continuous inx;

(iii) Y∈Rn_{is a given initial value withE}ˆ_[|Y|_{] <∞andL<Y<U.}

Then there exists a unique solution Yt∈ ˆM_G(,T;Rn_{)of G-SDE(.)such that Lt<Yt<Ut} for t∈[,T],q.s.

Proof We only need to prove that the solutionYt ofG-SDE (.) does satisfy inequality (.). Assume that there exists an arbitrary interval (t,t)⊂[,T] such thatYt=Ltand

which yields a contradiction. ThusYt ≥Lt fort∈[,T]. By using similar arguments as above, one can show thatYt≤Utfort∈[,T]. Thus the proof is finished.

Now we consider the followingG-SDE:

wheref(t,x),gj,k(t,x) andK(t,x,z) do not need to be Lipschitz continuous with respect to x, onlyσi(t,x) is Lipschitz continuous inx.

Theorem . Suppose that

(i) for eachx∈Rn_,f(·,x),g

j,k(·,x),σi(·,x)∈ ˆMG(,T;Rn),K(·,x,·)∈ ˆHG([,T]×Rd;Rn);

(ii) σi(t,x)is Lipschitz continuous inx,f(t,x),gj,k(t,x)andK(t,x,z)are increasing inx; (iii) Ut andLt are the respective upper and lower solutions ofG-SDE(.).Moreover,

K(·,Ut,·),K(·,Lt,·)∈ ˆH monotone convergence theorem in [], one can prove the convergence of a monotone se-quence that belongs toHinMˆ

G(,T;Rn). ThusHis a regularly ordered metric space with the norm ofMˆ

G(,T;Rn).

Sincef(t,x),gj,k(t,x) andK(t,x,z) are increasing inx, it is easy to see that for any process V∈H,UtandLtare the respective upper and lower solutions for theG-SDE

dYt=f(t,Vt)dt+gj,k(t,Vt)d Sincef,g,Kare increasing functions, then

Y_t =FV_t

which implies thatY_tis a lower solution of theG-SDE

Yt=Y+

However, this problem has an upper solutionUt. Then the solution ofG-SDE (.)Y t satisfiesY

fixed pointY∗=F(Y∗)∈Hsuch thatLt≤Y_t∗≤Ut, q.s. and

Thus the proof is finished.

Example . Consider the following scalar stochastic differential equation:

dYt=H(Yt)dt+{Yt}dBt+dBt+

Rd_

{Yt}L(dt,dz), (.)

where the Heaviside functionH:R→Ris defined by

H(x) =

This is an important function in science, and it is considered to be a fundamental func-tion in engineering. The fracfunc-tional part funcfunc-tion{x}:R→[, ) has discontinuities at the integers and is defined by

{x}=x– [x], x∈R,

where [x] is the floor function. The importance of this function is clear from the sawtooth waves which are used in music and computer graphics.

LetUt=U+_tdu+_tdBu+

thatUtis the upper solution of equation (.). In a similar way, one can show thatLt= L+_tdBuis a lower solution of equation (.). Then, by Theorem ., there exists at least one solution for equation (.).

For the following definition and theorem, see [].

Theorem . If[a,b]is a non-empty order interval in a regularly ordered metric space, then each increasing mapping F: [a,b]→[a,b]has the least and the greatest fixed points.

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).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both of the authors jointly worked on deriving the results and approved the final manuscript.

Author details

1_{School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, P.R. China.}2_{Jinling Institute of Technology,}

Nanjing, 211169, P.R. China. 3_{Jinling College, Nanjing University, Nanjing, 210089, P.R. China.}

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Received: 6 March 2017 Accepted: 13 June 2017

References

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2. Peng, S:G-Expectations,G-Brownian motion and related stochastic calculus of Itô’s type. In: Benth, FE, Di Nunno, G, Lindstrom, T, Øksendal, B, Zhang, T (eds.) Proceedings of the 2005 Abel Symposium, pp. 541-567. Springer, Berlin (2006). Preprint version in: arXiv:math/0601035v2

3. Hu, M, Peng, S:G-Lévy processes under sublinear expectations. arXiv:0911.3533v1

4. Ren, L: On representation theorem of sublinear expectation related toG-Lévy processes and paths ofG-Lévy processes. Stat. Probab. Lett.83, 1301-1310 (2013)

5. Paczka, K: Itô calculus and jump diffusions forG-Lévy processes. arXiv:1211.2973v3

6. Faizullah, F, Piao, D: Existence of solutions forG-SDEs with upper and lower solutions in the reverse order. Int. J. Phys. Sci.7, 432-439 (2012)

7. Faizullah, F: Existence of solutions for stochastic differential equations underG-Brownian motion with discontinuous coefficients. Z. Naturforsch. A67, 692-698 (2012)

8. Paczka, K:G-Martingale representation in theG-Lévy setting. arXiv:1404.2121v1

9. Denis, L, Hu, M, Peng, S: Function spaces and capacity related to a sublinear expectation: application toG-Brownian motion paths. Potential Anal.34, 139-161 (2011)

10. Soner, M, Touzi, N, Zhang, J: Martingale representation theorem for theG-expectation. Stoch. Process. Appl.121, 265-287 (2011)

11. Peng, S: Multi-dimensionalG-Brownian motion and related stochastic calculus underG-expectation. Stoch. Process. Appl.118, 2223-2253 (2008)

12. Peng, S: Nonlinear expectations and stochastic calculus under uncertainly. arXiv:1002.4546v1 (2010)

13. Li, X, Peng, S: Stopping times and related Itô’s calculus withG-Brownian motion. Stoch. Process. Appl.121, 1492-1508 (2011)