Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients

Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients

Faiz Faizullah

Department of Basic Sciences and Humanities, College of Electrical and Mechanical Engineering (EME), National University of Sciences and Technology (NUST), Islamabad, Pakistan and College of Physical and Environmental Oceanography, Ocean University of China, Qingdao 266100, PR China

Reprint requests to F. F.; E-mail:faiz math@yahoo.com

Z. Naturforsch.67a,692 – 698 (2012) / DOI: 10.5560/ZNA.2012-0092

Received May 15, 2012 / revised August 16, 2012 / published online November 28, 2012

The existence theory for the vector valued stochastic differential equations under G-Brownian mo- tion (G-SDEs) of the typeXt=X0+^R₀^tf(v,X_v)dv+^R₀^tg(v,X_v)dhBiv+^R₀^th(v,X_v)dB_v,t∈[0,T],with first two discontinuous coefficients is established. It is shown that the G-SDEs have more than one solution if the coefficientgor the coefficients f andgsimultaneously, are discontinuous functions.

The upper and lower solutions method is used and examples are given to explain the theory and its importance.

Key words:Stochastic Differential Equations; G-Brownian Motion; Discontinuous Coefficients;

Existence; Upper and Lower Solutions.

Mathematics Subject Classification 2000:60H10, 60H20 1. Introduction

Measuring risk in finance is an important problem, and in the last twelve years, many kind of risk mea- sures have been presented by various authors such as the coherent risk measures, the convex risk measures, and the law invariant risk measures [1–3]. In the re- cent time, the notion of a sublinear expectation is in- troduced by S. Peng for measuring risk in finance un- der volatile uncertainty [4]. He developed the notions of G-Brownian motion and the theory of stochastic cal- culus under sublinear expectation [4–6]. Since under volatile uncertainty the corresponding uncertain prob- abilities are singular from each other, they produce a serious problem for the related path analysis. The traditional classical techniques provide a limited un- derstanding of these types of problems. For example path-dependent derivatives under a classical probabil- ity space. For such type of problems, G-Brownian mo- tion provides a powerful tool and can be easily treated.

Under his stochastic calculus, Peng established the existence and uniqueness of solutions for the stochastic differential equations under G-Brownian motion (G- SDEs) with Lipschitz continuous coefficients [4,5].

Motivated from the importance of discontinuous func- tions, Faizullah and Piao established the existence of

solutions for the G-SDEs with a discontinuous drift co- efficient [7]. Now here, we develop the existence the- ory when the coefficientgor the coefficients f andg simultaneously are discontinuous functions such as in the following scalar G-SDEs:

dX_t=dt+H(X_t)dhBi_t+dB_t, dX_t=H(X_t)dt+{X_t}dhBi_t+dB_t,

where the unit step function or Heaviside functionH: R→Ris defined by [8,9]

H(x) =

(0, if x<0 ; 1, if x≥0.

This is an important function in science and is consid- ered to be a fundamental function in engineering; for example, the switching process of voltage in an elec- trical circuit is mathematically described by the unit step function and arises in many discontinuous ordi- nary differential equations [10]. The sawtooth function or fractional part function{x}:R→[0,1)has discon- tinuities at the integers and is defined by

{x}=x− bxc, x∈R,

wherebxcis the floor function [11]. The importance of this function is clear from the sawtooth waves which

© 2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

are used in music and computer graphics. Moreover, the impressed voltage on a circuit could be represented by the sawtooth function [12].

In this article, the following stochastic differential equation under G-Brownian motion is studied:

Xt=X0+ Z _t

0

f(v,Xv)dv+ Z _t

0

g(v,Xv)dhBi_v +

Z _t 0

h(v,X_v)dB_v, t∈[0,T],

(1)

whereX0∈Rⁿis a given constant initial condition and (hBi_t)t≥0 is the quadratic variation process of the G- Brownian motion(B_t)_t≥0. All the coefficients f(t,x), g(t,x), and h(t,x) are in the spaceM_G²(0,T;Rⁿ) [6].

A process X_t belonging to the mentioned space and satisfying the G-SDE (1) is said to be its solution. It is assumed that f(t,x) and g(t,x) are discontinuous functions whereas h(t,x) is Lipschitz continuous for allx∈Rⁿ.

This paper is organized in the following manner.

In Section2, some basic notions and definitions are given. In Section3, the upper and lower solutions method for the stochastic differential equations under G-Brownian motion is introduced. In Section4, the comparison theorem is established, while the existence of solutions for the G-SDEs with simultaneous discon- tinuous coefficients f andgis developed in Section5.

The proof is given in the appendix.

2. Preliminaries

The book [6] and papers [4,5,13–16] are good references for the material of this section. Let Ω be a (non-empty) basic space andHbe a linear space of real valued functions defined onΩ such that any arbi- trary constantc∈ Hand ifX ∈ Hthen|X| ∈ H. We consider thatHis the space of random variables.

Definition 1. (Sublinear Expectation). A functionalE: H →Ris called sublinear expectation, if∀X,Y ∈ H, c∈R, andλ ≥0 it satisfies the following properties:

(i) (Monotonicity): IfX≥Y thenE[X]≥E[Y].

(ii) (Constant Preserving):E[c] =c.

(iii) (Subadditivity):E[X+Y]≤E[X] +E[Y] orE[X]−E[Y]≤E[X−Y].

(iv) (Positive Homogeneity):E[λX] =λE[X].

The triple (Ω,H,E)is called a sublinear expecta- tion space.

Consider the space of random variablesHsuch that if X₁,X₂, . . . ,X_n∈ H then ϕ(X₁,X₂, . . . ,X_n)∈ H for eachϕ∈Cl.Lip(Rⁿ), whereCl.Lip(Rⁿ)is the space of linear functionsϕdefined as

Cl.Lip(Rⁿ) ={ϕ:Rⁿ→R| ∃C∈R⁺, m∈Ns. t.

|ϕ(x)−ϕ(y)| ≤C(1+|x|^m+|y|^m)|x−y|}, forx,y∈Rⁿ.

Definition 2. (Independence). Ann-dimensional ran- dom vector Y = (Y₁,Y₂, . . . ,Y_n) is said to be inde- pendent from an m-dimensional random vector X = (X₁,X₂, . . . ,X_m)if

E[ϕ(X,Y)] =E[E[ϕ(x,Y)]_x=X],

∀ϕ∈Cl.Lip(R^m×Rⁿ).

Definition 3. (Identical Distribution). Twon-dimen- sional random vectorsX and ˆX defined, respectively, on the sublinear expectation spaces (Ω,H,E) and (Ωˆ,H,ˆ Eˆ)are said to be identically distributed, denoted byX∼Xˆ orX=^dXˆ,if

E[ϕ(X)] =Eˆ[ϕ(Xˆ)] ∀ϕ∈Cl.Lip(Rⁿ). Definition 4. (G-Normal Distribution). Let(Ω,H,E) be a sublinear expectation space andX∈ Hwith

σ¯²=E[X²], σ²=−E[−X²].

ThenX is said to be G-distributed orN(0;[σ¯²,σ²])- distributed, if∀a,b≥0, we have

aX+bY∼p

a²+b²X

for eachY ∈ Hwhich is independent toXandY ∼X.

G-Expectation and G-Brownian Motion. LetΩ= C₀(R⁺), that is, the space of allR-valued continuous paths(w_t)_t∈R+withw₀=0 equipped with the distance

ρ(w¹,w²) =

∞ k=1

∑

1 2^k

max

t∈[0,k]|w_t¹−w_t²| ∧1

,

and consider the canonical processB_t(w) =w_t fort∈ [0,∞),w∈Ω, then for each fixedT∈[0,∞), we have L_ip(Ω_T) ={ϕ(B_t1,B_t2, . . . ,B_tn):

t₁, . . . ,t_n∈[0,T],ϕ∈Cl.Lip(Rⁿ),n∈N}, where L_ip(Ω_t)⊆L_ip(Ω_T) for t ≤T and L_ip(Ω) =

∪^∞_m=1L_ip(Ω_m).

Consider a sequence {ξ_i}^∞_i=1 of n-dimensional random vectors on a sublinear expectation space (Ωˆ,Hˆ_p,Eˆ) such that ξi+1 is independent of (ξ₁,ξ2, . . . ,ξi) for each i = 1,2, . . . ,n−1, and ξi

is G-normally distributed. Then a sublinear expecta- tionE[.]defined on L_ip(Ω)is introduced as follows.

For 0=t₀<t₁< . . . <t_n<∞,ϕ∈Cl.Lip(Rⁿ)and each

X=ϕ(B_t1−B_t0,B_t2−B_t1, . . . ,B_tn−B_tn−1)∈L_ip(Ω), E[ϕ(B_t1−B_t0,B_t2−B_t1, . . . ,B_tn−B_tn−1)]

=Eˆ[ϕ(√

t₁−t₀ξ1, . . . ,√

t_n−tn−1ξn)].

Definition 5. (G-Brownian Motion). The sublinear ex- pectationE: L_ip(Ω)→Rdefined above is called a G- expectation, and the corresponding canonical process (B_t)t≥0is called a G-Brownian motion.

The completion of L_ip(Ω)under the normkXk_p= (E[|X|^p])^1/p for p ≥ 1 is denoted by L_G^p(Ω) and L_G^p(Ω_t)⊆L_G^p(Ω_T)⊆L_G^p(Ω)for 0≤t≤T <∞. The filtration generated by the canonical process(B_t)_t≥0is denoted byF_t=σ{B_s,0≤s≤t},F={F_t}t≥0. Itˆo’s Integral of G-Brownian Motion. For anyT ∈ R⁺,a finite ordered subset πT ={t₀,t₁, . . . ,t_N} such that 0=t0<t1< . . . . <t_N =T is a partition of[0,T] and

µ(π_T) =max{|t_i+1−t_i|:i=0,1, . . . ,N−1}. A sequence of partitions of[0,T]is denoted byπ_T^N= {t₀^N,t₁^N, . . . ,t_N^N}such that lim

N→∞µ(π_T^N) =0.

Consider the following simple process: Let p≥1 be fixed. For a given partitionπT ={t₀,t1, . . . ,tN}of [0,T],

ηt(w) =

N−1 i=0

∑

ξi(w)I_[ti,ti+1](t), (2) where ξ_i ∈L_G^p(Ω_ti), i=0,1, . . . ,N−1. The collec- tion containing the above type of processes is de- noted byM_G^p,0(0,T). The completion ofM_G^p,0(0,T)un- der the normkηk={^R₀^TE[|η_v|^p]dv}^1/pis denoted by M_G^p(0,T)and for 1≤p≤q,M_G^p(0,T)⊃M_G^q(0,T).

Definition 6. (Itˆo’s Integral). For eachη_t∈M_G^2,0(0,T), the Itˆo’s integral of G-Brownian motion is defined as

I(η) = Z T

0 ηvdB_v=

N−1 i=0

∑

ξi(B_ti+1−B_ti), whereη_t is given by (2).

Definition 7. (Quadratic Variation Process). An in- creasing continuous process(hBi_t)t≥0withhBi₀=0, defined by

hBi_t=B²_t −2 Z _t

0

B_vdB_v,

is called the quadratic variation process of G-Brownian motion.

Let B(Ω) be the Borel σ-algebra of Ω. It was proved in [6,13] that there exists a weakly com- pact family P of probability measures P defined on (Ω,B(Ω))such that

E[X] =sub

P∈PE_P[X], ∀X∈L_ip(Ω). This makes the following definitions reasonable.

Definition 8. (Capacity). The capacity ˆc(.)associated to the familyPis defined by

c(A) =ˆ sub

P∈PP(A), A∈ B(Ω).

Definition 9. (Polar Set and Quasi-Sure Property).

A setAis said to be polar if its capacity is zero, that is, ˆc(A) =0 and a property holds quasi-surely (q. s. in short) if it holds outside a polar set.

Through out the paper forX = (x₁,x₂, . . . ,x_n),Y= (y₁,y₂, . . . ,y_n),X≤Y meansx_i≤y_i,i=1,2, . . . ,n.

3. Upper and Lower Solutions Method

Recall that the concept of upper and lower solutions for the classical stochastic differential equations was established in [8,9,17,18].

Definition 10. (Upper Solution). A process Ut ∈ M_G²(0,T) is said to be an upper solution of the G- SDE (1) on the interval[0,T]if for any fixedsthe in- equality (interpreted component wise)

U_t≥U_s+ Z t

s

f(v,U_v)dv+ Z t

s

g(v,U_v)dhBi_v +

Z t s

h(v,U_v)dB_v, 0≤s≤t≤T, (3)

holds q. s.

Definition 11. (Lower Solution). A process L_t ∈ M_G²(0,T) is said to be a lower solution of the

G-SDE (1) on the interval[0,T]if for any fixedsthe inequality (interpreted component wise)

L_t≤L_s+ Z t

s

f(v,L_v)dv+

Z t s

g(v,L_v)dhBi_v +

Z _t s

h(v,L_v)dB_v, 0≤s≤t≤T,

(4)

holds q. s.

Example 1. Consider the following scalar stochastic differential equation: (G-SDE)

dX_t=H(X_t)dt+{X_t}dhBi_t+dB_t, t∈[0,T], (5) where the respective Heaviside and sawtooth functions H(x)and{x}are defined in the introduction.

ThenU_t=U₀+^R₀^tdv+^R₀^tdhBi_v+^R₀^tdB_vandL_t= L₀+^R₀^tdB_vfort∈[0,T]are the upper and lower solu- tions of the G-SDE (5), respectively, which are shown as the following:

Ut=U0+ Z _t

0

dv+ Z _t

0

dhBi_v+ Z _t

0

dB_v

=U_s+ Z _t

s

dv+ Z _t

s

dhBi_v+ Z _t

s

dB_v

≥U_s+ Z t

s

{U_v}dv+ Z t

s

H(U_v)dhBi_v +

Z t s

dB_v, 0≤s≤t≤T,

whereUs=U0+^R₀^sdv+^R₀^sdhBi_v+^R₀^sdB_v for each fixedssuch that 0≤s≤t≤T. On similar arguments as above, one can show thatL_t=L₀+^R₀^tdB_vis a lower solution of the scalar G-SDE (5). The existence of so- lutions of the G-SDE (5) will be discussed later in Sec- tion5.

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

dX_t=f(t,w)dt+g(t,w)dhBi_t

+h(t,X_t)dB_t, t∈[0,T]. (6) Define two functionsp_L,U,q_L,U :[0,T]×Rⁿ×Ω→Rⁿ by

p_L,U(t,x,w) =max{L_t(w),min{U_t(w),x}}, q_L,U(t,x,w) =p_L,U(t,x,w)−x, (7) and consider the stochastic differential equation

dX_t=f˜(t,X_t)dt+g(t,X˜ _t)dhBi_t

+h(t˜ ,X_t)dB_t, t∈[0,T] (8)

with a given constant initial conditionX0∈Rⁿ, where f˜(t,x,w) =f(t,w) +q_L,U(t,x,w),

g(t,˜ x,w) =g(t,w) +q_L,U(t,x,w), h(t,˜ x,w) =h(p_L,U(t,x,w))

are Lipschitz continuous in x. It is known that the stochastic differential equation (8) has a unique solu- tionXt∈M²_G(0,T;Rⁿ)[4,5,14].

4. Comparison Theorem for the G-SDEs

Lemma 1. Assume that the respective upper and lower solutions U_tand L_tof the G-SDE (6) satisfy L_t≤U_tfor t∈[0,T]. Then U_tand L_tare upper and lower solutions of the G-SDE (8), respectively.

Proof. Since L_t ≤U_t yields p_L,U(t,U_t) = U_t and q_L,U(t,U_t) =0, therefore

U_s+ Z _t

s

f˜(v,U_v)dv+ Z _t

s

˜

g(v,U_v)dhBi_v+ Z _t

s

h(v,U˜ _v)dB_v

=U_s+ Z t

s

h

f(v,w) +q_L,U(v,U_v)i dv+

Z t s

h g(v,w) +q_L,U(v,U_v)i

dhBi_v+ Z t

s

h(v,p_L,U(v,U_v))dB_v

=U_s+ Z t

s

f(v,w)dv+ Z t

s

g(v,w)dhBi_v +

Z _t s

h(v,U_v)dB_v≤Ut.

HenceU_t for 0≤s≤t≤T is an upper solution of the G-SDE (8). In a similar way as above, one can show thatL_tis a lower solution of the G-SDE (8).

The following lemma can be found in [19]. For the proof see SectionAppendix A.

Lemma 2. Let X_t,Y_t∈M_G^1,0([0,T];Rⁿ). If X_t ≤Y_t for t∈[0,T]and any w∈Ω, then

Z T 0

X_tdhBi_t≤ Z T

0

Y_tdhBi_t.

Theorem 1. (Comparison Result). Suppose that (i) The functions f(t,x) and g(t,x) are measurable

with ^R₀^tE[|φ(v, .)|²]dv<∞ for φ = f and g, re- spectively, where h(t,x) is Lipschitz continuous in x.

(ii) The respective upper and lower solutions Ut

and L_t of the G-SDE (6) with E[|U_t|²] < ∞, E[|L_t|²]<∞satisfy L_t≤U_tfor t∈[0,T].

(iii) Also X₀ ∈ Rⁿ is a given initial value with E[|X₀|²]<∞and L₀≤X₀≤U₀.

Then there exists a unique solution X_t∈M_G²(0,T;Rⁿ) of the G-SDE (6) such that L_t≤X_t≤U_t for t∈[0,T] q. s.

Proof. Define the functions p_L,U,q_L,U :[0,T]×Rⁿ× Ω →Rⁿby (7) and consider the G-SDE (8).

Now the G-SDE (8) has a unique solution and by Lemma1 ifU_t andL_t are upper and lower solutions of the G-SDE (6), respectively, then they are the re- spective upper and lower solutions for the G-SDE (8).

Also it is clear that any solutionX_t of the modified G- SDE (8) such that

L_t≤X_t≤U_t, t∈[0,T], (9) q. s. is also a solution of the G-SDE (6). Thus we only need to show that any solutionX_t of the problem (8) does satisfy the inequality (9).

Assume that there exists an arbitrary interval (t₁,t2)⊂[0,T] such that Xt₁ =Lt₁ and Xt <Lt for t∈(t₁,t₂), then we have

X_t−L_t= Z t

t₁

f˜(v,X_v)dv+

Z t t₁

˜

g(v,X_v)dhBi_v +

Z t t₁

h(v,X˜ _v)dB_v− Z t

t₁

f˜(v,L_v)dv

− Z _t

t1

˜

g(v,L_v)dhBi_v− Z _t

t1

h(v,˜ Lv)dB_v

= Z _t

t1

h

f(v,w) +q_L,U(v,X_v)i dv+

Z _t t1

h g(v,w)

+q_L,U(v,X_v)i dhBi_v+

Z t t1

h(v,p_L,U(v,X_v))dB_v

− Z t

t₁

[f(v,w) +q_L,U(v,L_v)]dv− Z t

t₁

h g(v,w)

+q_L,U(v,L_v)i dhBi_v−

Z t t₁

h

v,p_L,U(v,L_v) dB_v. Since L_t ≤U_t gives p_L,U(t,L_t) =L_t and X_t <L_t implies X_t <U_t which gives p_L,U(t,X_t) = L_t. Also q_L,U(t,L_t) =0 andq_L,U(t,X_t) =L_t−X_t>0. Thus X_t−L_t=

Z t t₁

q_L,U(v,X_v)dv+ Z t

t₁

q_L,U(v,X_v)dhBi_v>0, which yields a contradiction. Thus X_t ≥L_t for t ∈ [0,T]. By using the similar arguments as above one can show thatX_t≤U_tfort∈[0,T].

5. G-SDEs with Simultaneous Discontinuous Coefficientsfandg

Now we take the G-SDE

dX_t=f(t,X_t)dt+g(t,X_t)dhBi_t+h(t,X_t)dB_t,

t∈[0,T], (10)

where f(t,x)andg(t,x)do not need to be continuous but suppose that they are increasing, that is, ifx≥y thenφ(t,x)≥φ(t,y)forφ(t, .) =f(t, .)andg(t, .), re- spectively (where the inequalities are interpreted com- ponent wise), andh(t,x)is Lipschitz continuous inx.

Theorem 2. Suppose that

(i) The functions f(t,x)and g(t,x)are increasing in x, where h(t,x)is Lipschitz continuous in x.

(ii) U_t and L_t are the respective upper and lower so- lutions of the G-SDE (10) with^R₀^tE[|φ(U_v)|²]dv<

∞,^R₀^tE[|φ(L_v)|²]dv<∞forφ=f , g, respectively, and Lt≤Utfor t∈[0,T].

Then there exists at least one solution X_t ∈ M_G²(0,T;Rⁿ)of the G-SDE (10) such that L_t≤X_t≤U_t for t∈[0,T]q. s.

Proof. Define the space of alld-dimensional stochas- tic processes by Hˆ₂, that is, ˆH₂ = {X = {X_t,t ∈ [0,T]} : E[|X_t|²] < ∞} with the norm kX_tk₂ = {^R₀^tE[|X_v|²]dv}^1/2for allt∈[0,T], which is a Banach space [4–6].

Denote the order interval[L,U]in ˆH₂byK, that is, K={X :X ∈Hˆ₂ and L_t ≤X_t ≤U_t} for t∈[0,T], which is closed and bounded by the above norm. By using the monotone convergence theorem [13], one can prove the convergence of a monotone sequence that be- longs toKin ˆH₂. ThusKis a regularly ordered metric space with the above norm. It is clear that for any pro- cessV ∈ K,Ut, andLt are the respective upper and lower solutions for the G-SDE

dX_t=f(t,V_t)dt+g(t,V_t)dhBi_t+h(t,X_t)dB_t,

t∈[0,T]. (11)

Hence by Theorem1, for anyX₀∈RⁿwithE[|X₀|²]<

∞and L0≤X0≤U0, the G-SDE (11) has a unique solutionX_t∈M_G²(0,T;Rⁿ)such thatL_t ≤X_t ≤U_t for t∈[0,T]q. s.

Define an operatorF:K → KbyF(V) =X, where X is the unique solution of the G-SDE (11). Now

suppose thatV_t⁽¹⁾≤V_t⁽²⁾ for allt ∈[0,T] and define X⁽¹⁾=F(V⁽¹⁾),X⁽²⁾=F(V⁽²⁾)whereV⁽¹⁾,V⁽²⁾∈ K.

Since it is given that f andgare increasing functions, thereforeX_t⁽¹⁾is a lower solution of the G-SDE X_t=X₀+

Z t 0

f(v,V_v⁽²⁾)dv+

Z t 0

g(v,V_v⁽²⁾)dhBi_v +

Z t 0

h(v,X_v)dB_v, t∈[0,T].

(12)

But this problem has an upper solutionU_t. Hence by Theorem1, the G-SDE (12) has a solutionX_t⁽²⁾ such that X_t⁽¹⁾ ≤X_t⁽²⁾≤U_t fort∈[0,T]. ThusF is an in- creasing mapping and by Theorem 3, it has a fixed pointX^(∗)=F(X^(∗))∈ Ksuch thatY_t≤X_t^(∗)≤U_tq. s., where

X_t^(∗)=X₀+ Z t

0

f(v,Xv^(∗))dv+ Z t

0

g(v,Xv^(∗))dhBi_v +

Z _t 0

h(v,X_v^(∗))dB_v, t∈[0,T].

Now continuing Example 1 by the above Theo- rem 2, there exists at least one solution X^(∗) of the G-SDE (5) such thatL₀+B_t≤X_t^(∗)≤U₀+t+hBi_t+B_t for t∈[0,T], whereL_t =L₀+B_t andU_t =U₀+t+ hBi_t+B_t are the respective lower and upper solutions of (5).

Remark 1. As the above results (i. e. Theorem1and Theorem 2) for the existence theory of G-SDEs are more general, we therefore give the following sketch for the existence of solutions for G-SDEs with a dis- continuous coefficientg. The rest of work can be ob- tained in a similar fashion. Assume thatUt andLt are the respective upper and lower solutions of the G-SDE dX_t=f(t,X_t)dt+g(t,w)dhBi_t+h(t,X_t)dB_t,

t∈[0,T].

Define the functions p_L,U andq_L,U by (7) and consider the stochastic differential equation

dX_t=f˜(t,X_t)dt+g(t,X˜ _t)dhBi_t+h(t,˜ X_t)dB_t, t∈[0,T],

with a given constant initial conditionX₀∈Rⁿ, where f˜(t,x,w) =f(p_L,U(t,x,w)),

g(t,˜ x,w) =g(t,w) +q_L,U(t,x,w), h(t,˜ x,w) =h(p_L,U(t,x,w)) are Lipschitz continuous inx.

6. Discussion

The existence theory for the G-SDEs with a discon- tinuous coefficienthis still open. We are unable to dis- cuss this case. Because like the classical Itˆo’s integral, the monotonicity condition does not hold in G-Itˆo’s integral with respect to the G-Brownian motion, i. e.

X_t≥Y_tdoes not imply^R₀^TX_tdB_t≥^R₀^TY_tdB_t. However its quadratic variation process is an increasing contin- uous process starting from zero and satisfy the above stated property, see Lemma2[19].

Acknowledgements

I am very grateful to Prof. Daxiong Piao for his mo- tivation and some useful suggestions for this work.

Appendix A Proof

For the following definition and theorem see [20].

Definition 12. An ordered metric space M is called regularly (resp. fully regularly) ordered, if each mono- tone and order (resp. metrically) bounded ordinary se- quence ofMconverges.

Theorem 3. 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 point.

Proof of Lemma (2).

Proof. Since{hBi_t:t≥0}is an increasing continuous process withhBi₀=0. Therefore for any fixedw∈Ω andti+1≥t_i, hBi_ti+1− hBi_ti ≥0, i=0,1, . . . ,N−1.

Also forX_t,Y_t ∈M_G^1,0([0,T];Rⁿ),X_t =∑^N−1_i=0 ξ_iI_[ti,ti+1) and Y_t =∑^N−1_i=0 ξ˜_iI_[ti,ti+1), where ξ_i,ξ˜_i ∈L¹_G(Ω_i), i= 0,1, . . . ,N−1. ThenX_t≤Y_t implies that

N−1 i=0

∑

ξiI_[t

i,t_i+1)≤

N−1 i=0

∑

ξ˜iI_[t

i,t_i+1), which yields

N−1 i=0

∑

ξi

hhBi_ti+1− hBi_tii

≤

N−1 i=0

∑

ξ˜i

hhBi_ti+1− hBi_tii .

Hence Z T

0

X_tdhBi_t≤ Z T

0

Y_tdhBi_t.

Remark 2. The above lemma shows that G-Ito’s in- tegral with respect to the quadratic variation process

satisfies the monotonicity property. Also ifXt≤0 then R_T

0 X_tdhBi_t≤0.

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