2. The Setup and a Comparison Theorem In our paper, we just consider a 1-dimensional case. We always assume that (Ω,F,P) is a completed probability space, W=:{Wt:t≥0} is a real-valued Brownian motion defined on (Ω,F,P), and {Ft:t≥0} is natural filtration generated by the Brownian motion W; i.e., for any t≥0(4)Ft=σWs:s≤t.

We consider SDE (1) with coefficients b(t,x):R+×R→R and σ(t,x):R+×R→R, where R+ and R are a positive real number and real number, respectively. And we use · to denote norm of R. The following is the definition of a strong solution for SDE.

Definition 1. An adapted continuous process Xt defined on (Ω,F,P) is said to be a strong solution for SDE (1) if it satisfies that(5)Xt=X0+∫0tbs,Xsds+∫0tσs,XsdWs, t≥0,holds with probability 1.

Moreover, Xt and X~t are two strong solutions of SDE (1); then P[Xt=X~t; 0≤t<∞]=1. Under this condition, the solution of SDE (1) is said to be unique.

The following is the conception of upper and lower solutions for stochastic differential equations, which are given by N. Halidias and P. E. Kloeden [12]. Many authors discussed the upper and lower solutions of the stochastic differential equation by using the other name which is the solutions of the stochastic differential inequality, for example, S. Assing and R. Manthey [13] and X. Ding and R. Wu [14].

Definition 2. An adapted continuous stochastic process Ut (resp., Lt) is an upper (resp., lower) solution of SDE (1) if the inequalities

(1) Ut≥Us+∫stb(u,Uu)du+∫stσ(u,Uu)dWu, t≥s≥0;

(2) Lt≤Ls+∫stb(u,Lu)du+∫stσ(u,Lu)dWu, t≥s≥0,

hold with probability 1.

Remark 3. It is not an easy thing to calculate the exact upper and lower solution of the general stochastic differential equations. However, one can discuss the existence of upper and lower solutions. S. Assing and R. Manthey [13] discussed the “maximal/minimal solution” of the stochastic differential inequality. They proved the existence of a “maximal/minimal solution” under some conditions. However, it is easy to show there exist the upper solutions of stochastic differential equations if the minimal solution of the stochastic differential inequality exists. In fact, the minimal solution is special upper solutions of stochastic differential equations. Similarly, we can show the existence of the lower solution by using the maximal solution of the stochastic differential inequality.

Usually, the existence and uniqueness of solutions of SDE (1) are investigated under the conditions in which the diffusion coefficient satisfies Lipschitz condition and liner growth condition. In fact, the Lipschitz condition can be generalized. In this paper, the diffusion coefficient satisfies the (Cσ) condition.

(Cσ): For N>0, there exist an increasing function ρN:R+→R+ and a predictable process GN(t,ω) such that(6)σt,ω,x-σt,ω,y≤GNt,ωρNx-y,∫0tGNt,ωdt<∞ a.s.,∫0+ρN-2udu=∞,for all t≥0, and x,y∈R with x,y≤N.

Note that the Lipschitz condition satisfies the (Cσ) condition. The following lemma is an important tool of this paper and had to be proved in proposition 2.3 of X. Ding and R. Wu [14].

Lemma 4. In SDE (1), we assume σ satisfies (Cσ) and b satisfies that, for each N>0, there exists a measurable process LN(t,ω) such that (7)bt,ω,x-bt,ω,y≤LNt,ωx-y,∫0tLNt,ωdt<∞, a.s.,for all t≥0 and x,y∈R with x,y≤N. Then SDE (1) has a unique local (explosion in the finite time) strong solution.

Remark 5. Moreover, if b and σ satisfy the liner growth condition (cf. J. Jacod and J. Memin [15]) (8)bt,ω,x+σt,ω≤Ht,ω1+x,where H(t,ω), t≥0, is a predictable process such that ∫0tH2(s,ω)ds<∞, a.s. Then SDE (1) has a unique global strong solution.

The following theorem can be considered as a comparison theorem, and we will use it to arrive at our main result.

Theorem 6. Let b:R+×Ω→R be predictable such that ∫0tb2(s,ω)ds<∞, a.s. for any t≥0, and let σ:R+×Ω×R→R be predictable. Suppose that σ satisfies (Cσ) and there exists a predictable process H(t,ω), t≥0 such that (9)σt,ω≤Ht,ω1+x,where ∫0tH2(s,ω)ds<∞, a.s. And suppose that Ut and Lt are upper and lower solutions of the following SDE:(10)Xt=X0+∫0tbs,ωds+∫0tσs,XsdWs, t≥0,such that L0≤X0≤U0,a.s.

Then there is a unique strong solution Xt which satisfies that Lt≤Xt≤Ut for any t≥0 holds with probability 1.

Proof. Obviously, we have that SDE (10) has a unique strong solution Xt by using Lemma 4 and Remark 5. In the following we will show (11)PLt≤Xt≤Ut, ∀t≥0=1.We only prove P{Xt≤Ut, ∀t≥0}=1, because we can prove P{Lt≤Xt, ∀t≥0}=1 by using the similar way.

Define the stopping time (12)TN≕inft∈0,∞:Xt∨Lt∨t>N∧N.Obviously, TN→∞ when N→∞. And define the stopping time τ≕inf{t∈[0,∞):Xt<Lt}. If P{τ<TN}=0 for N≥1, then P{τ<∞}=0; that is, P{Lt≤Xt, ∀t≥0}=1. Indeed, ∀q∈Q+ and N≥1, we define α≕(τ+q)∧TN and Ωα≕{Xα<Lα}. Note that (13)PΩα=0, ∀q∈Q+, N≥1⇒Pτ<TN=0.In fact, by P{Ωα}=0 and X,L being continuous and the denseness of the rational number in R, we have(14)Xτ+t∧TN≥Lτ+t∧TN a.s. on τ<TNfor all t≥0. That is for a.s. ω∈{τ<TN} and t∈[τ(ω),TN(ω)] one has Xt≥Lt. However, by the definition of τ and Lτ≤Xτ,a.s. we have P{τ<TN}=0.

In the following we shall prove P{Ωα}=0, ∀q∈Q+, N≥1. Set β≕sup{t∈[0,α):Lt≤Xt}. By continuity of X and L we have Xβ≥Lβ, a.s. Obviously, {Xα≥Lα}={β=α}. So, we have Ωα≕{Xα<Lα}={β<α}. Hence, for ω∈Ωα and t∈(β(ω),α(ω)] we have Xt<Lt. Using L as a lower solution of SDE (10), we have(15)Lt-Xt≤∫βtσs,Ls-σs,XsdWs≕Mt.Hence, (16)Lt-XtIΩαIβ,αt≤MtIΩαIβ,αt.Let us take M+≕max{M,0}. By the Tanaka formula (refer to [3]) we have (17)Mt+IΩα=Mβ+IΩα+IΩα∫βtIMs>0dMs+12IΩαLt0M-Lβ0M,where Ltx(M) denotes local time at the point x for M. By the definition of local time, one can prove easily that Lt0(M)-Lβ0(M)=0, for t∈(β,α] on Ωα. So, by Mβ+IΩα=0 (using the definition M) we have (18)M+IΩα=∫βtIMs>0IΩασs,Ls-σs,XsdWs≕Nt. Since for ω∈Ωα and t∈(β(ω),α(ω)] we have Xt<Lt, by (18) we have(19)M+IΩα≤Nt+∫βtIMs>0IΩαLs-Usds.Using (16), we have(20)M+IΩα≤Nt+∫βtIΩαM+ds.By the stochastic Gronwall inequality (e.g., Lemma 2.1 [14]), we have (21)IΩαMα+e-t≤Nβe-t+∫βαe-tdNs.By Nβ=0 we have(22)EIΩαMα+e-t≤E∫βαe-tdNs=0.So, using (16) once again we have (23)IΩαLα-Xα≤IΩαMα+=0 a.e.That is Lα≤Xα on Ωα a.s. Hence, P{Ωα}=0. The proof is completed.

3. Existence of Strong Solutions In this section, we will show the existence of the solution for SDEs with discontinuous drift coefficients. The method of the proof of our main result is based on Amann’s fixed point theorem (e.g., Theorem 11.D [16]), so we introduce it in the following.

Lemma 7. Suppose that

(1) the mapping f:X→X is monotone increasing on an ordered set X

(2) every chain in X has a supremum

(3) there is an element xo∈X for which x0≤f(x0)

Then f has a smallest fixed point in the set {x∈X:x0≤x}.

The following theorem is our main result.

Theorem 8. Let b,σ:R+×Ω×R→R be predictable. Suppose that b is an increasing function in x and σ satisfies (Cσ) and there exists a predictable process H(t,ω), t≥0, such that (24)bt,ω,x+σt,ω,x≤Ht,ω1+x,where ∫0tH2(s,ω)ds<∞, a.s. Moreover, suppose that Ut and Lt are upper and lower solutions of the SDE(25)Xt=X0+∫0tbs,Xsds+∫0tσs,XsdWs, t≥0,such that L0≤X0≤U0, a.s.

Then there is at least a strong solution Xt which satisfies that Lt≤Xt≤Ut for t≥0 holds with probability 1.

Proof. Let X be a space of adapted and continuous processes and define the order relation ⪯: (26)X⪯Y⇔PXt≤Yt, ∀t≥0=1,for X,Y∈X. We consider a subset of the space (X,⪯)(27)D≕L,U≕X∈X:PLt≤Xt≤Ut, ∀t≥0=1.For arbitrary fixed Z∈D, we consider the following equation: (28)Xt=X0+∫0tbs,Zsds+∫0tσs,XsdWs;by Theorem 6 there exists a unique strong solution Xt∗. Define a mapping S:D→X and S(Z)=X∗. To complete the proof it is enough to show S has a fixed point.

Since b is an increasing function and U is an upper solution of SDE (25), we have that(29)Ut≥Us+∫stbu,Zudu+∫stσu,UudWuholds with probability 1 for t≥s≥0. Then U is also an upper solution of SDE (28). Similarly, we have that (30)Lt≤Ls+∫stbu,Zudu+∫stσu,LudWuholds with probability 1 for t≥s≥0 such that L is also a lower solution of SDE (28). Hence, using Theorem 6 we have(31)PLt≤SZt≤Ut, ∀t≥0=1.Since Z is arbitrary, we have S:D→D and L⪯S(L) and S(U)⪯U. If S is an increasing mapping, by Lemma 7S has a fixed point on D. In fact, take Z1,Z2∈D and Z1⪯Z2 and set Xi≕S(Zi); that is, (32)Xti=X0+∫0tbs,Zsids+∫0tσs,XsidWs, i=1,2.Since b is an increasing function, we have that (33)Xt2≥Xs+∫stbu,Zu1du+∫stσu,Xu2dWuholds with probability 1 for t≥s≥0. Hence X2 is an upper solution of the following equation: (34)Xt=X0+∫0tbs,Zs1ds+∫0tσs,XsdWs. And by (29) U is an upper solution of (34). Using Theorem 6 again, we have(35)PSZt1≤SZt2≤Ut, t≥0=1;that is, S(Zt1)⪯S(Zt2). Hence S is an increasing function. The proof is completed.

Example 9. We consider the following SDE: (36)dXt=sgnXtdt+dWt, ∀t≥0,with initial value X0. Obviously, X0-t+Wt≤X0+∫0tsgn(Xs)ds+Wt≤X0+t+Wt. By Theorem 8, there exists at least one solution Xt such that X0-t+Wt≤Xt≤X0+t+Wt,t≥0 holds with probability 1.

Example 10. We have the SDE (37)dXt=fXt,tdt+σdWt, ∀t≥0,with initial value X0, where f(x,t) is a bounded function and is defined as(38)fx,t=M+1,x≥M;x+1,0≤x<M;x-1,-M≤x<0;-M-1,x≤-M.It is easy to show Xt=X0-(M+1)t+σWt and Xt=X0+(M+1)t+σWt are the lower solution and upper solution of (37), respectively. And f(x,t) is an increasing function in x but is not continuous in x, so we have that SDE (37) has a strong solution by using Theorem 8.