IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi 10.1155/2018/2059694 2059694 Research Article The Existence of Strong Solutions for a Class of Stochastic Differential Equations http://orcid.org/0000-0001-5745-1409 Zhang Junfei 1 Langa José A School of Statistics and Mathematics Central University of Finance and Economics Beijing 100081 China cufe.edu.cn 2018 15102018 2018 11 07 2018 25 09 2018 15102018 2018 Copyright © 2018 Junfei Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we will consider the existence of a strong solution for stochastic differential equations with discontinuous drift coefficients. More precisely, we study a class of stochastic differential equations when the drift coefficients are an increasing function instead of Lipschitz continuous or continuous. The main tools of this paper are the lower solutions and upper solutions of stochastic differential equations.

Fundamental Research Funds for the Central Universities Central University of Finance and Economics
1. Introduction

There are many works  about the existence and uniqueness of strong or weak solutions for the following stochastic differential equation (denoted briefly by SDE): (1)dXt=bt,Xtdt+σt,XtdWtt0,where b(t,x):R+×RR and σ(t,x):R+×RR are called drift and diffusion coefficients, respectively. Wt is standard Brownian motion. Usually, the drift and diffusion coefficients are Lipschitz or local Lipschitz continuous or at least are continuous with respect to x when the existence and uniqueness of solutions are investigated. In fact, the solutions of stochastic differential equations may exist when their drift and diffusion coefficients are discontinuous with respect to x. Therefore, many authors discussed the existence of solutions for SDE with discontinuous coefficients. For example, L. Karatzas and S. E. Shreve  (Proposition 3.6 of §5.3) considered the existence of a weak solution when the drift coefficient of SDE need not be continuous with respect to x. A. K. Zvonkin  considered the following stochastic differential equation with a discontinuous diffusion coefficient: (2)Xt=0tsgnXsdWs;0t<,where (3)sgnx=1,x>0;-1,x0.The weak solution of this stochastic differential equation exists, but there is not the strong solution. N. V. Ktylov  and N. V. Ktylov and R. Liptser  also discussed existence issues of SDE when their diffusion coefficients are discontinuous with respect to x. And many authors also considered the approximation solutions of SDE with discontinuous coefficients, such as .

In this paper, we will consider the existence of a strong solution of SDE (1) when the drift coefficient b(t,x) is an increasing function but need not be continuous with respect to x and the diffusion coefficient σ(t,Xt) satisfies (Cσ) condition. Section 1 is an introduction. In Section 2, we will show a comparison theorem by using the upper and lower solutions of SDE. We will prove our main result by using the above comparison theorem in Section 3.

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:t0} is a real-valued Brownian motion defined on (Ω,F,P), and {Ft:t0} is natural filtration generated by the Brownian motion W; i.e., for any t0(4)Ft=σWs:st.

We consider SDE (1) with coefficients b(t,x):R+×RR and σ(t,x):R+×RR, 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,t0,holds with probability 1.

Moreover, Xt and X~t are two strong solutions of SDE (1); then P[Xt=X~t;0t<]=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 . 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  and X. Ding and R. Wu .

Definition 2.

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

(1) UtUs+stb(u,Uu)du+stσ(u,Uu)dWu,ts0;

(2) LtLs+stb(u,Lu)du+stσ(u,Lu)dWu,ts0,

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  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,yGNt,ωρNx-y,0tGNt,ωdt<a.s.,0+ρN-2udu=,for all t0, and x,yR with x,yN.

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 .

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,yLNt,ωx-y,0tLNt,ωdt<,a.s.,for all t0 and x,yR with x,yN. 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 ) (8)bt,ω,x+σt,ωHt,ω1+x,where H(t,ω),t0, 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+×ΩR be predictable such that 0tb2(s,ω)ds<,a.s.foranyt0, and let σ:R+×Ω×RR be predictable. Suppose that σ satisfies (Cσ) and there exists a predictable process H(t,ω),t0 such that (9)σt,ωHt,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,t0,such that L0X0U0,a.s.

Then there is a unique strong solution Xt which satisfies that LtXtUt for any t0 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)PLtXtUt,t0=1.We only prove P{XtUt,t0}=1, because we can prove P{LtXt,t0}=1 by using the similar way.

Define the stopping time (12)TNinft0,:XtLtt>NN.Obviously, TN when N. And define the stopping time τinf{t[0,):Xt<Lt}. If P{τ<TN}=0 for N1, then P{τ<}=0; that is, P{LtXt,t0}=1. Indeed, qQ+ and N1, we define α(τ+q)TN and Ωα{Xα<Lα}. Note that (13)PΩα=0,qQ+,N1Pτ<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+tTNLτ+tTNa.s.onτ<TNfor all t0. That is for a.s.ω{τ<TN} and t[τ(ω),TN(ω)] one has XtLt. However, by the definition of τ and LτXτ,a.s. we have P{τ<TN}=0.

In the following we shall prove P{Ωα}=0,qQ+,N1. Set βsup{t[0,α):LtXt}. By continuity of X and L we have XβLβ,a.s. Obviously, {XαLα}={β=α}. So, we have Ωα{Xα<Lα}={β<α}. Hence, for ωΩα and t(β(ω),α(ω)] we have Xt<Lt. Using L as a lower solution of SDE (10), we have(15)Lt-Xtβtσs,Ls-σs,XsdWsMt.Hence, (16)Lt-XtIΩαIβ,αtMtIΩαIβ,αt.Let us take M+max{M,0}. By the Tanaka formula (refer to ) we have (17)Mt+IΩα=Mβ+IΩα+IΩαβtIMs>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,XsdWsNt. Since for ωΩα and t(β(ω),α(ω)] we have Xt<Lt, by (18) we have(19)M+IΩαNt+βtIMs>0IΩαLs-Usds.Using (16), we have(20)M+IΩαNt+βtIΩαM+ds.By the stochastic Gronwall inequality (e.g., Lemma 2.1 ), we have (21)IΩαMα+e-tNβe-t+βαe-tdNs.By Nβ=0 we have(22)EIΩαMα+e-tEβαe-tdNs=0.So, using (16) once again we have (23)IΩαLα-XαIΩαMα+=0a.e.That is LαXα 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 ), so we introduce it in the following.

Lemma 7.

Suppose that

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

(2) every chain in X has a supremum

(3) there is an element xoX for which x0f(x0)

Then f has a smallest fixed point in the set {xX:x0x}.

The following theorem is our main result.

Theorem 8.

Let b,σ:R+×Ω×RR be predictable. Suppose that b is an increasing function in x and σ satisfies (Cσ) and there exists a predictable process H(t,ω),t0, such that (24)bt,ω,x+σt,ω,xHt,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,t0,such that L0X0U0,a.s.

Then there is at least a strong solution Xt which satisfies that LtXtUt for t0 holds with probability 1.

Proof.

Let X be a space of adapted and continuous processes and define the order relation : (26)XYPXtYt,t0=1,for X,YX. We consider a subset of the space (X,)(27)DL,UXX:PLtXtUt,t0=1.For arbitrary fixed ZD, 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:DX 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)UtUs+stbu,Zudu+stσu,UudWuholds with probability 1 for ts0. Then U is also an upper solution of SDE (28). Similarly, we have that (30)LtLs+stbu,Zudu+stσu,LudWuholds with probability 1 for ts0 such that L is also a lower solution of SDE (28). Hence, using Theorem 6 we have(31)PLtSZtUt,t0=1.Since Z is arbitrary, we have S:DD and LS(L) and S(U)U. If S is an increasing mapping, by Lemma 7S has a fixed point on D. In fact, take Z1,Z2D and Z1Z2 and set XiS(Zi); that is, (32)Xti=X0+0tbs,Zsids+0tσs,XsidWs,i=1,2.Since b is an increasing function, we have that (33)Xt2Xs+stbu,Zu1du+stσu,Xu2dWuholds with probability 1 for ts0. 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)PSZt1SZt2Ut,t0=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,t0,with initial value X0. Obviously, X0-t+WtX0+0tsgn(Xs)ds+WtX0+t+Wt. By Theorem 8, there exists at least one solution Xt such that X0-t+WtXtX0+t+Wt,t0 holds with probability 1.

Example 10.

We have the SDE (37)dXt=fXt,tdt+σdWt,t0,with initial value X0, where f(x,t) is a bounded function and is defined as(38)fx,t=M+1,xM;x+1,0x<M;x-1,-Mx<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.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This paper was supported by the Fundamental Research Funds for the Central Universities and the School of Statistics and Mathematics of CUFE.

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