The Existence of Strong Solutions for a Class of Stochastic Differential Equations

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.Themain tools of this paper are the lower solutions and upper solutions of stochastic differential equations.


Introduction
There are many works [1][2][3] about the existence and uniqueness of strong or weak solutions for the following stochastic differential equation (denoted briefly by SDE): =  (,   )  +  (,   )    ≥ 0, where (, ) : R + × R → R and (, ) : R + × R → R are called drift and diffusion coefficients, respectively.  is standard Brownian motion.Usually, the drift and diffusion coefficients are Lipschitz or local Lipschitz continuous or at least are continuous with respect to  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 .Therefore, many authors discussed the existence of solutions for SDE with discontinuous coefficients.For example, L. Karatzas and S. E. Shreve [1] (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 . A. K. Zvonkin [4] considered the following stochastic differential equation with a discontinuous diffusion coefficient: where The weak solution of this stochastic differential equation exists, but there is not the strong solution.N. V. Ktylov [5] and N. V. Ktylov and R. Liptser [6] also discussed existence issues of SDE when their diffusion coefficients are discontinuous with respect to .And many authors also considered the approximation solutions of SDE with discontinuous coefficients, such as [7][8][9][10][11].
In this paper, we will consider the existence of a strong solution of SDE (1) when the drift coefficient (, ) is an increasing function but need not be continuous with respect to  and the diffusion coefficient (,   ) satisfies (  ) 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.

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,  =: {  :  ≥ 0} is a real-valued Brownian motion defined on (Ω, F, P), and {F  :  ≥ 0} is natural filtration generated by the Brownian motion ; i.e., for any  ≥ 0 We consider SDE (1) with coefficients (, ) : R + × R → R and (, ) : 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 .An adapted continuous process   defined on (Ω, F, P) is said to be a strong solution for SDE (1) if it satisfies that (5) holds with probability 1.
Moreover,   and X are two strong solutions of SDE (1); then [  = X ; 0 ≤  < ∞] = 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 .An adapted continuous stochastic process   (resp.,   ) is an upper (resp., lower) solution of SDE (1) if the inequalities (1) Remark .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 (  ) condition.
Note that the Lipschitz condition satisfies the (  ) 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].
for all  ≥ 0 and ,  ∈ R with ‖‖, ‖‖ ≤ .en SDE ( ) has a unique local (explosion in the finite time) strong solution.
en there is a unique strong solution   which satisfies that   ≤   ≤   for any  ≥ 0 holds with probability .

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 ( ) the mapping  :  →  is monotone increasing on an ordered set  ( ) every chain in  has a supremum ( ) there is an element   ∈  for which  0 ≤ ( 0 ) en  has a smallest fixed point in the set { ∈  :  0 ≤ }.
en there is at least a strong solution   which satisfies that   ≤   ≤   for  ≥ 0 holds with probability .
Proof.Let X be a space of adapted and continuous processes and define the order relation ⪯: International Journal of Differential Equations for ,  ∈ X.We consider a subset of the space (X, ⪯) For arbitrary fixed  ∈ D, we consider the following equation:  ( It is easy to show   =  0 −(+1)+  and   =  0 +(+ 1) +   are the lower solution and upper solution of (37), respectively.And (, ) is an increasing function in  but is not continuous in , so we have that SDE (37) has a strong solution by using Theorem 8.