International Journal of Differential Equations
Volume 2018, Issue 1 2059694
Research Article
Open Access

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

Junfei Zhang, 

Corresponding Author

Junfei Zhang

School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China cufe.edu.cn

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First published: 15 October 2018
https://doi.org/10.1155/2018/2059694
Citations: 7
Academic Editor: José A Langa

Abstract

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.

1. Introduction

There are many works [1–3] about the existence and uniqueness of strong or weak solutions for the following stochastic differential equation (denoted briefly by SDE):
()
where and 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 [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 x. 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 x. And many authors also considered the approximation solutions of SDE with discontinuous coefficients, such as [7–11].

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 is a completed probability space, W = :{Wt : t ≥ 0} is a real-valued Brownian motion defined on , and is natural filtration generated by the Brownian motion W; i.e., for any t ≥ 0
()

We consider SDE (1) with coefficients and , where and are a positive real number and real number, respectively. And we use ‖·‖ to denote norm of . The following is the definition of a strong solution for SDE.

Definition 1. An adapted continuous process Xt defined on is said to be a strong solution for SDE (1) if it satisfies that

()
holds with probability 1.

Moreover, Xt and are two strong solutions of SDE (1); then . 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) ;

(2) ,

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 and a predictable process GN(t, ω) such that
()
for all t ≥ 0, and 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

()
for all t ≥ 0 and 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])

()
where H(t, ω),   t ≥ 0, is a predictable process such that 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 be predictable such that , and let be predictable. Suppose that σ satisfies (Cσ) and there exists a predictable process H(t, ω),   t ≥ 0 such that

()
where And suppose that Ut and Lt are upper and lower solutions of the following SDE:
()
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

()
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

()
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
()
In fact, by P{Ωα} = 0 and X, L being continuous and the denseness of the rational number in , we have
()
for 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

()
Hence,
()
Let us take M+≕max⁡{M, 0}. By the Tanaka formula (refer to [3]) we have
()
where denotes local time at the point x for M. By the definition of local time, one can prove easily that , for t ∈ (β, α] on Ωα. So, by (using the definition M) we have
()
Since for ω ∈ Ωα and t ∈ (β(ω), α(ω)] we have Xt < Lt, by (18) we have
()
Using (16), we have
()
By the stochastic Gronwall inequality (e.g., Lemma 2.1 [14]), we have
()
By Nβ = 0 we have
()
So, using (16) once again we have
()
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 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

()
where Moreover, suppose that Ut and Lt are upper and lower solutions of the SDE
()
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 be a space of adapted and continuous processes and define the order relation ⪯:

()
for . We consider a subset of the space
()
For arbitrary fixed , we consider the following equation:
()
by Theorem 6 there exists a unique strong solution . Define a mapping 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

()
holds with probability 1 for t ≥ s ≥ 0. Then U is also an upper solution of SDE (28). Similarly, we have that
()
holds with probability 1 for t ≥ s ≥ 0 such that L is also a lower solution of SDE (28). Hence, using Theorem 6 we have
()
Since Z is arbitrary, we have and L⪯S(L) and S(U)⪯U. If S is an increasing mapping, by Lemma 7S has a fixed point on . In fact, take and Z1⪯Z2 and set Xi≕S(Zi); that is,
()
Since b is an increasing function, we have that
()
holds with probability 1 for t ≥ s ≥ 0. Hence X2 is an upper solution of the following equation:
()
And by (29) U is an upper solution of (34). Using Theorem 6 again, we have
()
that is, . Hence S is an increasing function. The proof is completed.

Example 9. We consider the following SDE:

()
with initial value X0. Obviously, . 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

()
with initial value X0, where f(x, t) is a bounded function and is defined as
()
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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