# The Existence of Strong Solutions for a Class 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):

d[X.sub.t] = b (t, [X.sub.t]) dt + [sigma] (t, [X.sub.t]) d[W.sub.t] t [greater than or equal to] 0, (1)

where b(t, x) : [R.sub.+] x R [right arrow] R and [sigma](t, x) : [R.sub.+] x R [right arrow] R are called drift and diffusion coefficients, respectively. [W.sub.t] 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 [section]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:

[X.sub.t] = [[integral].sup.t.sub.0] sgn ([[X.sub.s]) d[W.sub.s]; 0 [less than or equal to] t < [infinity], (2)

where

[mathematical expression not reproducible]. (3)

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 [sigma](t, [X.sub.t]) satisfies ([C.sub.[sigma]]) 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 ([OMEGA], F, P) is a completed probability space, W =: {[W.sub.t] : t [greater than or equal to] 0} is a real-valued Brownian motion defined on ([OMEGA], F, P), and {[F.sub.t] : t [greater than or equal to] 0} is natural filtration generated by the Brownian motion W; i.e., for any t [greater than or equal to] 0

[F.sub.t] = [sigma]{[W.sub.s] : s [less than or equal to] t}. (4)

We consider SDE (1) with coefficients b(t, x) : [R.sup.+] x R [right arrow] R and [sigma](t, x) : [R.sup.+] x R [right arrow] R, where [R.sup.+] and R are a positive real number and real number, respectively. And we use [parallel] x [parallel] to denote norm of R. The following is the definition of a strong solution for SDE.

Definition 1. An adapted continuous process [X.sub.t] defined on ([OMEGA], F, P) is said to be a strong solution for SDE (1) if it satisfies that

[mathematical expression not reproducible], (5)

holds with probability 1.

Moreover, [X.sub.t] and [[??].sub.t] are two strong solutions of SDE (1); then P[[X.sub.t] = [[??].sub.t]; 0 [less than or equal to] t < [infinity]] = 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 [U.sub.t] (resp., [L.sub.t]) is an upper (resp., lower) solution of SDE (1) if the inequalities

(1) [mathematical expression not reproducible];

(2) [mathematical expression not reproducible],

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 ([C.sub.[sigma]]) condition.

([C.sub.[sigma]]): For N > 0, there exist an increasing function [[rho].sub.N] : [R.sub.+] [right arrow] [R.sub.+] and a predictable process [G.sub.N](t, [omega]) such that

[mathematical expression not reproducible], (6)

for all t [greater than or equal to] 0, and x, y [member of] R with [parallel] x [parallel], [parallel] y [parallel] [less than or equal to] N.

Note that the Lipschitz condition satisfies the ([C.sub.[sigma]]) 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. InSDE(1), we assume [sigma] satisfies ([C.sub.[sigma]]) and b satisfies that, for each N > 0, there exists a measurable process [L.sub.N](t, [omega]) such that

[parallel]b (t, [omega], x) - b (t, [omega], y) [parallel] [less than or equal to] [L.sub.N] (t, [omega]) [parallel] x - y [parallel], [[integral].sup.t.sub.0] [L.sub.n] (t, [omega]) dt < [infinity], a.s., (7)

for all t [greater than or equal to] 0 and x, y [member of] R with [parallel] x [parallel], [parallel] y [parallel] [less than or equal to] N. Then SDE (1) has a unique local (explosion in the finite time) strong solution.

Remark 5. Moreover, if b and a satisfy the liner growth condition (cf. J. Jacod and J. Memin [15])

[parallel]b(t, [omega], x) [parallel] + [parallel][sigma](t, [omega]) [parallel] [less than or equal to] H(t, [omega])(1 + [parallel]x[parallel]), (8)

where H(t, [omega]), t [greater than or equal to] 0, is a predictable process such that [[integral].sup.t.sub.0] [H.sup.2] (s, [omega])ds < [infinity], 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.sup.+] x [OMEGA] [right arrow] R be predictable such that [[integral].sup.t.sub.0] [b.sup.2](s, [omega])ds < [infinity], a.s. for any t [greater than or equal to] 0, and let [sigma] : [R.sup.+] x [OMEGA] x R [right arrow] R be predictable. Suppose that [sigma] satisfies ([C.sub.[sigma]]) and there exists a predictable process H(t, [omega]), t [greater than or equal to] 0 such that

[parallel][sigma](t, [omega]) [parallel] [less than or equal to] H(t, [omega])(1 + [parallel]x[parallel]), (9)

where [[integral].sup.t.sub.0] [H.sup.2] (s, [omega])ds < [infinity], a.s. And suppose that [U.sub.t] and [L.sub.t] are upper and lower solutions of the following SDE:

[mathematical expression not reproducible], (10)

such that [L.sub.0] [less than or equal to] [X.sub.0] [less than or equal to] [U.sub.0], a.s.

Then there is a unique strong solution [X.sub.t] which satisfies that [L.sub.t] [less than or equal to] [X.sub.t] [less than or equal to] [U.sub.t] for any t [greater than or equal to] 0 holds with probability 1.

Proof. Obviously, we have that SDE (10) has a unique strong solution [X.sub.t] by using Lemma 4 and Remark 5. In the following we will show

P [[L.sub.t] [less than or equal to] [X.sub.t] [less than or equal to] [U.sub.t], [for all]t [greater than or equal to] 0} = 1. (11)

We only prove P{[X.sub.t] [less than or equal to] [U.sub.t], [for all]I [greater than or equal to] 0} = 1,because we can prove P{[L.sub.t] [less than or equal to] [X.sub.t], [for all]t [greater than or equal to] 0} = 1 by using the similar way.

Define the stopping time

[T.sub.N] = inf {t [member of] [0, [infinity]) : [absolute value of [X.sub.t]] [disjunction] [absolute value of [L.sub.t]] [disjunction] t > N([conjunction] N. (12)

Obviously, [T.sub.N] [right arrow] [infinity] when N [right arrow] [infinity]. And define the stopping timer [tau] = inf{t [member of] [0, [infinity]) : [X.sub.t] < [L.sub.t]. If P{[tau] < [T.sub.N]} = 0 for N > 1, then P|r < ot} = 0; that is, P|Lt < [X.sub.t], Vi > 0} = 1. Indeed, [for all]q [member of] [Q.sup.+] and N [greater than or equal to] 1, we define [alpha] =: ([tau] + q) [conjunction] [T.sub.N] and [[OMEGA].sub.[alpha]] = {[X.sub.[alpha]] < [L.sub.[alpha]]. Note that

P {[[OMEGA].sub.[alpha]]} = 0, [for all]q [member of] [Q.sup.+], N [greater than or equal to] 1 [??] P {[tau] < [T.sub.N]} = 0. (13)

In fact, by P{[[OMEGA].sub.[alpha]]} = 0 and X, L being continuous and the denseness of the rational number in R, we have

[mathematical expression not reproducible] (14)

for all t [greater than or equal to] 0. That is for a.s. [omega] [member of] {[tau] < [T.sub.N]} and t [member of] [[tau]([omega]), [T.sub.N]([omega])] one has [X.sub.t] [greater than or equal to] [L.sub.t]. However, by the definition of [tau] and [L.sub.[tau]] [less than or equal to] [x.sub.[tau]], a.s. we have P{[tau] < [T.sub.N]} = 0.

In the following we shall prove P{[[OMEGA].sub.[alpha]]} = 0, [for all]q [member of] [Q.sup.+], N [greater than or equal to] 1. Set [beta] = sup{t [member of] [0, [alpha]) : [L.sub.t] [less than or equal to] [X.sub.t]}. By continuity of X and L we have [X.sub.[beta]] [greater than or equal to] [L.sub.[beta]], a.s. Obviously, {[X.sub.[alpha]] [greater than or equal to] [X.sub.[alpha]]} = {[beta] = [alpha]}. So, we have [[OMEGA].sub.a] = {[X.sub.[alpha]] < [L.sub.[alpha]]} = {[beta] < [alpha]}. Hence, for [omega] [member of] [[OMEGA].sub.[alpha]] and t [member of] ([beta]([omega]), [alpha]([omega])] we have [X.sub.t] < [L.sub.t]. Using L as a lower solution of SDE (10), we have

[X.sub.t] - [X.sub.t] [less than or equal to] [integral].sup.t.sub.[beta]] [[sigma] (s, [L.sub.s]) - [sigma] (s, [X.sub.s])] d[W.sub.s] = [M.sub.t]. (15)

Hence,

[mathematical expression not reproducible]. (16)

Let us take [M.sup.t] = max{M, 0}. By the Tanaka formula (refer to [3]) we have

[mathematical expression not reproducible], (17)

where [L.sup.x.sub.t](M) denotes local time at the point x for M. By the definition of local time, one can prove easily that [mathematical expression not reproducible] (using the definition M) we have

[mathematical expression not reproducible]. (18)

Since for [omega] [member of] [[OMEGA].sub.a] and t [member of] ([beta]([omega]), [alpha]([omega])] we have [X.sub.t] < [L.sub.t], by (18) we have

[mathematical expression not reproducible]. (19)

Using (16), we have

[mathematical expression not reproducible]. (20)

By the stochastic Gronwall inequality (e.g., Lemma 2.1 [14]), we have

[mathematical expression not reproducible], (21)

By [N.sub.[beta]] = 0 we have

[mathematical expression not reproducible]. (22)

So, using (16) once again we have

[mathematical expression not reproducible]. (23)

That is [L.sub.[alpha]] < [X.sub.[alpha]] on [[OMEGA].sub.[alpha]] a.s. Hence, P{[[OMEGA].sub.[alpha]]} = 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 [right arrow] X is monotone increasing on an ordered set X

(2) every chain in X has a supremum

(3) there is an element [x.sub.0] [member of] X for which [x.sub.0] [less than or equal to] f([x.sub.0]) Then f has a smallest fixed point in the set {x [member of] X : [x.sub.0] [less than or equal to] X}.

The following theorem is our main result.

Theorem 8. Let b, [sigma] : [R.sup.+] x Q x R [right arrow] R be predictable. Suppose that b is an increasing function in x and a satisfies ([C.sub.[sigma]]) and there exists a predictable process H(i, [omega]), t [greater than or equal to] 0, such that

[parallel] b(t, [omega], x)[parallel] + [parallel][sigma](t, [omega], x)[parallel] [less than or equal to] H(t, [omega])(1 + [parallel] x [parallel]), (24)

where [[integral].sup.t.sub.0] [H.sup.2] (s, [omega]) < [infinity], a.s. Moreover, suppose that [U.sub.t] and [L.sub.t] are upper and lower solutions of the SDE

[mathematical expression not reproducible], (25)

such that [L.sub.0] [less than or equal to] [X.sub.0] [less than or equal to] [U.sub.0] a.s.

Then there is at least a strong solution [X.sub.t] which satisfies that [L.sub.t] [less than or equal to] [X.sub.t] [less than or equal to] [U.sub.t] for t [greater than or equal to] 0 holds with probability 1.

Proof. Let X be a space of adapted and continuous processes and define the order relation [less than or equal to]

X [less than or equal to] y [??] {[X.sub.t] [less than or equal to] [Y.sub.t], [for all]t [greater than or equal to] 0} = 1, (26)

for X, Y [member of] X. We consider a subset of the space (X, [less than or equal to])

[mathematical expression not reproducible]. (27)

For arbitrary fixed Z [member of] D, we consider the following equation:

[mathematical expression not reproducible]; (28)

by Theorem 6 there exists a unique strong solution [X.sup.*]. Define a mapping S : D [right arrow] X and S(Z) = [X.sup.*]. 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

[mathematical expression not reproducible] (29)

holds with probability 1 for t [greater than or equal to] s [greater than or equal to] 0. Then U is also an upper solution of SDE (28). Similarly, we have that

[mathematical expression not reproducible] (30)

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

P {[L.sub.t] [less than or equal to] S([Z.sub.t]) [less than or equal to] [U.sub.t], [for all]t > 0} = 1. (31)

Since Z is arbitrary, we have S : D [right arrow] D and L [less than or equal to] S(L) and S(U) [less than or equal to] U. If S is an increasing mapping, by Lemma 7 S has a fixed point on D. In fact, take [Z.sup.1], [Z.sup.2] [member of] D and [Z.suo.1] [less than or equal to] [Z.sup.2] and set [X.sup.i] =: S([z.sup.i]); that is,

[mathematical expression not reproducible]. (32)

Since b is an increasing function, we have that

[mathematical expression not reproducible] (33)

holds with probability 1 for t [greater than or equal to] s [greater than or equal to] 0. Hence [X.sup.2] is an upper solution of the following equation:

[mathematical expression not reproducible]. (34)

And by (29) U is an upper solution of (34). Using Theorem 6 again, we have

P [s([z.sup.1.sub.t]) [less than or equal to] s([z.sup.2.sub.t]) [less than or equal to] [U.sub.t], t [greater than or equal to] 0) = 1; (35)

that is, S([z.sup.1.sub.t]) [less than or equal to] S([z.sup.2.sub.t]). Hence S is an increasing function. The proof is completed.

Example 9. We consider the following SDE:

d[X.sub.t] = sgn ([X.sub.t]) dt + d[W.sub.t], [for all]t [greater than or equal to] 0, (36)

with initial value [X.sub.0]. Obviously, [X.sub.0] - t + [W.sub.t] [less than or equal to] [X.sub.0] + [[integral].sup.t.sub.0]] sgn([X.sub.s])ds + [W.sub.t] [less than or equal to] [X.sub.0] + t + [W.sub.t]. By Theorem 8, there exists at least one solution [X.sub.t] such that [X.sub.0] - t + [W.sub.t] < [X.sub.t] < [X.sub.0] + t + [W.sub.t], t > 0 holds with probability 1.

Example 10. We have the SDE

d[X.sub.t] = f([X.sub.t], t) dt + [sigma]d[W.sub.t], [for all]t [greater than or equal to] 0, (37)

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

[mathematical expression not reproducible]. (38)

It is easy to show [X.sub.t] = [X.sub.0] - (M+1) t + [sigma][W.sub.t] and [X.sub.t] = [X.sub.0] + (M+ 1)t + [sigma][W.sub.t] 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.

https://doi.org/10.1155/2018/2059694

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.

References

[1] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, NY, USA, 2nd edition, 1991.

[2] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 2nd edition, 1997.

[3] P. Protter, Stochastic Integration and Differential Equations, Springer, Berlin, Germany, 1990.

[4] A. K. Zvonkin, "A transformation of the phase space of a diffusion process that will remove the drift," Mathematics of the USSR-Sbornik, vol. 22, pp. 129-149, 1974.

[5] N. V. Krylov, "On weak uniqueness for some diffusions with discontinuous coefficients," Stochastic Processes and Their Applications, vol. 113, no. 1, pp. 37-64, 2004.

[6] N. V. Krylov and R. Liptser, "On diffusion approximation with discontinuous coefficients," Stochastic Processes and Their Applications, vol. 102, no. 2, pp. 235-264, 2002.

[7] N. Halidias and P. E. Kloeden, "A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient," BIT Numerical Mathematics, vol. 48, no. 1, pp. 51-59, 2008.

[8] A. Kohatsu-Higa, A. Lejay, and K. Yasuda, "On weak approximation of stochastic differential equations with discontinuous drift coefficient," Journal of Mathematical Economics, vol. 1788, pp. 94-106,2012.

[9] H.-L. Ngo and D. Taguchi, "Strong convergence for the Euler-CMaruyama approximation of stochastic differential equations with discontinuous coefficients," Statistics and Probability Letters, vol. 125, pp. 55-63,2017.

[10] G. Leobacher and M. Szolgyenyi, "Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient," Numerische Mathematik, vol. 138, no. 1, pp. 219-239, 2018.

[11] P. Przybylowicz, "Optimality of Euler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients," International Journal of Computer Mathematics, vol. 91, no. 7, pp. 1461-1479, 2014.

[12] N. Halidias and P. E. Kloeden, "A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient," Journal of Applied Mathematics and Stochastic Analysis, pp. 1-6, 2006.

[13] S. Assing and R. Manthey, "The behavior of solutions of stochastic differential inequalities," Probability Theory and Related Fields, vol. 103, no. 4, pp. 493-514, 1995.

[14] X. Ding and R. Wu, "A new proof for comparison theorems for stochastic differential inequalities with respect to semi-martingales," Stochastic Processes and Their Applications, vol. 78, no. 2, pp. 155-171, 1998.

[15] J. Jacod and J. Memin, "Weak and strong solutions of stochastic differential equations: existence and stability," Stochastic Integrals, vol. 851, pp. 169-212, 1980.

[16] E. Zeidler, Nonliner Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer, Berlin, Germany, 1986.

Junfei Zhang (iD)

School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China

Correspondence should be addressed to Junfei Zhang; zhangfei851115@163.com

Received 11 July 2018; Accepted 25 September 2018; Published 15 October 2018