Existence and uniqueness of solutions for stochastic differential equations with locally one-sided Lipschitz condition

1.   Introduction

Stochastic differential equations (SDEs) have been widely applied in many fields, such as biology, economics and physics for modeling (see, e.g., ^{[1,2,3,4,5,6]}). More and more people have showed their interests in SDEs. So far, many results of solutions for SDEs have been obtained, such as the existence and uniqueness of solutions ^{[7,8,9,10,11,12]}, Markov property ^[13], and even the long-term behavior ^[14]. In addition, to describe a wide variety of natural and man-made systems precisely, various types of SDEs are developed (see, e.g., ^[10,15]), and the theory of these SDEs has always been a focus.

One of the popular topics of SDEs is the existence and uniqueness of solutions. Generally, the classical existence and uniqueness theorem for SDEs requires the coefficients to satisfy the global Lipschitz condition (see, e.g., ^[16,17]). Under the local Lipschitz condition and the linear growth condition, Arnold ^[18] has showed the existence of the unique solutions for SDEs. However, there are many interesting SDEs such that their coefficients are only superlinear, for such SDEs, Mao ^[10] has derived that there exists a unique regular solution under the locally Lipschitz condition and the monotone condition. Based on these existence and uniqueness results for the classical SDEs, many authors have studied the existence and uniqueness problems for other types of SDEs. For instance, Zvonkin ^[19] has investigated the strong solutions of SDEs with singular coefficients. Mao and Yuan ^[20] have introduced the existence and uniqueness of solutions for SDEs with Markovian switching.

Furthermore, although many SDEs have been showed that they each have a unique solution, it is important to determine precisely under which conditions one obtains a unique solution for SDEs. Compared with more restrictive conditions, general conditions can provide the existence and uniqueness of solutions for a larger class of SDEs. By using the Euler method, Krylov ^[9] has established the existence and uniqueness theorem under the monotone condition and a more general condition which is known as the local one-sided Lipschitz condition. Then, Gy$\mathrm{\ddot{o}}$ngy and Sabanis ^[7] have developed this result to stochastic differential delay equations. Recently, Ji and Yuan ^[8] have established the existence and uniqueness result for neutral stochastic differential delay equations. In this paper, inspired by Li et al. ^[21] and Krylov ^[9], we aim to study the existence and uniqueness of solutions for SDEs under weaker conditions compared with what we have mentioned above. Also, we can obtain the $p$th moment boundedness. The main contribution of this paper is that we have included the case of $0 < p < 2$ in our conditions.

The rest organization of this paper is as follows: In Section 2, some notations and preliminaries are introduced. In Section 3, the existence and uniqueness of solutions is provided by deriving a localization lemma, and the $p$th moment is further estimated. An example is given to illustrate our results in Section 4.

2.   Notations and preliminaries

In this paper, we let $(\Omega, \; \mathcal{F}, \; \mathbb{P})$ be a complete probability space with a filtration $\{\mathcal{F}_{t}\}_{t\geq0}$ satisfying the usual conditions (that is, it is right continuous and increasing while ${\mathcal{F}}_{0}$ contains all P-null sets). Denote $\mathbb{N}$ as the set of natural numbers and $m, d\in\mathbb{N}$. Let $\{B(t)\}_{t\geq0}$ be a standard $m$-dimensional Brownian motion defined on the probability space. Let $\mathbb{R}_+ = \{x\in\mathbb{R}:x\geq0\}$, $\mathbb{R}^{d}$ be $d$-dimensional Euclidean space, and $\mathbb{R}^{d\times m}$ be the space of real $d\times m$-matrices. If $x\in\mathbb{R}^{d}$, then $|x|$ is the Euclidean norm. For any matrix $A$, define its trace norm by $\|A\| = \sqrt{\mathrm{trace}(AA^{\top})}$, where $A^{\top}$ denotes its transpose. Moreover, for any $a\in \mathbb{R}$ and $b\in\mathbb{R}$, define $a\wedge b = \min\{a, b\}$ and $a\vee b = \max\{a, b\}$. For a set $G$, let $I_{G}(x) = 1$ if $x\in G$ and otherwise $0$. Let $\inf\emptyset = \infty$ (as usual, $\emptyset$ denotes the empty set). For any $x\in\mathbb{R}$, let $\lfloor x\rfloor$ be the integer part of $x$. For any $p\in(0, +\infty)$, let $L^{p} = L^{p}(\Omega; \mathbb{R}^{d})$ be the family of $\mathbb{R}^{d}$-valued random variables $Z(\omega)$ with $\mathbb{E}\big[|Z(\omega)|^{p}\big] < +\infty$, and $\mathcal{L} = \mathcal{L}([0, T];\mathbb{R})$ denotes the set of $\mathbb{R}$-valued nonnegative integrable functions on $[0, T]$.

In this paper, we consider a $d$-dimensional SDE described by

with the initial value $X(0) = X_{0}\in L^{p}$, where

is Borel-measurable and has continuous mappings.

Moreover, in order for the main results, we impose the following assumptions.

Assumption 1. For any $R, \; T\in[0, \infty)$, and $x\in\mathbb{R}^{d}$,

Assumption 2. (Locally one-sided Lipschitz condition) For any $R, \; T\in[0, \infty)$, there exists a $K_{R}(t)\in\mathcal{L}$, which is dependent on $R$ such that

for all $t\in[0, T]$, $x_{1}, x_{2}\in\mathbb{R}^{d}$, and $|x_{1}|\vee|x_{2}|\leq R$, where the $K_{R}(t)$ are satisfying $\int_{0}^{T}K_{R}(t)\mathrm{d}t < \infty$, for any $R, \; T\in[0, \infty)$.

For the regularity and $p$th moment boundedness of the exact solution, we make the following assumption.

Assumption 3. For any $T\in[0, \infty)$ and $p\in(0, \infty)$, there exists a $K(t)\in\mathcal{L}$ such that

for all $t\in[0, T]$ and $x\in\mathbb{R}^{d}$.

Remark 2.1. If $p = 2$, we have

which is the assumption of the paper ^[9]. In this paper, for any $p > 0$, the $p$th moment boundedness of solutions is also provided.

For the sake of simplicity, throughout the paper, we will fix $T \in[0, +\infty)$ arbitrarily, and unless otherwise stated, $C$ denotes a generic positive real constant dependent on $T, R$ etc. Please note that the values of $C$ may change between occurrences.

3.   Existence and uniqueness of solution

In this section, we shall show that there exists a unique regular solution to $(2.1)$. According to ^[8,9,12], we prepare a localization lemma below.

Lemma 3.1. Suppose that $\{X^{n}(t)\}_{t\in[0, T]}$ are given continuous, $\mathbb{R}^{d}$-valued, and $\mathcal{F}_{t}$-adapted processes on $(\Omega, \; \mathcal{F}, \; \mathbb{P})$. For $n\in\mathbb{N}$ such that $X^{n}(0) = X(0)$, and

where $P^{n}(t)$ is a progressively measurable process. Moreover, for $n\in\mathbb{N}$ and $R\in[0, \infty)$, suppose that there exists a nonrandom function $r:[0, \infty)\rightarrow [0, \infty)$ such that $\lim\limits_{R\rightarrow \infty}r(R) = \infty$, and let $\tau_{n}(R)$ be $\mathcal{F}_{t}$-stopping times such that

$(i)$ $|X^{n}(t)|+|P^{n}(t)|\leq R$ for $t\in[0, \tau_{n}(R)]\; \; a.s.$

$(ii)$ $\lim\limits_{n\rightarrow \infty}\mathbb{E}\big[\int_{0}^{T\wedge \tau_{n}(R)}|P^{n}(t)|\mathrm{d}t\big] = 0$ for all $R, \; T\in[0, \infty)$.

$(iii)$ For any $T\in[0, \infty)$,

Then, also for any $T\in[0, \infty)$, we have

Proof. We borrow the techniques from ^[12] mainly and divide the proof into 2 steps.

Step 1. For $R\in[0, \infty)$ and $t\in[0, T]$, from Assumption 1 we assume that

(otherwise, we regard $K_{R}(t)$ as the maximum of $K_{R}(t)$ and the integrand in Assumption 1). Fix $R\in[0, \infty)$ and define the $\mathcal{F}_{t}$-stopping time

where $\alpha_{R}(t) = \int_{0}^{t}K_{R}(s)\mathrm{d}s < \infty$. Clearly, $\tau(R, u)\uparrow\infty$ as $u\rightarrow \infty$. In particular, there exists $u(R)\in(0, \infty)$ such that

Now, we let $\tau(R) = \tau(R, u(R))$, then $\tau(R)\rightarrow \infty$ in probability as $R\rightarrow \infty$ and $\alpha_{R}(t\wedge\tau(R))\leq u(R)$. Moreover, referring to ^[8,12], it is easy to prove that all three conditions $(i)$–$(iii)$ still hold if we replace $\tau_{n}(R)$ by $\tau_{n}(R)\wedge\tau(R)$. So we can further assume that $\tau_{n}(R)\leq\tau(R)$, then we have $\alpha_{R}(t\wedge\tau_{n}(R))\leq u(R)$. For a fixed $R\in[0, \infty)$, we define

and $\tau_{(n, m)}(R) = \tau_{n}(R)\wedge\tau_{m}(R)$ for $m, n\in\mathbb{N}$. Then we can obtain

Under Assumption 2, we have

We omit the proof of $(3.2)$ and $(3.3)$ there as the reader can refer to ^[8,12] for more details.

Step 2. In order for $(3.1)$, we need to show that

for any given $T\in[0, \infty)$. For $t\in[0, T]$, let $\kappa$ be a negative constant and define

where $\beta(t) = \int_{0}^{t}K(s)\mathrm{d}s$. For $t\in[0, T\wedge\tau_{n}(R)]$, applying the It$\hat{\mathrm{o}}$ formula, we have

where

Then, we further write that

where

By Assumption 3, we have

For $0 < p\leq4$, we have $\big(1+|X^{n}(t)|^{2}\big)^{\frac{p-4}{2}}\leq1$. For $t\in[0, T\wedge\tau_{n}(R)]$, using the condition $(i)$, we have $|X^{n}(t)|+|P^{n}(t)|\leq R\; a.s.$. Then we can derive that

where $C_{R}$ denotes a generic positive constant related to $R$ in this paper. Please note that the values of $C_{R}$ may change between occurrences.

While $p > 4$, using Young's inequality, we have

For $p > 0$, combining $(3.5)$ and $(3.6)$, we have

Next, we compute $J_{2}(t)$, that is,

Obviously, we also need to consider $(3.8)$ in two cases, respectively: $0 < p\leq4$ and $p > 4$. By the condition $(i)$, for $p > 0$, it is easy to show that

For $J_{3}(t)$, we can write that

In the same way as discussed above, we have

Repeating the similar procedures, we also have

and

Therefore, for $t\in[0, T\wedge\tau_{n}(R)]$ and $p > 0$, by virtue of the condition $(i)$, we derive that

and

Substituting $(3.7)$ and $(3.9)$–$(3.12)$ into $(3.4)$, we have

Choosing $\kappa = -C$ and then replacing $K_{R}(s)$ by $K(s)\vee K_{R}(s)$, we have

Furthermore, since $\psi(t)\leq1$ and $J_{n}^{R}(t)$ is a continuous local $\mathcal{F}_{t}$-martingale with $J_{n}^{R}(0) = 0$, according to ^[10], for any $R, \; T\in[0, \infty)$, taking expectations on both sides of $(3.14)$, it is easy to see that

where $\varsigma$ represents any $\mathcal{F}_{t}$-stopping time satisfying $\varsigma\leq T\wedge\tau_{n}(R)$. Then, based on [9, p. 584, Lemma 1], for any $l\in(0, \infty)$, we have

We then have

Thanks to $(3.2)$, it is easy to derive that

Recalling that $r(R)\rightarrow \infty\; as\; R\rightarrow \infty$ and choosing $l = r^{p}(R)\psi(t)$ in $(3.15)$, we have

which implies

Under condition $(iii)$, we obtain

Hence for any $\varepsilon > 0$, thanks to $(3.3)$, we have

which leads to $(3.1)$. □

We now give the theorem about the existence and uniqueness of the exact solution to $(2.1)$.

Theorem 3.1. Let Assumptions $1$–$3$ hold with $p > 0$. Then, for any $T\in[0, \infty)$, there exists a unique process $\{X(t)\}_{t\in[0, T]}$ that satisfies Eq $(2.1)$ with the property

Proof. Based on Euler's method, we construct a sequence $\{X^{n}(\cdot)\}, n\in\mathbb{N}$. For $n\in\mathbb{N}$, we define $\{X^{n}(t)\}_{t\geq0}$ as follows:

We further define $\iota(n, t) = \lfloor nt\rfloor/n$. Then, for $t\geq0$, we have

which can be written as

This is equivalent to

where $P^{n}(t) = X^{n}(\iota(n, t))-X^{n}(t) = -\int_{\iota(n, t)}^{t} f(X^{n}(\iota(n, s)), s)\mathrm{d}s-\int_{\iota(n, t)}^{t} g(X^{n}(\iota(n, s)), s)\mathrm{d}B(s)$. In order for the existence and uniqueness, we need to show that there exist an $\mathcal{F}_{t}$-adapted continuous process $\{X(t)\}_{t\in[0, T]}$ and

after taking limits on both sides of $(3.17)$. Also, define $\tau_{n}(R)$ as the first exit time of $X^{n}(t)$ from the sphere ($|x| < \frac{R}{3}$), and let nonrandom function $r(R) = \frac{R}{4}$. Then, $|P^{n}(t)|\leq \frac{2R}{3}$, $|X^{n}(t)|\leq \frac{R}{3}$ for $t\in[0, \tau_{n}(R)]\; a.s.$ and in terms of Lemma $3.1$, the proofs of these are same as ^[9,12], so we omit it there. Thus, there exists a unique solution of Eq $(2.1)$.

It remains to prove the $p$th moment boundedness. In fact, for an application of the It$\hat{\mathrm{o}}$ formula to Eq (2.1), we have

where

We recall that $\big(1+|X(t)|^{2}\big)^{\frac{p-4}{2}}\leq1$ for $0 < p\leq4$, and Young's inequality can be used in the case of $p > 4$. Therefore, by Assumption 3, $(3.16)$ follows directly from [21, p. 851, Theorem 2.3]. □

4.   Example

In this section, we consider an example that is a scalar SDE as follows:

with the initial data $X(0) = X_{0}\in\mathbb{R}$, where

and $B(t)$ is a Brownian motion. Clearly, Assumption $1$ holds for any $R, \; T\in[0, \infty)$ and $x\in\mathbb{R}$, and $f(x, t)$ does not satisfy the local Lipschitz condition. Therefore, the techniques in ^[10,21] can't be applied to the existence and uniqueness of the solution for $(4.1)$. However, by the Young inequality, we then have

for any $x_{1}, x_{2}\in\mathbb{R}$, and $|x_{1}|\vee|x_{2}|\leq R$. This means that $f(x, t)$ satisfies Assumption 2 (i.e., locals one-sided Lipschitz condition). Moreover, it should be noted that the monotone condition requiring $p\geq2$ in ^[9,12] doesn't hold there. Let $0 < p\leq1$. By computation, it is easy to verify that Assumption 3 holds, that is, for any $x\in\mathbb{R}$,

Then, by Theorem $3.1$ we can conclude that the SDE $(4.1)$ has a unique global solution $X(t)$ on $t\geq0$ with the boundedness of the $p$th moment on $[0, T]$, that is,

5.   Conclusions

The current focus lies in the existence and uniqueness of solutions for stochastic differential equations with locally one-sided Lipschitz condition, and we can obtain the pth moment boundedness. In future research, we are going to study the stability of the solution, furthermore, we shall investigate an implicit numerical scheme for these equations under a local one-sided Lipschitz condition.

Author contributions

Fangfang Shen and Huaqin Peng: Conceptualization, Methodology, Investigation, Writing-original draft, Writing-review and editing. All authors of this article have been contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Guangxi Natural Science Foundation (2023GXNSFFA026001), the Science and technology project of Guangxi (Guike AD21220114), Guangxi Basic Ability Promotion Project for Young and Middle-aged Teachers (2023KY0067, 2024KY0076), the National Natural Science Foundation of China (62363003, 12001125, 12061016) and the Key Laboratory of Mathematical Model and Application (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.

Conflict of interest

The author declares that there is no conflict of interests regarding the publication of this paper.