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Approximate solutions for a class of doubly perturbed stochastic differential equations
Advances in Difference Equations volume 2018, Article number: 37 (2018)
Abstract
In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed stochastic differential equations (DPSDEs). Based on the Carathéodory approximation procedure, we prove that DPSDEs have a unique solution and show that the Carathéodory approximate solution converges to the solution of DPSDEs under the global Lipschitz condition. Moreover, we extend the above results to the case of DPSDEs with non-Lipschitz coefficients.
1 Introduction
As the limit process from a weak polymers model, the following doubly perturbed Brownian motion
was discussed by Norris et al. [1], and it also arises as the scaling limit of some self-interacting random walks (see [2]). During the past few decades, equation (1.1) has attracted much interest from many scholars, for example, [3–9]. Following them, Doney et al. [10] studied the singly perturbed Skorohod equations
Using the Picard iterative procedure, they showed the existence and uniqueness of the solution to equation (1.2). Hu et al. [11] discussed the existence and uniqueness of the solution to doubly perturbed neutral stochastic functional equations, while Luo [12] obtained the existence and uniqueness of the solution to doubly perturbed jump-diffusion processes.
In fact, the Picard iterative method is a well-known procedure for approximating the solution of stochastic differential equations (SDEs). However, to obtain the Picard iterative sequence \(x^{n}(t)\), one needs to compute \(x^{i}(t)\), \(0\le i\le n-1\). And this brings us a lot of calculations on stepwise iterated Ito’s integrals. In the early twentieth century, Carathéodory [13] put forward the Carathéodory approximation scheme for ordinary differential equations. In this scheme, Carathéodory defined the approximate solution via a delay equation, and the delay equation can be solved explicitly by successive integrations over intervals of length \(\frac{1}{n}\). In other words, the Carathéodory approximation scheme avoids calculating \(x^{i}(t)\), \(0\le i\le n-1\).
Because of its advantage, this approximation procedure has received great attention, and many people have been devoted to the study of the Carathéodory scheme for SDEs. For example, Bell and Mohammed [14] extended the Carathéodory approximation scheme to the case of SDEs and showed the convergence of the Carathéodory approximate solution. Mao [15, 16] considered a class of SDEs with variable delays and studied the Carathéodory approximate solution of delay SDEs. Turo [17] discussed the Carathéodory approximate solution of stochastic functional differential equations (SFDEs) and established the existence theorem for SFDEs. Liu [18] investigated a class of semilinear stochastic evolution equations with time delays and proved that the Carathéodory approximate solution converges to the solution of stochastic delay evolution equations.
Motivated by the above mentioned papers, we will study the Carathéodory approximate scheme of doubly perturbed stochastic differential equations (DPSDEs)
To the best of our knowledge, so far little is known about the Carathéodory approximations for equation (1.3), and the aim of this paper is to close this gap. In this paper, we will prove that the Carathéodory approximate solution converges to the solution under the global Lipschitz condition. Moreover, we will replace the global Lipschitz condition by a more general condition proposed by [19, 20] and show that equation (1.3) has a unique solution under the non-Lipschitz condition.
This paper is organized as follows. In Section 2, we establish the existence theorem of equation (1.3) and show that the Carathéodory approximate solution converges to the solution of equation (1.3) under the global Lipschitz condition. While in Section 3, we extend the existence and convergence results of Section 2 to the case of equation (1.3) with non-Lipschitz coefficients.
2 Carathéodory approximation and global Lipschitz DPSDEs
Let \((\Omega , {\mathcal {F}}, \{{\mathcal {F}}_{t}\}_{t\ge0}, P)\) be a complete probability space with a filtration \(\{{\mathcal {F}}_{t}\}_{t\ge 0}\) satisfying the usual conditions (i.e., it is increasing and right continuous, while \({\mathcal {F}}_{0}\) contains all P-null sets). Let \(\{w(t)\}_{t\ge0}\) be a one-dimensional Brownian motion defined on the probability space \((\Omega , {\mathcal {F}}, P)\). Let \({\mathcal {L}}^{2}([a,b];R)\) denote the family of \(\mathcal {F}_{t}\)-measurable, R-valued processes \(f(t)=\{f(t,\omega)\}\), \(t\in [a,b]\) such that \(\int_{a}^{b}|f(t)|^{2}\,dt<\infty\) a.s.
Consider the following doubly perturbed stochastic differential equations:
where \(\alpha,\beta\in(0,1)\), the initial value \(x(0)=x_{0}\in R\) and \(f:[0,T]\times R\to R\), \(g:[0,T]\times R \to R\) are both Borel-measurable functions. In this paper, we assume that the initial value \(x_{0}\) is independent of w and satisfies \(E|x_{0}|^{2}<\infty\).
Now, we define the sequence of the Carathéodory approximate solutions \(x^{n}: [-1,T]\to R\). For all \(n\ge1\), we define
Note that \(x^{n}(t)\) can be calculated step by step on the intervals \([0,\frac{1}{n})\), \([\frac{1}{n},\frac{2}{n}),\ldots \) , etc.
To obtain the main results, we give the following conditions.
Assumption 2.1
For any \(x,y\in R\) and \(t\in[0,T]\), there exists a positive constant k such that
Assumption 2.2
For any \(t\in[0,T]\), there exists a positive constant k̄ such that
Assumption 2.3
The coefficients satisfy \(|\alpha|+|\beta|<1\).
Remark 2.1
Clearly, Assumptions 2.1 and 2.2 imply the linear growth condition. That is, for any \(x\in R\) and \(t\in[0,T]\),
where \(L=2\max(k,\bar{k})\). Similarly, we have \(|g(t,x)|^{2}\le L(1+|x|^{2})\).
Now, we state our main results.
Theorem 2.1
Let Assumptions 2.1-2.3 hold. Then there exists a unique \(\mathcal{F}_{t}\)-adapted solution \(\{x(t)\}_{t\ge0}\) to equation (2.1). Moreover, for any \(T>0\),
where C is a constant independent of n.
In the sequel, to prove our main results, we need some useful lemmas.
Lemma 2.1
(Gronwall’s inequality [21])
Let \(u_{0}\ge0\) and \(v(t)\ge0\), and let \(u(\cdot)\) be a real continuous function on \([0,T ]\). If
then we have
for all \(t\in[0,T]\).
Lemma 2.2
Under Assumptions 2.1-2.3, for all \(n\ge1\),
where \(C_{1}\) is a positive constant.
Proof
For any \(t\in[0,T]\), it follows from (2.2) that
By the basic inequality \(|a+b+c|^{2}\le3(|a|^{2}+|b|^{2}+|c|^{2})\), one has
for any \(t_{1}\in[0,T]\). Using the Hölder inequality and the Burkholder-Davis-Gundy inequality, we can easily see that
and
Then, by Assumptions 2.1, 2.2 and 2.3, we have
Finally, the Gronwall inequality implies that
The proof is therefore complete. □
Lemma 2.3
For all \(n\ge1\) and \(0\le s< t\le T\),
where \(C_{2}\) is a positive constant.
Proof
For all \(n\ge1\) and \(0\le s< t\le T\), it follows from (2.2) that
Note that \(\inf_{0\le\sigma\le t}x^{n}(\sigma-\frac{1}{n})\le\inf_{0\le\sigma\le s}x^{n}(\sigma-\frac{1}{n})\), we have
Next, let us consider the following two cases.
Case I. If \(\sup_{0\le\sigma\le t}x^{n}(\sigma-\frac{1}{n})=\sup_{0\le \sigma\le s}x^{n}(\sigma-\frac{1}{n})\), then we get from (2.13) that
Case II. If \(\sup_{0\le\sigma\le t}x^{n}(\sigma-\frac{1}{n})>\sup_{0\le \sigma\le s}x^{n}(\sigma-\frac{1}{n})\), then there exists \(r\in(s,t]\) such that \(x^{n}(r)=\sup_{0\le\sigma\le t}x^{n}(\sigma-\frac{1}{n})\). So we get from (2.13) that
We therefore have
Hence,
Then Lemma 2.2 yields
where \(C_{2}=\frac{2(T+4)L}{(1-\alpha)^{2}}(1+C_{1})\). The proof is therefore complete. □
Proof of Theorem 2.1
Firstly, we will show that the sequence \(\{x^{n}(t)\}\) is a Cauchy sequence in \({\mathcal {L}}^{2}([0,T];R)\). For any \(n>m\ge1\), it follows that
Noting that
and
one can have
By the basic inequality and Assumption 2.3, we obtain that
Then, using the Hölder inequality and the Burkholder-Davis-Gundy inequality again, we have
and
Substituting (2.17) and (2.18) into (2.16), one has
Then Lemma 2.3 yields
Hence,
where \(C_{3}=\frac{6(4+T)k}{(1-|\alpha|-|\beta|)^{2}}, C_{4}=\frac {3[2(4+T)kT+(|\alpha| +|\beta|)^{2}]C_{2}}{(1-|\alpha|-|\beta|)^{2}}\). By the Gronwall inequality, we have
which implies that
This shows that the sequence \(\{x^{n}(t)\}\) is a Cauchy sequence in \({\mathcal {L}}^{2}([0,T];R)\). Denote the limit by \(x(t)\). Letting \(m\to\infty\) in (2.20) yields
Then the Borel-Cantelli lemma can be used to show that \(x^{n}(t)\) converges to \(x(t)\) almost surely uniformly on [0,T] as \(n\to\infty\). Taking limits on both sides of (2.2) and letting \(n\to\infty\), we can obtain that \(x(t)\) is a solution of equation (2.1).
Now we show the uniqueness of the solution. Let \(x(t)\) and \(y(t)\) be any two solutions of equation (2.1). We can prove using the same procedure as (2.19) that
for all \(t\in[0,T]\). The Gronwall inequality gives that
i.e., for any \(t\in[0,T]\), \(x(t)\equiv y(t)\) a.s. This completes the proof. □
Remark 2.2
By (2.6), we conclude that the Carathéodory approximate solution converges to the true solution of equation (2.1) in the mean square sense, i.e., for any \(T>0\),
In fact, the proof of the convergence of the Carathéodory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution to delay DPSDEs. In other words, the Carathéodory approximation scheme is applicable to a class of DPSDEs.
3 Non-Lipschitz DPSDEs
In this section, we will replace the global Lipschitz condition (2.3) with a more general condition and show that the Carathéodory approximate solution still converges to the true solution of equation (2.1).
Assumption 3.1
For any \(x,y\in R\) and \(t\in[0,T]\), there exists a function \(k(\cdot)\) such that
where \(k(u)\) is a concave non-decreasing continuous function such that \(k(0)=0\) and \(\int_{0^{+}}\frac{u}{k^{2}(u)}\,du=\infty\).
Remark 3.1
Since \(k(\cdot)\) is concave and \(k(0)=0\), one can find a pair of positive constants a and b such that
Theorem 3.1
Let Assumptions 3.1, 2.2 and 2.3 hold. Then there exists a unique \(\mathcal{F}_{t}\)-adapted solution \(\{x(t)\}_{t\ge0}\) to equation (2.1). Moreover, for any \(T>0\),
To prove Theorem 3.1, we will need the following Bihari inequality.
Lemma 3.1
(Bihari’s inequality [22])
Let \(k: R_{+} \to R_{+}\) be a continuous, non-decreasing function satisfying \(k(0)=0\) and \(\int_{0^{+}} \frac{ds}{k(s)}=+\infty\). Let \(u(\cdot)\) be a Borel measurable bounded non-negative function defined on \([0,T ]\) satisfying
where \(u_{0}>0\) and \(v(\cdot)\) is a non-negative integrable function on \([0,T]\). Then we have
where \(G(t)=\int_{t_{0}}^{t}\frac{du}{k(u)}\) is well defined for some \(t_{0}>0\), and \(G^{-1}\) is the inverse function of G.
In particular, if \(u_{0}=0\), then \(u(t)=0\) for all \(t\in[0,T]\).
Lemma 3.2
Under Assumptions 3.1, 2.2 and 2.3, for all \(n\ge1\),
where \(\bar{C}_{1}\) is a positive constant.
Proof
By the Hölder inequality and the Burkholder-Davis-Gundy inequality, it follows from (2.8) that
By Assumptions 2.2 and 3.1, we have
Then the Jensen inequality implies that
Let \(\rho(x)=k^{2}(x^{\frac{1}{2}})\), it follows that
Since \(\frac{k(x)}{x}\) and \(k'_{+}(x)\) are non-negative, non-increasing functions, we have that
is a non-negative, non-increasing function which implies that ρ is a non-negative, non-decreasing concave function. Note that \(k(0)=0\), then \(\rho(0)=0\), and there exists a pair of positive constants a and b such that
We therefore have
Set
then \(r(\cdot)\) is the solution to the following ordinary differential equation:
By recurrence, it is easy to verify that, for each \(n\ge0\),
Note that \(r(t_{1})\) is continuous and bounded on \([0,T]\), one can have
for any \(n\ge1\). This completes the proof of Lemma 3.2. □
Lemma 3.3
For all \(n\ge1\) and \(0\le s< t\le T\),
where \(\bar{C}_{2}\) is a positive constant.
The proof is similar to Lemma 2.3, we omit its proof.
Now, let us apply the above lemmas to prove Theorem 3.1.
Proof of Theorem 3.1
By the Hölder inequality and the Burkholder-Davis-Gundy inequality, it follows from (2.16) that
By Assumption 3.1 and the Jensen inequality, we have
Similar to (3.6), one obtains
Since \(\rho(\cdot)\) is concave, we have \(\rho(a+b)\le\rho(a)+\rho(b)\). Then Lemma 3.3 yields
where
Inserting (3.12) into (3.11), we obtain that
where
Then the Bihari inequality gives that
where \(G(t)=\int_{1}^{t}\frac{ds}{\rho(s)}\). Obviously, G is a strictly increasing function, then G has an inverse function which is strictly increasing, and \(G^{-1}(-\infty)=0\). Note that when \(m,n\to\infty\), then \(\frac{3C(m,n)}{(1-|\alpha|-|\beta|)^{2}}\to0\). Recalling \(\int_{0^{+}}\frac{ds}{\rho(s)}=\int_{0^{+}}\frac {s}{k^{2}(s)}\,ds=\infty\), we have
and
We therefore have
which implies that \(\{x^{n}(t)\}_{n\ge1}\) is a Cauchy sequence. Denote the limit by \(x(t)\). Letting \(m\to\infty\) in (3.14) yields
Similar to (3.13), (3.14), we can show that \(x(t)\) is a unique solution of equation (2.1) under non-Lipschitz conditions. Then the proof is completed. □
Remark 3.2
To see the generality of our results, let us give a few examples of the function \(k(\cdot)\). Let \(\delta\in(0,1)\) be sufficiently small, define
and
They are all concave non-decreasing functions satisfying \(\int_{0^{+}}\frac{u}{k_{i}^{2}(u)}\,du=\infty\), \(i=1,2\).
Remark 3.3
In particular, if we let \(k(u)=ku\), \(u\ge0\), we see that the Lipschitz condition (2.3) is a special case of our proposed condition (3.1). In other words, we obtain a more general result than Theorem 2.1.
Remark 3.4
In fact, our theories developed can be applied to study doubly perturbed stochastic differential equations with jumps (DPSDEwJs) and doubly perturbed stochastic differential equations with Markovian switching (DPSDEwMS) respectively. If \(\alpha=\beta=0\), then DPSDEwJs and DPSDEwMS will become SDEs with jumps and SDEs with Markovian switching which were investigated by [23–34]. Similarly, we can also give the Carathéodory approximate solution and show that the Carathéodory approximate solution converges to the solution of SDEs with jumps and SDEs with Markovian switching under our assumptions.
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Acknowledgements
The authors would like to thank the editor and the anonymous referee for their careful reading and valuable comments. This work was supported by the Leverhulme Trust (RF-2015-385), the Royal Society Wolfson Research Merit Award (WM160014), the NSFC (Nos. 11401261, 11471071) and ‘333 High-level Personnel Training Project’ of Jiangsu Province.
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Mao, W., Hu, L. & Mao, X. Approximate solutions for a class of doubly perturbed stochastic differential equations. Adv Differ Equ 2018, 37 (2018). https://doi.org/10.1186/s13662-018-1490-5
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DOI: https://doi.org/10.1186/s13662-018-1490-5
Keywords
- Carathéodory approximate solution
- doubly perturbed stochastic differential equations
- global Lipschitz condition
- non-Lipschitz condition