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Mean square Hyers-Ulam stability of stochastic differential equations driven by Brownian motion
Advances in Difference Equations volume 2016, Article number: 271 (2016)
Abstract
In this paper, the Hyers-Ulam stability for a class of first order stochastic differential equations is studied by using the Ito formula. Furthermore, the research results are applied to a class of second order stochastic differential equations with constant coefficients by the substitution method. In the end, the Hyers-Ulam stability of general second order stochastic differential equations is considered by the solutions of two deterministic second order differential equation boundary value problems.
1 Introduction
The Hyers-Ulam stability of functional equations was introduced with the motivation of studying the stability of approximate solutions [1, 2]. Since then, much attention was given to the stability studies of functional equations; see [3–6] and the references therein. In 1993, Obloza introduced the notion of Hyers-Ulam stability for the studies of differential equations [7, 8]. Furthermore, the stability studies of differential equations have been considered in the recent decade; see [9–20] and the references therein. To the best of the author’s knowledge, after the success of the investigations of the Hyers-Ulam stability for deterministic differential equations, there are a few arguments about the Hyers-Ulam stability of stochastic differential equations in the literature. However, uncertainty is involved in all kinds of natural phenomena, and stochastic differential equations are the suitable mathematical models for the natural phenomena. Therefore, it is important to generalize the research results of deterministic differential equations to stochastic differential equations. In the paper, we will consider the Hyers-Ulam stability of the following stochastic differential equations in the mean square which are perturbed by the Brownian motion:
and
where \(t\geq0\), \(a,b,c,f,h,r,k:[0,+\infty)\rightarrow\mathbb{R}\) are continuous, \(B_{t}\) is a standard one-dimensional Brownian motion, \(X_{t}\) is a stochastic process which is adapted to the same filtration as \(B_{t}\). If \(h_{t}\equiv0\), \(k_{t}\equiv0\), equations (1.1) and (1.2) are deterministic equations, which had been considered by the method of integral factors in [13–15].
2 Preliminary
Now we introduce the fundamental definitions and a lemma, which are used later in the article. Throughout this paper, we consider a filtered probability space \((\Omega,\mathcal{F},P)\) with 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.
Definition 2.1
Assume that for any \(\varepsilon\geq0\) and any stochastic process
satisfies
where E is the expectation operator, then there exists a solution \(X_{t}\) of equation (1.1) such that \(|Y_{t}-X_{t}|\leq K\varepsilon\), \(t\in(0,T)\) with K is a positive real constant. We say that equation (1.1) is Hyers-Ulam stable on \((0,T)\) in the mean square.
Definition 2.2
Assume that for any \(\varepsilon>0\) and any stochastic process
satisfies the following inequality:
where E is the expectation operator, then there exists a solution \(X_{t}\) of equation (1.2) such that \(|Y_{t}-X_{t}|\leq K\varepsilon\), \(t\in(0,T)\) with K a positive real constant. We say that equation (1.2) is Hyers-Ulam stable on \((0,T)\) in the mean square.
To consider the integration of the stochastic process, we use the Ito formula as follows.
Lemma 2.1
[21]
Suppose \(dX_{t}=U_{t}\,dt+V_{t}\,dB_{t}\), where the vector \(U=(U_{1},\ldots,U_{m})\) and the matrix \(V=(V_{1},\ldots ,V_{m})\) have \(\mathcal{L}_{2}\) components and B is the vector of m independent Brownian motions. Let F be a twice continuously differentiable function from \(\mathbb{R}^{m}\) into \(\mathbb{R}\). Then \(Y_{t}=F(X_{t})\) is also an Ito process and
where \(dX_{i,t}\cdot dX_{j,t}\) is computed by using the rules \(dt\,dt=dt\,dB_{i,t}=dB_{i,t}\,dt=0\), \(dB_{i,t}\,dB_{j,t}=0\) for \(i\neq j\) and \((dB_{i,t})^{2}=dt\).
Let \(m=2\), \(F(X_{t})=X_{1,t}X_{2,t}\), then from Lemma 2.1, we see
and
3 Hyers-Ulam stability of (1.1)
In this section, we establish some criteria of the Hyers-Ulam stability of equation (1.1), by using the Ito formula.
Theorem 3.1
Let \(Y_{t}\) be an Ito process, \(a,f,h\in\mathcal{L}^{2}[0,T]\),
assume that \(Y_{t}\) satisfies \(E(g(t,Y_{t}))^{2}\leq\varepsilon\), for \(t\in(0,T)\), \(\varepsilon\geq0\). Then there exists a solution \(X_{t}\) of equation (1.1) such that \(X_{0}=Y_{0}\), \(E(X_{t}-Y_{t})^{2}\leq M\varepsilon\) with
That means equation (1.1) is Hyers-Ulam stable in the mean square on the interval \((0,T)\).
Proof
Multiplying two sides of (3.1) by the function \(e^{-\int _{0}^{t}a_{s}\,ds}\), we obtain
Applying Lemma 2.1, we have
From (3.2), we have
Integrating the two sides of (3.3) from 0 to t and multiplying the two sides of (3.3) by the function \(e^{\int_{0}^{t}a_{s}\,ds}\), we get
Define
then we have \(X_{0}=Y_{0}\) and
Hence \(X_{t}\) is a solution of equation (1.1). We rewrite (3.4) as
Applying Lemma 2.1, we have
where \(\int_{0}^{t}g(s,Y_{s})\,de^{-\int_{0}^{s}a_{\tau}\,d\tau}\) is a Stieltjes integral. Taking expectations on the two sides of (3.5), we see
on the interval \([0,T]\) by (3.6). Hence equation (1.1) is Hyers-Ulam stable in the mean square on the interval \([0,T]\). The proof is completed. □
4 Hyers-Ulam stability of (1.2)
First of all, we consider the Hyers-Ulam stability of equation (1.2) by using the substitution method for a special case. We assume that \(b_{t}\) and \(c_{t}\) are both constant functions and write b and c instead of \(b_{t}\) and \(c_{t}\).
Theorem 4.1
Let \(Y'_{t}\) be an Ito process,
Assume that \(E(G(t,Y_{t}))^{2}\leq\varepsilon\) for \(t\in(0,T)\), \(\varepsilon\geq0\). Then there exists a solution \(X_{t}\) of equation (1.2) such that
with
That means equation (1.2) is Hyers-Ulam stable in the mean square on the interval \([0,T]\).
Proof
Let
then (1.2) can be rewritten as
We write
instead of (4.1). Multiplying two sides of (4.3) by the matrix function \(e^{-At}\), we get
Since \(Y'_{t}\) is an Ito process, without loss of generality, we can define
By computing, we have
By Lemma 2.1, we see \(dt\,dU_{t}=dt(dY'_{t},dY_{t})=0\). Hence
From (4.4), we have
Integrating two sides of (4.5) from 0 to t and multiplying (4.5) by the matrix function \(e^{At}\), we see
Define
Then we have
Therefore \(Z_{t}=( X_{t}, X'_{t})^{T}\) is a solution of (4.2), that is, \(X_{t}\) is a solution of (1.2) with \(X_{0}=Y_{0}\), \(X'_{0}=Y'_{0}\). We rewrite (4.6) as
Similar to Theorem 3.1, by Lemma 2.1, we have
Assume
with E the identity matrix. Hence
By (4.8), (4.9), (4.10), (4.11), (4.12), (4.13), we have
We consider three possibilities for computing \(\alpha(t)\), \(\beta(t)\).
(i) If \(b^{2}+4c>0\), we see that
are different real eigenvalues of the matrix A. By (4.10), we have
Hence
(ii) If \(b^{2}+4c<0\), we see that
are two different complex eigenvalues. By (4.10), we have
Hence
(iii) If \(b^{2}+4c=0\), we see that \(\lambda_{1}=\lambda_{2}=\frac {b}{2}\). By (4.10), we have
Hence
Taking expectations on the two sides of (4.14), we have
with
Hence equation (1.2) is Hyers-Ulam stable in the mean square on the interval \([0,T]\). The proof is completed. □
Since matrix multiplication is, in general, not commutative, Theorem 4.1 is not suitable for equation (1.2), when \(b_{t}\) is not a constant function or \(c_{t}\) is not a constant function. Now, we consider equation (1.2) by the solutions of two deterministic boundary value problems.
Let u and v be the solutions of the boundary value problems
and
respectively. Define
Lemma 4.2
Let \(X'_{t}\) be an Ito process, \(B_{t}\) is a standard one-dimensional Brownian motion. Assume that \(p,b,c\in L^{2}[0,T]\), then
Proof
Let
By Lemma 2.1, we have
Multiplying by the function \(u'_{s}\), integrating two sides of (4.18) from 0 to t, we see
Similarly, multiplying by the function \(v'_{s}\), integrating two sides of (4.18) from t to T, we see
Therefore, by Abel’s differential equation identity, we have
That is,
The proof is completed. □
Lemma 4.3
Let \(B_{t}\) is a standard one-dimensional Brownian motion. C, D are two stochastic variables. Assume that \(p,r,k\in L^{2}[0,T]\), then the stochastic process
is a solution of equation (1.2) such that \(X_{0}=C\), \(X(T)=D\).
Proof
By Lemma 2.1, we obtain
Similarly, we have
Hence, by Abel’s differential equation identity, we have
Therefore (4.19) is a solution of equation (1.2). □
Theorem 4.4
Let \(Y'_{t}\) be an Ito process,
Assume that \(E(G(t,Y_{t}))^{2}\leq\varepsilon\) for \(t\in(0,T)\), \(\varepsilon\geq0\), \(b,c,r,k\in\mathcal{L}^{2}(0,T)\). Then there exists a solution \(X_{t}\) of equation (1.2) such that
with
That means equation (1.2) is Hyers-Ulam stable in the mean square on the interval \((0,T)\).
Proof
By Lemma 4.2, we have
Let
by Lemma 4.3, we obtain \(X_{t}\) as a solution of equation (1.2) such that \(X_{0}=Y_{0}\), \(X_{T}=Y_{T}\). By (4.20), (4.21), (4.22), we get
By computing, we have
Taking expectations on the two sides of (4.23), we have
by (4.23). Hence equation (1.2) is Hyers-Ulam stable in the mean square on the interval \([0,T]\). The proof is completed. □
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Supported by the Fundamental Research Funds for the Central Universities.
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Zhao, X. Mean square Hyers-Ulam stability of stochastic differential equations driven by Brownian motion. Adv Differ Equ 2016, 271 (2016). https://doi.org/10.1186/s13662-016-1002-4
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DOI: https://doi.org/10.1186/s13662-016-1002-4