Abstract
In this paper, the exponential stability of stochastic differential equations driven by multiplicative fractional Brownian motion (fBm) with Markovian switching is investigated. The quasi-linear cases with the Hurst parameter H ∈ (1/2, 1) and linear cases with H ∈ (0, 1/2) and H ∈ (1/2, 1) are all studied in this work. An example is presented as a demonstration.
1 Introduction
In the natural world, it is a common phenomena that many practical systems may face random abrupt changes in their structures and parameters, such as environmental variance, changing of subsystem interconnections and so on. To deal with these abrupt changes, Markovian switching systems, a particular class of hybrid systems, are investigated and widely used [1, 2]. Especially in signal processing, financial engineering, queueing networks, wireless communications and so on (see, e.g. [1, 3]).
In recent years, much attention has been paid to the stability of stochastic hybrid systems. For example, Mao [4] considers the exponential stability of general nonlinear stochastic hybrid systems. In [5], the criteria of moment exponential stability are obtained for stochastic hybrid delayed systems with Lévy noise in mean square. Zhou [6] investigates the pth moment exponential stability of the same systems. Some sufficient conditions for asymptotic stability in distribution of SDEs with Markovian switching are reported in [7]. See also [8, 9] for more results about Markovian switching.
On the other hand, it is generally known that if H ∈ (0, 1/2) and H ∈ (1/2, 1), has a long range dependence, which means if we putthen . Besides, the process is also self-similar for any H ∈ (0, 1). Since the pioneering work of Hurst [10, 11] and Mandelbrot [12], the fractional Brownian motion has been suggested as a useful tool in many fields such as mathematical finance [13, 14] and weather derivatives [15]. Even though fractional Brownian motion is not a semimartingale, more and more financial models have been extended to fBm (see, e.g. [16, 17]). Therefore, in this paper, the risk assets are described by hybrid stochastic systems driven by multiplicative fBm. Then it is a natural and interesting question that under what conditions, this stochastic systems have some exponential stability. For the sake of clarity, we only consider the one dimensional cases. For more details about fractional noise, we refer the reader to [18–21].
The main purpose of this paper is to discuss the exponential stability of a risky asset, with price dynamics:where g(Xt, t, rt) = σ(t, rt)Xt, is a Markov chain taking values in , is a standard fractional Brownian motion. Moreover, and . In this paper, the initial value x0 is assumed to be deterministic, otherwise more calculations about Wick product are required.
Equation 1 can be regarded as the result of the following N fractional stochastic differential equations:switching from one to another according to the movement of .
Throughout this paper, unless otherwise specified, we let C denote a general constant and p denote a non-negative constant. Let denote the family of all real value functions on which are continuously twice differentiable with respect to the first variables and once differentiable with respect to the second variables.
This paper is organized as follows. For the convenience of the reader, we briefly recall some of the basic results in Section 2. In Section 3, we investigate the solution and an extended Itô’s Formula for the general hybrid fractional stochastic differential Equation 1. Section 3 is devoted to the linear cases. In this section the moment exponential stability and almost sure exponential stability are discussed respectively. In Section 4, some useful criteria for the exponential stability with respect to quasi-linear cases are presented. Finally, a numerical example and graphical illustration are presented in Section 6.
2 Preliminaries
2.1 Markov Chain
Let be a right-continuous Markov chain taking values in a finite state space . The generator is given bywhere △ > 0.
Here qij is the transition rate from i to j if i ≠ j. According to [22, 23], a continuous-time Markov chain with generator can be represented as a stochastic integral with respect to a Poisson random measure. Then we havewith initial condition r0 = i0, where ν(dt × dy) is a Poisson random measure with intensity dt × m(dy). Here m(⋅) is the Lebesgue measure on .
Throughout this paper, unless otherwise specified, the Markov chain has the invariant probability measure and is assumed to be independent of . Almost every sample path of the Markov chain is assumed to be a right-continuous step function with a finite number of simple jumps in any finite time interval [0, T]. The generator is assumed to be irreducible and conservative, i.e., qi≔ − qii = ∑i≠jqij < ∞. For more details about Markovian switching we further refer the reader to [24–26].
2.2 Fractional Brownian Motion and Wick Product
We recall some of the basic results of fBm briefly, which will be needed throughout this paper. For more details about fBm we refer the reader to [16, 17, 27, 28]. If H ∈ (0, 1/2) ∪ (1/2, 1), then the (standard) fractional Brownian motion with Hurst parameter H is a continuous centered Gaussian process with and covariance function:
To simplify the representation, it is always assumed that .
Besides, has the following Wiener integral representation:where is a Wiener process and KH(t, s) is the kernel function defined byin which , where B(⋅, ⋅) is the Beta function, and s < t. In this paper, generates a filtration with . Denote the complete probability space, with the filtration described above.
Let be the set of all finite multi-indices α = (α1, …, αn) for some n ≥ 1 of non-negative integers. Denote |α| = α1 + ⋯ + αn, and α! = α1!⋯αn!.
Define the Hermite polynomials:and Hermite functions:
Let denote the Schwartz space of rapidly decreasing infinitely differentiable -valued functions. Denote the dual space of by . Definethe product of Hermite polynomials. Consider a square integrable random variable
According to [17, 29], every F(ω) has a unique representation:besides,
Definition 2.1(Wick Product) For , set and . Their Wick product is defined by
2.3 Malliavin Derivative
Let be the space of all random variables , such thatand letwhere ϕ(s, t) = H(2H − 1)|s − t|2H−2.
Definition 2.2The ϕ-derivative of F ∈ Lp in the direction of Φg is defined byif the limit exists in Lp. Moreover if there exists a process such thatfor all , then F is said to be ϕ-differentiable.According to [16, 30], let be the family of stochastic process on [0, T] such that if and F is ϕ-differentiable, the trace of exists and , and for each sequence of partitions such that |πn| → 0, as n → ∞. Moreoverandas n → ∞. Here , and .Now we define the -integral considered in [16].
Definition 2.3Let be a stochastic process such that . Define bywhere |π| = maxi∈{0,1,…,n−1}{ti+1 − ti}.
Remark 2.1: According to Theorem 3.6.1 in [16], if , then the stochastic integral satisfies , and What’s more, according to Definition 3.4.1 in [16], the stochastic integral can be extended bywhere is a given function such that Ft ⋄ WH(t) is dṭ − integrable in . Here is the fractional Hida distribution space defined by Definition 3.1.11 in [16]. In particular, the integral on [0, T] can be defined by
3 Hybrid Fractional Systems
In this section, firstly, we consider the existence and uniqueness of solution for Eq. 1. Then, an extended Itô’s Formula is presented.
3.1 Existence and Uniqueness
To ensure the existence and uniqueness, we impose the following assumptions.
Assumption 3.1
Let
satisfy the hypothesises:
1) For each fixed , f(x, t, i) is measurable in all the arguments.
2) For each fixed , there exists a constant C > 0, such that .
3) For each fixed , there exists a constant C > 0, such that
Assumption 3.2
Let
satisfy the hypothesises:
1) For each fixed , σ(t, i) is nonrandom;
2) For each fixed , .
Lemma 3.1: Let Assumptions 3.1,3.2 hold. Then Eq. 1 has a unique solution.Proof: The existence and uniqueness can be proved similar to that for Theorem 2.6 in [31], so we omit it here.
3.2 The Itô Formula
Next, we first review the results in [16, 30] on the Itô formula with respect to fBm. Then we extend it to SDEs driven by fBm with Markovian switching.
Lemma 3.2[16] (The Itô Formula) Let (Fu, 0 ≤ u ≤ T) be a stochastic process in . Assume that there exists an α > 1 − H and C > 0 such thatwhere |u − v| ≤ δ for some δ > 0 andLet sup0≤s≤T|Gs| < ∞ and with bounded derivatives. Moreover, for , it is assumed that and is in . Denote , for t ∈ [0, T]. Let ,. Then for t ∈ [0, T],Here is the Malliavin derivative defined in Definition 2.2.In particular, for the process , with each fixed , we have thatFormally,LetSubstituting Eq. 3 into Eq. 2, we getIn the sequel of this paper, unless otherwise specified, we let the coefficients of Eq. 1 satisfy the conditions in Lemma 3.2, for each fixed . Set . Next we consider the Itô formula which reveals how V maps (Xt, t, rt) into a new process V(Xt, t, rt), where is a stochastic process with the stochastic differential Eq. 1.
4 Linear Hybrid Fractional Systems
There are many models for financial markets with fBm (see, e.g. [16]). The simplest nontrivial type of market is the fBm version of the classical Black Scholes market, in which linear fractional SDEs is used. Thus, we would like to give some new criteria for switching linear fractional SDEs with or . At first, we present a definition and a useful lemma.
Definition 4.1Let H ∈ (0, 1). The operator M is defined on functions bywhereHere Γ(⋅) denotes the classical Gamma function.According to [16], Eq. 6 can be restated as follows.For H ∈ (0, 1/2), we haveFor H = 1/2, we haveFor H ∈ (1/2, 1), we have
Lemma 4.1Let be a right-continuous Markov chain which takes values in a finite state space . Assume that it is irreducible and positive recurrent with invarient measure μ. If is a function verifyingThen there exists constants C, c > 0 such that:for any initial condition r0 and every t ≥ 0.Proof: It is a consequence of Perron-Frobenius theorem and the study of eigenvalues. See Proposition 4.1 in [25], Proposition 4.2 in [25], and Lemma 2.7 in [26], for further details.In Eq. 1, let us consider the case g(x, t, rt) = σ(t, rt)x = thb(rt)x, f(x, t, rt) = α(rt)x, where α(i) and b(i) are constants for each . This means that we are considering the following linear equation:Set and . x0 is the deterministic initial value. For the sake of clarity, we firstly set h = 1/2 − H.
4.1 pth Moment Exponential Stability
Proof. According to [16], without too many calculations, we obtain that has the following form:where Ms is the operator M acting on the variable s. Let x0 ≠ 0. It follows from Eq. 8 that
We then see from Eq. 9 thatwhere
Noting that ζt is the solution to the equationwith initial value ζ0 = |x0|p. Thuswhich yields
Substituting Eq. 11 into Eq. 10 gives
Note that
Consequently, by Definition 4.1 and [16], one has
Making use of Eqs 12, 13, we obtain that
Therefore, by Lemma 4.1 and Eq. 12, the required assertions follow. The proof is complete.
Proof: Similar to Theorem 4.1, we write the solution as follows.
Note that Ms is the operator M acting on the variable s, where
According to [16], we also have that
Consequently, by Lemma 4.1, the result follows. The proof is complete.
Remark 4.1In the above Theorems 4.1,4.2, the parameter h is supposed to be H − 1/2. Noting that by Eqs 13, 15 and together with the Definition 4.1, the stability of solution for Eq. 7 with h < 1/2 − H or h > 1/2 − H can be deduced respectively without too many difficulties.
Remark 4.2Take H = 1/2. It’s easy to show that if , then , and if , then , which coincide with the results of SDEs driven by Brownian motion in [4, 32].
4.2 Almost Sure Exponential Stability
To proceed, we need to introduce the definition of almost sure stability and a useful lemma.
Definition 4.2The equilibrium point x = 0 is said to be almost surely exponential stable iffor any .
Lemma 4.2(Law of the iterated logarithm) For a standard fBm , we have thatwhere CH > 0 is a suitable constant.Proof: By [33], we havewhere cH is a suitable constant. Then the thesis follows by the self-similarity of fBm and a change of variable t → 1/t.For the sake of clarity, we firstly set h = 0. Namely, let us considerNoting that Eq. 17 is exactly the geometry fBm with Markovian Switching. We proceed to discuss the almost sure exponential stability about it.
1) If 0 < H < 1/2, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable when , but unstable when ; 2) If H = 1/2, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable when , but unstable when ; 3) If 1/2 < H < 1, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable for all parameters α(i) and σ(i),.
Proof: Define
By Definition 4.1 and [16], one has
Making use of Eq. 18, we get
Especially, when H = 1/2, we have that
Therefore, the required results follows. The proof is complete.
Remark 4.3Making use of Eq. 18, one can discuss the almost sure exponential stability for Eq. 7 with h ≠ 0. The proofs are similar to Theorem 4.3 and are omitted.
5 Quasi-Linear Hybrid Fractional Systems
We now apply the extended Itô Formula in Section 3 to discuss the stability for quasi-linear fractional SDEs with Markovian switching.
: Let Assumptions 3.1,3.2 hold. If there exists a function and positive constants a1, a2, b and p ≥ 1, such thatfor all , t ≥ t0,.
Then the solution of Eq. 1 is pth moment exponential stable. More precisely,
Proof: According to Lemma 3.1, Eq. 1 has a unique solution. Denote it . Setwhere , η > 0. Making use of Definition 2.3 and Lemma 3.2, one has and .
Applying the conditions Eq. 19, 20, together with the generalized ItôEq. 5 and Remark 2.1, we obtain that for any t ∈ [0, T]
Thus we obtain that
Dividing both sides of Eq. 21 by a1eηt, noting that λa2 − b < 0, we get
Consequently,
Letting T → ∞ givesand the required assertion follows. The proof is complete.
In the sequel of this section, we give another useful criterion and prove it briefly.
Assume that Eq. 1 has a unique solution and there exist a function and positive constants b1, b2, p ≥ 1 and such that for all , t ≥ t0, ,and Then Eq. 1 is pth moment exponential stable.
Proof: Set , where θ ∈ (0, 1). Let . Let denote the vector which all elements are 1. Then,
By [1], Eq. 22 implies the Poisson equation:
Note that Eq. 23 has the solution . Hence,
For each , set U(x, t, i) = (1 − θci)V(x, t, i), where θ ∈ (0, 1) is already defined and sufficiently small satisfying 1 − θci > 0.
Then, for any t ∈ [0, T] we get
Making use of Eqs 25, 26, we obtain that
Substituting Eq. 24 into Eq. 27, we getwhere κ < 0. Making use of Theorem 5.1, the desired criterion follows.
On the other hand, we can prove it in another way. Set η > 0 and λ ∈ (η, − κ). Define
Compute
Thus we obtain that
Dividing both sides of Eq. 28 by b1eηt, noting that b2(λ + κ) < 0, we get
Therefore, we obtain the required assertion
The proof is complete.
6 Example
In this section we give a numerical example to illustrate our results.
Example 1Let be a right-continuous Markov chain taking values in with invariant probability measure .Consider a risky asset, with the price dynamics:on t ≥ 0. Here we take H = 0.7 andNote that for all , satisfy the hypothesises (i)-(v). Then, by Lemma 3.1, it is easy to show that Eq. 29 has a unique solution as well. Set V(x, t, i) = x2, for i = 1, 2.Noting that for some t0 > 0 sufficiently large and all t > t0, we haveandCompute By Theorem 5.2, it’s clear that the solution of Eq. 29 is second moment exponential stable. Figures 1, 2 show a single path of the solution and the solution’s norm square, respectively.
FIGURE 1
A single path of solution.
FIGURE 2
Norm square trajectory.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
WP contributed to conception and design of the study. WP wrote the first draft of the manuscript. ZZ and WP wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.
Funding
The research of WP was supported by the Characteristic and Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics).
Acknowledgments
The authors are grateful to thank the reviewers for careful reading of the paper and for helpful comments that led to improvement of the first version of this paper.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
stochastic differential equation (SDEs), stability, fractional brownian motion, markovian switching, hybrid system
Citation
Pei W and Zhang Z (2021) Stability of Hybrid SDEs Driven by fBm. Front. Phys. 9:783434. doi: 10.3389/fphy.2021.783434
Received
26 September 2021
Accepted
13 October 2021
Published
02 November 2021
Volume
9 - 2021
Edited by
Ming Li, Zhejiang University, China
Reviewed by
Xichao Sun, Bengbu University, China
Yaozhong Hu, University of Alberta, Canada
Updates
Copyright
© 2021 Pei and Zhang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Wenyi Pei, peiwenyi@163.com
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.