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International Journal of Stochastic Analysis
VolumeΒ 2012Β (2012), Article IDΒ 185474, 17 pages
http://dx.doi.org/10.1155/2012/185474
Research Article

Asymptotic Stability of Semi-Markov Modulated Jump Diffusions

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

Received 28 February 2012; Revised 6 July 2012; Accepted 12 July 2012

Academic Editor: LukaszΒ Stettner

Copyright Β© 2012 Amogh Deshpande. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations 𝑑𝑋𝑑/𝑑𝑑=𝑓(𝑋𝑑) is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion.

1. Introduction

The stability of stochastic differential equations (SDEs) has a long history with some key works being those of Arnold [1], Khasminskii [2], and Ladde and Lakshmikantham [3]. SDEs with switching have been applied in diverse areas such as finance (Deshpande and Ghosh [4]) and biology (Hanson [5]). On the same note, the stability of these processes has been much studied, in particular by Ji and Chizeck [6] and Mariton [7], who both studied the stability of a jump-linear system of the form Μ‡π‘₯𝑑=𝐴(π‘Ÿπ‘‘)π‘₯𝑑, where π‘Ÿπ‘‘ is a Markov chain. Basak et al. [8] discussed the stability of a semilinear SDE with Markovian-regime switching of the form Μ‡π‘₯𝑑=𝐴(π‘Ÿπ‘‘)π‘₯𝑑𝑑𝑑+𝜎(π‘Ÿπ‘‘,π‘₯𝑑)π‘‘π‘Šπ‘‘. Mao [9] studied the exponential stability of a general nonlinear diffusion with Markovian switching of the form 𝑑π‘₯𝑑=𝑓(π‘₯𝑑,𝑑,π‘Ÿπ‘‘)𝑑𝑑+𝑔(π‘₯𝑑,𝑑,π‘Ÿπ‘‘)π‘‘π‘Šπ‘‘. Yin and Xi [10] studied the stability of Markov-modulated jump-diffusion processes (MMJDs).

Consider the following jump-diffusion equation in which the coefficients are modulated by an underlying semi-Markov process: 𝑑𝑋𝑑𝑋=𝑏𝑑,πœƒπ‘‘ξ€Έξ€·π‘‹π‘‘π‘‘+πœŽπ‘‘,πœƒπ‘‘ξ€Έπ‘‘π‘Šπ‘‘+𝑑𝐽𝑑,𝑑𝐽𝑑=ξ€œΞ“π‘”ξ€·π‘‹π‘‘,πœƒπ‘‘ξ€Έπ‘π‘‹,𝛾(𝑑𝑑,𝑑𝛾),0=π‘₯,πœƒ0=𝑖,(1.1) where 𝑋(β‹…) takes values in β„π‘Ÿ and πœƒπ‘‘ is a finite-state semi-Markov process taking values in 𝒳={1,…,𝑀}. Let Ξ“ be a subset of β„π‘Ÿβˆ’0; it is the range space of impulsive jumps. For any set 𝐡 in Ξ“, 𝑁(𝑑,𝐡) counts the number of jumps on [0,𝑑] with values in 𝐡 and is independent of the Brownian motion π‘Šπ‘‘, 𝑏(β‹…,β‹…)βˆΆβ„π‘ŸΓ—π’³β†’β„π‘Ÿ,𝜎(β‹…,β‹…)βˆΆβ„π‘ŸΓ—π’³β†’β„π‘ŸΓ—β„π‘‘,𝑔(β‹…,β‹…,β‹…)βˆΆβ„π‘ŸΓ—π’³Γ—Ξ“β†’β„π‘Ÿ. For future use we define the compensated Poisson measure 𝑁(𝑑𝑑,𝑑𝛾)=𝑁(𝑑𝑑,𝑑𝛾)βˆ’πœ†πœ‹(𝑑𝛾)𝑑𝑑, where πœ‹(β‹…) is the jump distribution and 0<πœ†<∞ is the jump rate. Equation (1.1) can be regarded as the result of the following 𝑀 equations: 𝑑𝑋𝑑𝑋=𝑏𝑑𝑋,𝑖𝑑𝑑+πœŽπ‘‘ξ€Έ,π‘–π‘‘π‘Šπ‘‘+ξ€œΞ“π‘”ξ€·π‘‹π‘‘ξ€Έπ‘‹,𝑖,𝛾𝑁(𝑑𝑑,𝑑𝛾)0=π‘₯,πœƒ0=πœƒ,(1.2) which switch from one state to another according to the underlying movement of the semi-Markov process.

Unlike the special Markov-modulated case in which the π‘₯-dependent diffusion is a partial differential operator, the semi-Markov case is characterized by an integro-partial differential operator. In this paper we study the asymptotic stability of sMMJDs. We also investigate the perturbation of the nonlinear differential equation 𝑑𝑋𝑑/𝑑𝑑=𝑓(𝑋𝑑) by an sMMJD. We determine the conditions under which the perturbed system is almost surely exponentially stable. We show that for a one-dimensional state space, the deterministic linear unstable system of differential equations that can be stabilized by the addition of a jump component of the process 𝑋𝑑, surprisingly can never be destabilized by an addition of a Brownian motion. An interesting question we may ask here is, can the similar inference hold true for 𝑋𝑑 in higher dimension? The answer is surprisingly no. We show that for a state space with dimension greater than or equal to 2, a corresponding nonlinear system that is stabilized by the jump component of the process 𝑋𝑑 can in fact be destabilized by addition of the Brownian motion part. We organize the paper as follows.

In Section 2, we briefly establish a representation of a class of semi-Markov processes as a stochastic integral with respect to a Poisson random measure. We define the concepts of almost sure exponential stability and moment exponential stability. In Section 3, we present conditions that guarantee almost sure exponential stability and moment exponential stability of the trivial solution of (1.1). In general, there is no connection between these two stability criteria. However, under additional conditions one can say when does the moment exponential stability guarantees or implies almost sure exponential stability. We elaborate on this aspect while concluding this section. In Section 4, we provide some examples to illustrate these two stability criterion in our context. In Section 5, we investigate the conditions for which a nonlinear system of differential equation of the type 𝑑𝑋𝑑/𝑑𝑑=𝑓(𝑋𝑑) is almost surely exponentially stable. We then investigate its behavior in higher-dimensional state space, as mentioned earlier. The paper ends with concluding remarks.

2. Preliminaries

We assume that the probability space (Ξ©,β„±,{ℱ𝑑},β„™) is complete with filtration {ℱ𝑑}𝑑β‰₯0 and is right-continuous and β„±0 contains all β„™ null sets. If 𝑣 is some vector, then |𝑣| is its Euclidean norm and π‘£ξ…ž is its transpose, while if 𝐴 is a matrix then its trace norm is denoted as √||𝐴||=tr(π΄ξ…žπ΄). ℝ+ stands for positive part of the real line while π‘Ÿ is a positive integer. Let π’ž2,1(β„π‘ŸΓ—π’³Γ—β„+) denote the family of all functions on β„π‘ŸΓ—π’³Γ—β„+, which are twice continuously differentiable in π‘₯ and continuously differentiable in 𝑦. Consider {πœƒπ‘‘}𝑑β‰₯0 as a semi-Markov process taking values in 𝒳 with transition probability 𝑝𝑖,𝑗 and conditional holding time distribution 𝐹(π‘‘βˆ£π‘–). Thus if 0≀𝑑0≀𝑑1≀⋯ are times when jumps occur, then π‘ƒξ‚€πœƒπ‘‘π‘›+1=𝑗,𝑑𝑛+1βˆ’π‘‘π‘›β‰€π‘‘βˆ£πœƒπ‘‘π‘›ξ‚=𝑖=𝑝𝑖𝑗𝐹(π‘‘βˆ£π‘–).(2.1) Matrix [𝑝𝑖𝑗]{𝑖,𝑗=1,…,𝑀} is irreducible and for each 𝑖, 𝐹(β‹…βˆ£π‘–) has continuously differentiable and bounded density 𝑓(β‹…βˆ£π‘–). Embed 𝒳 in β„π‘Ÿ by identifying 𝑖 with π‘’π‘–βˆˆβ„π‘Ÿ. For π‘¦βˆˆ[0,∞)𝑖,π‘—βˆˆπ’³, let πœ†π‘–π‘—(𝑦)=𝑝𝑖𝑗𝑓(𝑦/𝑖)1βˆ’πΉ(𝑦/𝑖)β‰₯0,βˆ€π‘–β‰ π‘—,πœ†π‘–π‘–(𝑦)=βˆ’π‘—βˆˆπ’³,π‘—β‰ π‘–πœ†π‘–π‘—(𝑦)βˆ€π‘–βˆˆπ’³.(2.2) Let the stationary distribution of the semi-Markov process be defined as πœˆπ‘–βˆ«β‰œ(1/𝑑)𝑑0π•€πœƒπ‘ =𝑖𝑑𝑠 where 𝕀⋅ takes value 1 if πœƒπ‘ =𝑖 and 0 otherwise for any π‘–βˆˆπ’³.

For π‘–β‰ π‘—βˆˆπ’³, π‘¦βˆˆβ„+, let Λ𝑖𝑗(𝑦) be consecutive (with respect to lexicographic ordering on 𝒳×𝒳) left-closed, right-open intervals of the real line, each having length πœ†π‘–π‘—(𝑦). Define the functions β„ŽβˆΆπ’³Γ—β„+×ℝ→ℝ and π‘”βˆΆπ’³Γ—β„+×ℝ→ℝ+ by ξ‚»β„Ž(𝑖,𝑦,𝑧)=π‘—βˆ’π‘–ifπ‘§βˆˆΞ›π‘–π‘—(𝑦),0otherwise,𝑔(𝑖,𝑦,𝑧)=𝑦ifπ‘§βˆˆΞ›π‘–π‘—(𝑦),𝑗≠𝑖,0otherwise.(2.3)

Let β„³(ℝ+×ℝ) be the set of all nonnegative integer-valued 𝜎-finite measures on a Borel 𝜎-field of (ℝ+×ℝ). Define the process {πœƒξ…žπ‘‘,π‘Œπ‘‘} described by the following stochastic integral equations: πœƒξ…žπ‘‘=πœƒξ…ž0+ξ€œπ‘‘0ξ€œβ„β„Žξ€·πœƒπ‘’βˆ’,π‘Œπ‘’βˆ’ξ€Έπ‘,𝑧1(π‘Œπ‘‘π‘’,𝑑𝑧),π‘‘ξ€œ=π‘‘βˆ’π‘‘0ξ€œβ„π‘”ξ€·πœƒπ‘’βˆ’,π‘Œπ‘’βˆ’ξ€Έπ‘,𝑧1(𝑑𝑒,𝑑𝑧),(2.4) where 𝑁1(𝑑𝑑,𝑑𝑧) is an β„³(ℝ+×ℝ)-valued Poisson random measure with intensity π‘‘π‘‘π‘š(𝑑𝑧) independent of the 𝒳-valued random variable πœƒξ…ž0, where π‘š(β‹…) is a Lebesgue measure on ℝ. We define the corresponding compensated or centred Poisson measure as 𝑁1(𝑑𝑠,𝑑𝑧)=𝑁1(𝑑𝑠,𝑑𝑧)βˆ’π‘‘π‘ π‘š(𝑑𝑧). It was shown in Theorem  2.1 of Ghosh and Goswami [11] that πœƒξ…žπ‘‘ is a semi-Markov process with transition probability matrix [𝑝𝑖𝑗]{𝑖,𝑗=1,…,𝑀} with conditional holding time distributions 𝐹(π‘¦βˆ£π‘–). Therefore, one can write πœƒξ…žπ‘‘=πœƒπ‘‘. We assume that 𝑁(β‹…,β‹…),𝑁1(β‹…,β‹…), and πœƒ0,π‘Šπ‘‘,𝑆0 defined on (Ξ©,β„±,β„™) are independent.

To ensure that zero is the only equilibrium point of (1.1), we need the following assumption.

Assumption 2.1. Assume 𝑔(π‘₯,𝑖,𝛾) is ℬ(β„π‘ŸΓ—π’³Γ—(β„βˆ’{0}))-measurable and that constants 𝐢>0 exist such that for each π‘–βˆˆπ’³,π‘₯1,π‘₯2 being β„π‘Ÿ-valued and for each π›ΎβˆˆΞ“ we have ||𝑏π‘₯1ξ€Έξ€·π‘₯,π‘–βˆ’π‘2ξ€Έ||+||πœŽξ€·π‘₯,𝑖1ξ€Έξ€·π‘₯,π‘–βˆ’πœŽ2ξ€Έ||||π‘₯,𝑖≀𝐢1βˆ’π‘₯2||,||𝑔π‘₯1ξ€Έξ€·π‘₯,𝑖,π›Ύβˆ’π‘”2ξ€Έ||||π‘₯,𝑖,𝛾≀𝐢1βˆ’π‘₯2||.(2.5) We also need the condition that the generator matrix 𝑄(β‹…) is bounded and continuous. 𝑏(0,𝑖)=0, 𝜎(0,𝑖)=0 and 𝑔(π‘₯,𝑖,0)=0 and 𝑔(0,𝑖,𝛾)=0 for each π‘₯βˆˆβ„π‘Ÿ, π‘–βˆˆπ’³ and each π›ΎβˆˆΞ“.

The process (𝑋𝑑,πœƒπ‘‘,π‘Œπ‘‘) defined on (Ξ©,β„±,β„™) in (1.1) and (2.4) is jointly Markov and has a generator 𝐺 given as follows. For π‘“βˆˆπ’ž2,1(β„π‘Ÿ,𝒳,ℝ+), we have 1𝐺𝑓(π‘₯,𝑖,𝑦)=2π‘Ÿξ“π‘˜,𝑙=1π‘Žπ‘˜π‘™(π‘₯,𝑖)πœ•π‘“(π‘₯,𝑖,𝑦)πœ•π‘₯π‘˜πœ•π‘₯𝑙+π‘Ÿξ“π‘˜=1π‘π‘˜(π‘₯,𝑖)πœ•π‘“(π‘₯,𝑖,𝑦)πœ•π‘₯π‘˜+πœ•π‘“(π‘₯,𝑖,𝑦)+πœ•π‘¦π‘“(π‘¦βˆ£π‘–)1βˆ’πΉ(π‘¦βˆ£π‘–)𝑗≠𝑖,π‘—βˆˆπ’³π‘π‘–π‘—[]ξ€œπ‘“(π‘₯,𝑗,0)βˆ’π‘“(π‘₯,𝑖,𝑦)+πœ†Ξ“(𝑓(π‘₯+𝑔(π‘₯,𝑖,𝛾),𝑖,𝑦)βˆ’π‘“(π‘₯,𝑖,𝑦))πœ‹(𝑑𝛾),(2.6) where π‘₯βˆˆβ„π‘Ÿ,π‘Ž(π‘₯,𝑖)=𝜎(π‘₯,𝑖)πœŽβ€²(π‘₯,𝑖) is an β„π‘ŸΓ—π‘Ÿ matrix and π‘Žπ‘˜π‘™(π‘₯,𝑖) is the (π‘˜,𝑙)th element of the matrix π‘Ž while π‘π‘˜(π‘₯,𝑖) is the π‘˜th element of the vector 𝑏(π‘₯,𝑖).

We define the jump times, that is, time epochs when jumps occur by {πœπ‘π‘›}, where πœπ‘1<πœπ‘2<β‹…β‹…β‹…<πœπ‘π‘›<β‹…β‹…β‹…, to be the enumeration of all elements in the domain 𝐷𝑝 of the point process 𝑝(𝑑) corresponding to the stationary ℱ𝑑-Poisson point process 𝑁(𝑑𝑑,𝑑𝛾). It is easy to see that {πœπ‘π‘›} is an ℱ𝑑-stopping time for each 𝑛. Moreover, we have limπ‘›β†’βˆžπœπ‘π‘›=+∞ since the characteristic measure π‘š(β‹…) is finite. Next, let us denote the successive switching instants of the second component, which is the semi-Markov process πœƒπ‘‘ that switches from one point on the space 𝒳 to another and is denoted by πœπœƒ0=0,πœπœƒπ‘›=inf{π‘‘βˆΆπ‘‘>πœπœƒπ‘›βˆ’1,π‘‹π‘‘β‰ π‘‹πœπœƒπ‘›βˆ’1},𝑛β‰₯1. Since the Poisson random measure 𝑁(β‹…,β‹…) is independent of 𝑁1(β‹…,β‹…), one could adapt the proof of Xi ([12]) to show that with probability 1, {πœπ‘π‘›βˆΆπ‘›β‰₯1} and {πœπœƒπ‘›βˆΆπ‘›β‰₯1} are mutually disjoint. Hence between two chain-switching epochs, the process 𝑋𝑑 behaves like an ordinary jump-diffusion process without switching, a fact that we will use below to show the existence and uniqueness of the sMMJD process 𝑋𝑑. Accordingly, we describe next the existence-uniqueness theorem for (1.1).

Theorem 2.2. Assume that Assumption 2.1 holds. Then there exists a unique solution (𝑋𝑑,𝑑β‰₯0) with initial data (𝑋0,πœƒ0,π‘Œ0) to (1.1).

Proof. We only provide a sketch of the proof here. Consider [𝑠,𝑑],πœπœƒ1,…,πœπœƒπ‘β‰€π‘‘. Then as described above, on each of the intervals between the chain switching times, that is, [𝑠,πœπœƒ1),…,(πœπœƒπ‘,𝑑], the sMMJD process 𝑋𝑑 behaves like a jump-diffusion process. We can then use the standard Picard iteration argument in Applebaum [13] to show the existence-uniqueness of solution 𝑋𝑑.

Before we proceed with our main analysis concerning these two stability issues we introduce a key Lemma.

Lemma 2.3. {𝑃(𝑋𝑑≠0,𝑑≠0)}=1 for any 𝑋0=π‘₯β‰ 0, and πœƒ0=πœƒβˆˆπ’³. Thus almost all sample paths of any solutions of (1.1) starting from a nonzero state will never reach the origin.

Proof. We show this in a simple way. From the condition on the coefficients, 𝑏(0,𝑖)=0,𝜎(0,𝑖)=0, and 𝑔(0,𝑖,0)=0. So (1.1) admits a trivial solution 𝑋𝑑=0. From Theorem 2.2 above, due to the uniqueness of the solution of (1.1) the conclusion now follows.

We next have the following generalized Ito’s formula.

Lemma 2.4. Utilizing the operator 𝐺 in (2.1), the generalized Ito’s formula is given by 𝑓𝑋𝑑,πœƒπ‘‘,π‘Œπ‘‘ξ€Έξ€œβˆ’π‘“(π‘₯,πœƒ,𝑦)=𝑑0𝑋𝐺𝑓𝑠,πœƒπ‘ ,π‘Œπ‘ ξ€Έξ€œπ‘‘π‘ +𝑑0ξ€·ξ€·π‘‹βˆ‡π‘“π‘ ,πœƒπ‘ ,π‘Œπ‘ ξ€Έξ€Έξ…žπœŽξ€·π‘‹π‘ ,πœƒπ‘ ξ€Έπ‘‘π‘Šπ‘ +ξ€œπ‘‘0ξ€œΞ“ξ€Ίπ‘“ξ€·π‘‹π‘ βˆ’ξ€·π‘‹+π‘”π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ,𝛾,πœƒπ‘ ,π‘Œπ‘ βˆ’ξ€Έξ€·π‘‹βˆ’π‘“π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ‚+ξ€œξ€Έξ€»π‘(𝑑𝑠,𝑑𝛾)𝑑0ξ€œβ„ξ‚ƒπ‘“ξ‚€π‘‹π‘ βˆ’,πœƒπ‘ βˆ’+β„Žξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έ,π‘Œ,π‘§π‘ βˆ’βˆ’π‘”ξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚ξ€·π‘‹,π‘§βˆ’π‘“π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚„ξ‚π‘1(𝑑𝑠,𝑑𝑧),(2.7) where the local martingale terms are explicitly defined as 𝑑𝑀1𝑋(𝑑)∢=βˆ‡π‘“π‘‘,πœƒπ‘‘,π‘Œπ‘‘ξ€Έξ€Έξ…žπœŽξ€·π‘‹π‘‘,πœƒπ‘‘ξ€Έπ‘‘π‘Šπ‘‘,𝑑𝑀2ξ€œ(𝑑)∢=Ξ“ξ€Ίπ‘“ξ€·π‘‹π‘ βˆ’ξ€·π‘‹+π‘”π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ,𝛾,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ€·π‘‹βˆ’π‘“π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ‚ξ€Έξ€»π‘(𝑑𝑠,𝑑𝛾),𝑑𝑀3ξ€œ(𝑑)∢=β„ξ‚ƒπ‘“ξ‚€π‘‹π‘ βˆ’,πœƒπ‘ βˆ’+β„Žξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έ,𝑧,π‘Œπ‘ βˆ’βˆ’π‘”ξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚ξ€·π‘‹,π‘§βˆ’π‘“π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚„ξ‚π‘1(𝑑𝑠,𝑑𝑧).(2.8)

Proof. For details refer to Ikeda and Watanabe [14].

We now discuss the two criteria for stochastic stability that we intend to consider.

Definition 2.5 (Almost sure exponential stability). The trivial solution of (1.1) is almost surely exponentially stable if limsupπ‘‘β†’βˆž1𝑑||𝑋log𝑑||<0a.s.βˆ€π‘‹0βˆˆβ„π‘Ÿa.s.(2.9) The quantity on the left-hand side of the above equation is termed as the sample Lyapunov exponent.

Definition 2.6 (Moment exponential stability). Let 𝑝>0. The trivial solution of (1.1) is said to be 𝑝th moment exponentially stable if there exists a pair of constants πœ†>0 and 𝐢>0, such that for any 𝑋0βˆˆβ„π‘ŸπΈξ€Ί||𝑋𝑑||𝑝||𝑋≀𝐢0||𝑝exp(βˆ’πœ†π‘‘)βˆ€π‘‘β‰€0.(2.10)

In the next section, we detail the proofs for obtaining the conditions under which the trivial solution of (1.1) is almost surely exponentially stable and moment exponentially stable.

3. Almost Sure Stability and Moment Exponential Stability

In the sequel we will always, as standing hypotheses, assume that Assumption 2.1 holds. From Theorem 2.2 we deduce that there exists a unique solution to (1.1). By Lemma 2.3, we know that 𝑋𝑑 will never reach zero whenever 𝑋0β‰ 0. So in what follows we will only need a function 𝑉(π‘₯,𝑖,𝑦)βˆˆπ’ž2,1(β„π‘ŸΓ—π’³Γ—β„+) defined on the domain of the deleted neighborhood of zero. Our first main result provides conditions under which the trivial solution to (1.1) is almost surely exponentially stable.

Theorem 3.1. Assume that there exist a function π‘‰βˆˆπ’ž2,1(β„π‘ŸΓ—π’³Γ—β„+) in any deleted neighborhood of zero. Moreover, assume that there exist positive constants 𝛼,𝛽,𝜌1,𝜌2,𝜌1, and 𝜌2 for each π‘₯βˆˆβ„π‘Ÿ,π‘–βˆˆπ’³ and for each π›ΎβˆˆΞ“ such that |||ξ€·βˆ‡πΊlog𝑉(π‘₯,𝑖,𝑦)β‰€βˆ’π›Ό,π‘₯𝑉(π‘₯,𝑖,𝑦)ξ…ž|||𝜌𝜎(π‘₯,𝑖)≀𝛽𝑉(π‘₯,𝑖,𝑦),1≀𝑉(π‘₯+𝑔(π‘₯,𝑖,𝛾),𝑖,𝑦)𝑉(π‘₯,𝑖,𝑦)β‰€πœŒ2,𝜌1β‰€βŽ›βŽœβŽœβŽπ‘‰ξ‚€π‘₯,𝑖+β„Ž(𝑖,𝑦,𝑧),π‘¦βˆ’ξ‚π‘”(𝑖,𝑦,𝑧)βŽžβŽŸβŽŸβŽ β‰€π‘‰(π‘₯,𝑖,𝑦)𝜌2,(3.1) then the solution to (1.1) is almost surely exponentially stable.

Proof. Note that 𝑋log𝑉𝑑,πœƒπ‘‘,π‘Œπ‘‘ξ€Έξ€·π‘‹=log𝑉0,πœƒ0,π‘Œ0ξ€Έ+ξ€œπ‘‘0𝑋𝐺log𝑉𝑠,πœƒπ‘ ,π‘Œπ‘ ξ€Έπ‘‘π‘ +𝑀1(𝑑)+𝑀2(𝑑)+𝑀3(𝑑).(3.2) Here the local martingale terms 𝑀1(𝑑),𝑀2(𝑑), and 𝑀3(𝑑) are, respectively, 𝑀1ξ€œ(𝑑)=𝑑0ξ€·βˆ‡π‘₯𝑉𝑋𝑠,πœƒπ‘ ,π‘Œπ‘ ξ€Έξ€Έξ…žπœŽξ€·π‘‹π‘ ,πœƒπ‘ ξ€Έπ‘‰ξ€·π‘‹π‘ ,πœƒπ‘ ,π‘Œπ‘ ξ€Έπ‘‘π‘Šπ‘ ,𝑀2ξ€œ(𝑑)=𝑑0ξ€œΞ“ξƒ©π‘‰ξ€·π‘‹logπ‘ βˆ’ξ€·π‘‹+π‘”π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ,𝛾,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έπ‘‰ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξƒͺ𝑀𝑁(𝑑𝑠,𝑑𝛾),3ξ€œ(𝑑)=𝑑0ξ€œβ„ξ‚ƒξ‚€π‘‹logπ‘‰π‘ βˆ’,πœƒπ‘ βˆ’+β„Žξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έ,𝑧,π‘Œπ‘ βˆ’βˆ’π‘”ξ€·πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚ξ€·π‘‹,π‘§βˆ’logπ‘‰π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξ‚„ξ‚π‘1(𝑑𝑠,𝑑𝑧).(3.3) We deal with (3.2) term by term to derive an upper bound on limsupπ‘‘β†’βˆžlog𝑉(𝑋𝑑,𝑖,π‘Œπ‘‘)/𝑑. Consider first the drift term of (3.2). It is easy to see from the assumptions made that βˆ«π‘‘0𝐺log𝑉(𝑋𝑠,πœƒπ‘ ,π‘Œπ‘ )𝑑𝑠 will be bounded above by βˆ’π›Όπ‘‘. Secondly, we now concentrate on the local martingale terms of (3.2). First consider the quadratic variation of the 𝑀1(𝑑) term. By Ito’s isometry, we have βŸ¨π‘€1(𝑑),𝑀1ξ€œ(𝑑)⟩=𝑑0|||||ξ€·βˆ‡π‘₯𝑉𝑋𝑠,πœƒπ‘ ,π‘Œπ‘ ξ€Έξ€Έξ…žπœŽ(𝑋𝑠,πœƒπ‘ )𝑉𝑋𝑠,πœƒπ‘ ,π‘Œπ‘ ξ€Έ|||||2β‰€ξ€œπ‘‘π‘ π‘‘0𝛽2𝑑𝑠≀𝛽2𝑑.(3.4) Next consider the quadratic variation of the local martingale term 𝑀2(𝑑). Based on the following result presented in Kunita [15, page 323], and noting that the jump distribution πœ‹ is a probability measure that is, βˆ«Ξ“πœ‹(𝑑𝛾)=1 we have βŸ¨π‘€2(𝑑),𝑀2ξ€œ(𝑑)⟩=𝑑0ξ€œΞ“ξƒ©ξƒ¬π‘‰ξ€·π‘‹logπ‘ βˆ’ξ€·π‘‹+π‘”π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ,𝛾,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έπ‘‰ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’,π‘Œπ‘ βˆ’ξ€Έξƒ­ξƒͺ2ξ‚ƒξ€·πœ‹(𝑑𝛾)𝑑𝑠≀maxlog𝜌1ξ€Έ2,ξ€·log𝜌2ξ€Έ2𝑑.(3.5) On very similar lines, one can easily show that the quadratic variation of the local martingale term 𝑀3(𝑑) is given by βŸ¨π‘€3(𝑑),𝑀3(𝑑)βŸ©β‰€maxlog𝜌1ξ€Έ2,ξ€·log𝜌2ξ€Έ2𝑑.(3.6) Thus by SLLN for local martingales (refer to Lipster and Shiryayev [16, page 140–141]), we can say that limsupπ‘‘β†’βˆžπ‘€1𝑑=limsupπ‘‘β†’βˆžπ‘€2𝑑=limsupπ‘‘β†’βˆžπ‘€3𝑑=0.(3.7) Thus from (3.2) and the above discussion, one can infer that limsupπ‘‘β†’βˆžlog𝑉(π‘₯,𝑖,𝑦)π‘‘β‰€βˆ’π›Ό.(3.8) Thus, since by assumption 𝛼>0, from the definition of almost sure exponential stability, the trivial solution to (1.1) is almost surely exponentially stable.

We now provide conditions under which the trivial solution to (1.1) is moment exponentially stable.

Theorem 3.2. Let 𝑝,𝛼,𝛼1,𝛼2>0. Assume that there exists a function 𝑉(π‘₯,𝑖,𝑦)βˆˆπ’ž2,1(β„π‘Ÿ,𝒳,ℝ+) such that 𝛼1|π‘₯|𝑝≀𝑉(π‘₯,𝑖,𝑦)≀𝛼2|π‘₯|𝑝,𝐺𝑉(π‘₯,𝑖,𝑦)β‰€βˆ’π›Ό|π‘₯|𝑝.(3.9) Then, limsupπ‘‘β†’βˆž1𝑑||𝑋log𝐸𝑑||π‘β‰€βˆ’π›Όπ›Ό2||𝑋0||𝑝.(3.10) As a result the trivial solution of (1.1) is 𝑝th-moment exponentially stable under the conditions discussed above and the 𝑝th-moment Lyapunov exponent should not be greater than βˆ’π›Ό/𝛼2.

Proof. The proof is omitted as it is a simple extension of the Markov-modulated SDE case discussed in Mao [9].

In the next theorem, we provide criteria to connect these two seemingly disparate stabilty criteria. Specifically, we provide conditions under which the 𝑝th-moment exponential stability for 𝑝β‰₯2 always implies almost sure exponential stability for (1.1).

Theorem 3.3. Assume that there exists a positive constant 𝐢 such that for each π‘–βˆˆπ’³||||∨||||∨||||𝑏(π‘₯,𝑖)𝜎(π‘₯,𝑖)𝑔(π‘₯,𝑖,𝛾)≀𝐢|π‘₯|.(3.11) If for all 𝑋0=π‘₯0βˆˆβ„π‘Ÿ, limsupπ‘‘β†’βˆž1𝑑||𝑋log𝐸𝑑||π‘ξ€Έβ‰€βˆ’π‘Ž,(3.12) then limsupπ‘‘β†’βˆž1𝑑||𝑋log𝑑||ξ€Έπ‘Žβ‰€βˆ’π‘a.s.(3.13) Then 𝑝th-moment exponential stability implies almost sure exponential stability.

We need the Burkholder-Davis-Gundy inequality which is detailed in the following remark below.

Remark 3.4. Let us recall that [𝑋] denotes the quadratic variation of a process say 𝑋, and π‘‹βˆ—π‘‘β‰‘sup𝑠≀𝑑|𝑋𝑠| is its maximum process. Then the Burkholder-Davis-Gundy theorem states that for any 1≀𝑝<∞, there exist positive constants 𝑐𝑝,𝐢𝑝 such that, for all local martingales 𝑋 with 𝑋0=0 and stopping times 𝜏, the following inequality holds: 𝑐𝑝𝐸[𝑋]πœπ‘/2ξ‚„π‘‹β‰€πΈξ€Ίξ€·βˆ—πœξ€Έπ‘ξ€»β‰€πΆπ‘πΈξ‚ƒ[𝑋]πœπ‘/2ξ‚„.(3.14) Furthermore, for continuous local martingales, this statement holds for all 0<𝑝<∞. For its proof refers to Theorem  3.28 page 166 in Karatzas and Shreve [17].

Proof of Theorem 3.3. Let 𝑋0βˆˆβ„π‘Ÿ. Let πœ– be arbitrarily small positive number. By the definition of 𝑝th-moment exponential stability of (3.15), there exists a constant 𝐾 such that 𝐸||𝑋𝑑||𝑝≀𝐾expβˆ’(π‘Žβˆ’πœ–)𝑑,𝑑β‰₯0.(3.15) Let 𝛿>0 be sufficiently small such that, 5𝑝𝐢𝑝𝛿𝑝+𝐢𝑝𝛿𝑝/2ξ€Έ<14.(3.16) From (1.1) we have 𝑋𝑑=𝑋0+ξ€œπ‘‘0𝑏𝑋𝑠,πœƒπ‘ ξ€Έξ€œπ‘‘π‘ +𝑑0πœŽξ€·π‘‹π‘ ,πœƒπ‘ ξ€Έπ‘‘π‘Šπ‘ +ξ€œπ‘‘0ξ€œΞ“π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έξ‚ξ€œ,𝛾𝑁(𝑑𝑠,𝑑𝛾)+πœ†π‘‘0ξ€œΞ“π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ,π›Ύπœ‹(𝑑𝛾)𝑑𝑠.(3.17) Noting that for π‘Ž,𝑏,𝑐,𝑑,𝑒β‰₯0(π‘Ž+𝑏+𝑐+𝑑+𝑒)𝑝≀[]5(π‘Žβˆ¨π‘βˆ¨π‘βˆ¨π‘‘βˆ¨π‘’)𝑝=5𝑝(π‘Žπ‘βˆ¨π‘π‘βˆ¨π‘π‘βˆ¨π‘‘π‘βˆ¨π‘’π‘)≀5𝑝(π‘Žπ‘+𝑏𝑝+𝑐𝑝+𝑑𝑝+𝑒𝑝),(3.18) we have 𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώ||𝑋𝑑||𝑝≀5𝑝𝐸||𝑋(π‘˜βˆ’1)𝛿||𝑝+5π‘πΈξ‚΅ξ€œπ‘˜π›Ώ(π‘˜βˆ’1)𝛿||𝑏𝑋𝑠,πœƒπ‘ ξ€Έ||𝑑𝑠𝑝+5𝑝𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώξ€œπ‘‘(π‘˜βˆ’1)𝛿||πœŽξ€·π‘‹π‘ ,πœƒπ‘ ξ€Έπ‘‘π‘Šπ‘ ||𝑝ξƒͺ+5𝑝𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώξ€œπ‘‘(π‘˜βˆ’1)π›Ώξ€œΞ“||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έξ‚π‘||,𝛾(𝑑𝑠,𝑑𝛾)𝑝ξƒͺ+5π‘πœ†π‘πΈξ‚΅ξ€œπ‘˜π›Ώ(π‘˜βˆ’1)π›Ώξ€œΞ“||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ||ξ‚Ά,π›Ύπœ‹(𝑑𝛾)𝑑𝑠𝑝.(3.19) Noting that βˆ«Ξ“πœ‹(𝑑𝛾)=1, we have 𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώξ€œπ‘‘(π‘˜βˆ’1)π›Ώξ€œΞ“||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έξ‚||,𝛾𝑁(𝑑𝑠,𝑑𝛾)𝑝ξƒͺβ‰€πΆπ‘πΈξ‚΅ξ€œπ‘˜π›Ώ(π‘˜βˆ’1)𝛿||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ||,𝛾2𝑑𝑠(𝑝/2)≀𝐢𝑝𝐸𝛿sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ||,𝛾2ξƒͺ(𝑝/2)≀𝐢𝑝𝐢𝑝𝛿𝑝/2𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||𝑋𝑠||𝑝.(3.20) Similarly, 𝐸||||ξ€œπ‘˜π›Ώ(π‘˜βˆ’1)π›Ώξ€œΞ“||||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έξƒͺ,π›Ύπœ‹(𝑑𝛾)𝑑𝑠𝑝≀𝐸𝛿sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||π‘”ξ€·π‘‹π‘ βˆ’,πœƒπ‘ βˆ’ξ€Έ||ξƒ­,𝛾𝑝≀𝐢𝑝𝛿𝑝𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||𝑋𝑠||𝑝ξƒͺ.(3.21) From (3.15), one can easily show that 𝐸||𝑋(π‘˜βˆ’1)𝛿||𝑝≀𝐾expβˆ’(π‘Žβˆ’πœ–)(π‘˜βˆ’1)π›ΏπΈξ‚΅ξ€œ,(3.22)π‘˜π›Ώ(π‘˜βˆ’1)𝛿||𝑏𝑋𝑠,πœƒπ‘ ξ€Έ||𝑑𝑠𝑝≀𝐢𝑝𝛿𝑝𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||𝑋𝑠||𝑝𝐸,(3.23)sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώξ€œπ‘‘(π‘˜βˆ’1)𝛿||πœŽξ€·π‘‹π‘ ,πœƒπ‘ ξ€Έ||π‘‘π‘Šπ‘ ξƒͺ𝑝≀𝐢𝑝𝐢𝑝𝛿𝑝/2𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘ β‰€π‘˜π›Ώ||𝑋𝑝𝑠||ξƒ­.(3.24) Hence, substituting (3.20)–(3.23) in (3.19), we obtain 𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώ||𝑋𝑑||𝑝1βˆ’5𝑝𝐢𝑝𝛿𝑝+𝐢𝑝𝐢𝑝𝛿𝑝/2+𝐢𝑝𝐢𝑝𝛿𝑝/2+𝐢𝑝𝛿𝑝≀𝐾5𝑝expβˆ’(π‘Žβˆ’πœ–)(π‘˜βˆ’1)𝛿.(3.25) From (3.16) we obtain that 𝐸sup(π‘˜βˆ’1)π›Ώβ‰€π‘‘β‰€π‘˜π›Ώ||𝑋𝑑||𝑝≀2Γ—5𝑝𝐾expβˆ’(π‘Žβˆ’πœ–)(π‘˜βˆ’1)𝛿,(3.26) and utilizing the Borel-Cantelli Lemma as in Mao [9] we deduce the desired implication that 𝑝th-moment stability implies almost sure exponential stability.

4. Examples

We now provide some simple examples to illustrate both the almost surely exponential stability and moment exponential stability. We start with an example on almost surely exponential stability.

Consider a two state semi-Markov modulated Jump-diffusion problem with π‘‹π‘‘βˆˆβ„π‘Ÿ and 𝑉(𝑋𝑑,𝑖,π‘Œπ‘‘)=|𝑋𝑑| where the generator matrix is given by||||||||||||𝑄=βˆ’221βˆ’1.(4.1) Let the holding time in each regime be assumed to follow 𝑓(π‘¦βˆ£π‘–)=πœ†π‘–π‘’βˆ’πœ†π‘–π‘¦,𝑦>0,π‘–βˆˆ{1,2}. Note that with the choice of the holding time distribution, the sMMJD collapses to the MMJD case in which case the generator 𝐺 acting on 𝑉(π‘₯,𝑖,𝑦) is given by 1𝐺𝑉(π‘₯,𝑖,𝑦)=2𝐼traceξ‚Έξ‚΅βˆ’|π‘₯|π‘₯π‘₯ξ…ž|π‘₯|3ξ‚ΆπœŽ(π‘₯,𝑖)πœŽξ…žξ‚Ή+π‘₯(π‘₯,𝑖)ξ…žξ€œ|π‘₯|𝑏(π‘₯,𝑖)+πœ†Ξ“ξ€Ί||||ξ€»π‘₯+𝑔(π‘₯,𝑖,𝛾)βˆ’|π‘₯|πœ‹(𝑑𝛾).(4.2) Now from Assumption 2.1 as ||||=||||||||=||||𝜎(π‘₯,𝑖)𝜎(π‘₯,𝑖)βˆ’πœŽ(0,𝑖)≀𝐢|π‘₯|𝑏(π‘₯,𝑖)𝑏(π‘₯,𝑖)βˆ’π‘(0,𝑖)≀𝐢|π‘₯|(4.3) and |𝑔(π‘₯,𝑖,𝛾)|≀𝐢|π‘₯|, we have =𝐺𝑉(π‘₯,𝑖,𝑦)≀𝐢|π‘₯|+𝐢|π‘₯|+πœ†(2+𝐢)|π‘₯|(2𝐢+πœ†πΆ+2πœ†)|π‘₯|.(4.4) If we choose 𝐢 and πœ† such that for any π‘₯βˆˆβ„π‘Ÿβˆ’{0}, there exists π›ΌβˆΆ=(2+πœ†)𝐢+2πœ†)β‰₯0 such that 𝐺log𝑉(π‘₯,𝑖,𝑦)β‰€βˆ’π›Ό. Also |βˆ‡π‘₯𝑉(π‘₯,𝑖,𝑦)ξ…žπœŽ(π‘₯,𝑖)/𝑉(π‘₯,𝑖,𝑦)|≀𝐢. Similarly, if there exist a positive constant 𝛽 such that for any π‘₯βˆˆβ„π‘Ÿ, 𝐢≀𝛽, then |βˆ‡π‘₯𝑉(π‘₯,𝑖,𝑦)ξ…žπœŽ(π‘₯,𝑖)/𝑉(π‘₯,𝑖,𝑦)|≀𝛽. If there exist constants 𝜌1 and 𝜌2 such that 𝜌1≀𝑔(π‘₯,𝑖,𝛾)β‰€πœŒ2 for any π‘₯βˆˆβ„π‘Ÿ, π‘–βˆˆπ’³ and π›ΎβˆˆΞ“, then it is easy to see that (𝜌1)≀(𝑉(π‘₯+𝑔(π‘₯,𝑖,𝛾),𝑖,𝑦)/𝑉(π‘₯,𝑖,𝑦))≀(𝜌2). Thus in brief for certain conditions on the growth of the drift, diffusion and the integrand of the jump component of the process given by (1.1), we satisfy the conditions of Theorem 3.1 for the solution to (1.1) to be almost surely exponentially stable.

We next provide a simple example to illustrate Theorem 3.2. Consider that π‘₯βˆˆβ„ and 𝑉(π‘₯,𝑖,𝑦)=π‘₯2. Also assume that the conditional holding time distribution be 𝑓(π‘¦βˆ£π‘–)=πœ†π‘–π‘’βˆ’πœ†π‘–π‘¦ for π‘–βˆˆ{1,2}. Let 𝑔(π‘₯,𝑖,𝛾)=π‘₯, πœ†π‘–=1, 𝑏(π‘₯,𝑖)=π‘Ž1π‘₯, 𝜎(π‘₯,𝑖)=π‘Ž2π‘₯ for π‘–βˆˆ{1,2}. Then from (2.4) we have 𝐺𝑉(π‘₯,𝑖,𝑦)=(2π‘Ž1+π‘Ž2+3)π‘₯2. If 2π‘Ž1+π‘Ž2+3<0 and π‘₯β‰ 0, then condition (ii) of Theorem 3.2 for 𝑝=2 is satisfied. Moreso if we assume that there exist constants 𝛼1 and 𝛼2 such that 𝛼1|π‘₯|2≀𝑉(π‘₯,𝑖,𝑦)≀𝛼2|π‘₯|2 is true, then condition (i) of Theorem 3.2 is satisfied. Thus both conditions (i) and (ii) now guarantee that the solution of (1.1) is moment exponentially stable.

Next we discuss the issue of stochastic stabilization and destabilization of nonlinear systems.

5. Stochastic Stabilization and Destabilization of Nonlinear Systems

We now investigate the stability of the nonlinear deterministic system of differential equations given by the following dynamics: 𝑑𝑋𝑑𝑋𝑑𝑑=𝑓𝑑(5.1) on 𝑑β‰₯0 with 𝑋0=π‘₯0βˆˆβ„π‘Ÿ where 𝑓(π‘₯)βˆΆβ„π‘Ÿβ†’β„π‘Ÿ is locally Lipschitz continuous and furthermore there exists some constant 𝐾>0 such that |𝑓(π‘₯)|≀𝐾|π‘₯|forallπ‘₯βˆˆβ„π‘Ÿ. When perturbed by noise, the nonlinear system (5.1) is either stable if it originally unstable, in the sense that by adding noise we can force the solution of the stochastic differential equation to converge to the trivial solution as time increases indefinitely. This is the aim of stochastic stabilization. Likewise if our original system in stable, then this system is said to destabilize when perturbed by noise if the sample paths of the process escapes to infinity almost surely instead of converging to the trivial solution as time tends to infinity. This is termed as stochastic destabilization. Consequently, the system then becomes what is known as unstable. Mao [9] and Applebaum and Siakalli [18] have established a general theory of stochastic stabilization/destabilization of (5.1) using a Brownian motion and the general Levy process, respectively. However, no specific work has been done so far for the case where 𝑋𝑑 is an sMMJD. In this paper we focus on the first-order nonlinear system of ODEs that is perturbed by an sMMJD. In the following section, we show that an unstable linear system counterpart of (5.1) wherein 𝑑𝑋𝑑/𝑑𝑑=π‘Žπ‘‹π‘‘ for π‘Ž>0 can be stabilized just by the addition of a jump component to the dynamics of the one-dimensional process 𝑋𝑑. We observe that such a jump-stabilized system of DEs cannot be destabilized by further addition of a Brownian motion. On the contrary, we show that such a jump-stabilized nonlinear system of differential equations can surprisingly be destabilized by addition of Brownian motion if the dimension of the state space is at least two. Before we go into the proofs of these statements, we begin by mentioning the key dynamics of the sMMJD process {𝑋𝑑,𝑑β‰₯0} and some assumptions that follow. Suppose we have an π‘š-dimensional standard ℱ𝑇-adapted Brownian motion process 𝐡=(𝐡1(𝑑),…,π΅π‘š(𝑑)) for each 𝑑β‰₯0. The system (5.1) is perturbed by the following sMMJD dynamics of 𝑋𝑑 given by 𝑑𝑋𝑑𝑋=𝑓𝑑𝑑𝑑+π‘šξ“π‘˜=1πΊπ‘˜ξ€·πœƒπ‘‘ξ€Έπ‘‹π‘‘π‘‘π΅π‘˜ξ€œ(𝑑)+πœ†Ξ“π·ξ€·πœƒπ‘‘βˆ’ξ€Έπ‘‹,π›Ύπ‘‘βˆ’π‘(𝑑𝑑,𝑑𝛾)βˆ€π‘‘β‰₯0,(5.2) where πΊπ‘˜(𝑖) is β„π‘ŸΓ—π‘Ÿ for each π‘–βˆˆπ’³. Likewise 𝐷(𝑖,𝛾) is an β„π‘ŸΓ—π‘Ÿ-valued matrix for each π‘–βˆˆπ’³ and π›ΎβˆˆΞ“βŠ‚β„π‘Ÿβˆ’{0}. We refer to a system (5.1) perturbed by the dynamics of 𝑋𝑑 as in (5.2) as just a perturbed system. We make the following key assumption that remains valid until the end of this section.

Assumption 5.1. Assume that for each π‘–βˆˆπ’³ and π›ΎβˆˆΞ“ we have βˆ«Ξ“(||𝐷(𝑖,𝛾)||∨||𝐷(𝑖,𝛾)||2)πœ‹(𝑑𝛾)<∞ and that 𝐷(𝑖,𝛾) does not have an eigenvalue equal to βˆ’1πœ‹ almost surely.

In the following, we will establish the conditions on the coefficients of (5.2) for the trivial solution of the perturbed system to be almost surely exponentially stable. In particular, this surprisingly demonstrates that the jump process can have a stabilizing effect, as for the Brownian motion part as has been shown by Mao [9]. We state this formally as one of our main theorems.

Theorem 5.2. Assume that Assumption 5.1 holds. Suppose that the following conditions are satisfied for π‘Ž(𝑖)>0, 𝑏(𝑖)β‰₯0:(i)βˆ‘π‘šπ‘˜=1|πΊπ‘˜(𝑖)π‘₯|2β‰€π‘Ž(𝑖)|π‘₯|2, (ii)βˆ‘π‘šπ‘˜=1|π‘₯ξ…žπΊπ‘˜(𝑖)π‘₯|2β‰₯𝑏(𝑖)|π‘₯|4 for each π‘–βˆˆπ’³ and π‘₯βˆˆβ„π‘Ÿ.Then the sample Lyapunov exponent of the solution of (5.2) exists and satisfies limsupπ‘‘β†’βˆžlog|π‘‹π‘‘βˆ‘|β‰€πΎβˆ’π‘–βˆˆπ’³[(𝑏(𝑖)βˆ’π‘Ž(𝑖)/2βˆ’πœ†log(1+||𝐷(𝑖,𝛾)||))]πœˆπ‘– for any 𝑋0β‰ 0. If βˆ‘βˆ’πΎ+π‘–βˆˆπ’³[𝑏(𝑖)βˆ’π‘Ž(𝑖)/2βˆ’πœ†log(1+||𝐷(𝑖,𝛾)||)]πœˆπ‘–>0, then the trivial solution to the system in (5.2) is almost surely exponentially stable.

Proof.  
Step  1. Define 𝑉(π‘₯,𝑖,𝑦)=log|π‘₯| for all π‘–βˆˆπ’³. As 𝑉(π‘₯,𝑖,𝑦) is independent of states 𝑖 and 𝑦, the following terms in (2.4) are zero: 𝑓(π‘¦βˆ£π‘–)1βˆ’πΉ(π‘¦βˆ£π‘–)𝑗≠𝑖,π‘—βˆˆπ’³π‘π‘–π‘—[]𝑉(π‘₯,𝑗,0)βˆ’π‘‰(π‘₯,𝑖,𝑦)=0,πœ•π‘‰(π‘₯,𝑖,𝑦)πœ•π‘¦=0.(5.3) Hence as an application of the generalized Ito’s formula, we have for 𝑑>0||𝑋log𝑑||||𝑋=log0||+ξ€œπ‘‘0π‘‹ξ…žπ‘ ||𝑋𝑠||2𝑓𝑋𝑠1𝑑𝑠+2π‘šξ“π‘˜=1ξ€œπ‘‘0||πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||2βˆ’2||π‘‹ξ…žπ‘ πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||4ξƒ­ξ€œπ‘‘π‘ +πœ†π‘‘0ξ€œΞ“ξƒ©||𝑋logπ‘ βˆ’ξ€·πœƒ+π·π‘ βˆ’ξ€Έπ‘‹,π›Ύπ‘ βˆ’||||π‘‹π‘ βˆ’||ξƒͺπœ‹(𝑑𝛾)𝑑𝑠+𝑀1(𝑑)+𝑀2(𝑑),(5.4) where 𝑀1βˆ‘(𝑑)=π‘šπ‘˜=1βˆ«π‘‘0|π‘‹ξ…žπ‘ πΊπ‘˜(πœƒπ‘ )𝑋𝑠/|𝑋𝑠|2|π‘‘π΅π‘˜(𝑠) and 𝑀2∫(𝑑)=𝑑0βˆ«Ξ“log((|π‘‹π‘ βˆ’+𝐷(πœƒπ‘ βˆ’,𝛾)π‘‹π‘ βˆ’|)/|π‘‹π‘ βˆ’ξ‚|)𝑁(𝑑𝑠,𝑑𝛾) are the two local martingale terms.
Step  2. Consider now the quadratic variation of the two martingale terms. From Ito’s isometry and noting that ||π‘‹ξ…žπ‘ πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||4=||π‘‹ξ…žπ‘ ξ€·πΊξ…žπ‘˜ξ€·πœƒπ‘ ξ€Έ+πΊπ‘˜ξ€·πœƒπ‘ π‘‹ξ€Έξ€Έπ‘ ||24||𝑋𝑠||4ξ€·πΊβ‰€πœŒπ‘˜ξ€·πœƒπ‘ ξ€Έξ€Έ2,(5.5) where 𝜌(πΊπ‘˜(πœƒπ‘ )) is the spectral radius of the symmetric π‘ŸΓ—π‘Ÿ matrix (πΊπ‘˜(πœƒπ‘ )+πΊξ…žπ‘˜(πœƒπ‘ ))/2, βŸ¨π‘€1(𝑑),𝑀1(𝑑)βŸ©β‰€π‘šξ“π‘˜=1ξ€œπ‘‘0||π‘‹ξ…žπ‘ πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||4π‘‘π‘ β‰€π‘‘π‘šmax1β‰€π‘˜β‰€π‘š,𝑖=1,…,π‘€πœŒξ€·πΊπ‘˜ξ€Έ.(𝑖)(5.6) Next, the quadratic variation of the process 𝑀2(𝑑) is given by βŸ¨π‘€2(𝑑),𝑀2ξ€œ(𝑑)⟩=2Ξ“ξ€œπ‘‘0||𝑋logπ‘ βˆ’ξ€·πœƒ+π·π‘ βˆ’ξ€Έπ‘‹,π›Ύπ‘ βˆ’||ξ€Έ||π‘‹π‘ βˆ’||ξƒ­ξ‚΅π‘‘π‘ πœ‹(𝑑𝛾)≀2𝑑log1+max1≀𝑖≀𝑀.‖𝐷(𝑖,𝛾)β€–(5.7)
Step  3. We work with the rest of the terms in the following way: limsupπ‘‘β†’βˆž|||||1π‘‘ξ€œπ‘‘0π‘‹ξ…žπ‘ π‘“ξ€·π‘‹π‘ ξ€Έ||𝑋𝑠||2|||||𝑑𝑠≀𝐾,(5.8) also limsupπ‘‘β†’βˆž1π‘‘ξ€œπ‘‘012ξƒ¬π‘šξ“π‘˜=1||πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||2βˆ’2||π‘‹π‘ πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ ||2||𝑋𝑠||4≀1ξƒͺξƒ­π‘‘π‘ π‘‘ξ“π‘–βˆˆπ’³ξ€œπ‘‘0ξ‚Έπ‘Ž(𝑖)2ξ‚Ήπ•€βˆ’π‘(𝑖)πœƒπ‘ =π‘–β‰€ξ“π‘‘π‘ π‘–βˆˆπ’³ξ‚Έπ‘Ž(𝑖)2ξ‚Ήπœˆβˆ’π‘(𝑖)𝑖,limsupπ‘‘β†’βˆžπœ†π‘‘ξ€œπ‘‘0ξ€œΞ“ξƒ©||𝑋logπ‘ βˆ’ξ€·πœƒ+π·π‘ βˆ’ξ€Έπ‘‹,π›Ύπ‘ βˆ’||||π‘‹π‘ βˆ’||ξƒͺξ“πœ‹(𝑑𝛾)π‘‘π‘ β‰€πœ†π‘–βˆˆπ’³log(1+‖𝐷(𝑖,𝛾)β€–)πœˆπ‘–.(5.9) Thus, limsupπ‘‘β†’βˆž(1/𝑑)log|𝑋𝑑|<0 if βˆ‘πΎ+π‘–βˆˆπ’³[(π‘Ž(𝑖)/2βˆ’π‘(𝑖)+πœ†log(1+||𝐷(𝑖,𝛾)||))]πœˆπ‘–<0.

Remark 5.3. Consider a 1-D sMMJD with the dynamics 𝑑𝑋𝑑=π‘Žπ‘‹π‘‘π‘‘π‘‘+𝑏(𝑖)𝑋𝑑𝑑𝐡𝑑+𝑐(𝑖,𝛾)𝑋𝑑𝑑𝑁𝑑,(5.10) where 𝑏(π‘₯,𝑖)>0 and 𝑐(𝑖,𝛾)>βˆ’1 for each π‘₯βˆˆβ„, π‘–βˆˆ{1,…,𝑀} and π›ΎβˆˆΞ“. 𝐡𝑑 is a 1D Brownian motion and {𝑁𝑑,𝑑β‰₯0} is a compensated Poisson process with 𝑁𝑑=π‘π‘‘βˆ’πœ†π‘‘, where πœ†>0 is the intensity of the Poisson process. Assume that the processes 𝐡𝑑 and 𝑁𝑑 are independent. Then one can show from the SLLN for a Brownian motion and for a Poisson process (refer to Applebaum [13]) that for each π‘–βˆˆ{1,2,…,𝑀}limsupπ‘‘β†’βˆž1𝑑||𝑋log𝑑||1=π‘Ž+βˆ’πœ†π‘(𝑖,𝛾)βˆ’2𝑏2ξ‚„(𝑖)+πœ†log(1+𝑐(𝑖,𝛾))<0a.s.(5.11) Note that 𝑏2(𝑖)β‰₯0 for all π‘–βˆˆπ’³ and has a negative sign attached to it. Hence when the one-dimensional perturbed system 𝑑𝑋𝑑/𝑑𝑑=π‘Žπ‘‹π‘‘ for π‘Ž>0 is stabilized by the addition of a jump process infact can never be destabilized by the addition of a Brownian motion. An interesting question we may ask here is: can the same inference hold true in higher dimensions? The answer is surprisingly no. In the following theorem, we show that for a state space of dimension greater than or equal to two, an unstable nonlinear system of differential equation stabilized by a jump component can still be destabilized by the addition of the Brownian motion. This surprising phenomenon was also observed by Applebaum and Siakalli [19] for the Levy process case.

To prove this assertion, let us now consider system of nonlinear differential equation (5.1) stabilized by (5.2) but with πΊπ‘˜(𝑖)=0 for each 𝑖=1,…,𝑀 and π‘˜=1,…,π‘š. We now show that it gets destabilized by further addition of the π‘š-dimensional Brownian motion to (5.1). This corresponds to πΊπ‘˜(𝑖)β‰ 0 for each 𝑖=1,…,𝑀 and π‘˜=1,…,π‘š.

Theorem 5.4. Assume that matrix 𝐷 is an π‘ŸΓ—π‘Ÿ symmetric positive definite matrix. Now let(i)βˆ‘π‘šπ‘˜=1|πΊπ‘˜(𝑖)π‘₯|2β‰₯π‘Ž(𝑖)|π‘₯|2,(ii)βˆ‘π‘šπ‘˜=1|π‘₯ξ…žπΊπ‘˜(𝑖)π‘₯|2≀𝑏(𝑖)|π‘₯|4,for π‘Ž(𝑖)>0,𝑏(𝑖)β‰₯0 for each π‘–βˆˆπ’³, π‘₯βˆˆβ„π‘Ÿ. Hence liminfπ‘‘β†’βˆž(1/𝑑)log|π‘‹π‘‘βˆ‘|β‰₯βˆ’πΎ+π‘–βˆˆπ’³[(π‘Ž(𝑖)/2βˆ’π‘(𝑖)+πœ†log(1+min1≀𝑖≀𝑀||𝐷(𝑖,𝛾)||))]πœˆπ‘– for any 𝑋0β‰ 0. In particular if βˆ‘βˆ’πΎ+π‘–βˆˆπ’³[π‘Ž(𝑖)/2βˆ’π‘(𝑖)+πœ†log(1+min1≀𝑖≀𝑀||𝐷(𝑖,𝛾)||)]πœˆπ‘–>0, then the trivial solution of (5.2), tends to be infinity almost surely exponentially fast.

Proof. Fix 𝑋0β‰ 0. From Lemma  2.2, 𝑋𝑑≠0 for all 𝑑β‰₯0. Applying Ito’s lemma to log|𝑋𝑑|, for 𝑑>0 and for each π‘–βˆˆπ’³, ||𝑋log𝑑||||𝑋=log0||+ξ€œπ‘‘0π‘‹ξ…žπ‘ ||𝑋𝑠||2𝑓𝑋𝑠1𝑑𝑠+2π‘šξ“π‘˜=1ξ€œπ‘‘0||πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ βˆ’||2||π‘‹π‘ βˆ’||2βˆ’2||π‘‹π‘ βˆ’πΊπ‘˜ξ€·πœƒπ‘ ξ€Έπ‘‹π‘ βˆ’||2||π‘‹π‘ βˆ’||4ξƒ­ξ€œπ‘‘π‘ +πœ†π‘‘0ξ€œΞ“ξƒ©ξ€·||𝑋logπ‘ βˆ’ξ€·πœƒ+π·π‘ βˆ’ξ€Έπ‘‹=𝑖,π‘Œπ‘ βˆ’||ξ€Έ||π‘‹π‘ βˆ’||ξƒͺπœ‹(𝑑𝛾)𝑑𝑠+𝑀1(𝑑)+𝑀2(𝑑),(5.12) where 𝑀1βˆ‘(𝑑)=π‘šπ‘˜=1βˆ«π‘‘0(|π‘‹ξ…žπ‘ βˆ’πΊπ‘˜(πœƒπ‘ )π‘‹π‘ βˆ’|/|π‘‹π‘ βˆ’|2)π‘‘π΅π‘˜(𝑠) and 𝑀2∫(𝑑)=Ξ“βˆ«π‘‘0log((|π‘‹π‘ βˆ’+𝐷(πœƒπ‘ βˆ’=𝑖,π‘Œ)π‘‹π‘ βˆ’|)/|π‘‹π‘ βˆ’ξ‚|)𝑁(𝑑𝑠,𝑑𝛾) are the two local martingale terms. Now using methodology similar to Theorem 5.2 we find liminfπ‘‘β†’βˆž1𝑑||𝑋log𝑑||β‰₯βˆ’πΎ+π‘–βˆˆπ’³ξ‚Έξ‚΅π‘Ž(𝑖)2ξ‚΅βˆ’π‘(𝑖)+πœ†log1+min1β‰€π‘–β‰€π‘€β€–πœˆβ€–π·(𝑖,𝛾)𝑖(5.13) for any 𝑋0β‰ 0. In particular, if βˆ‘βˆ’πΎ+π‘–βˆˆπ’³[π‘Ž(𝑖)/2βˆ’π‘(𝑖)+πœ†log(1+min1≀𝑖≀𝑀||𝐷(𝑖,𝛾)||)]πœˆπ‘–>0, then the trivial solution of the 𝑋𝑑-perturbed system given by (5.2) tends to be infinity almost surely exponentially fast.

6. Concluding Remarks

We presented conditions under which the solution of a semi-Markov Modulated jump diffusion is almost surely exponentially stable and moment exponentially stable. We also provide conditions that connect these two notions of stability. We further determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equation 𝑑𝑋𝑑/𝑑𝑑=𝑓(𝑋𝑑) is almost surely exponentially stable. We show that an unstable deterministic system can be stabilized by adding jumps. Such jump stabilized system, however, can get de-stabilized by Brownian motion if the dimension of the state space is at least two.

Acknowledgments

The author is very grateful to the anonymous referees and the editor for their careful reading, valuable comments, and helpful suggestions, which have helped him to improve the presentation of this paper significantly.

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