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 𝑑𝑋𝑡/𝑑𝑡=𝑓(𝑋𝑡) 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 ̇𝑥𝑡=𝐴(𝑟𝑡)𝑥𝑡, where 𝑟𝑡 is a Markov chain. Basak et al. [8] discussed the stability of a semilinear SDE with Markovian-regime switching of the form ̇𝑥𝑡=𝐴(𝑟𝑡)𝑥𝑡𝑑𝑡+𝜎(𝑟𝑡,𝑥𝑡)𝑑𝑊𝑡. Mao [9] studied the exponential stability of a general nonlinear diffusion with Markovian switching of the form 𝑑𝑥𝑡=𝑓(𝑥𝑡,𝑡,𝑟𝑡)𝑑𝑡+𝑔(𝑥𝑡,𝑡,𝑟𝑡)𝑑𝑊𝑡. 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: 𝑑𝑋𝑡𝑋=𝑏𝑡,𝜃𝑡𝑋𝑑𝑡+𝜎𝑡,𝜃𝑡𝑑𝑊𝑡+𝑑𝐽𝑡,𝑑𝐽𝑡=Γ𝑔𝑋𝑡,𝜃𝑡𝑁𝑋,𝛾(𝑑𝑡,𝑑𝛾),0=𝑥,𝜃0=𝑖,(1.1) where 𝑋() takes values in 𝑟 and 𝜃𝑡 is a finite-state semi-Markov process taking values in 𝒳={1,,𝑀}. Let Γ be a subset of 𝑟0; it is the range space of impulsive jumps. For any set 𝐵 in Γ, 𝑁(𝑡,𝐵) counts the number of jumps on [0,𝑡] with values in 𝐵 and is independent of the Brownian motion 𝑊𝑡, 𝑏(,)𝑟×𝒳𝑟,𝜎(,)𝑟×𝒳𝑟×𝑑,𝑔(,,)𝑟×𝒳×Γ𝑟. For future use we define the compensated Poisson measure 𝑁(𝑑𝑡,𝑑𝛾)=𝑁(𝑑𝑡,𝑑𝛾)𝜆𝜋(𝑑𝛾)𝑑𝑡, where 𝜋() is the jump distribution and 0<𝜆< is the jump rate. Equation (1.1) can be regarded as the result of the following 𝑀 equations: 𝑑𝑋𝑡𝑋=𝑏𝑡𝑋,𝑖𝑑𝑡+𝜎𝑡,𝑖𝑑𝑊𝑡+Γ𝑔𝑋𝑡𝑋,𝑖,𝛾𝑁(𝑑𝑡,𝑑𝛾)0=𝑥,𝜃0=𝜃,(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 𝑥-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 𝑑𝑋𝑡/𝑑𝑡=𝑓(𝑋𝑡) 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 𝑋𝑡, 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 𝑋𝑡 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 𝑋𝑡 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 𝑑𝑋𝑡/𝑑𝑡=𝑓(𝑋𝑡) 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 (Ω,,{𝑡},) is complete with filtration {𝑡}𝑡0 and is right-continuous and 0 contains all null sets. If 𝑣 is some vector, then |𝑣| is its Euclidean norm and 𝑣 is its transpose, while if 𝐴 is a matrix then its trace norm is denoted as ||𝐴||=tr(𝐴𝐴). + stands for positive part of the real line while 𝑟 is a positive integer. Let 𝒞2,1(𝑟×𝒳×+) denote the family of all functions on 𝑟×𝒳×+, which are twice continuously differentiable in 𝑥 and continuously differentiable in 𝑦. Consider {𝜃𝑡}𝑡0 as a semi-Markov process taking values in 𝒳 with transition probability 𝑝𝑖,𝑗 and conditional holding time distribution 𝐹(𝑡𝑖). Thus if 0𝑡0𝑡1 are times when jumps occur, then 𝑃𝜃𝑡𝑛+1=𝑗,𝑡𝑛+1𝑡𝑛𝑡𝜃𝑡𝑛=𝑖=𝑝𝑖𝑗𝐹(𝑡𝑖).(2.1) Matrix [𝑝𝑖𝑗]{𝑖,𝑗=1,,𝑀} is irreducible and for each 𝑖, 𝐹(𝑖) has continuously differentiable and bounded density 𝑓(𝑖). Embed 𝒳 in 𝑟 by identifying 𝑖 with 𝑒𝑖𝑟. For 𝑦[0,)𝑖,𝑗𝒳, let 𝜆𝑖𝑗(𝑦)=𝑝𝑖𝑗𝑓(𝑦/𝑖)1𝐹(𝑦/𝑖)0,𝑖𝑗,𝜆𝑖𝑖(𝑦)=𝑗𝒳,𝑗𝑖𝜆𝑖𝑗(𝑦)𝑖𝒳.(2.2) Let the stationary distribution of the semi-Markov process be defined as 𝜈𝑖(1/𝑡)𝑡0𝕀𝜃𝑠=𝑖𝑑𝑠 where 𝕀 takes value 1 if 𝜃𝑠=𝑖 and 0 otherwise for any 𝑖𝒳.

For 𝑖𝑗𝒳, 𝑦+, 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 𝜆𝑖𝑗(𝑦). Define the functions 𝒳×+× and 𝑔𝒳×+×+ by (𝑖,𝑦,𝑧)=𝑗𝑖if𝑧Λ𝑖𝑗(𝑦),0otherwise,𝑔(𝑖,𝑦,𝑧)=𝑦if𝑧Λ𝑖𝑗(𝑦),𝑗𝑖,0otherwise.(2.3)

Let (+×) be the set of all nonnegative integer-valued 𝜎-finite measures on a Borel 𝜎-field of (+×). Define the process {𝜃𝑡,𝑌𝑡} described by the following stochastic integral equations: 𝜃𝑡=𝜃0+𝑡0𝜃𝑢,𝑌𝑢𝑁,𝑧1(𝑌𝑑𝑢,𝑑𝑧),𝑡=𝑡𝑡0𝑔𝜃𝑢,𝑌𝑢𝑁,𝑧1(𝑑𝑢,𝑑𝑧),(2.4) where 𝑁1(𝑑𝑡,𝑑𝑧) is an (+×)-valued Poisson random measure with intensity 𝑑𝑡𝑚(𝑑𝑧) independent of the 𝒳-valued random variable 𝜃0, where 𝑚() is a Lebesgue measure on . We define the corresponding compensated or centred Poisson measure as 𝑁1(𝑑𝑠,𝑑𝑧)=𝑁1(𝑑𝑠,𝑑𝑧)𝑑𝑠𝑚(𝑑𝑧). It was shown in Theorem  2.1 of Ghosh and Goswami [11] that 𝜃𝑡 is a semi-Markov process with transition probability matrix [𝑝𝑖𝑗]{𝑖,𝑗=1,,𝑀} with conditional holding time distributions 𝐹(𝑦𝑖). Therefore, one can write 𝜃𝑡=𝜃𝑡. We assume that 𝑁(,),𝑁1(,), and 𝜃0,𝑊𝑡,𝑆0 defined on (Ω,,) are independent.

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

Assumption 2.1. Assume 𝑔(𝑥,𝑖,𝛾) is (𝑟×𝒳×({0}))-measurable and that constants 𝐶>0 exist such that for each 𝑖𝒳,𝑥1,𝑥2 being 𝑟-valued and for each 𝛾Γ we have ||𝑏𝑥1𝑥,𝑖𝑏2||+||𝜎𝑥,𝑖1𝑥,𝑖𝜎2||||𝑥,𝑖𝐶1𝑥2||,||𝑔𝑥1𝑥,𝑖,𝛾𝑔2||||𝑥,𝑖,𝛾𝐶1𝑥2||.(2.5) We also need the condition that the generator matrix 𝑄() is bounded and continuous. 𝑏(0,𝑖)=0, 𝜎(0,𝑖)=0 and 𝑔(𝑥,𝑖,0)=0 and 𝑔(0,𝑖,𝛾)=0 for each 𝑥𝑟, 𝑖𝒳 and each 𝛾Γ.

The process (𝑋𝑡,𝜃𝑡,𝑌𝑡) defined on (Ω,,) in (1.1) and (2.4) is jointly Markov and has a generator 𝐺 given as follows. For 𝑓𝒞2,1(𝑟,𝒳,+), we have 1𝐺𝑓(𝑥,𝑖,𝑦)=2𝑟𝑘,𝑙=1𝑎𝑘𝑙(𝑥,𝑖)𝜕𝑓(𝑥,𝑖,𝑦)𝜕𝑥𝑘𝜕𝑥𝑙+𝑟𝑘=1𝑏𝑘(𝑥,𝑖)𝜕𝑓(𝑥,𝑖,𝑦)𝜕𝑥𝑘+𝜕𝑓(𝑥,𝑖,𝑦)+𝜕𝑦𝑓(𝑦𝑖)1𝐹(𝑦𝑖)𝑗𝑖,𝑗𝒳𝑝𝑖𝑗[]𝑓(𝑥,𝑗,0)𝑓(𝑥,𝑖,𝑦)+𝜆Γ(𝑓(𝑥+𝑔(𝑥,𝑖,𝛾),𝑖,𝑦)𝑓(𝑥,𝑖,𝑦))𝜋(𝑑𝛾),(2.6) where 𝑥𝑟,𝑎(𝑥,𝑖)=𝜎(𝑥,𝑖)𝜎(𝑥,𝑖) is an 𝑟×𝑟 matrix and 𝑎𝑘𝑙(𝑥,𝑖) is the (𝑘,𝑙)th element of the matrix 𝑎 while 𝑏𝑘(𝑥,𝑖) is the 𝑘th element of the vector 𝑏(𝑥,𝑖).

We define the jump times, that is, time epochs when jumps occur by {𝜏𝑁𝑛}, where 𝜏𝑁1<𝜏𝑁2<<𝜏𝑁𝑛<, to be the enumeration of all elements in the domain 𝐷𝑝 of the point process 𝑝(𝑡) corresponding to the stationary 𝑡-Poisson point process 𝑁(𝑑𝑡,𝑑𝛾). It is easy to see that {𝜏𝑁𝑛} is an 𝑡-stopping time for each 𝑛. Moreover, we have lim𝑛𝜏𝑁𝑛=+ since the characteristic measure 𝑚() is finite. Next, let us denote the successive switching instants of the second component, which is the semi-Markov process 𝜃𝑡 that switches from one point on the space 𝒳 to another and is denoted by 𝜏𝜃0=0,𝜏𝜃𝑛=inf{𝑡𝑡>𝜏𝜃𝑛1,𝑋𝑡𝑋𝜏𝜃𝑛1},𝑛1. Since the Poisson random measure 𝑁(,) is independent of 𝑁1(,), one could adapt the proof of Xi ([12]) to show that with probability 1, {𝜏𝑁𝑛𝑛1} and {𝜏𝜃𝑛𝑛1} are mutually disjoint. Hence between two chain-switching epochs, the process 𝑋𝑡 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 𝑋𝑡. 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 (𝑋𝑡,𝑡0) with initial data (𝑋0,𝜃0,𝑌0) to (1.1).

Proof. We only provide a sketch of the proof here. Consider [𝑠,𝑡],𝜏𝜃1,,𝜏𝜃𝑁𝑡. Then as described above, on each of the intervals between the chain switching times, that is, [𝑠,𝜏𝜃1),,(𝜏𝜃𝑁,𝑡], the sMMJD process 𝑋𝑡 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 𝑋𝑡.

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

Lemma 2.3. {𝑃(𝑋𝑡0,𝑡0)}=1 for any 𝑋0=𝑥0, and 𝜃0=𝜃𝒳. 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 𝑋𝑡=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 𝑓𝑋𝑡,𝜃𝑡,𝑌𝑡𝑓(𝑥,𝜃,𝑦)=𝑡0𝑋𝐺𝑓𝑠,𝜃𝑠,𝑌𝑠𝑑𝑠+𝑡0𝑋𝑓𝑠,𝜃𝑠,𝑌𝑠𝜎𝑋𝑠,𝜃𝑠𝑑𝑊𝑠+𝑡0Γ𝑓𝑋𝑠𝑋+𝑔𝑠,𝜃𝑠,𝛾,𝜃𝑠,𝑌𝑠𝑋𝑓𝑠,𝜃𝑠,𝑌𝑠+𝑁(𝑑𝑠,𝑑𝛾)𝑡0𝑓𝑋𝑠,𝜃𝑠+𝜃𝑠,𝑌𝑠,𝑌,𝑧𝑠𝑔𝜃𝑠,𝑌𝑠𝑋,𝑧𝑓𝑠,𝜃𝑠,𝑌𝑠𝑁1(𝑑𝑠,𝑑𝑧),(2.7) where the local martingale terms are explicitly defined as 𝑑𝑀1𝑋(𝑡)=𝑓𝑡,𝜃𝑡,𝑌𝑡𝜎𝑋𝑡,𝜃𝑡𝑑𝑊𝑡,𝑑𝑀2(𝑡)=Γ𝑓𝑋𝑠𝑋+𝑔𝑠,𝜃𝑠,𝛾,𝜃𝑠,𝑌𝑠𝑋𝑓𝑠,𝜃𝑠,𝑌𝑠𝑁(𝑑𝑠,𝑑𝛾),𝑑𝑀3(𝑡)=𝑓𝑋𝑠,𝜃𝑠+𝜃𝑠,𝑌𝑠,𝑧,𝑌𝑠𝑔𝜃𝑠,𝑌𝑠𝑋,𝑧𝑓𝑠,𝜃𝑠,𝑌𝑠𝑁1(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 limsup𝑡1𝑡||𝑋log𝑡||<0a.s.𝑋0𝑟a.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 𝜆>0 and 𝐶>0, such that for any 𝑋0𝑟𝐸||𝑋𝑡||𝑝||𝑋𝐶0||𝑝exp(𝜆𝑡)𝑡0.(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 𝑋𝑡 will never reach zero whenever 𝑋00. So in what follows we will only need a function 𝑉(𝑥,𝑖,𝑦)𝒞2,1(𝑟×𝒳×+) 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 𝑉𝒞2,1(𝑟×𝒳×+) in any deleted neighborhood of zero. Moreover, assume that there exist positive constants 𝛼,𝛽,𝜌1,𝜌2,𝜌1, and 𝜌2 for each 𝑥𝑟,𝑖𝒳 and for each 𝛾Γ such that |||𝐺log𝑉(𝑥,𝑖,𝑦)𝛼,𝑥𝑉(𝑥,𝑖,𝑦)|||𝜌𝜎(𝑥,𝑖)𝛽𝑉(𝑥,𝑖,𝑦),1𝑉(𝑥+𝑔(𝑥,𝑖,𝛾),𝑖,𝑦)𝑉(𝑥,𝑖,𝑦)𝜌2,𝜌1𝑉𝑥,𝑖+(𝑖,𝑦,𝑧),𝑦𝑔(𝑖,𝑦,𝑧)𝑉(𝑥,𝑖,𝑦)𝜌2,(3.1) then the solution to (1.1) is almost surely exponentially stable.

Proof. Note that 𝑋log𝑉𝑡,𝜃𝑡,𝑌𝑡𝑋=log𝑉0,𝜃0,𝑌0+𝑡0𝑋𝐺log𝑉𝑠,𝜃𝑠,𝑌𝑠𝑑𝑠+𝑀1(𝑡)+𝑀2(𝑡)+𝑀3(𝑡).(3.2) Here the local martingale terms 𝑀1(𝑡),𝑀2(𝑡), and 𝑀3(𝑡) are, respectively, 𝑀1(𝑡)=𝑡0𝑥𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠𝜎𝑋𝑠,𝜃𝑠𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠𝑑𝑊𝑠,𝑀2(𝑡)=𝑡0Γ𝑉𝑋log𝑠𝑋+𝑔𝑠,𝜃𝑠,𝛾,𝜃𝑠,𝑌𝑠𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠𝑀𝑁(𝑑𝑠,𝑑𝛾),3(𝑡)=𝑡0𝑋log𝑉𝑠,𝜃𝑠+𝜃𝑠,𝑌𝑠,𝑧,𝑌𝑠𝑔𝜃𝑠,𝑌𝑠𝑋,𝑧log𝑉𝑠,𝜃𝑠,𝑌𝑠𝑁1(𝑑𝑠,𝑑𝑧).(3.3) We deal with (3.2) term by term to derive an upper bound on limsup𝑡log𝑉(𝑋𝑡,𝑖,𝑌𝑡)/𝑡. Consider first the drift term of (3.2). It is easy to see from the assumptions made that 𝑡0𝐺log𝑉(𝑋𝑠,𝜃𝑠,𝑌𝑠)𝑑𝑠 will be bounded above by 𝛼𝑡. Secondly, we now concentrate on the local martingale terms of (3.2). First consider the quadratic variation of the 𝑀1(𝑡) term. By Ito’s isometry, we have 𝑀1(𝑡),𝑀1(𝑡)=𝑡0|||||𝑥𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠𝜎(𝑋𝑠,𝜃𝑠)𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠|||||2𝑑𝑠𝑡0𝛽2𝑑𝑠𝛽2𝑡.(3.4) Next consider the quadratic variation of the local martingale term 𝑀2(𝑡). Based on the following result presented in Kunita [15, page 323], and noting that the jump distribution 𝜋 is a probability measure that is, Γ𝜋(𝑑𝛾)=1 we have 𝑀2(𝑡),𝑀2(𝑡)=𝑡0Γ𝑉𝑋log𝑠𝑋+𝑔𝑠,𝜃𝑠,𝛾,𝜃𝑠,𝑌𝑠𝑉𝑋𝑠,𝜃𝑠,𝑌𝑠2𝜋(𝑑𝛾)𝑑𝑠maxlog𝜌12,log𝜌22𝑡.(3.5) On very similar lines, one can easily show that the quadratic variation of the local martingale term 𝑀3(𝑡) is given by 𝑀3(𝑡),𝑀3(𝑡)maxlog𝜌12,log𝜌22𝑡.(3.6) Thus by SLLN for local martingales (refer to Lipster and Shiryayev [16, page 140–141]), we can say that limsup𝑡𝑀1𝑡=limsup𝑡𝑀2𝑡=limsup𝑡𝑀3𝑡=0.(3.7) Thus from (3.2) and the above discussion, one can infer that limsup𝑡log𝑉(𝑥,𝑖,𝑦)𝑡𝛼.(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(𝑥,𝑖,𝑦)𝒞2,1(𝑟,𝒳,+) such that 𝛼1|𝑥|𝑝𝑉(𝑥,𝑖,𝑦)𝛼2|𝑥|𝑝,𝐺𝑉(𝑥,𝑖,𝑦)𝛼|𝑥|𝑝.(3.9) Then, limsup𝑡1𝑡||𝑋log𝐸𝑡||𝑝𝛼𝛼2||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 𝛼/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 𝑝2 always implies almost sure exponential stability for (1.1).

Theorem 3.3. Assume that there exists a positive constant 𝐶 such that for each 𝑖𝒳||||||||||||𝑏(𝑥,𝑖)𝜎(𝑥,𝑖)𝑔(𝑥,𝑖,𝛾)𝐶|𝑥|.(3.11) If for all 𝑋0=𝑥0𝑟, limsup𝑡1𝑡||𝑋log𝐸𝑡||𝑝𝑎,(3.12) then limsup𝑡1𝑡||𝑋log𝑡||𝑎𝑝a.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 𝑋𝑡sup𝑠𝑡|𝑋𝑠| is its maximum process. Then the Burkholder-Davis-Gundy theorem states that for any 1𝑝<, there exist positive constants 𝑐𝑝,𝐶𝑝 such that, for all local martingales 𝑋 with 𝑋0=0 and stopping times 𝜏, the following inequality holds: 𝑐𝑝𝐸[𝑋]𝜏𝑝/2𝑋𝐸𝜏𝑝𝐶𝑝𝐸[𝑋]𝜏𝑝/2.(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 𝑋0𝑟. Let 𝜖 be arbitrarily small positive number. By the definition of 𝑝th-moment exponential stability of (3.15), there exists a constant 𝐾 such that 𝐸||𝑋𝑡||𝑝𝐾exp(𝑎𝜖)𝑡,𝑡0.(3.15) Let 𝛿>0 be sufficiently small such that, 5𝑝𝐶𝑝𝛿𝑝+𝐶𝑝𝛿𝑝/2<14.(3.16) From (1.1) we have 𝑋𝑡=𝑋0+𝑡0𝑏𝑋𝑠,𝜃𝑠𝑑𝑠+𝑡0𝜎𝑋𝑠,𝜃𝑠𝑑𝑊𝑠+𝑡0Γ𝑔𝑋𝑠,𝜃𝑠,𝛾𝑁(𝑑𝑠,𝑑𝛾)+𝜆𝑡0Γ𝑔𝑋𝑠,𝜃𝑠,𝛾𝜋(𝑑𝛾)𝑑𝑠.(3.17) Noting that for 𝑎,𝑏,𝑐,𝑑,𝑒0(𝑎+𝑏+𝑐+𝑑+𝑒)𝑝[]5(𝑎𝑏𝑐𝑑𝑒)𝑝=5𝑝(𝑎𝑝𝑏𝑝𝑐𝑝𝑑𝑝𝑒𝑝)5𝑝(𝑎𝑝+𝑏𝑝+𝑐𝑝+𝑑𝑝+𝑒𝑝),(3.18) we have 𝐸sup(𝑘1)𝛿𝑡𝑘𝛿||𝑋𝑡||𝑝5𝑝𝐸||𝑋(𝑘1)𝛿||𝑝+5𝑝𝐸𝑘𝛿(𝑘1)𝛿||𝑏𝑋𝑠,𝜃𝑠||𝑑𝑠𝑝+5𝑝𝐸sup(𝑘1)𝛿𝑡𝑘𝛿𝑡(𝑘1)𝛿||𝜎𝑋𝑠,𝜃𝑠𝑑𝑊𝑠||𝑝+5𝑝𝐸sup(𝑘1)𝛿𝑡𝑘𝛿𝑡(𝑘1)𝛿Γ||𝑔𝑋𝑠,𝜃𝑠𝑁||,𝛾(𝑑𝑠,𝑑𝛾)𝑝+5𝑝𝜆𝑝𝐸𝑘𝛿(𝑘1)𝛿Γ||𝑔𝑋𝑠,𝜃𝑠||,𝛾𝜋(𝑑𝛾)𝑑𝑠𝑝.(3.19) Noting that Γ𝜋(𝑑𝛾)=1, we have 𝐸sup(𝑘1)𝛿𝑡𝑘𝛿𝑡(𝑘1)𝛿Γ||𝑔𝑋𝑠,𝜃𝑠||,𝛾𝑁(𝑑𝑠,𝑑𝛾)𝑝𝐶𝑝𝐸𝑘𝛿(𝑘1)𝛿||𝑔𝑋𝑠,𝜃𝑠||,𝛾2𝑑𝑠(𝑝/2)𝐶𝑝𝐸𝛿sup(𝑘1)𝛿𝑠𝑘𝛿||𝑔𝑋𝑠,𝜃𝑠||,𝛾2(𝑝/2)𝐶𝑝𝐶𝑝𝛿𝑝/2𝐸sup(𝑘1)𝛿𝑠𝑘𝛿||𝑋𝑠||𝑝.(3.20) Similarly, 𝐸||||𝑘𝛿(𝑘1)𝛿Γ||||𝑔𝑋𝑠,𝜃𝑠,𝛾𝜋(𝑑𝛾)𝑑𝑠𝑝𝐸𝛿sup(𝑘1)𝛿𝑠𝑘𝛿||𝑔𝑋𝑠,𝜃𝑠||,𝛾𝑝𝐶𝑝𝛿𝑝𝐸sup(𝑘1)𝛿𝑠𝑘𝛿||𝑋𝑠||𝑝.(3.21) From (3.15), one can easily show that 𝐸||𝑋(𝑘1)𝛿||𝑝𝐾exp(𝑎𝜖)(𝑘1)𝛿𝐸,(3.22)𝑘𝛿(𝑘1)𝛿||𝑏𝑋𝑠,𝜃𝑠||𝑑𝑠𝑝𝐶𝑝𝛿𝑝𝐸sup(𝑘1)𝛿𝑠𝑘𝛿||𝑋𝑠||𝑝𝐸,(3.23)sup(𝑘1)𝛿𝑡𝑘𝛿𝑡(𝑘1)𝛿||𝜎𝑋𝑠,𝜃𝑠||𝑑𝑊𝑠𝑝𝐶𝑝𝐶𝑝𝛿𝑝/2𝐸sup(𝑘1)𝛿𝑠𝑘𝛿||𝑋𝑝𝑠||.(3.24) Hence, substituting (3.20)–(3.23) in (3.19), we obtain 𝐸sup(𝑘1)𝛿𝑡𝑘𝛿||𝑋𝑡||𝑝15𝑝𝐶𝑝𝛿𝑝+𝐶𝑝𝐶𝑝𝛿𝑝/2+𝐶𝑝𝐶𝑝𝛿𝑝/2+𝐶𝑝𝛿𝑝𝐾5𝑝exp(𝑎𝜖)(𝑘1)𝛿.(3.25) From (3.16) we obtain that 𝐸sup(𝑘1)𝛿𝑡𝑘𝛿||𝑋𝑡||𝑝2×5𝑝𝐾exp(𝑎𝜖)(𝑘1)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 𝑋𝑡𝑟 and 𝑉(𝑋𝑡,𝑖,𝑌𝑡)=|𝑋𝑡| where the generator matrix is given by||||||||||||𝑄=2211.(4.1) Let the holding time in each regime be assumed to follow 𝑓(𝑦𝑖)=𝜆𝑖𝑒𝜆𝑖𝑦,𝑦>0,𝑖{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(𝑥,𝑖,𝑦) is given by 1𝐺𝑉(𝑥,𝑖,𝑦)=2𝐼trace|𝑥|𝑥𝑥|𝑥|3𝜎(𝑥,𝑖)𝜎+𝑥(𝑥,𝑖)|𝑥|𝑏(𝑥,𝑖)+𝜆Γ||||𝑥+𝑔(𝑥,𝑖,𝛾)|𝑥|𝜋(𝑑𝛾).(4.2) Now from Assumption 2.1 as ||||=||||||||=||||𝜎(𝑥,𝑖)𝜎(𝑥,𝑖)𝜎(0,𝑖)𝐶|𝑥|𝑏(𝑥,𝑖)𝑏(𝑥,𝑖)𝑏(0,𝑖)𝐶|𝑥|(4.3) and |𝑔(𝑥,𝑖,𝛾)|𝐶|𝑥|, we have =𝐺𝑉(𝑥,𝑖,𝑦)𝐶|𝑥|+𝐶|𝑥|+𝜆(2+𝐶)|𝑥|(2𝐶+𝜆𝐶+2𝜆)|𝑥|.(4.4) If we choose 𝐶 and 𝜆 such that for any 𝑥𝑟{0}, there exists 𝛼=(2+𝜆)𝐶+2𝜆)0 such that 𝐺log𝑉(𝑥,𝑖,𝑦)𝛼. Also |𝑥𝑉(𝑥,𝑖,𝑦)𝜎(𝑥,𝑖)/𝑉(𝑥,𝑖,𝑦)|𝐶. Similarly, if there exist a positive constant 𝛽 such that for any 𝑥𝑟, 𝐶𝛽, then |𝑥𝑉(𝑥,𝑖,𝑦)𝜎(𝑥,𝑖)/𝑉(𝑥,𝑖,𝑦)|𝛽. If there exist constants 𝜌1 and 𝜌2 such that 𝜌1𝑔(𝑥,𝑖,𝛾)𝜌2 for any 𝑥𝑟, 𝑖𝒳 and 𝛾Γ, then it is easy to see that (𝜌1)(𝑉(𝑥+𝑔(𝑥,𝑖,𝛾),𝑖,𝑦)/𝑉(𝑥,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 𝑥 and 𝑉(𝑥,𝑖,𝑦)=𝑥2. Also assume that the conditional holding time distribution be 𝑓(𝑦𝑖)=𝜆𝑖𝑒𝜆𝑖𝑦 for 𝑖{1,2}. Let 𝑔(𝑥,𝑖,𝛾)=𝑥, 𝜆𝑖=1, 𝑏(𝑥,𝑖)=𝑎1𝑥, 𝜎(𝑥,𝑖)=𝑎2𝑥 for 𝑖{1,2}. Then from (2.4) we have 𝐺𝑉(𝑥,𝑖,𝑦)=(2𝑎1+𝑎2+3)𝑥2. If 2𝑎1+𝑎2+3<0 and 𝑥0, 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|𝑥|2𝑉(𝑥,𝑖,𝑦)𝛼2|𝑥|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: 𝑑𝑋𝑡𝑋𝑑𝑡=𝑓𝑡(5.1) on 𝑡0 with 𝑋0=𝑥0𝑟 where 𝑓(𝑥)𝑟𝑟 is locally Lipschitz continuous and furthermore there exists some constant 𝐾>0 such that |𝑓(𝑥)|𝐾|𝑥|forall𝑥𝑟. 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 𝑋𝑡 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 𝑑𝑋𝑡/𝑑𝑡=𝑎𝑋𝑡 for 𝑎>0 can be stabilized just by the addition of a jump component to the dynamics of the one-dimensional process 𝑋𝑡. 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 {𝑋𝑡,𝑡0} and some assumptions that follow. Suppose we have an 𝑚-dimensional standard 𝑇-adapted Brownian motion process 𝐵=(𝐵1(𝑡),,𝐵𝑚(𝑡)) for each 𝑡0. The system (5.1) is perturbed by the following sMMJD dynamics of 𝑋𝑡 given by 𝑑𝑋𝑡𝑋=𝑓𝑡𝑑𝑡+𝑚𝑘=1𝐺𝑘𝜃𝑡𝑋𝑡𝑑𝐵𝑘(𝑡)+𝜆Γ𝐷𝜃𝑡𝑋,𝛾𝑡𝑁(𝑑𝑡,𝑑𝛾)𝑡0,(5.2) where 𝐺𝑘(𝑖) is 𝑟×𝑟 for each 𝑖𝒳. Likewise 𝐷(𝑖,𝛾) is an 𝑟×𝑟-valued matrix for each 𝑖𝒳 and 𝛾Γ𝑟{0}. We refer to a system (5.1) perturbed by the dynamics of 𝑋𝑡 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 𝑖𝒳 and 𝛾Γ we have Γ(||𝐷(𝑖,𝛾)||||𝐷(𝑖,𝛾)||2)𝜋(𝑑𝛾)< and that 𝐷(𝑖,𝛾) does not have an eigenvalue equal to 1𝜋 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 𝑎(𝑖)>0, 𝑏(𝑖)0:(i)𝑚𝑘=1|𝐺𝑘(𝑖)𝑥|2𝑎(𝑖)|𝑥|2, (ii)𝑚𝑘=1|𝑥𝐺𝑘(𝑖)𝑥|2𝑏(𝑖)|𝑥|4 for each 𝑖𝒳 and 𝑥𝑟.Then the sample Lyapunov exponent of the solution of (5.2) exists and satisfies limsup𝑡log|𝑋𝑡|𝐾𝑖𝒳[(𝑏(𝑖)𝑎(𝑖)/2𝜆log(1+||𝐷(𝑖,𝛾)||))]𝜈𝑖 for any 𝑋00. If 𝐾+𝑖𝒳[𝑏(𝑖)𝑎(𝑖)/2𝜆log(1+||𝐷(𝑖,𝛾)||)]𝜈𝑖>0, then the trivial solution to the system in (5.2) is almost surely exponentially stable.

Proof.
Step  1. Define 𝑉(𝑥,𝑖,𝑦)=log|𝑥| for all 𝑖𝒳. As 𝑉(𝑥,𝑖,𝑦) is independent of states 𝑖 and 𝑦, the following terms in (2.4) are zero: 𝑓(𝑦𝑖)1𝐹(𝑦𝑖)𝑗𝑖,𝑗𝒳𝑝𝑖𝑗[]𝑉(𝑥,𝑗,0)𝑉(𝑥,𝑖,𝑦)=0,𝜕𝑉(𝑥,𝑖,𝑦)𝜕𝑦=0.(5.3) Hence as an application of the generalized Ito’s formula, we have for 𝑡>0||𝑋log𝑡||||𝑋=log0||+𝑡0𝑋𝑠||𝑋𝑠||2𝑓𝑋𝑠1𝑑𝑠+2𝑚𝑘=1𝑡0||𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||22||𝑋𝑠𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||4𝑑𝑠+𝜆𝑡0Γ||𝑋log𝑠𝜃+𝐷𝑠𝑋,𝛾𝑠||||𝑋𝑠||𝜋(𝑑𝛾)𝑑𝑠+𝑀1(𝑡)+𝑀2(𝑡),(5.4) where 𝑀1(𝑡)=𝑚𝑘=1𝑡0|𝑋𝑠𝐺𝑘(𝜃𝑠)𝑋𝑠/|𝑋𝑠|2|𝑑𝐵𝑘(𝑠) and 𝑀2(𝑡)=𝑡0Γlog((|𝑋𝑠+𝐷(𝜃𝑠,𝛾)𝑋𝑠|)/|𝑋𝑠|)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 ||𝑋𝑠𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||4=||𝑋𝑠𝐺𝑘𝜃𝑠+𝐺𝑘𝜃𝑠𝑋𝑠||24||𝑋𝑠||4𝐺𝜌𝑘𝜃𝑠2,(5.5) where 𝜌(𝐺𝑘(𝜃𝑠)) is the spectral radius of the symmetric 𝑟×𝑟 matrix (𝐺𝑘(𝜃𝑠)+𝐺𝑘(𝜃𝑠))/2, 𝑀1(𝑡),𝑀1(𝑡)𝑚𝑘=1𝑡0||𝑋𝑠𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||4𝑑𝑠𝑡𝑚max1𝑘𝑚,𝑖=1,,𝑀𝜌𝐺𝑘.(𝑖)(5.6) Next, the quadratic variation of the process 𝑀2(𝑡) is given by 𝑀2(𝑡),𝑀2(𝑡)=2Γ𝑡0||𝑋log𝑠𝜃+𝐷𝑠𝑋,𝛾𝑠||||𝑋𝑠||𝑑𝑠𝜋(𝑑𝛾)2𝑡log1+max1𝑖𝑀.𝐷(𝑖,𝛾)(5.7)
Step  3. We work with the rest of the terms in the following way: limsup𝑡|||||1𝑡𝑡0𝑋𝑠𝑓𝑋𝑠||𝑋𝑠||2|||||𝑑𝑠𝐾,(5.8) also limsup𝑡1𝑡𝑡012𝑚𝑘=1||𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||22||𝑋𝑠𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||41𝑑𝑠𝑡𝑖𝒳𝑡0𝑎(𝑖)2𝕀𝑏(𝑖)𝜃𝑠=𝑖𝑑𝑠𝑖𝒳𝑎(𝑖)2𝜈𝑏(𝑖)𝑖,limsup𝑡𝜆𝑡𝑡0Γ||𝑋log𝑠𝜃+𝐷𝑠𝑋,𝛾𝑠||||𝑋𝑠||𝜋(𝑑𝛾)𝑑𝑠𝜆𝑖𝒳log(1+𝐷(𝑖,𝛾))𝜈𝑖.(5.9) Thus, limsup𝑡(1/𝑡)log|𝑋𝑡|<0 if 𝐾+𝑖𝒳[(𝑎(𝑖)/2𝑏(𝑖)+𝜆log(1+||𝐷(𝑖,𝛾)||))]𝜈𝑖<0.

Remark 5.3. Consider a 1-D sMMJD with the dynamics 𝑑𝑋𝑡=𝑎𝑋𝑡𝑑𝑡+𝑏(𝑖)𝑋𝑡𝑑𝐵𝑡+𝑐(𝑖,𝛾)𝑋𝑡𝑑𝑁𝑡,(5.10) where 𝑏(𝑥,𝑖)>0 and 𝑐(𝑖,𝛾)>1 for each 𝑥, 𝑖{1,,𝑀} and 𝛾Γ. 𝐵𝑡 is a 1D Brownian motion and {𝑁𝑡,𝑡0} is a compensated Poisson process with 𝑁𝑡=𝑁𝑡𝜆𝑡, where 𝜆>0 is the intensity of the Poisson process. Assume that the processes 𝐵𝑡 and 𝑁𝑡 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 𝑖{1,2,,𝑀}limsup𝑡1𝑡||𝑋log𝑡||1=𝑎+𝜆𝑐(𝑖,𝛾)2𝑏2(𝑖)+𝜆log(1+𝑐(𝑖,𝛾))<0a.s.(5.11) Note that 𝑏2(𝑖)0 for all 𝑖𝒳 and has a negative sign attached to it. Hence when the one-dimensional perturbed system 𝑑𝑋𝑡/𝑑𝑡=𝑎𝑋𝑡 for 𝑎>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 𝐺𝑘(𝑖)=0 for each 𝑖=1,,𝑀 and 𝑘=1,,𝑚. We now show that it gets destabilized by further addition of the 𝑚-dimensional Brownian motion to (5.1). This corresponds to 𝐺𝑘(𝑖)0 for each 𝑖=1,,𝑀 and 𝑘=1,,𝑚.

Theorem 5.4. Assume that matrix 𝐷 is an 𝑟×𝑟 symmetric positive definite matrix. Now let(i)𝑚𝑘=1|𝐺𝑘(𝑖)𝑥|2𝑎(𝑖)|𝑥|2,(ii)𝑚𝑘=1|𝑥𝐺𝑘(𝑖)𝑥|2𝑏(𝑖)|𝑥|4,for 𝑎(𝑖)>0,𝑏(𝑖)0 for each 𝑖𝒳, 𝑥𝑟. Hence liminf𝑡(1/𝑡)log|𝑋𝑡|𝐾+𝑖𝒳[(𝑎(𝑖)/2𝑏(𝑖)+𝜆log(1+min1𝑖𝑀||𝐷(𝑖,𝛾)||))]𝜈𝑖 for any 𝑋00. In particular if 𝐾+𝑖𝒳[𝑎(𝑖)/2𝑏(𝑖)+𝜆log(1+min1𝑖𝑀||𝐷(𝑖,𝛾)||)]𝜈𝑖>0, then the trivial solution of (5.2), tends to be infinity almost surely exponentially fast.

Proof. Fix 𝑋00. From Lemma  2.2, 𝑋𝑡0 for all 𝑡0. Applying Ito’s lemma to log|𝑋𝑡|, for 𝑡>0 and for each 𝑖𝒳, ||𝑋log𝑡||||𝑋=log0||+𝑡0𝑋𝑠||𝑋𝑠||2𝑓𝑋𝑠1𝑑𝑠+2𝑚𝑘=1𝑡0||𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||22||𝑋𝑠𝐺𝑘𝜃𝑠𝑋𝑠||2||𝑋𝑠||4𝑑𝑠+𝜆𝑡0Γ||𝑋log𝑠𝜃+𝐷𝑠𝑋=𝑖,𝑌𝑠||||𝑋𝑠||𝜋(𝑑𝛾)𝑑𝑠+𝑀1(𝑡)+𝑀2(𝑡),(5.12) where 𝑀1(𝑡)=𝑚𝑘=1𝑡0(|𝑋𝑠𝐺𝑘(𝜃𝑠)𝑋𝑠|/|𝑋𝑠|2)𝑑𝐵𝑘(𝑠) and 𝑀2(𝑡)=Γ𝑡0log((|𝑋𝑠+𝐷(𝜃𝑠=𝑖,𝑌)𝑋𝑠|)/|𝑋𝑠|)𝑁(𝑑𝑠,𝑑𝛾) are the two local martingale terms. Now using methodology similar to Theorem 5.2 we find liminf𝑡1𝑡||𝑋log𝑡||𝐾+𝑖𝒳𝑎(𝑖)2𝑏(𝑖)+𝜆log1+min1𝑖𝑀𝜈𝐷(𝑖,𝛾)𝑖(5.13) for any 𝑋00. In particular, if 𝐾+𝑖𝒳[𝑎(𝑖)/2𝑏(𝑖)+𝜆log(1+min1𝑖𝑀||𝐷(𝑖,𝛾)||)]𝜈𝑖>0, then the trivial solution of the 𝑋𝑡-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 𝑑𝑋𝑡/𝑑𝑡=𝑓(𝑋𝑡) 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.