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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

The stability of stochastic differential equations (SDEs) has a long history with some key works being those of Arnold [

Consider the following jump-diffusion equation in which the coefficients are modulated by an underlying semi-Markov process:

Unlike the special Markov-modulated case in which the

In Section

We assume that the probability space (

For

Let

To ensure that zero is the only equilibrium point of (

Assume

The process (

We define the jump times, that is, time epochs when jumps occur by

Assume that Assumption

We only provide a sketch of the proof here. Consider

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

We show this in a simple way. From the condition on the coefficients,

We next have the following generalized Ito’s formula.

Utilizing the operator

For details refer to Ikeda and Watanabe [

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

The trivial solution of (

Let

In the next section, we detail the proofs for obtaining the conditions under which the trivial solution of (

In the sequel we will always, as standing hypotheses, assume that Assumption

Assume that there exist a function

Note that

We now provide conditions under which the trivial solution to (

Let

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

In the next theorem, we provide criteria to connect these two seemingly disparate stabilty criteria. Specifically, we provide conditions under which the

Assume that there exists a positive constant

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

Let us recall that [

Let

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

We next provide a simple example to illustrate Theorem

Next we discuss the issue of 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:

Assume that for each

In the following, we will establish the conditions on the coefficients of (

Assume that Assumption

Consider a 1-D sMMJD with the dynamics

To prove this assertion, let us now consider system of nonlinear differential equation (

Assume that matrix

Fix

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

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.