In this section, we consider the stochastic differential equation with regime switching (1.4). If stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any initial value, the coefficients of the equation are required to obey the linear growth condition and local Lipschitz condition. It is easy to see that the coefficients of (1.4) satisfy the local Lipschitz condition; therefore, there is a unique local solution
on
with initial value
,
, where
is the explosion time.
And since our purpose is to reveal the effect of environmental noises, we impose the following hypothesis on intensities of environmental noises.
Assumption 4.1 (
).
By virtue of comparison theorem, we will demonstrate that the local solution to (1.4) is global, which is motivated by [12]
Thus,
is the unique solution of
with
. By the comparison theorem, we get
for
a.s. It is easy to see that
with
, has a unique solution
Obviously,
,
a.s.
Moreover,
So, we get
,
a.s., where
Besides,
Then,
where
To sum up, we obtain
It can be easily verified that
,
,
,
all exist on
, hence
Theorem 4.2.
There is a unique positive solution
of (1.4) for any initial value
,
, and the solution has the properties
where
,
,
,
are defined as (4.1),(4.4),(4.6), and (4.9).
Theorem 4.2 tells us that (1.4) has a unique global solution, which makes us to further discuss its properties.
Now, we will investigate certain asymptotic limits of the population model (1.4). Referred to [12], it is not difficult to imply that
Then, we give the following essential theorems which will be used.
Theorem 4.3.
Let Assumption 4.1 hold. Then, the solution
has the property
Proof.
The proof is motivated by [12]. By (4.12) and (4.13), then
Thus, it remains to show that
Note the quadratic variation of
is
, thus the strong law of large numbers for local martingales yields that
Therefore, for any
, there exists some positive constant
such that for any
Then, for any
, we have
Moreover, it follows from (4.13) that for the above
and
, we get
By virtue of (4.6), we can derive for
Using the generalized Itô Lemma, we can conclude that
Consequently,
Then, for
Denote
and
. It is obvious that
. Hence,
So, we obtain
That is,
Therefore,
Note the fact that
is arbitrary, we obtain that
By (4.12), we have
a.s. Consequently,
So, we complete the proof.
Theorem 4.4.
Let Assumption 4.1 and
hold. Then,
has the property
Proof.
From (4.12) and (4.13), we have
Now, we only show that
. By virtue of (4.12), it remains to demonstrate that
. From the proof of Theorem 4.3, we know that for any
, there exists some positive constant
such that for any
It follows from (4.34) that
For above
and
, we get for
a.s. By the generalized Itô Lemma, then
Therefore,
Thus, it is easy to imply that
where
and
. By (4.9), we imply for
Consequently,
It follows from (4.40) that
Then,
For
is arbitrary, we imply
Finally, we obtain
Theorem 4.5.
Let Assumption 4.1 and
hold. Then, the solution
to (1.4) obeys
Proof.
Using the generalized Itô Lemma, we have
Therefore,
Thus, let
, by the ergodic properties of Markov chains, we have
Hence, by virtue of the strong law of large numbers of local martingales, we get
The proof is complete.