A Stochastic System and Its Corresponding Deterministic System 329
Define g(c) =
l
X
j=1
v
j
f
j
(c). Then by (8.32), Lemma 8.3.1 and Gronwall’s in-
equality, one can s how that with some mild co nditio ns on g the processes
{C
M
(t) : t ≥ 0} converge to a deterministic process that is the solution of the
following system of o rdinary differential equations :
˙
c(t) = g(c), c(0) = c
0
. (8.33)
This result was originally shown by Kurtz [34, 35] (and hence it is referred
to as Kurtz’s limit t heorem in this monograph), and is summarized in the
following theorem (e.g., see [9, 36] for details on the proof).
Theorem 8.3.3 Suppose that lim
M→∞
C
M
(0) = c
0
and for any compact set
Ω ⊂ R
n
there exists a positive constant η
Ω
such that
|g(c) − g(
˜
c)| ≤ η
Ω
|c −
˜
c|, c,
˜
c ∈ Ω.
Then we ...