We will prove comparison result on a system of linearly coupled HJB equations. Such a system usually arises from stochastic control problem with regime-switching diffusion.
Consider a system of Dirichlet PDE given by
In the above, is bounded open domain, and
, and the matrix
is a generator of a continuous Markov chain, i.e.
Solution of the system is a function set . Our goal is to show comparison result in viscosity sense. Note that, since (1) is not proper by (2), we can not directly apply the result of [CIL92].
Definition 1 A continuous function
is a viscosity subsolution of (1), if, for each
of
, for every
and every smooth test function
satisfying
we have
Similarly, define a viscosity supersolution.
Following two assumptions will be imposed for , which is satisfied by most HJB equations.
Assumption 1
satisfies, for any
and
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for some
.
for some function
with
, whenever
, and
Theorem 2 (Comparison result) Let
be bounded open subset of
,
be proper and satisfy Assumption 1 and Assumption 2 for each
. Let
(respectively
) be a subsolution (respectively supersolution) of (1) and
on
. Then
in
.
Such a maximizer exists due to USC property of and finiteness of
. It is enough to show
. To to contrary, if
, then maximizer must be interior point, i.e.
. Define
Let the maximizer of (exists) be
, writing
By Ishii’s lemma, there also exists , such that
-
,
-
,
-
and
satisfies (5), in particular,
.
By the above first two properties, we can write
and
If we write , then above two inequalities yield
Now, we are ready to find a contradiction as follows, by the fact of (7), (8), Assumption 1, and Assumption 2,
Since as ,
Therefore, if we force the limsup
, which leads to contradiction to
.