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.
Definition 1 A continuous function is a viscosity subsolution of (1), if, for each of , for every and every smooth test function satisfying
Similarly, define a viscosity supersolution.
Following two assumptions will be imposed for , which is satisfied by most HJB equations.
for some .
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
Since as ,
Therefore, if we force the limsup , which leads to contradiction to .