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 *

*
** for some . *

**Assumption 2** * satisfies *

*
* 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 . *

*Proof:* Let be maximizer of

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

Then, satisfies

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 ,

- by the choice of (6), property of (2), and USC property of .
- due to (7).

Therefore, if we force the limsup , which leads to contradiction to .

### Like this:

Like Loading...