In this below, we assume that
- is bounded open set in ;
- For and , ;
- For and , for a Levy measure .
Recall that a Levy measure on is a non-negative measure satisfying
Now we observe that
Proof: It’s the consequence of Taylor expansion.
Proof: Let . If , then it’s done. If , then the function is CDF of the probability distribution on the sample space given by
Therefore, is RCLL, so is . Since , we conclude
Proof: Since , we have for all . By mean value theorem, there exists such that
Finally, we can take and apply Proposition 2 to the last inequality to obtain the proof.
Next, if we set
If we define
then (1) can be rewritten by
for any .
We say, is a subsolution if on , and is a supersolution if . Next, we will show the comparison principle. The proof given below may be not the simplest one, but may be easy to be adapted to the proof of comparison principle in the context of the viscosity solution.
Proof: Because , there exists . Assuming , then shall satisfy
and we try to find a contradiction below. From the sub and super solution property of , we can write
The last inequality is the consequence of monotonicity of on the third variable together with properties of . This leads to an inequality
which makes a contradiction to Proposition 3.