The goal of this note is to show the uniqueness of solution of

with boundary value

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

where

and

**Proposition 1** * For any , and . *

*Proof:* It’s the consequence of Taylor expansion.

**Proposition 2** * as . *

*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

**Proposition 3** * Suppose and is the global maximum of , then *

*
** *

*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

then and

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.

**Theorem 4** * Let , and also let and be sub and super solutions, respectively. If on , then on . *

*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.

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