1
$\begingroup$

I'm reading a chapter of a book on Levy processes and it states:

"Any Levy process $X$ enjoys the following property: For all $t\geq0$ $$\mathbb{E}[e^{iu X_t}]=e^{t\phi(u)},$$ where $\phi$ is the characterstic function of $X$ of the process $X$".

I read somewhere else that this is a consequence of stationary increment property of Levy processes. Can somoneone provide me with more insight on this? I want to understand why this holds

$\endgroup$
0
$\begingroup$

If you denote the left side by $f(t)$ then $f(t+s)=f(t)f(s)$. This is an easy consequence of the fact that the process has stationary independent increments. Since $f$ is a continuous function and $f(0)=1$ we can show that $f$ must of the type specified.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.