Let $X$ be a one-dimensional Levy process with generating triplet $(\gamma,\sigma^2,\nu)$. Is there a formula for the quadratic variation for this process without further restrictions?
In the book Mathematical Finance by Eberlein and Kallsen (2019) it is stated that $$ [X,X]_t = \sigma^2t + \sum_{s\leq t}(\Delta X_s)^2 $$ for the above defined Levy process X, but in all other resources I have found so far, only ''weaker'' examples were given, such as quadratic variations of Levy processes without a Brownian part.
It makes sense to me, that if $X$ has only finitely many jumps, by the Levy-Ito decomposition, the quadratic variation is just the sum of the quadratic variation of the Brownian part and the quadratic variation of a compound Poisson process, which then yields the above expression. But in the case of infinitely many jumps, does the sum even exist/converge?