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I have two questions regarding the $p$-Variation of a Semimartingale:

Let $X_t$ be a semimartingale on $[0,1]$ and $\Pi_n = \{t^n _k = \frac{k}{n}: 0 \leq k \leq n\} $ a partition of $[0,1]$.

For $p >0$ we define the $p$-Variation-sum $V_{n,p}$ as: $$ V_{n,p} := \sum _{k=0} ^{n-1} |X(t^n _{k+1}) - X(t^n _k)|^p $$ and the $p$-Variation $V_p$ as the limit $$ V_p := \lim_{n\to \infty} V_{n,p} $$ How do we know that the p-Variation is well defined for a semimartingale?

It is often stated that the quadratic Variation of a semimartingale (i.e. $V_2$) is finite. Why is that?

K123
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    Karatzas & Shreve treat $p$-variation in the first chapter of their book Brownian Motion and Stochastic Calculus. I recommend Problem 5.11 for which the even provide the solution. Regarding your last question: spell out the definition of a semimartingale and remember what the $p$ is for the martingale part. What values of $p$ are then ruled out more or less by definition ? – Kurt G. Jan 05 '23 at 21:14
  • As a rule we don't want a Question to pose multiple problems, but an exception exists for multiple problems that are closely related in that a solution to one part is useful to the other. This is a borderline case, but I'll vote to leave open (hoping that you can edit the body of the Question to make the connection between parts explicit, or edit to narrow the scope to avoid that rule). – hardmath Jan 06 '23 at 16:16

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