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We have a jump diffusion:

$X_t=bt + \sigma W_t + Y_t$

where b is the drift parameter, $\sigma$ the diffusion parameter, $W_t$ a Wiener process and $Y_t$ a CPP (compound Poisson process). We know that $W_t$ has infinite total variation (since its quadratic variation equals $t$ and its paths are continuous with probability 1). It seems obvious that the remaining two parts can't really make quadratic variation infinite, but how to prove this formally?

Thanks

BGa
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1 Answers1

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Suppose that $(X_t)_{t \geq 0}$ was of finite total variation. Since we know that $bt$ and $Y_t$ are of finite total variation, this implies that

$$\sigma W_t = X_t - bt - Y_t$$

is of finite total variation. Contradiction!

saz
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  • Reading the body of the question, it seems the question is to show that X has finite quadratic variation (I agree, this is not the title and the question is not very clearly formulated). – Did Jun 15 '14 at 17:57
  • @Did I thought that the OP mentioned the quadratic variation, because it provides a sufficient criterion for having an infinite total variation. (E.g. in the case of BM: quadratic variation equals $t$ $\Rightarrow$ infinite total variation.) – saz Jun 15 '14 at 18:02
  • Hmmm... I see. This might be so. – Did Jun 15 '14 at 18:04
  • @Did It seems that the OP is not even willing to answer this simple question ... – saz Jun 18 '14 at 12:31
  • Yep--although they visited the site since the question was posted. :-( – Did Jun 18 '14 at 12:46