How to prove that a jump diffusion has infinite total variation?

Question

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

Answer 1

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!