càdlàg adapted process of finite variation

Question

$X$ is a semimartingale with $X_0=0$. I have to show, that $S_t:=\prod^{}_{s\le t}(1+\Delta X_s)\exp(-\Delta X_s)$ is a càdlàg adapted process of finite variation. Could you please help me?

Answer 1

First note that $S_t$ is only dependent on the jump part $\Delta X_t= X_s-X_{s^-}$ of the semimartingale $X$.

Now consider the function $S(x)=(1+x)\exp(-x)$. Note that $$\lim_{x->\infty}S(x) = 0.$$ Moreover $S(x) \in \mathcal{C^2}$ and so, as long as $\Delta X$ is a càdlàg process of finite variation, $S_t=\prod_{s\le t}S(\Delta X_t)$ will also be càdlàg of finite variation. The same is true for adaptedness, so we're left with verifying these properties for the jump process $\Delta{X_t}$.

Now if $\Delta X_t$ is of finite variation, the proof is complete.

If $\Delta X_t$ has unbounded variation on $[0,t]$, there must be a set of random times $\{\tau_i \le t, i>0\}$ such that the quadratic variation $[X]_{\tau_i}>0$. Let $\tau_1$ be the first of these occurrences. Even in this case, using the convergence of S(x), we have that $S_s=0$, $s \in [\tau_1, t]$, thus the process $S_t$ is indeed of finite variation.