Let $S$ be a positive cádlág process.
Let $L$ be a cádlág process of bounded variation, non-decreasing and predictable with respect to the filtration $\{\mathcal{F}^S_t\}_{t>0}$ generated by S. We can assume $L_0 = 0$.
If we assume that the processes have finite expectation, $\mathbb{E}[S_t] < \infty$ and $\mathbb{E}[L_t] < \infty$ for all $t>0$.
Is it possible to prove that
$$ \mathbb{E}\biggl[ \int_{0}^T S_{t^-} dL_t \biggr] < \infty $$
is true? (or false?)
Thanks a lot.
I tried to solve this problem. Please let me know if the answer is correct. Thank you.
We can write:
$$ \mathbb{E} \bigl[ \int_{0}^T S_{t^-} dL_t \bigr] \leq \mathbb{E} \biggl[ \biggl( \max_{t\in[0,T]} S_{t} \biggr) (L_T - L_0) \biggr] $$
Let us define the running maximum $M_T = \max_{t\in[0,T]} S_{t}$. The expectation on the right is: $$ \mathbb{E} \bigl[ M_T L_T \bigr] $$
Now we can consider the process $Z_T = \frac{M_T}{\mathbb{E}[M_T]}$. I don't know if $\mathbb{E}[M_T]< \infty$ is true in general, but for jump diffusion type processes it works.
The conditions $Z_T > 0$ and $ \mathbb{E} \bigl[ Z_T \bigr] = 1 $ say that we can change the measure to $$ \mathbb{E}[M_T] \; \mathbb{E} \bigl[ Z_T L_T \bigr] = \mathbb{E}[M_T] \;\tilde{\mathbb{E}} \bigl[ L_T \bigr] < \infty. $$
Alternative method
Simply apply Schwartz inequality.
Any comments would be helpful. Thanks