Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \, \mathrm{d}s < \infty$. I'm trying to find answers to the following questions:

Question 1: If $\operatorname{E} [ |A_t|^p] < \infty$ for some $p \geq 1$, is it possible that $\int_0^{t} |b_s|^p \, \mathrm{d}s$ and/or $\operatorname{E} \left[ \left( \int_0^{t} |b_s|^p \, \mathrm{d}s \right)^q \right] $ is infinite for some $q \geq 1$?

Question 2: If $\operatorname{E} [ |A_t|^p] < \infty$ for all $p \geq 1$, is it possible that $\int_0^{t} |b_s|^p \, \mathrm{d}s$ and/or $\operatorname{E} \left[ \left( \int_0^{t} |b_s|^p \, \mathrm{d}s \right)^q \right] $ is infinite for some $q \geq 1$?

What about the questions above when $t$ is replaced by $\infty$?

Some context: I'm studying the Banach spaces $H^p$ of continuous semimartingales $S_t = M_t + A_t$ indexed by $[0,\infty)$ such that $\| S_t \|_{H^p} = \| \sup_t S_t \|_p < \infty$, which, by Burkholder-Davis-Gundy, is equivalent to $\| \langle M,M \rangle_{\infty}^{1/2} + A_{\infty} \|_p < \infty$. I would like to know how "wild" the finite variation kernels can be, especially in the intersection $H = \bigcap_{p \geq 1} H_p$. It seems to me, that at least for question 2 the answer is no (for both, $t$ and $\infty$), but I can't find a proof or counterexample.

I would also be thankful for references where these spaces are studied.

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \, \mathrm{d}s < \infty$. I'm trying to find answers to the following questions:

Question 1: If $\operatorname{E} [ |A_t|^p] < \infty$ for some $p \geq 1$, is it possible that $\int_0^{t} |b_s|^p \, \mathrm{d}s$ and/or $\operatorname{E} \left[ \left( \int_0^{t} |b_s|^p \, \mathrm{d}s \right)^q \right] $ is infinite for some $q \geq 1$?

Question 2: If $\operatorname{E} [ |A_t|^p] < \infty$ for all $p \geq 1$, is it possible that $\int_0^{t} |b_s|^p \, \mathrm{d}s$ and/or $\operatorname{E} \left[ \left( \int_0^{t} |b_s|^p \, \mathrm{d}s \right)^q \right] $ is infinite for some $q \geq 1$?

What about the questions above when $t$ is replaced by $\infty$?

Some context: I'm studying the Banach spaces $H^p$ of continuous semimartingales $S_t = M_t + A_t$ indexed by $[0,\infty)$ such that $\| S_t \|_{H^p} = \| \sup_t S_t \|_p < \infty$, which, by Burkholder-Davis-Gundy, is equivalent to $\| \langle M,M \rangle_{\infty}^{1/2} + \int_0^{\infty} \, |\mathrm{d} A_t| \|_p < \infty$. I would like to know how "wild" the finite variation kernels can be, especially in the intersection $H = \bigcap_{p \geq 1} H_p$. It seems to me, that at least for question 2 the answer is no (for both, $t$ and $\infty$), but I can't find a proof or counterexample.

I would also be thankful for references where these spaces are studied.