I have been looking at a proof on page 299 of the following paper:
- Promel, D. J., and Scheffels, D. Stochastic volterra equations with Hölder diffusion coefficients. Stochastic Processes and their Applications, 3 (2023), 291–315.
While I think I understand most of the proof, there is an element which I do not quite grasp. I have tried to capture it in the following setting. Hopefully this is clear enough.
Le $T>0$ and $p>2$. Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},\mathbb{P})$ be some filtered probability space. Let further $X = (X_t)_{t\in [0,T]}$ and $Y = (Y_t)_{t\in [0,T]}$ be some $(\mathcal{F}_t)$-progressively measurable stochastic processes. Suppose further that $X$ is in $L^p(\Omega \times [0,T])$ and let $f:[0,T] \to \mathbb{R}$ be some continuous function.
Suppose that, for some $C>0$, we have that
$$\mathbb{E}\big[\lvert Y_t\rvert ^p\big] \leq C \bigg(1 + \lvert f(t) \rvert + \int_0^t\mathbb{E}\big[\lvert X_s\rvert ^p\big]ds, \bigg),$$
holds for all $t\in[0,T]$.
Then
- Is the map $t\to \mathbb{E}[\lvert Y_t\rvert ^p]$ bounded?
I would argue that yes, it is. Indeed for $t\in[0,T]$, we have \begin{equation} \begin{aligned} \mathbb{E}\big[\lvert Y_t\rvert ^p\big] & \leq C \bigg(1 + \lvert f(t) \rvert+ \int_0^t\mathbb{E}\big[\lvert X_s\rvert ^p\big]ds, \bigg) \\ & \leq C \bigg(1 + \sup\lvert f(t) \rvert + \int_0^t\mathbb{E}\big[\lvert X_s\rvert ^p\big]ds, \bigg) \\ & \leq C \bigg(1 + \sup\lvert f(t) \rvert + \int_0^T\mathbb{E}\big[\lvert X_s\rvert ^p\big]ds, \bigg). \end{aligned} \end{equation} Since $X$ is in $L^p$, then there is some $M>0$, such that $$\mathbb{E}\big[\lvert Y_t\rvert ^p\big]<M, \quad \hbox{for all $t\in[0,T]$}.$$
- Is $Y$ in $L^p(\Omega \times [0,T])$ ?
I would argue that, by integrating on $[0,T]$ we obtain that $Y$ is in $L^p$.
Is my development correct?
In the paper the authors have that $X = Y$ and wish to use Grönwall's lemma to obtain some bound on $X$. However, they use a sequence of stopping times seemingly to ensure that $t\to \mathbb{E}[\lvert Y_t\rvert ^p]$ is bounded. If my development above is correct, then using the stopping times seems unnecessary.
Some help would be much appreciated.
Thanks in advance.