Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form:
$$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$
So taking expectations gives:
$$E[X(t)^p] \leq a(t) + E\left[\int_0^t X(s)^pY(s) ds\right] = E[X(t)^p] \leq a(t) + \int_0^t E\left[X(s)^pY(s)\right] ds$$
If $X$ and $Y$ were independent, I could use Fubini's Theorem and get a bound easily using Gronwall's Inequality. However, because $X$ is described from an SDE, $X$ and $Y$ are dependent. The problem with Holder's Inequality is that it will raise the power of $X(s)^p$ inside the integral and then Gronwall's Lemma can't be used. Does anyone know a way around this? I know $X$ is a positive r.v. and $Y$ has a normal distribution, but I don't those facts help much
You can try \begin{align*} E[X(t)^p] &\leq a(t) + \int_0^t E\left[X(s)^pY(s)\right] \,ds\\ &= \int_0^t E\left[X(s)^pY(s)(1_{|Y| \leq M} +1_{|Y| \geq M})\right] \,ds\\ &\leq \int_0^t ME\left[X(s)^pY(s)1_{|Y| \leq M}\right] +E\left[X(s)^pY(s)1_{|Y| \geq M}\right] \,ds\\ &\leq \int_0^t ME\left[X(s)^p\right] + e(M) \,ds\\ &\leq \int_0^t ME\left[X(s)^p\right] \,ds + t + e(M)\\ \end{align*}
Now use gronwall $$E[X(t)^p] \leq (a(t) + e(M)t)e^{Mt} = \phi(M) $$
you can improve your inequality by considering the infimum in $M$ of $\phi(M)$