I have been studying stochastic differential equations (SDE) and came across the following questions.
Let $T \in (0,\infty)$. Let also $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be any filtered probability space with a complete filtration and $B = (B_t)_{t\in [0,T]}$ be any $(\mathbb{F},\mathbb{P})$-Brownian motion. Let some $p\geq 1$. Suppose that $X = (X_t)_{t\in [0,T]}$ is an $\mathbb{F}$-progressively measurable process in $L^p := L^p(\Omega\times[0,T],\mathbb{P}\otimes\lambda)$ that solves the SDE
$$dX_t = \mu(t,X_t)dt + \sigma(t,X_t)dB_s.$$
Then the SDE admits a strong solution in $L^p$ and $X$ is called a strong solution in $L^p$. If for any two strong solutions $X$ and $Y$ in $L^p$, $X$ and $Y$ are indistinguishable, then the SDE admits pathwise uniqueness in $L^p$. If the SDE admits a strong solution and pathwise uniqueness in $L^p$, the SDE admits a unique strong solution in $L^p$.
My questions are:
If the SDE admits a strong solution in $L^q$ for $q>2$, does it imply that the SDE admits a strong solution in $L^2(\Omega\times[0,T],\mathbb{P}\otimes\lambda)$? I would argue that it is the case by the finiteness of the measure $\mathbb{P}\otimes\lambda$ on $\Omega\times[0,T]$ and the embedding property of $L^p$ spaces.
Suppose that $X$ is pathwise unique in $L^p$, i.e. unique up to indistinguishability in $L^p$. Does that imply that pathwise uniqueness holds in $L^q$ for $q>p$? More precisely, if there exists a strong solution $Y$ in $L^q$, then it is indistinguishable with $X$. Again I would argue that it is the case by the same argument as in 1).
Suppose the SDE admits a strong solution $X$ in $L^p$ for some $p\geq 1$ and that for each $q \in [1,\infty)$, $\sup_{t\in[0,T]}E[|{X}|^q]< \infty $. Suppose further that the SDE admits pathwise uniqueness in $L^2$ can I conclude that for each $p\geq 2$ the SDE admits a strong, pathwise unique solution in $L^p$?
Some help would be very much appreciated! Thanks in advance