Let $X_t$ be a process which satisfies the SDE $$dX_t=(aX_t+bM_t)dt+M_tdB_t$$ where $a,b$ are constants, $B$ is a standard Brownian motion and $M_t=\sup_{s\leq t} X_s$. Since the coefficients are linear in $X$ and $M$, I believe solution to the SDE exists. My question is whether we can say something about the moments $\mathbb{E}[M_t^2]$? I believe we can have some exponential bounds by replicating the proof of the ordinary Picard's iteration.
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