# A SDE depending on its running maximum

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.