Does $Y_t^-\geq a+bM_t^-$ and integrability of $M_t^-$ imply that $Y_t^-$ is integrable?

Question

Consider a martingale $(M_t)_{t \in T}$. Then $M_t$ is integrable $\forall t \in T.$ So $M_t^-$ is also integrable $\forall t \in T$.

Let $a \in \mathbb R$ and $b\geq0$.

Assume $Y_t^-\geq a+bM_t^-$

Does this imply that $Y_t^-$ is integrable or can the integral of $Y_t^-$ still be $\infty$?

Answer 1

Take $M_t=0$ and $a=0$. The hypothesis becomes $Y_t^{-} \geq 0$ which is always true. So you can get a counter-example by taking any positive random variable $Z$ with infinite expectation and taking $Y_t =-Z$.