Upper Bound for Martingale at a Stopping Time

For each $omega$, if $1 le n le tau(omega)$ then we have $|M_n-1(omega)| le h$ and so $$|M_n(omega)| le |M_n-1(omega)| |M_n(omega) - M_n-1(omega)| le h h 2h quad texta.s. $$

by your third condition. When $ell 0$ the result is trivial, and otherwise we have $1 le tau wedge ell le tau$, so applying this with $ntau(omega) wedge ell$ we get $|M_tau wedge ell| le 2h$ a.s. Squaring and taking expectations gives the desired result. (You can treat $ell0$ as a special case.)Your first condition seems to be unnecessary for this.For your last question, yes, $tau wedge ell$ is a stopping time. More generally, it is a good exercise to show that if $tau, sigma$ are stopping times then so is $tau wedge sigma$. Constants are also stopping times.

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Upper Bound for Martingale at a Stopping Time 1

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