Given that ${X_n}$ is a submartingale with $\left|X_{k}-X_{k-1}\right|\leq M<\infty$, and defining stopping times $\tau_{1}<\tau_{2}$ with $E(\tau_{2})<\infty$, eventually I want to show that $E(X_{\tau_{2}}) \geq E(X_{\tau_{1}})$. In the proof, in order to take the expectation into the infinite sum I need to check that $\sum_{k=1}^{\infty}E(\left|X_{k}-X_{k-1}\right|\mathbf{1}_{\tau_{1}
Unbounded stopping times and optional stopping theorem
2
$\begingroup$
probability
sequences-and-series
martingales
1 Answers
1
I don't understand where is you problem (your post is not very clear). If you want to prove that $E[\tau_2]\geq E[\tau_1]$, this follows by monotonicity of the Lebesgue integral.
In the summation you can try: $E\sum_{k=1}^{\infty}|X_k-X_{k-1}|1_{\tau_1
$\leq E M(\tau_2-\tau_1) < + \infty.$
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0Sorry, I had made a mistake in the statement of the problem; it is corrected now. Thanks for your help on the summation. – 2012-03-26