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Let $(X_n,F_n)$ (n = 1,2,...) be a martingale. Is it true that $X_n^2$ is a submartingale?

Kerry
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    You can use the fact that x^2 is a convex function, and Jensen's inequality for conditional expectation to get that it is indeed a submartingale. – Milind Hegde Sep 27 '15 at 18:17
  • @Milind - Thanks. I thought Jenson's inequality did not have any conditional component? Could I prove this "Conditional Jenson's Inequality" easily? How would one go about doing that? – Kerry Sep 27 '15 at 18:21
  • Jensen's inequality does hold with conditional expectations in the same was as it does with standard expectations. It can be proved by applying the standard Jensen's inequality to integrals over $\mathcal G$-measurable sets, i.e. by going to the definition of conditional expectation when conditioning on $\mathcal G$. – Milind Hegde Sep 27 '15 at 18:25
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    Actually I am not sure how straightforward what I said above is. Another way to do it is to write the convex function as a supremum of linear functions and use positivity and linearity of conditional expectation. – Milind Hegde Sep 27 '15 at 18:38

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Notice that $$\mathbb E\left[X_{n+1}^2\mid\mathcal F_n\right]=\mathbb E[(X_{n+1}-X_n+X_n)^2\mid\mathcal F_n]\geqslant 2\mathbb E\left[X_n(X_{n+1}-X_n)\mid\mathcal F_n\right] +X_n^2=X_n^2,$$ where the last "$=$" follows from the fact that $(X_n,\mathcal F_n)$ is a martingale.

Davide Giraudo
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