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operationsres
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Homework Statement
If M is a martingale, prove that |M| is a submartingale.
Let F be the filtration.
The definition of a submartingale is that [itex]E[M_t | F_s] \geq M_s[/itex]
My question: Is my "proof" correct?
2. The attempt at a solution
Let I be the indicator function.
[itex]E[|M_t||F_s] = E[I_{\{M_t \geq 0\}}M_t | F_s] - E[I_{\{M_t < 0\}}M_t | F_s][/itex]
[itex] \geq E[I_{\{M_t \geq 0\}}M_t | F_s] + E[I_{\{M_t < 0\}}M_t | F_s][/itex]
[itex] = E[M_t | F_s] = M_s[/itex]
Therefore [itex]|M_t|[/itex] is a submartingale.