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If M is a martingal, prove |M| is submartingale

  1. Nov 11, 2012 #1
    1. The problem statement, all variables and given/known data

    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.
  2. jcsd
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