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

1. Nov 11, 2012

### operationsres

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 $E[M_t | F_s] \geq M_s$

My question: Is my "proof" correct?

2. The attempt at a solution

Let I be the indicator function.

$E[|M_t||F_s] = E[I_{\{M_t \geq 0\}}M_t | F_s] - E[I_{\{M_t < 0\}}M_t | F_s]$

$\geq E[I_{\{M_t \geq 0\}}M_t | F_s] + E[I_{\{M_t < 0\}}M_t | F_s]$

$= E[M_t | F_s] = M_s$

Therefore $|M_t|$ is a submartingale.