Absolute Value of Martingale is Submartingale

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.

Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-martingale.

Then $\sequence {\size {X_t} }_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-submartingale.

Proof

From Characterization of Integrable Functions:

$\size {X_t}$ is integrable for each $t \in \hointr 0 \infty$.

From Absolute Value Function is Convex, $x \mapsto \size x$ is a convex function.

From Martingale Composed with Convex Function is Submartingale, we have:

$\sequence {\size {X_t} }_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-submartingale.

$\blacksquare$

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): $3.3$: Continuous Time Martingales and Supermartingales