Martingale Composed with Convex Function 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.

Let $f : \R \to \R$ be a convex function such that $\map f {X_t}$ is integrable for each $t \in \hointr 0 \infty$.

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

Proof

Since $\sequence {X_t}_{t \ge 0}$ is a martingale, we have:

$X_t$ is $\FF_t$-measurable

for each $t \in \hointr 0 \infty$.

From Convex Real Function is Measurable and Composition of Measurable Mappings is Measurable:

$\map f {X_t}$ is $\FF_t$-measurable

for each $t \in \hointr 0 \infty$.

So $\sequence {\map f {X_t} }_{t \ge 0}$ is $\sequence {\FF_t}_{t \ge 0}$-adapted.

Now let $s, t \in \hointr 0 \infty$ be such that $0 \le s < t$.

Let $\expect {\map f {X_t} \mid \FF_s}$ be a version of the conditional expectation of $\map f {X_t}$ given $\FF_s$.

We then have:

\(\ds \expect {\map f {X_t} \mid \FF_s}\)

\(\ge\)

\(\ds \map f {\expect {X_t \mid \FF_s} }\)

almost surely, by Conditional Jensen's Inequality

\(\ds \)

\(=\)

\(\ds \map f {X_s}\)

almost surely, Definition of Martingale

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

$\blacksquare$

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): Proposition $3.12$