Conditional Expectations of Integrable Random Variable with respect to Filtration forms Martingale/Continuous Time

Theorem

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

Let $Z$ be an integrable random variable.

For each $t \in \hointr 0 \infty$, let $\expect {Z \mid \FF_t}$ be a version of the conditional expectation of $Z$ given $\FF_t$.

For each $t \in \hointr 0 \infty$ set:

$X_t = \expect {Z \mid \FF_t}$

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

Proof

From the definition of the conditional expectation of $Z$ given $\FF_t$, we have that:

$X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.

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

Now let $s, t \in \hointr 0 \infty$ have $s \le t$.

Then we have:

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

\(=\)

\(\ds \expect {\expect {Z \mid \FF_t} \mid \FF_s}\)

\(\ds \)

\(=\)

\(\ds \expect {Z \mid \FF_s}\)

applying Tower Property of Conditional Expectation, since $\FF_s \subseteq \FF_t$

\(\ds \)

\(=\)

\(\ds X_s\)

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

$\blacksquare$

Sources

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