Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Continuous Time

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 $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process.

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

Proof
Since $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process:


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

From Pointwise Scalar Multiple of Measurable Function is Measurable:


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

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

We then just need to check conditional expectations.

Let $s, t \in \hointr 0 \infty$ with $0 \le s < t$.

From Conditional Expectation is Linear, we have:


 * $\expect {X_t \mid \FF_s} \le X_s$ almost surely $\expect {-X_t \mid \FF_s} \ge -X_s$ almost surely.

That is:


 * $\sequence {X_t}_{t \ge 0}$ is a supermartingale $\sequence {-X_t}_{t \ge 0}$ is a submartingale.