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

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

Then $\sequence {X_n}_{n \ge 0}$ is a supermartingale $\sequence {-X_n}_{n \ge 0}$ is a submartingale.

Proof
Since $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process:


 * $X_n$ is $\FF_n$-measurable for each $n \in \N$.

From Pointwise Scalar Multiple of Measurable Function is Measurable:


 * $-X_n$ is $\FF_n$-measurable for each $n \in \N$.

So $\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

We then just need to check conditional expectations.

From Conditional Expectation is Linear, we have:


 * $\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely $\expect {-X_{n + 1} \mid \FF_n} \ge -X_n$ almost surely.

That is:


 * $\sequence {X_n}_{n \ge 0}$ is a supermartingale $\sequence {-X_n}_{n \ge 0}$ is a submartingale.