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 discrete-time filtered probability space.

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

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

Proof

Since $\sequence {X_n}_{n \ge 0}$ is a discrete-time $\sequence {\FF_n}_{n \ge 0}$-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 discrete-time $\sequence {\FF_n}_{n \ge 0}$-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 if and only if $\expect {-X_{n + 1} \mid \FF_n} \ge -X_n$ almost surely.

That is:

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

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.3$: Martingale, supermartingale, submartingale