Adapted Stochastic Process is Supermartingale iff Negative is Submartingale
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Theorem
Discrete Time
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.
Continuous Time
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 if and only if $\sequence {-X_t}_{t \ge 0}$ is a submartingale.