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

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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 if and only if $\sequence {-X_t}_{t \ge 0}$ 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 if and only if $\expect {-X_t \mid \FF_s} \ge -X_s$ almost surely.

That is:

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

$\blacksquare$


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