Expected Value of Supermartingale is Decreasing in Time/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 $\sequence {\FF_t}_{t \ge 0}$-supermartingale.
Let $t, s \in \hointr 0 \infty$ with $0 \le s < t$.
Then, we have:
- $\expect {X_t} \le \expect {X_s}$
Proof
From the definition of a supermartingale, we have:
- $\expect {X_t \mid \FF_s} \le X_s$ almost surely.
From Expectation is Monotone, we have:
- $\expect {\expect {X_t \mid \FF_s} } \le \expect {X_s}$
From Expectation of Conditional Expectation, we have:
- $\expect {X_t} \le \expect {X_s}$
$\blacksquare$
Sources
- 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): $3.3$: Continuous Time Martingales and Supermartingales