Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale/Corollary

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 continuous-time martingale such that $\size {X_t}^2$ is integrable for each $t \in \hointr 0 \infty$.

Let $s, t \in \hointr 0 \infty$ be such that $0 \le s < t$.

Let:


 * $s = t_0 < t_1 < \ldots < t_n = t$

be a finite subdivision of $\closedint s t$.

Then:
 * $\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2} = \expect {X_t^2 - X_s^2} = \expect {\paren {X_t - X_s}^2}$

Proof
From Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale:


 * $\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2 \mid \FF_s} = \expect {X_t^2 - X_s^2 \mid \FF_s} = \expect {\paren {X_t - X_s}^2 \mid \FF_s}$ almost surely.

From Expectation of Conditional Expectation, we then obtain:


 * $\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2} = \expect {X_t^2 - X_s^2} = \expect {\paren {X_t - X_s}^2}$