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Quantitative Finance · शब्दावली

Conditional expectation क्या है?

Definition 1.2 Quantitative Methods · अध्याय 1 — Probability at Speed

Let $X$ be integrable and $\mathcal G \subseteq \mathcal F$ a sub- $\sigma$ -algebra. The conditional expectation $\E[X \mid \mathcal G]$ is the almost surely unique $\mathcal G$ -measurable, integrable random variable $Y$ with $\E[X\mathbf 1_A] = \E[Y\mathbf 1_A]$ for every $A \in \mathcal G$ . We write $\E_t[X] = \E[X \mid \mathcal F_t]$ .

Conditional expectation as averaging over what is still unknown. F_1 splits the four outcomes into two atoms (dashed); _1[X] is the average of X on the atom reached, and _0[X] is the average of _1[X]: the tower rule. The three values 0.5 1 2 along the top path form a Doob martingale. — Figure 1.1. Conditional expectation as averaging over what is still unknown. $\mathcal F_1$ splits the four outcomes into two atoms (dashed); $\E_1[X]$ is the average of $X$ on the atom reached, and $\E_0[X]$ is the average of $\E_1[X]$ : the tower rule. The three values $0.5 \to 1 \to 2$ along the top path form a Doob martingale.

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