Conditional Expectation and Martingales

The measure-theoretic background (σ-algebras, measures, random variables, Lebesgue integration, convergence theorems, modes of convergence) is in the probability theory notes.

Filtrations and Adapted Processes

A filtration is $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots$ (or $\mathcal{F}_s \subseteq \mathcal{F}_t$ for $s \le t$). A process $X_t$ is adapted if each $X_t$ is $\mathcal{F}_t$-measurable.

For $\mathcal{G} \subseteq \mathcal{F}$, $X$ is $\mathcal{G}$-measurable if $\{X \in B\} = X^{-1}(B) \in \mathcal{G}$ for all Borel $B$. The σ-algebra generated by $X$, $\sigma(X) = \{X^{-1}(B) : B \in \mathcal{B}\}$, is the smallest σ-algebra for which $X$ is measurable.

Conditional Expectation

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $\mathcal{G} \subseteq \mathcal{F}$, and $X$ non-negative or integrable. $E[X|\mathcal{G}]$ is any random variable such that

1. $E[X|\mathcal{G}]$ is $\mathcal{G}$-measurable,
2. $\int_A E[X|\mathcal{G}]\, d\mathbb{P} = \int_A X\, d\mathbb{P}$ for all $A \in \mathcal{G}$.

Properties For $X, Y$ integrable (or non-negative) and $\mathcal{H} \subseteq \mathcal{G}$:

• Linearity $E[c_1 X + c_2 Y | \mathcal{G}] = c_1 E[X|\mathcal{G}] + c_2 E[Y|\mathcal{G}]$: the right-hand side is $\mathcal{G}$-measurable, and its integral over any $A \in \mathcal{G}$ is $c_1\int_A X\, d\mathbb{P} + c_2\int_A Y\, d\mathbb{P} = \int_A (c_1 X + c_2 Y)\, d\mathbb{P}$.
• Taking out what is known If $X$ is $\mathcal{G}$-measurable, $E[XY|\mathcal{G}] = X\, E[Y|\mathcal{G}]$. For $X = \mathbf{1}_B$, $B \in \mathcal{G}$, $\int_A \mathbf{1}_B E[Y|\mathcal{G}]\, d\mathbb{P} = \int_{A \cap B} E[Y|\mathcal{G}]\, d\mathbb{P} = \int_{A \cap B} Y\, d\mathbb{P}$.
• Tower $E\big[E[X|\mathcal{G}] \,\big|\, \mathcal{H}\big] = E[X|\mathcal{H}]$, since for $A \in \mathcal{H} \subseteq \mathcal{G}$ both sides integrate to $\int_A X\, d\mathbb{P}$.
• Independence If $X$ is independent of $\mathcal{G}$, $E[X|\mathcal{G}] = E[X]$, since for $A \in \mathcal{G}$, $\int_A E[X]\, d\mathbb{P} = E[X]\,\mathbb{P}(A) = E[X\mathbf{1}_A] = \int_A X\, d\mathbb{P}$.

Independence lemma Let $X_1, \ldots, X_k$ be $\mathcal{G}$-measurable and $Y_1, \ldots, Y_L$ independent of $\mathcal{G}$. For a function $f(x_1, \ldots, x_k, y_1, \ldots, y_L)$ define

$$g(x_1, \ldots, x_k) = E\big[f(x_1, \ldots, x_k, Y_1, \ldots, Y_L)\big]$$

Then

$$E\big[f(X_1, \ldots, X_k, Y_1, \ldots, Y_L) \,\big|\, \mathcal{G}\big] = g(X_1, \ldots, X_k)$$

Change of Measure

Let $Z \ge 0$ a.s. with $E[Z] = 1$, and define

$$\tilde{\mathbb{P}}(A) = \int_A Z\, d\mathbb{P}, \qquad A \in \mathcal{F}$$

Then $\tilde{E}[X] = E[XZ]$ for $X \ge 0$, and if $Z > 0$ a.s., $E[Y] = \tilde{E}[Y/Z]$ for $Y \ge 0$.

Martingales and Markov Processes

Let $(\mathcal{F}_t)_{0 \le t \le T}$ be a filtration and $M_t$ an adapted, integrable process. $M_t$ is a martingale if

$$E[M_t | \mathcal{F}_s] = M_s \quad \text{for all } 0 \le s \le t \le T$$

An adapted process $X_t$ is Markov if for all $0 \le s \le t \le T$ and every non-negative Borel $f$ there is a Borel $g$ with

$$E[f(X_t) | \mathcal{F}_s] = g(X_s)$$

The binomial model and continuous-time pricing are in risk-neutral pricing.