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For an integrable random variable $Y$, the conditional expectation satisfies $|\mathbb{E}[Y\mid \mathcal{G}]|\le \mathbb{E}[|Y|\mid \mathcal{G}]$ almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to $Y_n-Y$ shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.

A continuous martingale $M$ is L2-bounded when ${\sup }_{t\ge 0}\mathbb{E}|M_t{|}^2<\infty$. The L2 martingale convergence theorem gives a terminal value $M_{\infty }$ with convergence in $L^2$ and almost sure convergence. The resulting conditional expectation identity is $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. Boundedness on each separate finite horizon is a weaker condition. The subspace with $M_0=0$ is commonly denoted ${\mathcal{M}}_0^2$.

For square-integrable random variables, expand $X$ and $Y$ into their conditional expectations and their centered conditional residuals. The cross terms vanish by the law of total expectation, giving the displayed covariance decomposition. Conditional independence can therefore coexist with positive unconditional covariance from a shared latent variable.

A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.

In the model $d{r}_t=r_t(r_t-1)(dt+d{W}_t)$ under the pricing measure, with $0\le r_t\le 1$, the discounted affine expression $D_t[A(T-t)+B(T-t)r_t]$ has zero drift when $A^{\prime }=0$ and $B^{\prime }=-A-B$. Terminal conditions $A(0)=1$, $B(0)=0$ yield the displayed unit bond price. The expression is bounded, so the bounded local martingale criterion justifies pricing by conditional expectation.

For a random objective with an easily computable conditional expectation, reveal one random choice at a time and choose an outcome whose conditional expected objective is at least the current value. Such an outcome exists because the current value is an average of the child values. When all choices are fixed, the actual objective is at least the initial expectation. Biased as well as fair choices work. Efficient expectation updates make the resulting deterministic procedure a polynomial-time algorithm.