Section 18.1

Gaussian Structure

Covariances, variances of sums, Brownian-bridge conditioning, orthant probabilities such as P(W_s > 0, W_t > 0).

Toolkit

Standard Brownian motion $(W_t)_{t\ge0}$ is a centred Gaussian process with $\operatorname{Cov}(W_s,W_t)=\min(s,t)$ ; any finite collection $(W_{t_1},\dots,W_{t_n})$ is jointly Gaussian, so all variances, covariances, conditional laws and orthant probabilities reduce to Gaussian linear algebra.

Key tools:

Independent increments: for $s<t$ , $W_t-W_s\sim\mathcal N(0,t-s)$ , independent of $\mathcal F_s$ ; write any linear combination $\sum a_iW_{t_i}$ in terms of disjoint increments to get its variance directly.
Bivariate Gaussian conditioning: if $(X,Y)$ jointly Gaussian, mean zero, then
$\mathbb{E}[X\mid Y]=\frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)}\,Y,\qquad \operatorname{Var}(X\mid Y)=\operatorname{Var}(X)-\frac{\operatorname{Cov}(X,Y)^2}{\operatorname{Var}(Y)}.$
Orthant probability: for mean-zero jointly Gaussian $(X,Y)$ with correlation $\rho$ ,
$\mathbb{P}(X>0,Y>0)=\frac14+\frac{\arcsin\rho}{2\pi}.$

Problems