Stochastic Calculus

Basics Recap

Definition

standard Gaussian density

g (x) = \frac{1}{\sqrt{2} π} e^{- \frac{x^{2}}{2}}

and the variable $Z \sim N (0, 1)$ is distributed according to this density

Gaussian processes and spaces

Definition

a d-dimensional Gaussian vector is an $R^{2}$ -valued random variable $X$ such that $⟨ X, u ⟩$ is one dimensional Gaussian variable for any $u \in R^{d}$

Proposition

The law of X is uniquely determined by the mean vector $μ = E [X] \in R^{d}$ and the covariance matrix $Σ = E [(X - μ) (X - μ)^{T}]$

Proof. Take any $ϕ \in R^{d}$ . By definition $⟨ X, θ ⟩$ is Gaussian with some mean and variance, but we can compute those from $μ$ and $Σ$ :

E [⟨ X, θ ⟩] = ⟨ E [X], θ ⟩ = ⟨ μ, θ ⟩

by linearity of expectation and

Var (⟨ X, θ ⟩) = Cov (⟨ X, θ ⟩, ⟨ X, θ ⟩) = θ^{T} Σ θ

because covariance is bi-linear. This means that we know the full distribution of $⟨ X, θ ⟩$ which tells us the characteristic function

ϕ_{x} (θ) = E [\exp (i ⟨ X, θ ⟩)] = \exp (i ⟨ μ, θ ⟩ - \frac{1}{2} θ^{T} Σ θ)

and as we saw in 18.675 knowing all of these characteristic functions is enough to determine the law

TESTCHANGES

f = \sum_{i = 1}^{n} c_{i} 1_{A_{i}} \to μ (A_{i}) \to \int_{Ω} f d μ = \sum_{i = 1}^{n} c_{i} μ (A_{i})