All Questions

Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$. Consider also its KL ...

Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying $$ \mathbb{E}\left[ \int_0^{\...

In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...

I feel confused with a particular step in the $L^2$ consturction of Brownian motion. Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...

Preliminaries Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$. These random functions are drawn from a Gaussian measure $\...

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...