Deterministic Ito integrals are Gaussian

Given: AC, the standing hypothesis (H), a horizon $T>0$, deterministic $h,h_1,\dots ,h_d\in L^2[0,T]$ represented by Borel functions, and real coefficients $c_1,\dots ,c_d$.

A deterministic step function is an elementary predictable integrand with deterministic coefficients, and its Ito integral is the finite sum ${\sum }_k{a}_k^m(B_{t_{k+1}^m}-B_{t_k^m})$; the Brownian increments over disjoint intervals are independent with laws $N(0,{\mathrm{\Delta }}_k)$ and mean $0$. Elementary predictable Brownian integrands Ito integral of an elementary predictable process Brownian motion

A finite linear combination of independent centered normal variables is centered normal, and a $N(0,{\sigma }^2)$ variable has characteristic function $t\mapsto e^{-{\sigma }^2{t}^2\mathrm{/}2}$, including $\sigma =0$; a Borel probability law on $\mathbb{R}$ is determined by its characteristic function. Characteristic functions under affine maps and independent sums Characteristic function of a normal law Uniqueness of a law from its characteristic function

Convergence in $L^2(P)$ implies convergence in probability, which implies weak convergence of the laws. Weak convergence tests bounded continuous real functions; applying it separately to $x\mapsto \cos (tx)$ and $x\mapsto \sin (tx)$ and then combining the two real limits gives convergence of the complex characteristic functions. Convergence in probability Convergence in probability implies convergence in distribution Weak convergence of borel probability measures

The integration map on predictable $L^2$ is linear and isometric, and the integral of $h$ is the $L^2(P)$-limit of the elementary integrals of any admissible elementary approximation; the approximation $h^m$ is admissible because $\mathrm{\mid }{h}^m-h\mathrm{\mid }$ in $L^2(\mathrm{d}t\otimes P)$ equals the deterministic $L^2[0,T]$ distance. Ito isometry and linearity in predictable L2 The general Ito integral is well defined Ito integral for square-integrable predictable processes

If $h^m\to h$ in $L^2[0,T]$, the reverse triangle inequality in the stated normed space gives $\mid \parallel h^m{\parallel }_2-\parallel h{\parallel }_2\mid \le \parallel h^m-h{\parallel }_2⟶0.$ Hence ${\int }_0^T(h^m{)}^{2}ds=\parallel h^m{\parallel }_2^2\to \parallel h{\parallel }_2^2={\int }_0^T{h}^{2}ds$. The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$

A Borel probability law on ${\mathbb{R}}^d$ is $N_d(0,\mathrm{\Sigma })$ exactly when every projection $u\cdot X$ has law $N(0,u^T\mathrm{\Sigma }u)$; such a vector has mean $0$ and covariance $\mathrm{\Sigma }$. Multivariate normal law, including singular covariance

Under AC, hence Countable Choice, finite linear combinations of box indicators are dense in $L^2(\mathbb{R})$. Extend $h$ by zero outside $[0,T]$ and approximate that extension by such a box-step function. Its restriction to $[0,T]$ agrees almost everywhere with a function ${\sum }_k{a}_k{1}_{(t_k,t_{k+1}]}$ after adjoining the finitely many box endpoints and $0,T$ to one partition. Thus deterministic step functions of the required elementary form are dense in $L^2[0,T]$. Finite linear combinations of box indicators are dense in $L^p({\mathbb{R}}^n)$ for $1\le p<\mathrm{\infty }$ AC supplies countable selections and prescribed serial paths The Axiom of Choice