Second-Moment Evolution for Processes of Integral Form

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(\Omega,\mathcal{F},P)$ be a probability space, let $a,b$ be real numbers with $0\le a<b$, let $m\ge1$ be a natural number, and let $W=(W^{1},\dots,W^{m})$ be an $m$-dimensional Brownian motion. Let $y,z$ be square-integrable random variables, let $(\alpha_t)_{t\in[a,b]}$ and $(\beta_t)_{t\in[a,b]}$ be mean-square continuous families, and let $f_j,h_j:[a,b]\to\mathbb{R}$ ($1\le j\le m$) be continuous.

Convention. For a continuous function $g:[a,b]\to\mathbb{R}$, a component index $j\in\{1,\dots,m\}$, and $t\in[a,b]$, write

where $g^{0}$ denotes the extension of $g$ to $[0,b]$ by the constant value $g(a)$ on $[0,a]$, and the right-hand integrals are Wiener integrals (for $a=0$ this is the usual Wiener integral, and in every case $\int_a^ag\,dW^{j}=0$ by the convention of Ito Integrable Process and the Ito Integral). This convention applies below to $f_j$, to $h_j$, and to the function $f$ of claim 2.

Fix versions of the mean-square Riemann integrals and Wiener integrals below and set, for $t\in[a,b]$,

Then $(Y_t)$ and $(Z_t)$ are mean-square continuous families, with $Y_a=y$ and $Z_a=z$ almost surely.

1. (Second-moment evolution) Suppose the orthogonality hypothesis: for every $j\in\{1,\dots,m\}$ and all $a\le s<t\le b$, with the covariance,

Then, with the expectation, the function $t\mapsto\mathbb{E}[Y_tZ_t]$ is continuous on $[a,b]$, the integrand below is continuous, and, with the Riemann integral,

2. (Verification of the hypothesis in independent-input models) Let $l\ge1$ be a natural number and $\xi=(\xi^{1},\dots,\xi^{l})$ a tuple of square-integrable random variables with $\sigma(\xi^{1},\dots,\xi^{l})$ independent of $\sigma(W^{j'}_t:1\le j'\le m,\ t\ge0)$, with the generated $\sigma$-algebras. Fix a component index $j\in\{1,\dots,m\}$, reals $a\le s<t\le b$, and a continuous function $f:[a,b]\to\mathbb{R}$. If a square-integrable random variable $Q$ is a limit, in the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product, of finite linear combinations of the constant $1$, of the components $\xi^{i}$ ($1\le i\le l$), and of the values $W^{j'}_r$ ($1\le j'\le m$, $0\le r\le s$), then