Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 6

Approximate $h$ in $L^2[0,t]$ by deterministic step functions $h_n$. Each integral $\int h_{n}dB$ is a linear combination of independent Gaussian increments and hence is Gaussian, with mean zero and variance $\int h_n^2$. The Itô isometry gives convergence in $L^2$ to $\int hdB$, so characteristic functions pass to the limit. Thus

The Itô product rule for $e^{\lambda t}{X}_t$ givesTherefore the Ornstein-Uhlenbeck process has the explicit form

Part a applied to the deterministic kernel gives

Every linear combination of $X_{t_1},\dots ,X_{t_n}$ is a deterministic constant plus one stochastic integral of a deterministic $L^2$ function against $B$. Part a makes every such combination Gaussian. By the linear-combination characterization of a multivariate normal distribution, $(X_{t_1},\dots ,X_{t_n})$ is jointly Gaussian, so $X$ is a Gaussian process.

For $0<s<t$, only the Brownian noise accumulated through time $s$ is shared. The Itô isometry for cross terms gives

If $X_0\sim N(0,(2\lambda {)}^{-1})$ independently of $B$, thenThus $X_t\sim N(0,(2\lambda {)}^{-1})$ for every $t$. For $0<s<t$, the Markov decompositionhas an increment independent of $X_s$, and henceThis is the stationary Ornstein-Uhlenbeck covariance.