Stationary Dilations

1Stationary Dilations

Definition 1. Let $(\Omega, \mathcal{F}, P)$ and $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ be probability spaces. We say that $(\Omega, \mathcal{F}, P)$ is a factor of $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ if there exists a measurable surjective map $\phi : \tilde{\Omega} \to \Omega$ such that:

1. For all $A \in \mathcal{F}$, $\phi^{- 1} (A) \in \tilde{\mathcal{F}}$

2. For all $A \in \mathcal{F}$, $P (A) = \tilde{P} (\phi^{- 1} (A))$

In other words, $(\Omega, \mathcal{F}, P)$ can be obtained from $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ by projecting the larger space onto the smaller one while preserving the probability measure structure.

Remark 2. In the context of stationary dilations, this means that the original nonstationary process $\{X_t \}$ can be recovered from the stationary dilation $\{Y_t \}$ through a measurable projection that preserves the probabilistic structure of the original process.

Definition 3. (Stationary Dilation) Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\{X_t \}_{t \in \mathbb{R}_+}$ be a nonstationary stochastic process. A stationary dilation of $\{X_t \}$ is a stationary process $\{Y_t \}_{t \in \mathbb{R}_+}$ defined on a larger probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ such that:

1. $(\Omega, \mathcal{F}, P)$ is a factor of $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$

2. There exists a measurable projection operator $\Pi$ such that:

   $\displaystyle X_t = \Pi Y_t \quad \forall t \in \mathbb{R}_+$

Theorem 4. (Representation of Nonstationary Processes) For a continuous-time nonstationary process $\{X_t \}_{t \in \mathbb{R}_+}$, its stationary dilation exists which has sample paths $t \mapsto X_t (\omega)$ which are continuous with probability one when $X_t$:

• is uniformly continuous in probability over compact intervals:

  $\displaystyle \lim_{s \to t} P (|X_s - X_t | > \epsilon) = 0 \quad \forall \epsilon > 0, t \in [0, T], T > 0$

• has finite second moments:

  $\displaystyle \mathbb{E} [|X_t |^2] < \infty \quad \forall t \in \mathbb{R}_+$

• has an integral representation of the form:

  $\displaystyle X_t = \int_0^t \eta (s) ds$

  where $\eta (t)$ is a measurable random function that is stationary in the wide sense (with $\int_0^t \mathbb{E} [| \eta (s) |^2] \hspace{0.17em} ds < \infty$ for all $t$)

• and has a covariance operator

  $\displaystyle R (t, s) =\mathbb{E} [X_t X_s]$

  which is symmetric $(R (t, s) = R (s, t))$, positive definite and continuous

Under these conditions, there exists a representation:

$\displaystyle X_t = M (t) \cdot S_t$

where:

• $M (t)$ is a continuous deterministic modulation function

• $\{S_t \}_{t \in \mathbb{R}_+}$ is a stationary process

This representation can be obtained through the stationary dilation by choosing:

$\displaystyle Y_t = \left( \begin{array}{c} M (t)\\ S_t \end{array} \right)$

with the projection operator $\Pi$ defined as:

$\displaystyle \Pi Y_t = M (t) \cdot S_t$

Proposition 5. (Properties of Dilation) The stationary dilation satisfies:

1. Preservation of moments:

   $\displaystyle \mathbb{E} [|X_t |^p] \leq \mathbb{E} [|Y_t |^p] \quad \forall p \geq 1$

2. Minimal extension: Among all stationary processes that dilate $X_t$, there exists a minimal one (unique up to isomorphism) in terms of the probability space dimension

Corollary 6. For any nonstationary process satisfying the above conditions, the stationary dilation provides a canonical factorization into deterministic time-varying components and stationary stochastic components.