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.