Stationarity Conditions for a Lévy-Driven Continuous-Time AR Process
Summary
The document discusses a proposed characterization of weak stationarity for a continuous-time autoregressive process driven by Brownian noise. The stated conditions require the matrix governing the dynamics to have eigenvalues with negative real parts, and the initial state to have zero mean and a covariance matching the integral of the system’s decaying impulse response.
The questioner tries to motivate the covariance formula by extending Brownian motion to negative times and expressing the process as an integral from the infinite past. This is a useful route to the stationary solution when the stability condition holds. However, the document contains no accepted answer or proof, and the proposed extension alone does not establish necessity or fully address the initial-state assumptions. It is a proof question, not a complete derivation.
Key ideas
- Negative real parts of all system eigenvalues are stated as a stationarity condition.
- The stated initial-state mean is zero, with covariance determined by an integral over the decaying dynamics.
- Extending the noise to negative time motivates a stationary representation using the infinite past.
- The document poses the proof problem but does not resolve it.
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# Weak stationarity of continuous ARMA process from Brockwell
# Weak stationarity of continuous ARMA process from Brockwell
I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction. So we have a continuous AR process (CAR(p)) \begin{align*} X_t=e^{At}X_0+\int_{0}^{t}e^{A(t-u)}e dB_u\ \end{align*} \begin{align*} \Leftrightarrow X_t=e^{A(t-s)}X_s+\int_{s}^{t}e^{A(t-u)}edB_u, \ \ \text{for} \ \ t>s\geq 0 \end{align*} where $A$ is a matrix, $e$ a vector and $B_u$ a multidimensional standard Brownian motion. Statement to prove:
> Brockwell states that the necessary and sufficient conditions for stationarity of the process $X_t$ are that the eigenvalues of $A$ all have negative real parts and the distribution of $X_0$ has mean $ E (X_0)=0$ and covariance matrix $E (X_{0}X_0^T)=\int_{0}^{\infty}e^{Au} \mathbf{e}_p \mathbf{e}_p^T e^{A^Tu}du$.
My attempt: My idea is to extend the stochastic integral to the negative half-line. To extend the standard Brownian motion $B$, we define a new process $B^0$ with two independent standard Brownian motions $B^1$ and $B^2$ by \begin{align*} B^0(t)=B^1(t) 1_{\{t\in [0, \infty) \}}- B^2(-t-)1_{\{t\in (-\infty,0] \}}, \ \ t \in R. \end{align*} Now the extended process $\{X_t, \ t\in R\}$ is given by \begin{align*} X_t=e^{A(t-s)}X_s+\int_{s}^{t}e^{A(t-u)}edB^0_u, \ \ t>s, \ \text{with} \ s\in\mathbb{R}. \end{align*} Since the eigenvalues of $A$ all have negative real parts, letting $s\rightarrow -\infty$ gives us \begin{align*} X_t=\int_{-\infty}^{t}e^{A(t-u)}edB^0_u, \end{align*} which has the same distribution as \begin{align*} \int_{0}^{\infty} e^{Au}edB^0_u. \end{align*} Now with the extended process $X_t$ I am able to calculate the correct expected value and convariance.
My question: Can I extend the process to prove the statement or is that not possible? Many thanks in advanceShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.