Generalised Ornstein-Uhlenbeck processes

Markov/AR(1)-like processes

January 10, 2022 — September 21, 2022

dynamical systems
Hilbert space
Lévy processes
signal processing
stochastic processes
time series

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Ornstein-Uhlenbeck-type autoregressive, stationary stochastic processes, e.g. stationary gamma processes, classic Gaussian noise Ornstein-Uhlenbeck processes… There is a family of such induced by every Lévy process via its bridge.

Figure 1

1 Classic Gaussian

1.1 Discrete time

Given a \(K \times K\) real matrix \(\Phi\) with all the eignevalues of \(\Phi\) in to the interval \((-1,1)\), and given a sequence \(\varepsilon_t\) of multivariate normal variables \(\varepsilon_t \sim \mathrm{N}(0, \Sigma)\), with \(\boldsymbol{\Sigma}\) a \(K \times K\) positive definite symmetric real matrix, the stationary distribution of the process \[ \mathbf{x}_t=\varepsilon_t+\boldsymbol{\Phi} \mathbf{x}_{t-1}=\sum_{h=0}^t \boldsymbol{\Phi}^h \varepsilon_{t-h} \quad ? \] is given by the Lyapunov equation, or just by basic variance identities. It is Gaussian with \(\mathcal{N}(0, \Lambda)\) where the following recurrence relation holds for \(\Lambda\), \[ \Lambda=\Phi \mathbf{x} \Phi^{\top}+\Sigma. \] The solution is also, apparently, the limit of a summation \[ \Lambda=\sum_{k=0}^{\infty} \Phi^k \Sigma\left(\Phi^{\top}\right)^k. \]

1.2 Continuous time

Suppose we use a Wiener process \(W\) as the driving noise in continuous time with some small increment \(\epsilon\), \[ d \mathbf{x}(t)=-\epsilon A \mathbf{x}(t) d t+ \epsilon B d W(t) \] This is the Ornstein-Uhlenbeck process. If stable, at stationarity it an analytic stationary density \(\mathbf{x}\sim\mathcal{N}(0, \Lambda)\) where \[ \Lambda A+A \Lambda =\epsilon B B^{\top}. \]

2 Gamma

Over at Gamma processes, Wolpert (2021) notes several example constructions which “look like” Ornstein-Uhlenbeck processes, in that they are stationary-autoregressive, but constructed by different means. Should we look at processes like those here?

For fixed \(\alpha, \beta>0\) these notes present six different stationary time series, each with Gamma \(X_{t} \sim \operatorname{Ga}(\alpha, \beta)\) univariate marginal distributions and autocorrelation function \(\rho^{|s-t|}\) for \(X_{s}, X_{t} .\) Each will be defined on some time index set \(\mathcal{T}\), either \(\mathcal{T}=\mathbb{Z}\) or \(\mathcal{T}=\mathbb{R}\)

Five of the six constructions can be applied to other Infinitely Divisible (ID) distributions as well, both continuous ones (normal, \(\alpha\)-stable, etc.) and discrete (Poisson, negative binomial, etc). For specifically the Poisson and Gaussian distributions, all but one of them (the Markov change-point construction) coincide— essentially, there is just one “AR(1)-like” Gaussian process (namely, the \(\operatorname{AR}(1)\) process in discrete time, or the Ornstein-Uhlenbeck process in continuous time), and there is just one \(\operatorname{AR}(1)\)-like Poisson process. For other ID distributions, however, and in particular for the Gamma, each of these constructions yields a process with the same univariate marginal distributions and the same autocorrelation but with different joint distributions at three or more times.

3 References

Ahn, Korattikara, and Welling. 2012. Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring.” In Proceedings of the 29th International Coference on International Conference on Machine Learning. ICML’12.
Alexos, Boyd, and Mandt. 2022. Structured Stochastic Gradient MCMC.” In Proceedings of the 39th International Conference on Machine Learning.
Chen, Tianqi, Fox, and Guestrin. 2014. Stochastic Gradient Hamiltonian Monte Carlo.” In Proceedings of the 31st International Conference on Machine Learning.
Chen, Zaiwei, Mou, and Maguluri. 2021. Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization.”
Mandt, Hoffman, and Blei. 2017. Stochastic Gradient Descent as Approximate Bayesian Inference.” JMLR.
Simoncini. 2016. Computational Methods for Linear Matrix Equations.” SIAM Review.
Wolpert. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math].