[Paper Review] Continuous-time models with an autoregressive structure

Andreas Basse-O’Connor, Mikkel Slot Nielsen+2 more|arXiv (Cornell University)|Apr 27, 2017

Stochastic processes and financial applicationsEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper introduces two continuous-time models with an autoregressive structure: a stochastic delay differential equation (SDDE) and a level model based on moving averages. It establishes existence and uniqueness of stationary solutions, links them to CARMA and discrete-time ARMA processes, and shows that long-range dependence arises through the noise process rather than the kernel, with invertible CARMA processes emerging as solutions to the SDDE framework.

ABSTRACT

In this paper we suggest two continuous-time models which exhibit an autoregressive structure. We obtain existence and uniqueness results and study the structure of the solution processes. One of the models, which corresponds to general stochastic delay differential equations, will be given particular attention. We use the obtained results to link the introduced processes to both discrete-time and continuous-time ARMA processes.

Motivation & Objective

• To develop continuous-time autoregressive models with a rigorous mathematical foundation.
• To establish existence and uniqueness of stationary solutions for stochastic delay differential equations (SDDEs) with general noise processes.
• To connect the proposed models to existing classes such as CARMA and discrete-time ARMA processes.
• To clarify the role of the noise process in generating long-range dependence in continuous-time moving averages.
• To introduce and analyze a novel level model formulation that directly specifies the process rather than its increments.

Proposed method

• Formulates a continuous-time autoregressive model via a stochastic delay differential equation (SDDE): $ dX_t = \left(\int_{[0,\infty)} X_{t-u} \eta(du)\right) dt + dZ_t $, where $ Z_t $ has stationary increments.
• Proposes a level model: $ X_t = \int_0^\infty X_{t-u} \phi(du) + \int_{-\infty}^t \theta(t-u) dL_u $, which defines the process directly through its past values and a Lévy-driven moving average.
• Uses the theory of moving averages and stochastic Fubini theorems to derive the solution in terms of convolution with a kernel $ x_0 $, ensuring integrability and stationarity.
• Applies Fourier transforms and $ L^2 $-convergence to show that the solution process is a moving average of the Lévy process $ L_t $, with kernel $ \psi = \theta * x_0 $.
• Establishes that invertible CARMA processes arise as solutions to the SDDE under appropriate conditions on the kernel and noise.
• Employs $ L^1 $ and $ L^2 $ convergence arguments to prove almost sure and mean-square convergence of approximating sequences, ensuring well-defined solutions.

Experimental results

Research questions

• RQ1 Can a continuous-time autoregressive process be defined via a stochastic delay differential equation with general delay measures and stationary increment noise?
• RQ2 How do the proposed models relate to classical CARMA and discrete-time ARMA processes?
• RQ3 What conditions ensure the existence and uniqueness of a stationary solution to the SDDE?
• RQ4 Can long-range dependence in continuous-time moving averages be generated through the noise process rather than the kernel?
• RQ5 Is it possible to represent the solution of the level model as a moving average of a Lévy process, and under what conditions?

Key findings

• The solution to the SDDE is a stationary continuous-time moving average process if the noise process $ Z_t $ is of the form $ \int_{\mathbb{R}} [\theta(t-u) - \theta_0(-u)] dL_u $, with $ \theta_0 $ chosen such that the kernel integrates to zero.
• Long-range dependence in the resulting process arises exclusively from the noise process $ Z_t $, not from the delay measure $ \eta $, which must be compactly supported for stationarity.
• Invertible CARMA processes are shown to be solutions to the SDDE framework, with the kernel $ \psi $ in the moving average representation given by $ \psi = \theta * x_0 $, where $ x_0 $ solves the Volterra equation associated with the delay.
• The level model formulation ensures that the solution is a stationary moving average process when the kernel $ \psi $ is the $ L^2 $ limit of the convolution series $ \sum_{k=0}^n \theta * \phi^{*k} $.
• The solution to the level model converges in $ L^2 $ to $ X_t = \int_{\mathbb{R}} \psi(t-u) dL_u $, where $ \psi $ is the $ L^2 $ limit of the series $ \sum_{k=0}^\infty \theta * \phi^{*k} $, proving the process is a moving average.
• The paper proves that the solution process $ X_t $ is stationary almost surely, by showing convergence of approximating Riemann sums in $ L^1 $ and almost sure convergence of the limit.