A note on the non-negativity of continuous-time ARMA and GARCH processes

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

[1]  George Tauchen,et al.  Simulation Methods for Lévy-Driven Continuous-Time Autoregressive Moving Average (CARMA) Stochastic Volatility Models , 2006 .

[2]  G. Boole.  treatise on differential equations , 1865 .

[3]  Peter J. Brockwell,et al.  Lévy-driven and fractionally integrated ARMA processes with continuous time parameter , 2005 .

[4]  A. R. Forsyth,et al.  A Treatise on Differential Equations , 1903, Nature.

[5]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[6]  Howell Tong,et al.  Statistics and finance: an interface , 2000 .

[7]  C. D. Perttunen,et al.  Lipschitzian optimization without the Lipschitz constant , 1993 .

[8]  George Tauchen,et al.  Simulation Methods for Levy-Driven Carma Stochastic Volatility Models , 2004 .

[9]  C. T. Kelley,et al.  A Locally-Biased form of the DIRECT Algorithm , 2001, J. Glob. Optim..

[10]  J. M. Gablonsky.  Direct version 2.0 user guide , 2001 .

[11]  Continuous-time GARCH processes , 2006, math/0607109.

[12]  F. Comte,et al.  Long memory in continuous‐time stochastic volatility models , 1998 .

[13]  P. Protter.  Stochastic integration and differential equations , 1990 .

[14]  A note on non‐negative continuous time processes , 2005 .

[15]  T. Andersen,et al.  Estimating continuous-time stochastic volatility models of the short-term interest rate , 1997 .

[16]  P. Brockwell.  HEAVY-TAILED AND NON-LINEAR CONTINUOUS-TIME ARMA MODELS FOR FINANCIAL TIME SERIES , 2000 .

[17]  D. Finkel,et al.  Convergence analysis of the direct algorithm , 2004 .

[18]  Peter J. Brockwell,et al.  Representations of continuous-time ARMA processes , 2004, Journal of Applied Probability.

[19]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[20]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[21]  C. Klüppelberg,et al.  A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour , 2004, Journal of Applied Probability.

[22]  R. Wolpert.  Lévy Processes , 2000 .

[23]  G. Roberts,et al.  Bayesian inference for non‐Gaussian Ornstein–Uhlenbeck stochastic volatility processes , 2004 .