Asymptotic properties of linear multivariable continuous-time tracking systems incorporating high-gain error-actuated controllers

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

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Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

Advances in Applied Probability ◽

10.1017/s0001867800040301 ◽

1975 ◽

Vol 7 (01) ◽

pp. 66-82 ◽

Cited By ~ 9

Author(s):

N. H. Bingham ◽

R. A. Doney

Keyword(s):

State Space ◽

Continuous Time ◽

Regular Variation ◽

Branching Processes ◽

Asymptotic Properties ◽

Random Variables ◽

The Other ◽

Continuous State Space ◽

Continuous State ◽

Integer Exponent

We obtain results connecting the distribution of the random variablesYandWin the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1,EYβandEWβconverge or diverge together and regular variation of the tail of one ofY, Wwith non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

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Optimal estimation for semimartingales

Journal of Applied Probability ◽

10.1017/s0021900200029703 ◽

1986 ◽

Vol 23 (02) ◽

pp. 409-417 ◽

Cited By ~ 1

Author(s):

A. Thavaneswaran ◽

M. E. Thompson

Keyword(s):

Continuous Time ◽

Asymptotic Properties ◽

Optimal Estimation ◽

Parametric Estimation ◽

Local Martingale ◽

Regularity Conditions ◽

Probability Measures ◽

Simple Process ◽

Special Cases ◽

Least Squares Estimates

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. LetP={P} be a family of probability measures such that (Ω,F, P) is complete, (Ft, t≧0) is a standard filtration, andX = (XtFt, t ≧ 0)is a semimartingale for everyP ∈ P. For a parameterθ(Ρ), supposeXt=Vt,θ+Ht,θwhere theVθprocess is predictable and locally of bounded variation and theHθprocess is a local martingale. Consider estimating equations forθof the formprocess is predictable. Under regularity conditions, an optimal form forαθin the sense of Godambe (1960) is determined. WhenVt,θis linear inθthe optimal, corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of, are discussed.

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