By Mathieu Kessler, Alexander Lindner, Michael Sorensen
The 7th quantity within the SemStat sequence, Statistical equipment for Stochastic Differential Equations provides present study traits and up to date advancements in statistical tools for stochastic differential equations. Written to be available to either new scholars and pro researchers, each one self-contained bankruptcy begins with introductions to the subject to hand and builds steadily in the direction of discussing contemporary learn.
The ebook covers Wiener-driven equations in addition to stochastic differential equations with jumps, together with continuous-time ARMA tactics and COGARCH tactics. It offers a spectrum of estimation tools, together with nonparametric estimation in addition to parametric estimation in line with probability equipment, estimating services, and simulation ideas. chapters are dedicated to high-frequency information. Multivariate versions also are thought of, together with partly saw platforms, asynchronous sampling, assessments for simultaneous jumps, and multiscale diffusions.
Statistical tools for Stochastic Differential Equations turns out to be useful to the theoretical statistician and the probabilist who works in or intends to paintings within the box, in addition to to the utilized statistician or monetary econometrician who wishes the the right way to examine organic or monetary time sequence.
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Additional info for Statistical Methods for Stochastic Differential Equations (Chapman & Hall/CRC Monographs on Statistics & Applied Probability)
58, P (θˆn = θˆn ) → 0 as n → ∞. Hence θˆn is eventually the unique Gn –estimator on K. 3 Martingale estimating functions In this section we consider observations X0 , Xt1 , . . 11) where σ is a d × d-matrix and W a d-dimensional standard Wiener process. Here and elsewhere in this chapter, the observation times ti are non-random. We denote the state space of X by D. When d = 1, the state space is an interval ( , r), where could possibly be −∞, and r might be ∞. The drift b and the diffusion matrix σ depend on a parameter θ which varies in a subset Θ of IRp .
To solve the estimating equation G◦n (α, β) = 0 we introduce the weights n wiκ = κ(X(i−1)∆ )−2 / κ(X(j−1)∆ )−2 , j=1 n i=1 n κ ¯κ = and define X = and X −1 i=1 wi X(i−1)∆ . These two quantities are conditional precision weighted sample averages of Xi∆ and X(i−1)∆ , respectively. The equation G◦n (α, β) = 0 has a unique explicit solution provided that the weighted sample autocorrelation n ¯ κ )(X(i−1)∆ − X ¯κ ) wκ (Xi∆ − X −1 rnκ = i=1 in κ κ 2 ¯ ) w (X(i−1)∆ − X ¯κ wiκ Xi∆ i=1 i −1 is positive.
Xr ) for all (x1 , . . , xr , θ) ∈ Dr × Uθ . 5) ¯ hold. 8) ¯ If, moreover, the function g(x1 , . . , xr ; θ) is lounder P , where V = V (θ). 9) then the estimator θˆn is the unique Gn –estimator on any bounded subset of Θ containing θ¯ with probability approaching one as n → ∞. P ¯ Remark: By a θ-consistent estimator is meant that θˆn −→ θ¯ as n → ∞. e. 2 holds with θ¯ = θ0 . Note ¯ that because θ¯ ∈ int Θ, a θ-consistent Gn –estimator θˆn will satisfy Gn (θˆn ) = 0 with probability approaching one as n → ∞.
Statistical Methods for Stochastic Differential Equations (Chapman & Hall/CRC Monographs on Statistics & Applied Probability) by Mathieu Kessler, Alexander Lindner, Michael Sorensen