Nonparametric estimation of scalar diffusions based on low frequency data
By Emmanuel Gobet, Marc Hoffmann, Markus Reiß. 2002
135 citations recorded by the source corpus.
We study the problem of estimating the coefficients of a diffusion (X t , t ≥ 0); the estimation is based on discrete data X n Δ, n = 0, 1,..., N. The sampling frequency Δ -1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X nΔ , n = 0, 1,..., N) in a suitable Sobolev norm, together with an estimation of its invariant density.
Subfields: Finance, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/0b2c45c9d7cfedd1a19e6f16e26977f4a4dc703a
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Finance
- Fabienne Comte
- Valentine Genon-Catalot
- Yves Rozenholc
- Richard Nickl
- Jakob Sohl
- Christian M. Hafner
- Evarist Gin'e
- Helmut Herwartz
In Mathematical Physics
In Statistics and Probability
- Fabienne Comte
- Jakob Sohl
- Richard Nickl
- Valentine Genon-Catalot
- Yves Rozenholc
- Kolyan Ray
- I. Castillo
- Evarist Gin'e
Citations in the corpus
- Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, methodology): “This model is considered in [14], and under the above conditions there exists a weak solution (Xt : t > 0) of the SDE, see [26].”
- Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions (2014, unknown): “√ K with K = 2J (see, e.g., p. 2240 of Gobet, Hoffmann, and Reiss (2004) or p. 2377 of Kato (2013)).”
- Penalized nonparametric mean square estimation of the coefficients of diffusion processes (2007, methodology): “A nonparametric approach using spectral methods is investigated in Gobet et al. (2004), where non standard nonparametric rates are exhibited.”