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Diffusions with measurement errors. I. Local Asymptotic Normality

Arnaud Gloter, Jean Jacod (2010)

ESAIM: Probability and Statistics

We consider a diffusion process X which is observed at times i/n for i = 0,1,...,n, each observation being subject to a measurement error. All errors are independent and centered Gaussian with known variance pn. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when X is indeed a Gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information....

Diffusions with measurement errors. II. Optimal estimators

Arnaud Gloter, Jean Jacod (2001)

ESAIM: Probability and Statistics

We consider a diffusion process $X$ which is observed at times $i / n$ for $i = 0, 1, ..., n$ , each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance $ρ_{n}$ . There is an unknown parameter to estimate within the diffusion coefficient. In this second paper we construct estimators which are asymptotically optimal when the process $X$ is a gaussian martingale, and we conjecture that they are also optimal in the general case.

Diffusions with measurement errors. II. Optimal estimators

Arnaud Gloter, Jean Jacod (2010)

ESAIM: Probability and Statistics

We consider a diffusion process X which is observed at times i/n for i = 0,1,...,n, each observation being subject to a measurement error. All errors are independent and centered Gaussian with known variance pn. There is an unknown parameter to estimate within the diffusion coefficient. In this second paper we construct estimators which are asymptotically optimal when the process X is a Gaussian martingale, and we conjecture that they are also optimal in the general case.

Dirichlet forms, quasiregular functions and Brownian motion.

Bernt Oksendal (1988)

Inventiones mathematicae

Discounted optimal stopping for maxima in diffusion models with finite horizon.

Gapeev, Pavel V. (2006)

Electronic Journal of Probability [electronic only]

Discrete Models of Time-Fractional Diffusion in a Potential Well

Gorenflo, R., Abdel-Rehim, E. (2005)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 45K05, 60J60, 60G50, 65N06, 80-99.By generalization of Ehrenfest’s urn model, we obtain discrete approximations to spatially one-dimensional time-fractional diffusion processes with drift towards the origin. These discrete approximations can be interpreted (a) as difference schemes for the relevant time-fractional partial differential equation, (b) as random walk models. The relevant convergence questions as well as the behaviour for time tending to infinity...