Abstract
This paper is concerned with the problem of estimation of the drift parameter ϑ in a linear stochastic differential equation, with constant coefficients. When continuous sampling of the solution process is available, the maximum likelihood estimate \(\hat \theta _\tau\), based on observation in [0, T] is defined in terms of stochastic and ordinary integrals. So, in practice, to compute \(\hat \theta _\tau\)one has to approximate these integrals by appropriate finite sums which only depend on some discrete sampling in {t 0, t 1,…,t N } ⊂[0, T]. If \(\hat \theta _{N, T}\)(resp. \(\tilde \theta _{N, T}\)) denotes the resulting estimate (resp, maximum likelihood estimate based on observations at t 0,…,t N ), we show that, when Max‖t i+1−t i‖=δN goes to zero, \(P - \lim \hat \theta _{N, T} = P - \lim \tilde \theta _{N, T} = \hat \theta _{\rm T} , \delta _N^{ - 1/2} (\hat \theta _{N, T} )\) and \(\delta _N^{ - 1/2} (\tilde \theta _{N, T} - \hat \theta _T )\) both bounded in probability.
Preview
Unable to display preview. Download preview PDF.
References
A. Le Breton, “Estimation des paramètres d’une équation différentielle stochastique vectorielle linéaire”, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, Série A, t. 279 (1974) 289–292.
A. Le Breton, “Parameter estimation in a linear stochastic differential equation” in: Transactions of 7th Prague Conference and 1974 E.M.S., to appear.
J. Aitchinson and S. D. Silvey, “Maximum likelihood estimation of parameters subject to constraint”, The Annals of Mathematical Statistics 29 (1955) 813–828.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1976 The Mathematical Programming Society
About this chapter
Cite this chapter
Le Breton, A. (1976). On continuous and discrete sampling for parameter estimation in diffusion type processes. In: Wets, R.J.B. (eds) Stochastic Systems: Modeling, Identification and Optimization, I. Mathematical Programming Studies, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120770
Download citation
DOI: https://doi.org/10.1007/BFb0120770
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00783-5
Online ISBN: 978-3-642-00784-2
eBook Packages: Springer Book Archive