scholarly journals Stochastic differential equations as a tool to regularize the parameter estimation problem for continuous time dynamical systems given discrete time measurements

2014 ◽  
Vol 251 ◽  
pp. 54-62 ◽  
Author(s):  
Jacob Leander ◽  
Torbjörn Lundh ◽  
Mats Jirstrand
1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.


Author(s):  
Yakup Ari

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.


1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.


1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.


1996 ◽  
Vol 33 (4) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.


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