Hybrid multi-step estimators for stochastic differential equations based on sampled data

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

“…While the computation of the Bayes type estimator is generally free from a choice of the initial value, there is a serious problem that it takes so much time to compute the Bayes type estimator when the sample size is large. Kamatani and Uchida (2015) proposed the multi-step ML type estimator based on the initial Bayes type estimator with the full data, and Kaino et al (2017) and Kaino and Uchida (2018a,b) studied the adaptive ML type estimator with the initial Bayes type estimator derived from the reduced data by applying the result of Kutoyants (2017) to high frequency data analysis. In this paper, we have proposed the initial Bayes type estimator with the reduced data based on the local means obtained from the high frequency data with noise and hybrid multi-step estimator with the initial Bayes type estimator.…”

“…In the next place, we define the hybrid multi-step estimators. We introduce the quasi likelihood functions in Nakakita and Uchida (2018c) such that Kamatani and Uchida (2015) and Kamatani et al (2016), let us denote…”

“…In general, however, the optimisation of quasi likelihood functions for diffusion processes, regardless of noise existence, is strongly dependent on initial values, especially in the case where the volatility function a or drift function b are nonlinear with respect to parameters. Hence, Kaino et al (2017) and Kaino and Uchida (2018a,b) propose hybrid multi-step estimation procedure for diffusion processes where initial values in optimisation are derived from Bayes type estimation with reduced sample sizes and the sequential optimisation with these initial values is implemented, which inherits the idea of hybrid multi-step estimation for diffusion processes with full sample sizes in Kamatani and Uchida (2015) (see also Kutoyants, 2017). In this research, we also consider hybrid multi-step estimation and apply the idea into inference problem, in particular, PLDI for the quasi likelihood functions and the convergence of moments of estimators for discretely and noisily observed ergodic diffusion processes since PLDI and convergence of moment of estimators are key tools to show the mathematical validity of information criteria for model selection problems (see Uchida, 2010;Fujii and Uchida, 2014;Eguchi and Masuda, 2018).…”

On approximation of BSDE and multi-step MLE-processes

“…Here max i |t i − t i−1 | → 0 and T n → ∞. Such a model of parameter estimation was studied, e.g., in Kamatani and Uchida (2015) and Uchida and Yoshida (2014). It is possible to consider the mixture of discrete-time and small noise models, to consider the model with X t → ±∞ or the models with null recurrent forward equation etc.…”

“…Moreover, this device can be applied even in the case of preliminary estimator with the rate of convergence worse than √ n (see, e.g., Robinson (1988) and Kamatani and Uchida (2015)). For continuous-time stochastic processes such a construction was used, for example, in Skorohod and Khasminskii (1996).…”