*Yacine Aïıt-Sahalia and Jean Jacod*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161433
- eISBN:
- 9781400850327
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161433.003.0001
- Subject:
- Economics and Finance, Econometrics

This chapter presents a quick review of the theory of semimartingales, which are processes for which statistical methods are considered in this book. Topics covered include diffusions, Lévy ...
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This chapter presents a quick review of the theory of semimartingales, which are processes for which statistical methods are considered in this book. Topics covered include diffusions, Lévy processes, Itô semimartingales, and processes with conditionally independent increments.Less

This chapter presents a quick review of the theory of semimartingales, which are processes for which statistical methods are considered in this book. Topics covered include diffusions, Lévy processes, Itô semimartingales, and processes with conditionally independent increments.

*Tatjana Lemke and Simon J. Godsill*

- Published in print:
- 2015
- Published Online:
- January 2016
- ISBN:
- 9780199683666
- eISBN:
- 9780191763298
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199683666.003.0009
- Subject:
- Economics and Finance, Econometrics

This chapter begins with a simple general framework for inference in the presence of α-stable processes, where the stable processes are represented as conditionally Gaussian distributions, relying ...
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This chapter begins with a simple general framework for inference in the presence of α-stable processes, where the stable processes are represented as conditionally Gaussian distributions, relying on (exact) series representations of the stable laws and the corresponding stochastic integrations in terms of infinite summations of random Poisson process arrival times. Inference can therefore be carried out using techniques including auxiliary variables, Rao-Blackwellized particle filtering, and Markov chain Monte Carlo. The Poisson series representation is further enhanced by introducing an approximation of the series residual terms based on exact moment calculations. Extensions to the discrete-time asymmetric stable case and to continuous-time areLess

This chapter begins with a simple general framework for inference in the presence of **α**-stable processes, where the stable processes are represented as conditionally Gaussian distributions, relying on (exact) series representations of the stable laws and the corresponding stochastic integrations in terms of infinite summations of random Poisson process arrival times. Inference can therefore be carried out using techniques including auxiliary variables, Rao-Blackwellized particle filtering, and Markov chain Monte Carlo. The Poisson series representation is further enhanced by introducing an approximation of the series residual terms based on exact moment calculations. Extensions to the discrete-time asymmetric stable case and to continuous-time are

*Yacine Aïıt-Sahalia and Jean Jacod*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161433
- eISBN:
- 9781400850327
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161433.003.0005
- Subject:
- Economics and Finance, Econometrics

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes. It then introduces ...
More

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes. It then introduces a general notion of identifiability for a parameter, in a semi-parametric setting. A parameter can be a number (or a vector), as in classical statistics; it can also be a random variable, such as the integrated volatility. The analysis is first conducted for Lévy processes, because in this case parameters are naturally non-random, and then extended to the more general situation of semimartingales. It also considers the problem of testing a hypothesis which is “random,” such as testing whether a discretely observed path is continuous or discontinuous: the null and alternative are not the usual disjoint subsets of a parameter space, but rather two disjoint subsets of the sample space, which leads to an ad hoc definition of the level, or asymptotic level, of a test in such a context. Finally, the chapter returns to the question of efficient estimation of a parameter, which is mainly analyzed from the viewpoint of “Fisher efficiency.”Less

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes. It then introduces a general notion of identifiability for a parameter, in a semi-parametric setting. A parameter can be a number (or a vector), as in classical statistics; it can also be a random variable, such as the integrated volatility. The analysis is first conducted for Lévy processes, because in this case parameters are naturally non-random, and then extended to the more general situation of semimartingales. It also considers the problem of testing a hypothesis which is “random,” such as testing whether a discretely observed path is continuous or discontinuous: the null and alternative are not the usual disjoint subsets of a parameter space, but rather two disjoint subsets of the sample space, which leads to an ad hoc definition of the level, or asymptotic level, of a test in such a context. Finally, the chapter returns to the question of efficient estimation of a parameter, which is mainly analyzed from the viewpoint of “Fisher efficiency.”