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5 - Introduction to Stochastic Processes
Timothy DelSole, George Mason University, Virginia, Michael Tippett, Columbia University, New York
Book:

Statistical Methods for Climate Scientists

Published online:

03 February 2022

Print publication:

24 February 2022, pp 94-125
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Summary

Climate data are correlated over short spatial and temporal scales. For instance, today’s weather tends to be correlated with tomorrow’s weather, and weather in one city tends to be correlated with weather in a neighboring city. Such correlations imply that weather events are not independent. This chapter discusses an approach to accounting for spatial and temporal dependencies based on stochastic processes. A stochastic process is a collection of random variables indexed by a parameter, such as time or space. A stochastic process is described by the moments at a single time (e.g., mean and variance), and also by the degree of dependence between two times, often measured by the autocorrelation function. This chapter presents these concepts and discusses common mathematical models for generating stochastic processes, especially autoregressive models. The focus of this chapter is on developing the language for describing stochastic processes. Challenges in estimating parameters and testing hypotheses about stochastic processes are discussed.

Moderate deviations for the Durbin–Watson statistic related tothe first-order autoregressive process
S. Valère Bitseki Penda, Hacène Djellout, Frédéric Proïa
Journal:

ESAIM: Probability and Statistics / Volume 18 / 2014

Published online by Cambridge University Press:

03 October 2014, pp. 308-331

Print publication:

2014
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The purpose of this paper is to investigate moderate deviations for the Durbin–Watsonstatistic associated with the stable first-order autoregressive process where the drivennoise is also given by a first-order autoregressive process. We first establish a moderatedeviation principle for both the least squares estimator of the unknown parameter of theautoregressive process as well as for the serial correlation estimator associated with thedriven noise. It enables us to provide a moderate deviation principle for theDurbin–Watson statistic in the case where the driven noise is normally distributed and inthe more general case where the driven noise satisfies a less restrictive Chen–Ledoux typecondition.

Mean variance and goal achieving portfoliofor discrete-time market with currently observable source of correlations
Nikolai Dokuchaev
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 16 / Issue 3 / July 2010

Published online by Cambridge University Press:

18 June 2009, pp. 635-647

Print publication:

July 2010
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The paper studies optimal portfolio selection for discrete timemarket models in mean-variance and goal achieving setting. Theoptimal strategies are obtained for models with an observed processthat causes serial correlations of price changes. The optimalstrategies are found to be myopic for the goal-achieving problem andquasi-myopic for the mean variance portfolio.