Abstract
This chapter reviews some basic material. We collect some elementary concepts and properties in connection with random variables, expected values, multivariate and conditional distributions. Then we define stochastic processes, both discrete and continuous in time, and discuss some fundamental properties. For a successful study of the remainder of this book, the reader is required to be familiar with all of these principles.
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Notes
- 1.
- 2.
Sometimes also called a σ-field, which motivates the symbol F.
- 3.
An example for this is the Cauchy distribution, i.e. the t-distribution with one degree of freedom. For the Pareto distribution, as well, the existence of moments is dependent on the parameter value; this is shown in Problem 2.2.
- 4.
Then σ describes the square root of Var(X) with positive sign.
- 5.
The traditional German spelling is Gauß. Carl Friedrich Gauß lived from 1777 to 1855 and was a professor in Göttingen. His name is connected to many discoveries and inventions in theoretical and applied mathematics. His portrait and a graph of the density of the normal distribution decorated the 10-DM-bill in Germany prior to the Euro.
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- 7.
Up to this point a superscript prime at a function has denoted its derivative. In the rare cases in which we are concerned with matrices or vectors, the symbol will also be used to indicate transposition. Bearing in mind the respective context, there should not occur any ambiguity.
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- 9.
This is not a really rigorous way of introducing expectations conditioned on random variables. A mathematically correct exposition, however, requires measure theoretical arguments not being available at this point; cf. for example Davidson (1994, Ch. 10), or Klebaner (2005, Ch. 2). More generally, one may define expectations conditioned on a σ-algebra, \(\mbox{ E}(X\vert \mathcal{G})\), where \(\mathcal{G}\) could be the σ-algebra generated by Y: \(\mathcal{G} =\sigma (Y )\).
- 10.
- 11.
The convention of using upper case letters for continuous-time process is not universal.
- 12.
The acronym stands for “independently identically distributed”.
- 13.
By assumption, the information at an earlier point in time is contained in the information set at a subsequent point in time: \(\mathcal{I}_{t} \subseteq \mathcal{I}_{t+s}\) for s ≥ 0. A family of such nested σ-algebras is also called “filtration”.
- 14.
We cannot prove the first statement rigorously, which would require a generalization of Proposition 2.1(a). The more general statement taken e.g. from Breiman (1992, Prop. 4.20) or Davidson (1994, Thm. 10.26) reads in our setting as
$$\displaystyle{\mbox{ E}\left [\mbox{ E}(x_{t}\vert \mathcal{I}_{t-1})\vert x_{t-h}\right ] = \mbox{ E}(x_{t}\vert x_{t-h})\,.}$$
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Hassler, U. (2016). Basic Concepts from Probability Theory. In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_2
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