Abstract
In this chapter we deal with stochastic Riemann integrals, i.e. with ordinary Riemann integrals with a stochastic process as the integrand. Mathematically, these constructs are relatively unsophisticated, they can be defined pathwise for continuous functions as in conventional (deterministic) calculus. However, this pathwise definition will not be possible any longer for e.g. Ito integrals in the chapter after next. Hence, at this point we propose a way of defining integrals as a limit (in mean square) which will be useful later on. If the stochastic integrand is in particular a Wiener process, then the Riemann integral follows a Gaussian distribution with zero expectation and the familiar formula for the variance. A number of examples will facilitate the understanding of this chapter.
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Notes
- 1.
Bernhard Riemann (1826–1866) studied with Gauss in Göttingen where he himself became a professor. Already before his day, integration had been used as a technique which reverses differentiation by forming an antiderivative. However, Riemann explained for the first time under which conditions a function possesses an antiderivative at all.
- 2.
A definition and discussion of this mode of convergence can be found in the fourth section of this chapter.
- 3.
In particular in econometrics, one often writes alternatively plimX n  = X as \(n \rightarrow \infty \).
- 4.
See e.g. sections 5.7 through 5.10 in Grimmett and Stirzaker (2001) for an introduction to the theory and application of characteristic functions. In particular, it holds for the characteristic function of a random variable with a Gaussian distribution, \(Y \sim \mathcal{N}(\mu,\sigma ^{2})\), that:
$$\displaystyle{ \phi _{y}(u) =\exp \left \{i\,u\,\mu -\frac{u^{2}\sigma ^{2}} {2} \right \}\,,\quad i^{2} = -1\,,\ u \in \mathbb{R}\,. }$$
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Hassler, U. (2016). Riemann Integrals. In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_8
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DOI: https://doi.org/10.1007/978-3-319-23428-1_8
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