Abstract
For simplicity again, in this chapter we will only consider processes with just one component. We already remarked in the Sect. 6.3 that a white noise is a singular process whose main properties can be traced back to the non differentiability of some processes. As a first example we have shown indeed that the Poisson impulse process (6.63) and its associated compensated version (6.65) are white noises entailed by the formal derivation respectively of a simple Poisson process N(t) and of its compensated variant \(\widetilde{N}(t)\).
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Notes
- 1.
This is not a very restrictive hypothesis: since our processes will turn out to be Gaussian, independence and non correlation happen to be quite equivalent.
- 2.
A function w(x) defined on [a, b] is said of bounded variation if it exists \(C>0\) such that
$$\begin{aligned} \sum _{k=1}^n| w(x_k)- w(x_{k-1})|<C \end{aligned}$$for every finite partition \(a=x_0<x_1<\cdots <x_n=b\) of [a, b]; in this case the quantity
$$\begin{aligned} \mathcal {V}[ w]=\sup _{\mathcal {D}}\sum _{k=1}^n| w(x_k)- w(x_{k-1})| \end{aligned}$$where \(\mathcal {D}\) is the set of the finite partitions of [a, b], is called the total variation of w. It is known that the Lebesgue-Stieltjes integral
$$\begin{aligned} \int _a^bf(x)\,d w(x) \end{aligned}$$can be coherently defined when w(x) is a function of bounded variation. Remark that every function of bounded variation is (almost everywhere) differentiable: for further details see [1], pp. 328–332.
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Cufaro Petroni, N. (2020). An Outline of Stochastic Calculus. In: Probability and Stochastic Processes for Physicists. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-48408-8_8
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