Abstract
This chapter gathers various tools and results from measure and probability theory, martingale theory, uniform integrability, essential extrema and stochastic calculus (Itô’s formula) – some with a complete proof, others simply with precise references – which are used throughout the book. We also included the proofs of two specific mathematical results (discrepancy of the Halton sequence and Pitman-Yor identity) which are not essential in the context of numerical applications but give the mathematical flavor of the underlying theories we use at several places in the book.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A stochastic process \((Y_t)_{\ge 0}\) defined on \((\Omega , \mathcal{A}, {\mathbb P}\) is \(({\mathcal F}_t)\)-progressively measurable if for every \(t\!\in {\mathbb R}_+\), the mapping \((s,\omega )\mapsto Y_s(\omega )\) defined on \([0,t]\times \Omega \) is \(\mathcal{B}or([0,t])\otimes {\mathcal F}_t\)-measurable.
- 2.
This means that W is \(({\mathcal F}_t)\)-adapted and, for every \(s,\, t\ge 0\), \(s\le t\), \(W_t-W_s\) is independent of \({\mathcal F}_s\).
- 3.
The stochastic integral is defined by \(\int _0^s H_s\, dW_s = \left[ \sum _{j=1}^d H^{ij}_s dW^j_s\right] _{1\le i\le d}\).
- 4.
An \({\mathcal F}_t\)-adapted continuous process is a local martingale if there exists an increasing sequence \((\tau _n)_{n\ge 1}\) of \(({\mathcal F}_t)\)-stopping times, increasing to \(+\infty \), such that \((M_{t\wedge \tau _n}-M_0)_{t\ge 0}\) is an \(({\mathcal F}_t)\)-martingale for every \(n\ge 1\).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Pagès, G. (2018). Miscellany. In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-90276-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90274-6
Online ISBN: 978-3-319-90276-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)