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.
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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 map** \((s,\omega )\mapsto Y_s(\omega )\) 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 \(({\mathcal F}_t)\)-stop** times, increasing to \(+\infty \), such that \((M_{t\wedge \tau _n}-M_0)_{t\ge 0}\) is an \(({\mathcal F}_t)\)-martingale for every \(n\ge 1\).
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Pagès, G. (2018). Miscellany. In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_12
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DOI: https://doi.org/10.1007/978-3-319-90276-0_12
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