Название | Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics |
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Автор произведения | Patrick Muldowney |
Жанр | Математика |
Серия | |
Издательство | Математика |
Год выпуска | 0 |
isbn | 9781119595526 |
Figure 3.1 represents the graph of
To see the significance of these alternate perspectives1, suppose
But domains
The book can be read as a collection of standalone accounts of topics which are suggested, or introduced, or touched upon, in [MTRV]. Equally, it can be read as a supplement to [MTRV], developing the ideas of stochastic sums and path integrals which were introduced in [MTRV].
The formal theorem‐proof layout is avoided in favour of concrete illustrations. In almost cases, the relevant theorems and proofs are cited from [MTRV]. But exceptions are made for a few fundamental results such as analogues or versions of theorems 4, 62, 64, and 65 of [MTRV]. The theorem‐proof format is applied in these few exceptional cases.
Also, some of the illustrative examples are “thought experiments”, exercises of imagination intended to clarify, not finance or physics as such, but certain mathematical issues which arise in these subjects.
Notes and additional material for the book can be found at:
https://sites.google.com/site/StieltjesComplete/
—also used to record typos, errors, and corrections; and cited as [ website
] in this book.
Sincere thanks to colleagues and readers who help by sharing their thoughts via:
[email protected]
Note
1 1 In [MTRV], usually denotes , with denoting . Also, is used in [MTRV] to denote , or .
Introduction
The gauge integral is a version of the Riemann integral, with much improved convergence properties. Convergence properties are conditions which ensure integrability of a function; in particular, integrability of the limit of a convergent sequence of integrable functions, with integral of the limit equal to the limit of the integrals—the limit theorems.
Another notable property of the gauge integral