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References for Lebesgue-Stieltjes Integration in Stochastic Integration

Article Quant Q&A · Author: Evan Aad
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Summary

The document seeks a thorough and orderly treatment of Lebesgue-Stieltjes integration, bounded variation, and Stieltjes integrals as background for stochastic integration. It notes that one cited stochastic integration text lists relevant results without proving them or pointing readers to further explanations, and asks for references covering material useful to stochastic integration.

The answer recommends Royden's real analysis text for a general graduate-level treatment of integration topics and Williams's probability text for a more probability-oriented presentation. These are suggested reading paths rather than summaries of the books, and the document provides no comparison of coverage, specific chapters, or assessment of how completely either source addresses the requested results.

Key ideas

  • Lebesgue-Stieltjes integration provides mathematical groundwork for stochastic integration.
  • The question seeks proofs and an organized account of bounded variation and Stieltjes integral results.
  • Royden's real analysis text is recommended for general integration coverage.
  • Williams's probability text is suggested for a more probability-focused treatment.

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# Lebesgue-Stieltjes integration and related topics


# Lebesgue-Stieltjes integration and related topics












The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail.

Take, for instance, Chung and Williams' textbook "Introduction to Stochastic Integration", 2nd edition (Birkhaeuser 2014). Section 1.3, titled "Functions of Bounded Variation and Stieltjes Integrals", lists facts, but does not prove them nor gives reference for further reading.

Where can I turn to (a textbook, online class notes, etc.) to fill in the gaps and read an orderly exposition of the subject matter that accounts for all the results pertinent to stochastic integration, and, in particular, those cited in Chung and Williams' section 1.3?

P.S. The reference need not be in English.

## Answer by LazyCat (score 1)

https://quant.stackexchange.com/a/26132

Royden's "Real Analysis" is a standard textbook for a first year grad course in Real Analysis. It covers all the integration topics nicely. A more stochastically oriented book would be "Probability with Martingales" by Williams, which covers integration as well.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.