Jordan Measure and Riemann Integration

Orrin Frink

doi:10.2307/1968175

Applications of Squeeze Theorem to Limiting Processes Involving Riemann Integration

College Mathematics Journal ◽

10.1080/07468342.2021.1909980 ◽

2021 ◽

Vol 52 (3) ◽

pp. 224-226

Author(s):

Brian Becsi ◽

Solomon Huang ◽

Verenalei Schoenfeld ◽

Bogdan D. Suceavă ◽

Ashley Thune-Aguayo

Keyword(s):

Riemann Integration

Download Full-text

Riemann Integration

Real Analysis: With Proof Strategies ◽

10.1201/9781003091363-6 ◽

2020 ◽

pp. 151-188

Author(s):

Daniel W. Cunningham

Keyword(s):

Riemann Integration

Download Full-text

Riemann Integration on R

A Course in Real Analysis ◽

10.1201/b18085-11 ◽

2015 ◽

pp. 131-186

Keyword(s):

Riemann Integration

Download Full-text

Note on Newtonian Force-Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1949-019-7 ◽

1949 ◽

Vol 1 (2) ◽

pp. 199-208 ◽

Cited By ~ 1

Author(s):

Garrett Birkhoff ◽

Lindley Burton

Keyword(s):

Force Fields ◽

Volume Density ◽

Newtonian Force ◽

Riemann Integration

Although the behaviour of Newtonian potentials inside n-dimensional distributions of mass or charge has been discussed in the sense of Lebesgue-Stieltjes integrals by various authors, the discussion of various important theorems seems to have been made only in the sense of Riemann integration, and assuming the Hälder conditions (or at least piecewise continuity) for the volume density p. We shall generalize these theorems below.

Download Full-text

The Theory of Riemann Integration

A Course of Modern Analysis ◽

10.1017/cbo9780511608759.005 ◽

1996 ◽

pp. 61-81

Keyword(s):

Riemann Integration

Download Full-text

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

Proceedings of the Singapore National Academy of Science ◽

10.1142/s2591722621400068 ◽

2021 ◽

Vol 15 (01) ◽

pp. 45-59

Author(s):

E. M. Bonotto ◽

M. Federson ◽

P. Muldowney

Keyword(s):

Extension Theorem ◽

Continuous Functions ◽

Sample Space ◽

Asset Valuation ◽

Itô Calculus ◽

Black Scholes ◽

Pricing Theory ◽

Simple Sets ◽

Random Times ◽

Riemann Integration

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

Download Full-text