The Itô Formula and the Martingale Representation Theorem

Example 3.6 illustrates that the basic definition of Ito integrals is not very useful when we try to evaluate a given integral. This is similar to the situation for ordinary Riemann integrals, where we do not use the basic definition but rather the fundamental theorem of calculus plus the chain rule in the explicit calculations.

In the introduction to Chap. 10 the notion of the definite integral of a function f on an interval [a,b] has already been mentioned. It arises from summing up expressions of the form f(x)Δx and taking limits. Such sums appear in many applications including the calculation of areas, surface areas and volumes as well as the calculation of lengths of curves. This chapter employs the notion of Riemann integrals as the basic concept of definite integration. Riemann’s approach provides an intuitive concept in many applications, as will be elaborated in examples at the end of the chapter.The main part of Chap. 11 is dedicated to the properties of the integral. In particular, the two fundamental theorems of calculus are proven. The first theorem allows one to calculate a definite integral from the knowledge of an antiderivative. The second fundamental theorem states that the definite integral of a function f on an interval [a,x] with variable upper bound provides an antiderivative of f. Since the definite integral can be approximated, for example by Riemann sums, the second fundamental theorem offers a possibility to approximate the antiderivative numerically. This is of importance, for example, for the calculation of distribution functions in statistics.KeywordsCantilever BeamFundamental TheoremPiecewise ConstantPiecewise Continuous FunctionIntuitive ApproachThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In Chap. 9 we evaluated a few definite integrals by using Riemann sums, a method that is generally not easy, even for simple integrands. Then we evaluated a few more, by interpreting the definite integral as a certain familiar area. Generally however, computing definite integrals which routinely appear in problems with no apparent notion of area or of average value in sight, can be very difficult. Coming to the rescue in many cases is the Fundamental Theorem of Calculus. With it, many more definite integrals can be computed relatively easily. But this—the most important theorem in all of Calculus—gives us a great deal more.

1. Introduction 2. The Riemann integral 3. Explorations of R 4. Nowhere dense sets and the problem with the fundamental theorem of calculus 5. The development of measure theory 6. The Lebesgue integral 7. The fundamental theorem of calculus 8. Fourier series 9. Epilogue: A. Other directions B. Hints to selected exercises.

Three important calculus ideas –– tangent lines, the derivative definition, and the fundamental theorem of calculus –– are explored using Cabri Geometry II.

"Simultaneous Proof of the First Fundamental Theorem of Calculus and Integrability of Continuous Functions." The American Mathematical Monthly, 122(1), p. 23

SummaryWe provide an easy argument, using the second fundamental theorem of calculus, for computing the area of the unit circle.

The paper wants to show how it is possible to develop based on an adequate basic idea (so-called “Grundvorstellung”) of the derivative a visual understanding of the (first) Fundamental theorem of Calculus.

Fundamental properties of intuitionistic fuzzy calculus

This paper is intended to offer calculus students and teachers the opportunity to motivate and discover the (first) Fundamental Theorem of Calculus in an experimental, experiential, inductive, intuitive, vernacular-based manner. Starting from the observation that a distance travelled at a constant speed corresponds to the area inside a rectangle, a precursor of the FTC is quickly and painlessly motivated and discovered. Its validity is then investigated both experimentally and theoretically. Once its plausibility is established, pictures and words are wedded to generalize this discovery to a standard statement of the fundamental theorem.

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.

It is proved that the martingale representation theorem for set-valued martingales proposed by Kisielewicz (2014b) holds only in the degenerate case. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.

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Empirical data for all planets in the solar system were analyzed to determine the kinetic, potential, and total energy of each, as well as the area each sweep out per second. As expected, it is the conservation of angular momentum, and not the conservation of energy, that ensures planets sweep out equal areas in equal times through any orbit region. Planets all follow the Second Fundamental Theorem of the Calculus while orbiting, where swept area increases progressively as a function of distance from the sun. The potential energy for each is twice the kinetic energy which is consistent with theoretic explanation. The Universal Gravitation constant G for a quantity of mass is not altered by huge changes in planet temperatures which clarify a property of gravity. Unique features of the planetary orbits further expand the Newton explanation that the system behaves as a vast wound clock, rather than a random assemblage of planets.