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This chapter analyses a multidimensional model using the martingale machinery of Chapter 10. This model is more general than the one used in the previous chapter to obtain more general results, and enhance the understanding of the pricing and hedging problems for Wiener driven models. A self-contained proof of the Second Fundamental Theorem is presented. A practice exercise is included.

This chapter develops a model of adaptive evolution that is founded upon the genetical theory of natural selection, and which justifies the adaptationist approach of analysing traits at organismal level. Fisher’s Fundamental Theorem is derived from first principles, redefining its terms in a way that is consistent with the modern gene-centric view of evolution, and is shown to define the first of two evolutionary fluxes characterising selective turnover in a population. The first of these fluxes represents the spread of selective advantage through the population, as the direct result of changes in allele frequency; the second represents the inevitable diminishment in the selective advantage of the same alleles as they spread through the population. The effects of these fluxes are accurately captured by redefining the adaptive landscape with the ‘vertical’ axis representing the fitness of an individual relative to the population mean, and the ‘horizontal’ axes representing performance objectives for selection. A performance objective is defined as any quantity whose increase would be expected to enhance the selective advantage of an allele conferring that increase, in the hypothetical case that the increase could be effected without impacting performance in any other dimension. Selection necessarily causes an increasing proportion of individuals to occupy the higher parts of the landscape, but this in turn brings their own fitness closer to the population mean, which is represented by sea level. As a result, the adaptive landscape itself subsides under natural selection, in what is therefore called the drowning landscape model of adaptive evolution.

In this chapter the theoretical level is substantially increased, and we discuss in detail the deep connection between financial pricing theory and martingale theory. The first main result of the chapter is the First Fundamental Theorem which says that the market is free of arbitrage if and only if there exists an equivalent martingale measure. We provide a guided tour through the Delbaen–Schachemayer proof and we then apply the theory to derive a general risk neutral pricing formula for an arbitrary financial derivative. We also discuss the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. We define the stochastic discount factor and use it to provide an alternative form of the pricing formula. Finally, we provide a summary for the reader who wishes to go lighter on the (rather advanced) theory.

This chapter clarifies some unfamiliar concepts of Euclidean number theory and examines the bricks, constituents, and formative elements of numbers. It also considers three famous propositions from Euclid’s Elements. The best-known theorem is 9. 20, which states that there are more prime numbers than any assigned multitude of prime numbers. The last theorem, 9. 36, deals with perfect numbers – ones that equal the sum of its proper divisors. The third famous theorem, 9. 14, states that if a number is the least that is measured by (some) prime numbers, it will not be measured by any other prime number except those originally measuring it. Thus, 9. 14 is equivalent to the Fundamental Theorem of Arithmetic (FTA). When multiplication broke loose from addition and evolved its own algebra, it was fairly easy for Johan Carl Friedrich Gauss to realise that the FTA must be true, and set about to prove it along the lines that Euclid had drawn.

The author describes one of the breakthrough concepts of modern finance: the use of the no arbitrage principle in complete markets as the basis for the powerful mathematics of “risk neutral” or “equivalent martingale” pricing. This neoclassical finance model relies on two intertwined assumptions: the existence of complete markets, and the assumption that market participants will act to ensure that no arbitrage profits are possible. The author then presents strong evidence that both of these assumptions are lacking for private businesses and their investors, because markets for the equity in these firms are incomplete. The author argues that this severely undermines this model as a practical valuation tool. As with other principles, this assertion is tested by applying it to three actual companies.

In this chapter we study a general one period model living on a finite sample space. The concepts of no arbitrage and completeness are introduced, as well as the concept of a martingale measure. We then prove the First Fundamental Theorem, stating that absence of arbitrage is equivalent to the existence of an equivalent martingale measure. We also prove the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. Using this theory, we derive pricing and hedging formulas for financial derivatives.

This chapter presents the proof for the Fundamental Theorem of Descent in buildings: that if Γ‎ is a descent group, the set of residues of a building Δ‎ that are stabilized by a subgroup Γ‎ of Aut(Γ‎) forms a thick building. It begins with the hypothesis: Let Π‎ be an arbitrary Coxeter diagram, let S be the vertex set of Π‎ and let (W, S) be the corresponding Coxeter system. It then defines a Γ‎-residue and a Γ‎-chamber as well as a descent group of Δ‎ before concluding with the main result about the fixed point building of Γ‎.

The Pythagoreans developed many of the ideas related to numbers that have become so familiar to us, including even and odd numbers, square numbers, triangular numbers, and so on. They also discovered that some integers cannot be decomposed into factors. These are called prime numbers and they constitute the building blocks of all the other integers, called composite. This chapter deals with the prime numbers in a general non-technical way, since much of the writing about them is quite specialized. The prime numbers have remarkable properties, many of which are still resistant to being proved. Prime numbers matter deeply to mathematics, not to mention to the progress of human knowledge generally. Pythagoras believed that prime numbers were part of a secret code which, if deciphered, would allow us to unlock the mysteries of the cosmos itself.