In general, one can define a stochastic integral with respect to any martingale or semimartingale. However, for financial applications we only require integrals with respect to Brownian motion, so our focus will be on this case.

The essential purpose of European Union (EU) integration is, as Article 1, paragraph 2 of the Treaty on European Union (TEU) makes clear, to create ‘an ever closer union among the peoples of Europe’. The EU legal order is therefore shaped to support a dynamic process of European integration. Its legal construct relies on a number of foundational objectives, principles and values – such as equality between participating States and their peoples – that also act as compasses (Section 1.1). The ambitious nature of the process, going well beyond a simple form of economic collaboration among States, distinguishes it from other forms of regional integration while enabling multiple interactions with non-member States and other organisations (Section 1.2). This first chapter stresses that the legal specificities of the EU legal order are best understood in light of its ongoing process of change, constantly seeking to better match evolving political goals with an appropriate legal machinery (Section 1.3). The resulting decisive changes and predominant features of the EU legal order are first and foremost crystallised in the primary legal framework on which it relies, in the form of the two main treaties shaping the EU.

The computations in Section 5.6 suggest that it is reasonable to (try to) extend the discrete Black–Scholes model from Section 5.2 to a continuous-time model on [0, T].

This chapter discusses two different approaches to describing fluid flow: a Lagrangian approach (following a fluid element as it moves) and a Eulerian approach (watching fluid pass through a fixed volume in space). Understanding each of these descriptions of flow is needed to fully understand the dynamics of fluids. This chapter is devoted to diving into the differences between the two descriptions of fluid motion. Understanding this chapter will help tremendously in the understanding of the upcoming chapters when the Navier–Stokes equations and energy equation are discussed. This chapter will introduce the material derivative. It is extremely important to understand this derivative before the Navier–Stokes equations themselves are tackled.

The legal order of the European Union can be distinguished from other international law systems because its legal acts have autonomous effects that are independent of the specific laws of its members. These effects result in EU law being uniquely integrated with the laws of its Member States (Section 4.1), and this is also enabled by the particularly sophisticated machinery put in place at European level for the appropriate implementation and enforcement of EU law across the Member States of the EU (Section 4.2). To ensure adherence to its founding values, as well as its perennity, the powerful legal order that results is itself subject to the rule of law (Section 4.3), as well as to ongoing reflections on its limitations (Section 4.4).

This chapter examines some of the major protections contained in investment treaties: Fair and Equitable Treatment, Full Protection and Security and associated difficulties in interpretation. It also considers the now unpopular Umbrella clauses and increasingly common transparency provisions.

This chapter looks at the guarantees against expropriation found in most investment treaties. It discusses the expansive concept of indirect expropriation under which an investor retains title to its property yet loses much of its practical value. The chapter also considers the situations under which an expropriation of an investor’s property will be viewed as illegal.

In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

This chapter looks at the wide topic of public interest as a justification for breaches of investment treaty protections. It considers non-economic matters such as environmental protection, human rights and culture as well as the concept of the right to regulate which has afforded states more policy space in their interactions with foreign investors.

This chapter discusses the role of domestic foreign investment screening legislation and its link to investment treaties. The chapter proceeds to examine the treaty definitions of the concepts of investment and investor and associated controversies. It concludes with an explanation of performance requirements which may be imposed on foreign investors.

As outlined in Chapter 1, the EU Treaties play a central role in the process of European integration. They set out the boundaries of the European Union legal order. Within these boundaries, it is not only the EU Treaties, but also a vast array of legal acts that may be adopted and that produce legal effects. The European Union thereby autonomously generates a structured set of instruments complementing the Treaty framework. With a view to shedding light on this system of legal norms, this chapter further elaborates on the primary legal framework of the EU legal order (Section 3.1), on the basis of which the system of allocation of competences between the EU and its Member States may then be set out. Indeed, the rules and principles governing the existence of an EU competence (Section 3.2), the various types of competences (Section 3.3) and the principles governing the exercise of these competences (Section 3.4) all derive therefrom. It is within this framework that further legal acts (Section 3.5) may be adopted by EU actors, in accordance with procedures established for that purpose (Section 3.6).