Brownian motion is by far the most important stochastic process. It is the archetype of Gaussian processes, of continuous time martingales, and of Markov processes. It is basic to the study of stochastic differential equations, financial mathematics, and filtering, to name only a few of its applications.

In this chapter we define Brownian motion and consider some of its elementary aspects. Later chapters will take up the construction of Brownian motion and properties of Brownian motion paths.

Definition and basic properties

Let (Ω, ℱ, ℙ) be a probability space and let {ℱt} be a filtration, not necessarily satisfying the usual conditions.

Definition 2.1Wt = Wt(ω) is a one-dimensional Brownian motion with respect to {ℱt} and the probability measure ℙ, started at 0, if

1. (1)Wt is ℱt measurable for each t ≥ 0.

2. (2)W0 = 0, a.s.

3. (3)Wt − Ws is a normal random variable with mean 0 and variance t − s whenever s < t.

4. (4)Wt − Ws is independent of ℱs whenever s < t.

5. (5)Wt has continuous paths.

If instead of (2) we have W0 = x, we say we have a Brownian motion started at x. Definition 2.1(4) is referred to as the independent increments property of Brownian motion. The fact that Wt1 – Ws has the same law as Wt–s, which follows from Definition 2.1(3), is called the stationary increments property. When no filtration is specified, we assume the filtration is the filtration generated by W, i.e., ℱt = σ (Ws; s ≤ t). Sometimes a one-dimensional Brownian motion started at 0 is called a standard Brownian motion.

Often a Markov process is specified in terms of its behavior at each point, and one wants to form a global picture of the process. This means one is given the infinitesimal generator, which is a linear operator that is an unbounded operator in general, and one wants to come up with the semigroup for the Markov process.

We will begin by looking further at semigroups and resolvents, and then define the infinitesimal generator of a semigroup. We will prove the Hille–Yosida theorem, which is the primary tool for constructing semigroups from infinitesimal generators. Then we will look at two important examples: elliptic operators in nondivergence form and Lévy processes.

Semigroup properties

Let S be a locally compact separable metric space. We will take ℬ to be a separable Banach space of real-valued functions on S. For the most part, we will take ℬ to be the continuous functions on S that vanish at infinity (with the supremum norm), although another common example is to let ℬ be the set of functions on S that are in L2 with respect to some measure. We use ∥·∥ for the norm on ℬ.

For the duration of this chapter we will make the following assumption.

Assumption 37.1Suppose that Pt, t ≥ 0, are operators acting on ℬ such that

1. (1) the Ptare contractions: ∥Ptf∥≤∥f∥ for all t ≥ 0 and all f ∈ ℬ,

2. (2) the Ptform a semigroup: PsPt = Pt+sfor all s, t ≥ 0, and

3. (3) the Ptare strongly continuous: if f ∈ ℬ, then Ptf → f as t → 0.

At the opposite extreme from Brownian motion is the Poisson process. This is a process that only changes value by means of jumps, and even then, the jumps are nicely spaced. The Poisson process is the prototype of a pure jump process, and later we will see that it is the building block for an important class of stochastic processes known as Lévy processes.

Definition 5.1 Let {ℱt} be a filtration, not necessarily satisfying the usual conditions. A Poisson process with parameter λ > 0 is a stochastic process X satisfying the following properties:

1. (1) X0 = 0, a.s.

2. (2) The paths of Xt are right continuous with left limits.

3. (3) If s < t, then Xt – Xs is a Poisson random variable with parameter λ(t − s).

4. (4) If s < t, then Xt – Xs is independent of ℱs.

Define Xt− = lims→t,s<tXs, the left-hand limit at time t, and ΔXt = Xt – Xt–, the size of the jump at time t. We say a function f is increasing if s < t implies f(s) ≤ f(t). We use “strictly increasing” when s < t implies f(s) < f(t). We have the following proposition.

Proposition 5.2Let X be a Poisson process. With probability one, the paths of Xt are increasing and are constant except for jumps of size 1. There are only finitely many jumps in each finite time interval.

Suppose S is a metric space.We use Sℕ for the product space S×S×… furnished with the product topology. We may view Sℕ as the set of sequences (x1, x2, …) of elements of S. We use the σ-field on Sℕ generated by the cylindrical sets. Given an element x = (x1, x2, …) of Sℕ, we define πn(x) = (x1, …, xn) ∈ Sn.

We suppose we have a Radon probability measure μn defined on Sn for each n. (Being a Radon measure means that we can approximate μn(A) from below by compact sets; see

Folland (1999) for details.) The μn are consistent if μn+1(A × S) = μn(A) whenever A is a Borel subset of Sn. The Kolmogorov extension theorem is the following.

Theorem D.1Suppose for each n we have a probability measure μnon Sn. Suppose the μn's are consistent. Then there exists a probability measure μ Sℕ such that μ(A × Sℕ) = μn(A) for all A ⊂ Sn.

Proof Define μ on cylindrical sets by μ(A × Sℕ) = μn(A) if A ⊂ Sn. By the consistency assumption, μ is well defined. By the Carathéodory extension theorem, we can extend μ to the σ-field generated by the cylindrical sets provided we show that whenever An are cylindrical sets decreasing to ∅, then μ(An) → 0.

Suppose An are cylindrical sets decreasing to ∅ but μ(An) does not tend to 0; by taking a subsequence we may assume without loss of generality that there exists ε > 0 such that μ(An) ≥ ε for all n. We will obtain a contradiction.

A European call option is the option to buy a share of stock at a given price at some particular time in the future. For example, I might buy a call option to purchase one share of Company X for $40 three months from today. When the three months is up, I check the price of Company X. If, say, it is $35, then my option is worthless, because why would I buy a share for $40 using the option when I could buy it on the open market for $35? But if three months from now, the share price is, say, $45, then I can exercise my option, which means I buy a share for $40, and I can then turn around immediately and sell that share for $45 and make a profit of $5. Thus, today, there is a potential for a profit if I have a call option, and so I should pay something to purchase that option. A significant part of financial mathematics is devoted to the question of what is the fair price I should pay for a call option.

Options originated in the commodities market, where farmers wanted to hedge their risks. Since then many types of options have been developed (options are also known as derivatives), and the amount of money invested in options has for the past several years exceeded the amount of money invested in stocks.

Introduction

The nature of an organisation is important to the risk management context. However, there is no such thing as a simple, featureless institution, nor do any operate in a vacuum. The nature of each organisation and what surrounds it influences its operation fundamentally.

Understanding the internal environment is crucial for understanding the way in which risk management should be approached. An analysis of the various aspects of an organisation's internal risk environment helps risk managers within an organisation to appreciate what they need to do to carry out their roles effectively. It also helps external analysts to determine the risks that an organisation is taking – even if the organisation itself does not appreciate these risks.

Internal stakeholders

The only internal stakeholders with a principal relationship with an organisation are owner-managers – all other internal stakeholders are agents, acting on behalf of the an organisation's shareholders, customers, clients and so on. Their views of risk form an important aspect of the risk management environment, and they are discussed together with external stakeholders in the next chapter. However, as well as their individual views of risk, the ways in which they interact are an important determinant of the ways in which organisations behave. At the head of a firm, this means the board of directors. This group includes executive directors who have a day-to-day role in managing the firm and who are led by the chief executive, and non-executives who are are responsible for representing the interests of shareholders. The board of directors is led by the chairman.

Definitions and concepts of risk

The word ‘risk’ has a number of meanings, and it is important to avoid ambiguity when risk is referred to. One concept of risk is uncertainty over the range of possible outcomes. However, in many cases uncertainty is a rather crude measure of risk, and it is important to distinguish between upside and downside risks.

Risk can also mean the quantifiable probability associated with a particular outcome or range of outcomes; conversely, it can refer to the unquantifiable possibility of gains or losses associated with different future events, or even just the possibility of adverse outcomes.

Rather than the probability of a particular outcome, it can also refer to the likely severity of a loss, given that a loss occurs. When multiplied, the probability and the severity give the expected value of a loss.

A similar meaning of risk is exposure to loss, in effect the maximum loss that could be suffered. This could be regarded as the maximum possible severity, although the two are not necessarily equal. For example, in buildings insurance, the exposure is the cost of clearing the site of a destroyed house and building a replacement; however, the severity might be equivalent only to the cost of repairing the roof.