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Stochastic Calculus and Applications

Authors: Samuel N. Cohen, Robert J. Elliott

Publisher: Springer New York

Book Series : Probability and its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Completely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry.

New features of this edition include:

End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index.

"Such a self-contained and complete exposition of stochastic calculus and applications fills an existing gap in the literature. The book can be recommended for first-year graduate studies. It will be useful for all who intend to work with stochastic calculus as well as with its applications."–Zentralblatt (from review of the First Edition)

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Table of Contents

Frontmatter

Measure Theoretic Probability

Frontmatter

1. Measure and Integral

Abstract

In the first two chapters, we outline definitions and results from basic real analysis and measure theory, and their application to probability. These concepts form the foundation for all that follows.

Samuel N. Cohen, Robert J. Elliott

2. Probabilities and Expectation

Abstract

We now see how general measure theory specializes when we consider applications to probability.

Samuel N. Cohen, Robert J. Elliott

Stochastic Processes

Frontmatter

3. Filtrations, Stopping Times and Stochastic Processes

Abstract

In many situations, we have more than a single random variable to consider. In particular, we may have new observations at different points in time, each of which is random. Our goal in this section is to build a mathematical understanding of these ‘stochastic processes’, that is, of collections of random variables, the values of which become revealed through time.

Samuel N. Cohen, Robert J. Elliott

4. Martingales in Discrete Time

Abstract

In this chapter and the next, we consider one of the most important classes of stochastic processes, the class of martingales. Their significance was first emphasized in the now classical book of Doob [62].

Samuel N. Cohen, Robert J. Elliott

5. Martingales in Continuous Time

Abstract

In this chapter we extend our discussion of martingales to allow a continuous-time processes. Throughout we take \(\mathbb{T} = [0,\infty [\) or \([0,\infty ]\).

Samuel N. Cohen, Robert J. Elliott

6. The Classification of Stopping Times

Abstract

We now wish to classify stochastic processes more finely, which will be invaluable for the development of the theory of stochastic integration. In order to do this, we first study classes of stopping times.

Samuel N. Cohen, Robert J. Elliott

7. The Progressive, Optional and Predictable σ-Algebras

Abstract

We now move from looking at different types of stopping times to different types of processes. Recall that we defined a real-valued process Y to be progressive (Definition 3.2.25) if, for every t, the map (s, ω) ↦ X _s(ω) of [0, t] ×Ω into \((\mathbb{R},\mathcal{B}(\mathbb{R}))\) is measurable, when [0, t] ×Ω is given the product σ-algebra \(\mathcal{B}([0,t]) \otimes \mathcal{F}_{t}\). Essentially, this states that the process X is adapted and is Borel measurable with respect to time.

Samuel N. Cohen, Robert J. Elliott

Stochastic Integration

Frontmatter

8. Processes of Finite Variation

Abstract

Given our understanding of general stochastic processes, we now set our sights on establishing a theory of stochastic integration. We do this in stages, beginning with the simple case where we take the integral with respect to a process which does not vary ‘too much’, that is, where its paths are of finite variation for almost all ω. This first step is deceptively simple, as we can establish our integral pathwise, simply by appealing to the Lebesgue–Stieltjes integral considered in Chapter 1 We then use this theory to establish the stochastic integral for more general processes, over the coming chapters.

Samuel N. Cohen, Robert J. Elliott

9. The Doob–Meyer Decomposition

Abstract

In the previous chapter, we have seen that, for any process \(X \in \mathcal{A}\), we can find a predictable process Y = Π _p ^∗ X such that X − Y is a martingale. This is a fundamentally useful property, and in this chapter we show that a similar decomposition holds for all right-continuous local supermartingales (and hence local submartingales). To obtain this, we first consider the particularly ‘nice’ class of processes given by class (D). Recall that a right-continuous uniformly integrable supermartingale X is said to be of class (D) if the set of random variables \(\{X_{T}\}_{T\in \mathcal{T}}\) is uniformly integrable (where \(\mathcal{T}\) is the set of all stopping times). These were introduced in Section 5.6

Samuel N. Cohen, Robert J. Elliott

10. The Structure of Square Integrable Martingales

Abstract

We assume, as in previous chapters, that we are working on a probability space \((\varOmega,\mathcal{F},P)\) which has a filtration \(\{\mathcal{F}_{t}\}_{t\in [0,\infty ]}\) satisfying the usual conditions and, for simplicity, \(\mathcal{F}_{\infty } =\bigvee _{t<\infty }\mathcal{F}_{t}\). Furthermore, indistinguishable processes will be identified, so that when we speak of a process, we really mean an equivalence class of indistinguishable processes. When we speak of a martingale, we shall invariably mean its càdlàg version.

Samuel N. Cohen, Robert J. Elliott

11. Quadratic Variation and Semimartingales

Abstract

We now come to one of the key objects in stochastic analysis, and what fundamentally distinguishes the theory from classical calculus. This is the notion of the quadratic variation of a process.

Samuel N. Cohen, Robert J. Elliott

12. The Stochastic Integral

Abstract

We have now established enough basic theory to construct the stochastic integral in full generality. In this chapter, we develop the integral with respect to semimartingales, and prove some of its properties. As in the previous chapters, we assume we have a filtered probability space, with filtration satisfying the usual conditions, \(\mathcal{F}_{\infty } = \mathcal{F}_{\infty -}\), a martingale will always refer to its càdlàg version, and statements should be read as ‘up to an evanescent set’.

Samuel N. Cohen, Robert J. Elliott

13. Random Measures

Abstract

When dealing with jump processes, it is sometimes useful to have a theory of integration which distinguishes between jumps of different sizes. Particularly for processes with many jumps, this is most easily accomplished by treating the jump process as generating a ‘random measure’, that is a stochastic measure over time and the sizes of the jumps, such that the integrals with respect to this measure correspond, in some sense, to the stochastic integrals with respect to the original process. Formalizing this idea, in a general setting, is the purpose of this chapter.

Samuel N. Cohen, Robert J. Elliott

Stochastic Differential Equations

Frontmatter

14. Itô’s Differential Rule

Abstract

In order to use the theory of stochastic integration, much like in classical integration, certain rules are of fundamental importance. The most famous of these, ‘Itô’s Differential Rule’, generalizes the chain rule from classical calculus. Deriving this rule and exploring its consequences are the aims of this chapter.

Samuel N. Cohen, Robert J. Elliott

15. The Exponential Formula and Girsanov’s Theorem

Abstract

In this chapter, we consider a particularly important example of a stochastic differential equation, the ‘stochastic exponential’. The solutions to this type of equation have many applications, as they form the basis of understanding how to use stochastic processes to change from one probability measure to another.

Samuel N. Cohen, Robert J. Elliott

16. Lipschitz Stochastic Differential Equations

Abstract

As is now usual, all (in)equalities in this chapter should be read as ‘up to an evanescent set’, unless otherwise specified, martingales are càdlàg and we assume we have a filtered probability space satisfying the usual conditions. In this chapter, we consider stochastic differential equations (SDEs), that is, m-dimensional processes X satisfying an equation of the form

Samuel N. Cohen, Robert J. Elliott

17. Markov Properties of SDEs

Abstract

In the previous chapter, we have considered SDEs where the integral is with respect to a general semimartingale. In this chapter, we focus our attention on a much more specialized setting, where the integral is taken with respect to time (i.e. Lebesgue measure), a Brownian motion and a compensated Poisson random measure. Working in this setting allows the Markovian properties of the Brownian motion and the Poisson process to be inherited by the SDE solution. A full treatment of this topic would require consideration of general Markov processes. For this, see Ethier and Kurtz [77], or the more specialized treatments in Karatzas and Shreve [117] or Revuz and Yor [155]. We shall instead present only a selection of these issues.

Samuel N. Cohen, Robert J. Elliott

18. Weak Solutions of SDEs

Abstract

So far, we have focussed on solutions of SDEs where we are simply given a filtration, and with it the Brownian motion W and the random measure μ. We then construct the solution to our equation (17.2). In essence, we have used no properties of the filtration except the fact that W and μ are adapted. As we shall see, there are occasions where this approach is insufficient, and we require that the filtration is slightly richer.

Samuel N. Cohen, Robert J. Elliott

19. Backward Stochastic Differential Equations

Abstract

In this chapter, we consider a different type of stochastic differential equation. In the setting of Chapter 17, we specified a solution process X through its dynamics and its initial value, as in (17.6). In this chapter, we specify a solution process Y through its dynamics and its terminal value, at a fixed, deterministic time \(T \in ]0,\infty [\). The difficulty with this is that the terminal value is allowed to be a random variable, but we look for a solution which is adapted to a given filtration.

Samuel N. Cohen, Robert J. Elliott

Applications

Frontmatter

20. Control of a Single Jump

Abstract

In this and the coming chapter, we use the mathematical machinery we have developed to consider problems related to the optimal control of a random process. To begin with, we consider the simple case of a single jump process, as in Chapter 13, where a controller can determine the rate at which the jump occurs, but faces some cost for doing so. This example will allow us to demonstrate the main methods used in optimal control, before moving on to more technically demanding problems.

Samuel N. Cohen, Robert J. Elliott

21. Optimal Control of Drifts and Jump Rates

Abstract

We now discuss the optimal control of a stochastic differential equation, of a type similar to those considered in Chapter 17.

Samuel N. Cohen, Robert J. Elliott

22. Filtering

Abstract

In this chapter we suppose there is a signal process X which describes the state of a system, but which cannot be observed directly. Instead we can only observe some process Y with dynamics dependent on the value of X. Our object is to obtain an expression for the “best estimate” of X _t, (or of ϕ(X _t) for ϕ in a large enough class of functions), given the observations up to time t, that is, given the σ-algebra

$$\displaystyle{\mathcal{Y}_{t} =\sigma (Y _{s}: s \leq t).}$$

This problem is known as ‘filtering’, as we attempt to filter out the state of the hidden ‘signal’ X given our (noisy) observations of Y.

Samuel N. Cohen, Robert J. Elliott

Backmatter

Title: Stochastic Calculus and Applications
Authors: Samuel N. Cohen
Robert J. Elliott
Copyright Year: 2015
Publisher: Springer New York
Electronic ISBN: 978-1-4939-2867-5
Print ISBN: 978-1-4939-2866-8
DOI: https://doi.org/10.1007/978-1-4939-2867-5