LECTURE NOTES. Introduction to Probability Theory and Stochastic Processes (STATS)

Transcription

1 VIENNA GRADUATE SCHOOL OF FINANCE (VGSF) LECTURE NOTES Introduction to Probability Theory and Stochastic Processes (STATS) Helmut Strasser Department of Statistics and Mathematics Vienna University of Economics and Business Administration November 5, 27 Copyright c 26 by Helmut Strasser All rights reserved. No part of this text may be reproduced, stored in a retrieval system,

2 2 or transmitted, in any form or by any means, electronic, mechanical, photocoping, recording, or otherwise, without prior written permission of the author.

3 Contents Preliminaries i.1 Introduction i.2 Literature ii 1 Foundations of mathematical analysis Sets Set operations Cartesian products Uncountable sets Functions Real numbers Real-valued functions Simple functions Regulated functions Riemannian approximation Functions of bounded variation Banach spaces Hilbert spaces Measures and measurable functions Sigma-fields The concept of a sigma-field How to construct sigma-fields Borel sigma-fields Measurable functions The idea of measurability The basic abstract assertions The structure of real-valued measurable functions Measures The concept of measures The abstract construction of measures Measures on the real line

4 4 CONTENTS Point measures The Lebesgue measure Measure defining functions Discrete measures Integrals The integral of simple functions The extension process Extension to nonnegative functions Integrable functions Convergence of integrals The theorem of monotone convergence The infinite series theorem The dominated convergence theorem Stieltjes integration The notion of the Stieltjes integral Integral calculus Proofs of the main theorems More on integration The image of a measure Measures with densities Product measures and Fubini s theorem Spaces of integrable functions Integrable functions Square integrable functions Fourier transforms Probability Basic concepts of probability theory Probability spaces Random variables Distributions of random variables Expectation Independence Convergence and limit theorems Convergence in probability Convergence in distribution The causality theorem Random walks The ruin problem One player

5 CONTENTS Two players Optional stopping Wald s equation Improving chances First passage of a one-sided boundary First passage of a two-sided boundary Gambling systems Conditioning Conditional expectation Martingales Some theorems on martingales Continuous time processes Basic concepts The Poisson process Point processes Levy processes The Wiener Process Continuous time martingales From independent increments to martingales A technical issue: Augmentation Stopping times Hitting times The optional stopping theorem Application: First passage times of the Wiener process One-sided boundaries Two-sided boundaries The reflection principle The Markov property The stochastic integral Integrals along stochastic paths The integral of simple processes Semimartingales Extending the stochastic integral The Wiener integral Stochastic calculus The associativity rule Quadratic variation and the integration-by-parts formula Ito s formula

6 6 CONTENTS 12 Stochastic differential equations Introduction The abstract linear equation Wiener driven models Martingales and stochastic integrals Locally square integrable martingales Square integrable martingales Levy s theorem Martingale representation Change of probability measures Equivalent probability measures The exponential martingale Likelihood processes Girsanov s theorem

7 Preliminaries.1 Introduction The goal of this course is to give an introduction into some mathematical concepts and tools which are indispensable for understanding the modern mathematical theory of finance. Let us give an overview of historic origins of some of the mathematical tools. The central topic will be those probabilistic concepts and results which play an important role in mathematical finance. Therefore we have to deal with mathematical probability theory. Mathematical probability theory is formulated in a language that comes from measure theory and integration. This language differs considerably from the language of classical analysis, known under the label of calculus. Therefore, our first step will be to get an impression of basic measure theory and integration. We will not go into the advanced problems of measure theory where this theory becomes exciting. Such topics would be closely related to advanced set theory and topology which also differs basically from mere set theoretic language and topologically driven slang which is convenient for talking about mathematics but nothing more. Similarly, our usage of measure theory and integration is sort of a convenient language which on this level is of little interest in itself. For us its worth arises with its power to give insight into exciting applications like probability and mathematical finance. Therefore, our presentation of measure theory and integration will be an overview rather than a specialized training program. We will become more and more familiar with the language and its typical kind of reasoning as we go into those applications for which we are highly motivated. These will be probability theory and stochastic calculus. In the field of probability theory we are interested in probability models having a dynamic structure, i.e. a time evolution governed by endogeneous correlation properties. Such probability models are called stochastic processes. Probability theory is a young theory compared with the classical cornerstones of mathematics. It is illuminating to have a look at the evolution of some fundamental ideas of defining a dynamic structure of stochastic processes. i

8 ii PRELIMINARIES One important line of thought is looking at stationarity. Models which are themselves stationary or are cumulatives of stationary models have determined the econometric literature for decades. For Gaussian models one need not distinguish between strict and weak (covariance) stationarity. As for weak stationarity it turns out that typical processes follow difference or differential equations driven by some noise process. The concept of a noise process is motivated by the idea that it does not transport any information. From the beginning of serious investigation of stochastic processes (about 19) another idea was leading in the scientific literature, i.e. the Markov property. This is not the place to go into details of the overwhelming progress in Markov chains and processes achieved in the first half of the 2th century. However, for a long time this theory failed to describe the dynamic behaviour of continuous time Markov processes in terms of equations between single states at different times. Such equations have been the common tools for deterministic dynamics (ordinary difference and differential equations) and for discrete time stationary stochastic sequences. In contrast, continuous time Markov processes were defined in terms of the dynamic behaviour of their distributions rather than of their states, using partial difference and differential equations. The situation changed dramatically about the middle of the 2th century. There were two ingenious concepts at the beginning of this disruption. The first is the concept of a martingale introduced by Doob. The martingale turned out to be the final mathematical fixation of the idea of noise. The notion of a martingale is located between a process with uncorrelated increments and a process with independent increments, both of which were the competing noise concepts up to that time. The second concept is that of a stochastic integral due to K. Ito. This notion makes it possible to apply differential reasoning to stochastic dynamics. At the beginning of the stochastic part of this lecture we will present an introduction to the ideas of martingales and stopping times at hand of stochastic sequences (discrete time processes). However, the main subject of the second half of the lecture will be continuous time processes with a strong focus on the Wiener process. However, the notions of martingales, semimartingales and stochastic integrals are introduced in a way which lays the foundation for the study of more general process theory. The choice of examples is governed by be the needs of financial applications (covering the notion of gambling, of course)..2 Literature Let us give some comments to the bibliography. The popular monograph by Bauer, [1], has been for a long time the standard textbook

9 .2. LITERATURE iii in Germany on measure theoretic probability. However, probability theory has many different faces. The book by Shiryaev, [21], is much closer to those modern concepts we are heading to. Both texts are mathematically oriented, i.e. they aim at giving complete and general proofs of fundamental facts, preferable in abstract terms. A modern introduction into probability models containing plenty of fascinating phenomena is given by Bremaud, [6] and [7]. The older monograph by Bremaud, [5], is not located at the focus of this lecture but contains as appendix an excellent primer on probability theory. Our topic in stochastic processes will be the Wiener process and the stochastic analysis of Wiener driven systems. A standard monograph on this subject is Karatzas and Shreve, [15]. The Wiener systems part of the probability primer by Bremaud gives a very compact overview of the main facts. Today, Wiener driven systems are a very special framework for modelling financial markets. In the meanwhile, general stochastic analysis is in a more or less final state, called semimartingale theory. Present and future research applies this theory in order to get a much more flexible modelling of financial markets. Our introduction to semimartingale theory follows the outline by Protter, [2] (see also [19]). Let us mention some basic literature on mathematical finance. There is a standard source by Hull, [11]. Although this book heavily tries to present itself as not demanding, nevertheless the contrary is true. The reason is that the combination of financial intuition and the appearently informal utilization of advanced mathematical tools requires on the reader s side a lot of mathematical knowledge in order to catch the intrinsics. Paul Wilmott, [22] and [23], tries to cover all topics in financial mathematics together with the corresponding intuition, and to make the analytical framework a bit more explicit and detailed than Hull does. I consider these books by Hull and Wilmott as a must for any beginner in mathematical finance. The books by Hull and Wilmott do not pretend to talk about mathematics. Let us mention some references which have a similar goal as this lecture, i.e. to present the mathematical theory of stochastic analysis aiming at applications in finance. A very popular book which may serve as a bridge from mathematical probability to financial mathematics is by Björk, [4]. Another book, giving an introduction both to the mathematical theory and financial mathematics, is by Hunt and Kennedy, [12]. Standard monographs on mathematical finance which could be considered as cornerstones marking the state of the art at the time of their publication are Karatzas and Shreve, [16], Musiela and Rutkowski, [17], and Bielecki and Rutkowski, [3]. The present lecture should lay some foundations for reading books of that type.

10 iv PRELIMINARIES

11 Chapter 1 Foundations of mathematical analysis 1.1 Sets Set operations Let Ω be a basic set and let A, B, C... be subsets. Remember the basic set operations A B (intersection), A B (union), A c (complementation) and their rules. Denote the difference of sets by A \ B := A B c. PROBLEM 1.1: Describe in words de Morgan s laws: (A B) c = A c B c, (A B) c = A c B c. PROBLEM 1.2: Show that A \ (B C) = (A \ B) (A \ C). PROBLEM 1.3: Expand A \ (B C). We denote by N, Q, R the sets of natural numbers, of rational numbers and of real numbers, respectively. Set operations can also be applied to infinite families of sets, e.g. to a sequence (A i ) i=1 of sets. PROBLEM 1.4: Describe in words: A i, i=1 PROBLEM 1.5: Explain De Morgan s laws: ( ) c A i = A c i, i=1 i=1 i=1 A i ( ) c A i = i=1 PROBLEM 1.6: Describe the elements of the sets lim inf A i := A i, lim sup A i := i i k=1 i=k 1 i=1 A c i k=1 i=k A i

12 2 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS by the properties: is contained in at most finitely many A i, is contained in infinitely many A i, is contained in all but finitely many A i. PROBLEM 1.7: Establish the subset relations between the sets considered in the preceding problems. A sequence (A i ) i=1 of sets is increasing (A i ) if A 1 A 2 A 3... and it is decreasing (A i ) if A 1 A 2 A 3... A sequence of sets is a monotone sequence if it is either increasing or decreasing. PROBLEM 1.8: Find the union and the intersection of monotone sequences of sets. PROBLEM 1.9: Find lim inf and lim sup of monotone sequences of sets. The preceding problems explain why the union of an increasing sequence is called its limit. Similarly the intersection of a decreasing sequence is called its limit. PROBLEM 1.1: Let a < b. Find the limits of (a, b + 1/n], (a, b 1/n], (a, b + 1/n), (a, b 1/n) [a + 1/n, b), [a 1/n, b), (a + 1/n, b), (a 1/n, b] PROBLEM 1.11: Find the limits of {x : x < 1/n}, {x : x 1/n}, {x : x > 1/n}, {x : x 1/n} {x : x < 1 1/n}, {x : x < 1 + 1/n}, {x : x 1 1/n}, {x : x 1 + 1/n} PROBLEM 1.12: Let (A i ) i=1 be any sequence of sets. Determine the limits of n n B n := A i, C n := A i, for n. i=1 i=1 The set of all subsets of a set A is the power set of A. PROBLEM 1.13: Let A be a set with N elements. Explain why the power set contains 2 N elements. The preceding problem explains the name of the power set and why the power set of set A is denoted by 2 A.

13 1.1. SETS Cartesian products Let A and B be sets. Then the (Cartesian) product A B is the set of all ordered pairs (a, b) where a A and b B. This notion is extended in an obvious way to products of any finite or infinite collection of sets. We write A 2 := A A, A 3 := A A A etc. The elements of a product A n are lists (vectors) a = (a 1, a 2,..., a n ) whose elements a i are called components. For every product of sets there are coordinate functions X i : A n A : a = (a 1, a 2,..., a n ) a i In this way subsets of A n can be described by (X i = b) = {a A n : a i = b}, (X i = b 1, X j = b 2 ) = {a A n : a i = b 1, a j = b 2 } PROBLEM 1.14: Let Ω = {, 1} n. Find the number of elements of (max X i = 1) and (min X i = 1). PROBLEM 1.15: (1) Let Ω = {, 1} 3. Find (X 1 = ), (X 1 + X 3 = 1). (2) Let Ω = {, 1} n. Find the number of elements of (X X n = k). PROBLEM 1.16: (1) Let Ω = {, 1} 3. Find 2 Ω. (2) Let Ω = {, 1} n. Find the number of elements in Ω and in 2 Ω. (3) Let Ω = {ω 1,..., ω N }. Find the number of elements of Ω n. The symbol A N denotes the set of all infinite sequences consisting of elements of A. PROBLEM 1.17: Let Ω = {, 1} N. Describe by formula: (1) The set of all sequences in Ω containing no components equal to 1. (2) The set of all sequences in Ω containing at least one component equal to 1. (3) The set of all sequences in Ω containing at most finitely many components equal to 1. (4) The set of all sequences in Ω containing at least infinitely many components equal to 1. (5) The set of all sequences in Ω where all but at most finitely many components are equal to 1. (6) The set of all sequences in Ω where all components are equal to Uncountable sets An infinite set is countable if its elements can be arranged as a sequence. Otherwise it is called uncountable. Two sets A and B are called equivalent (have equal cardinality) if there is one-to-one correspondence between the elements of A and of B. It is clear that equivalent infinite sets are either both countable or uncountable.

14 4 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS PROBLEM 1.18: Explain why Ω = {, 1} N uncountable. PROBLEM 1.19: Explain that R is equivalent to Ω = {, 1} N and thus uncountable. PROBLEM 1.2: Explain why the power set of a countable set is equivalent to Ω = {, 1} N and thus uncountable. 1.2 Functions Let X and Y be non-empty sets. A function f : X Y is a set of pairs (x, f(x)) X Y such that for every x X there is exactly one f(x) Y. X is the domain of f and Y is the range of f. A function f : X Y is injective if f(x 1 ) = f(x 2 ) implies x 1 = x 2. It is surjective if for every y Y there is x X such that f(x) = y. If a function is injective and surjective then it is bijective. If A X then f(a) := {f(x) : x A} is the image of A under f. If B Y then f 1 (B) := {x : f(x) B} is the inverse image of B under f. PROBLEM 1.21: f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). PROBLEM 1.22: f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). PROBLEM 1.23: f 1 (B c ) = (f 1 (B)) c PROBLEM 1.24: Extend the preceding formulas to families of sets. PROBLEM 1.25: f(a 1 A 2 ) = f(a 1 ) f(a 2 ). PROBLEM 1.26: (1) f(a 1 A 2 ) f(a 1 ) f(a 2 ). (2) Give an example where inequality holds in (b). (3) Show that for injective functions equality holds in (b). (4) Extend (a) and (b) to families of sets. PROBLEM 1.27: f(f 1 (B)) = f(x) B PROBLEM 1.28: f 1 (f(a)) A Let f : X Y and g : Y Z. Then the composition g f is the function from X to Z such that (g f)(x) = g(f(x)). PROBLEM 1.29: Let f : X Y and g : Y Z. Show that (g f) 1 (C) = f 1 (g 1 (C)), C Z.

15 1.3. REAL NUMBERS Real numbers The set R of real numbers is well-known, at least regarding its basic algebraic operations. Let us talk about topological properties of R. The following is not intended to be an introduction to the subject, but a checklist which should be well understood or otherwise an introductory textbook has to be consulted. A subset M R is bounded from above if there is an upper bound of M. It is bounded from below if there is a lower bound. It is bounded if it is bounded both from above and from below. The simplest subsets of R are intervals. There are open intervals (a, b) where the boundary points a and b are not included, closed intervals [a, b] where the boundary points are included, half-open intervals [a, b) or (a, b], and so on. Intervals which are bounded and closed are called compact. Unbounded intervals are written as (a, ), (, b], and so on. If a set M is bounded from above then there is always a uniquely determined least upper bound sup M which is called the supremum of M. This is not a theorem but the completeness axiom. It requires an advanced mathematical construction to show that there exists R, i.e. a set having the familiar properties of real numbers including completeness. Any set M R which has a maximal element max M is bounded from above since the maximum is an upper bound. The maximum is also the least upper bound. A set M need not have a maximum. The existence of a maximum is equivalent to sup M M. If M is bounded from below then there is a gretest lower bound inf M called the infimum of M. A (open and connected) neighborhood of x R is an open interval (a, b) which contains x. Note that neighborhoods can be very small, i.e. can have any length ɛ >. An (infinite) sequence is a function form N R, denoted by n x n, for short (x n ), where n = 1, 2,.... When we say that an assertion holds for almost all x n then we mean that it is true for all x n, beginning with some index N, i.e. for x n with n N for some N. A number x R is called a limit of (x n ) if every neighborhood of x contains almost all x n. In other words: The sequence (x n ) converges to x: lim n x n = x or x n x. A sequence can have at most one limit since two different limits could be put into disjoint neighborhoods. A fundamental property of R is the fact that any bounded increasing sequence has a limit which implies that every bounded monotone sequence has a limit. An increasing sequences (x n ) which is not bounded is said to diverge to (x n

16 6 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS ), i.e. for any a we have x n > a for almost all x n. Thus, we can summarize: An increasing sequence either converges to some real number (iff it is bounded) or diverges to (iff it is unbounded). A similar assertion holds for decreasing sequences. A simple fact which is an elementary consequence of the order structure says that every sequence has a monotone subsequence. Putting terms together we arrive at a very important assertion: Every bounded sequence (x n ) has a convergent subsequence. The limit of a subsequence is called an accumulation point of the original sequence (x n ). In other words: Every bounded sequence has at least one accumulation point. An accumulation point x can also be explained in the follwing way: Every neighborhood of x contains infinitely many x n, but not necessarily almost all x n. A sequence can have many accumulation points, and it is not necessarily bounded to have accumulation points. A sequence has a limit iff it is bounded and has only one accumulation point, which then is necessarily the limit. If a sequence is bounded from above then the set of accumulation points is also bounded from above. It is a remarkable fact that in this case there is even a maximal accumulation point lim sup n x n called limit superior. Similarly a sequence bounded from from below has a minimal accumulation point lim inf n x n called limit inferior. A sequence has a limit iff both limit inferior and limit superior exist and are equal. There is a popular criterion for convergence of a sequence which is related to the assertion just stated. Call a sequence (x n ) a Cauchy-sequence if there exist arbitrarily small intervals containing almost all x n. Cleary every convergent sequence is a Cauchy-sequence. But also the converse is true in view of completeness. Indeed, every Cauchy-sequence is bounded and can have at most one accumulation point. By completeness it has at least one accumulation point, and is therefore convergent. The set R = [, ] is called the extended real line. If a sequence (x n ) R diverges to then we say that lim n x n =. If it has a subsequence which diverges to then we say that lim sup n x n =. In both cases we have sup x n =. There is a interesting convergence criterion which is important for martingale theory. 1.1 THEOREM. A sequence (x n ) R is convergent in R iff it crosses every interval (a, b) at most a finite number of times. PROOF: Note, that we always have lim inf x n lim sup x n where equality holds iff the sequence is convergent in R. Thus, the sequence is not convergent in R iff lim inf x n < lim sup x n. The last inequality means that for any a < b such that lim inf x n < a < b < lim sup x n the interval (a, b) is crossed infinitely often.

17 1.4. REAL-VALUED FUNCTIONS Real-valued functions In this section we give an overview over basic facts on real-valued functions as far these are required for understanding the ideas of integration theory Simple functions Let Ω by any set. For a subset A Ω the indicator function of A is defined to be 1 A (x) = { 1 if x A, if x A A function is a simple function if it has only finitely many different values. Every linear combination of indicator functions is a simple function. A linear combination of indicator functions is canonical if the sets supporting the indicators are a partition of Ω and the coefficients are pairwise different. PROBLEM 1.3: Show that every simple function has a uniquely determined canonical representation. PROBLEM 1.31: Let f and g be simple functions. Express the canonical representation of f + g in terms of the canonical representations of f and g. PROBLEM 1.32: Show that the set of all simple functions is a vector space (closed under linear combinations). Many facts of integration theory rely on approximation arguments where complicated functions are approximated by simple functions. There are a lot of different kinds of approximation. 1.2 DEFINITION. A sequence of functions f n : Ω R is pointwise convergent to f : Ω R if lim n f n(x) = f(x) for every x Ω. A sequence of functions f n : Ω R is uniformly convergent to f : Ω R if lim sup f n (x) f(x) =. n x Ω It is convenient to define f u = sup f(x) x Ω This is called the uniform norm (or the norm of uniform convergence).

18 8 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS A level set of a function f : Ω R is a set of the form {ω : a < f(ω) b}. 1.3 FUNDAMENTAL APPROXIMATION THEOREM. (1) Every real valued function f is the pointwise limit of simple functions based on level sets of f. (2) Every bounded real valued function f is the uniform limit of simple functions based on level sets of f. PROOF: The fundamental statement is (b). Let f. For every n N define { (k 1)/2 n whenever (k 1)/2 f n := n f < k/2 n, k = 1, 2,..., n2 n n whenever f n Then f n f. If f is bounded then (f n ) converges uniformly to f. Part (a) follows from f = f + f. PROBLEM 1.33: Draw a diagram illustrating the construction of the proof of PROBLEM 1.34: Show: If f is bounded then the approximating sequence can be chosen to be uniformly convergent. A simple functions f : [a, b] R is a step-function if its canonical partition consists of intervals (including single points). PROBLEM 1.35: Explain why every monotone function f : [a, b] R is the pointwise limit of step-functions Regulated functions 1.4 DEFINITION. A function f : [a, b) R has a limit from right at x [a, b) if for every sequence x n x the function values (f(x n )) converge to a common limit f(x+) := lim n f(x n ) R A function f : (a, b] R has a limit from left at x (a, b] if for every sequence x n x the function values (f(x n )) converge to a common limit f(x ) := lim n f(x n ) R

19 1.4. REAL-VALUED FUNCTIONS 9 Note, that function limits need not coincide with function values. 1.5 DEFINITION. A function is continuous from right (right-continuous) at x [a, b) if f(x+) = f(x) for every x [a, b). A function is continuous from left (left-continuous) at x (a, b] if f(x ) = f(x) for every x (a, b]. A function is continuous at x (a, b) if f(x+) = f(x ) = f(x). PROBLEM 1.36: Give the canonical representation of a left-continuous stepfunction. PROBLEM 1.37: Give the canonical representation of a right-continuous stepfunction. A function can be discontinuous in many ways. It may be that function limits exits but are not equal, or they are equal but don t coincide with the function value. It may also happen that function limits don t exist at all. A point where function limits exist but where f is not continuous is called a jump of f. 1.6 DEFINITION. [Regulated functions] A function is regulated on [a, b] it has limits from right on [a, b) and from left on (a, b]. Regulated functions have nice properties. 1.7 THEOREM. Let f : [a, b] R be a regulated function. Then (1) f is bounded on [a, b]. (2) All discontinuities of f are jumps. (3) For every positive number a > a regulated function can have only finitely many jumps with size exceeding a. PROBLEM 1.38: Explain via diagrams which kinds of discontinuities can happen for a regulated function. PROBLEM 1.39: Give an example of a regulated function which is neither rightcontinuous nor left-continuous. PROBLEM 1.4: Construct a regulated function with infinitely many jumps. PROBLEM 1.41: Show that a regulated function can have only countable many jumps. Regulated functions can be adjusted to be right-continuous or left-continuous. This is done by replacing the function values f(x) at every point x [a, b] by f(x+) and f(x ), respectivley. The resulting functions f + : x f(x+), f : x f(x )

20 1 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS are called the right- resp. left-continuous modifications of f. They are still regulated functions. 1.8 DEFINITION. It is cadlag (continuous from right with limits from left) if it is regulated and continuous from right on [a, b]. It is caglad (continuous from left with limits from right) if it is regulated and continuous from left on [a, b] Riemannian approximation For integration theory it is important to know how to approximate regulated functions by step-functions. Let f : [a, b] R be any function. The basic idea is to use subdivisions a = t < t 1 <... < t k = b and to define linear combinations of the form g = k f(ξ i )1 Ii i=1 where the intervals I i form an interval partition of [a, b] with separating points t i and ξ [t i 1, t i ]. Let us call such a stepfunction a Riemannian approximator of f. Of special importance are left-adjusted approximators g = and right-adjusted approximators g = k f(t i 1 )1 (ti 1,t i ] i=1 k f(t i )1 [ti 1,t i ) i=1 A sequence of subdivisions a = t < t 1 <... < t kn = b is called a Riemannian sequence of subdivisions if k n and max t i t i THEOREM. Let f : [a, b] R be a regulated function and consider a Riemannian sequence of subdivisions. (1) The sequence of left-adjusted Riemannian approximators converges pointwise to f, i.e. k f(t i 1 )1 (ti 1,t i ] f i=1 (2) The sequence of right-adjusted Riemannian approximators converges pointwise to f +, i.e. k f(t i )1 [ti 1,t i ) f + i=1

21 1.4. REAL-VALUED FUNCTIONS 11 PROBLEM 1.42: Explain via diagrams the idea of Riemannian approximation Functions of bounded variation Let f : [a, b] R be any function. 1.1 DEFINITION. The variation of f on the interval [s, t] [, T ] is sup n f(t j ) f(t j 1 ) j=1 where the supremum is taken over all subdivisions a = t < t 1 <... < t n = b and all n N. A function f is of bounded variation on [a, b] if V b a (f) <. The set of all functions of bounded variation is denoted by BV ([a, b]). PROBLEM 1.43: Show that monotone functions are in BV ([a, b]) and calculate their variation. PROBLEM 1.44: Show that BV ([a, b]) is a vector space (is stable under linear combinations). PROBLEM 1.45: Explain why BV ([a, b]) is a subset of the set of regulated functions. PROBLEM 1.46: Let f be differentiable on [a, b] with continuous derivative and finitley many critical points. Show that f BV )[a, b]) and V b a (f) = b a f (u) du PROBLEM 1.47: Show that any function f BV ([a, b]) can be written as f = g h where g, h are increasing and satisfy Va t (f) = g(t) + h(t). Hint: Let g(t) := (Va t (f) + f(t))/2 and h(t) := (Va t (f) f(t))/2. PROBLEM 1.48: Construct a continuous functions on a compact interval which is not of bounded variation.

22 12 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS 1.5 Banach spaces Let V be a vector space (a set which is closed under linear combinations) DEFINITION. A norm on V is a function v v, v V, satisfying the following conditions: (1) v, v = v = o, (2) v + w v + w, v, w V, (3) λv λ v, λ R, v V. A pair (V,. ) consisting of a vector space V and a norm. is a normed space EXAMPLE. (1) V = R is a a normed space with v = v. (2) V = R d is a normed space under several norms. E.g. v 1 = d ( d ) 1/2 v i, v 2 = vi 2 (Euclidean norm), v = max v i 1 i d i=1 i=1 (3) Let V = C([, 1]) be the set of all continuous functions f : [, 1] R. This is a vector space. Popular norms on this vector space are f = max s 1 f(s) and f 1 = 1 f(s) ds The distance of two elements of V is defined to be d(v, w) := v w This function has the usual properties of a dstance, in particular satisfies the triangle inequality. A set of the form B(v, r) := {w V : w v < r} is called an open ball around v with radius r. As sequence (v n ) V is convergent with limit v if v n v. A sequence (v n ) is a Cauchy-sequence if there exist arbitrarily small balls containing almost all members of the sequence, i.e. ɛ > N(ɛ) N such that v n v m < ɛ whenever n, m N(ɛ) 1.13 DEFINITION. A normed space is a Banach space if it is complete, i.e. if every Cauchy sequence is convergent.

23 1.6. HILBERT SPACES 13 It is clear that R and R d are complete under the usual norms. Actually they are complete under any norm. The situation is different with infinite dimensional normed spaces EXAMPLE. The space of continuous functions C([, 1]) is complete under. (under uniform convergence). However it is not complete under. 1. The latter fact is one of the reasons for extending the notion and the range of the elementary integral. 1.6 Hilbert spaces A special class of normed spaces are inner product spaces. Let V be a vector space DEFINITION. An inner product on V is a function (v, w) < v, w >, v, w V, satisfying the following conditions: (1) (v, w) < v, w > is linear in both variables, (2) < v, v), < v, v >= v = o. A pair (V, <.,. >) consisting of a vector space V and an inner product <.,. > is an inner product space. An inner product gives rise to a norm according to v :=< v, v > 1/2, v V. PROBLEM 1.49: Show that v :=< v, v > 1/2 is a norm EXAMPLE. (1) V = R is an inner product space with < v, w >= vw. The corresponding norm is v = v. (2) V = R d is an inner product space with The corresponding norm is v 2. < v, w >= d v i w i (3) Let V = C([, 1]) be the set of all continuous functions f : [, 1] R. This is an inner product space with The corresponding norm is < f, g >= 1 ( 1 f 2 = i=1 f(s)g(s) ds ) 1/2 f(s) 2 ds

24 14 CHAPTER 1. FOUNDATIONS OF MATHEMATICAL ANALYSIS 1.17 DEFINITION. An inner product space is a Hilbert space if it is complete under the norm defined by the inner product. Inner product spaces have a geometric structure which is very similar to that of R d endowed with the usual inner product. In particular, the notions of orthogonality and of projections are available on inner product spaces. The existence of orthogonal projections depends on completeness, and therefore requires Hilbert spaces. PROBLEM 1.5: Let C be a closed convex subset of an Hilbert space (V, <.,. >) and let v C. Show that there exists v C such that v v = min{ v w : w C} Hint: Let α := inf{ v w : w C} and choose a sequence (w n ) C such that v w n α. Apply the parallelogram equality to show that (w n ) is a Cauchy sequence. The following is an extension theorem which plays a central role in many parts of integration and stochastic integration. It is concerned with the extension of a linear function between two Hiblert spaces. Let (H 1, < > 1 ) and (H 2, < > 2 ) be two Hilbert spaces. Let D H 1 be a dense subspace, i.e. a subspace (closed under linear combinations) from which each element of H 1 can be reached as a limit. Assume that we have a linear function T : D H 2 which is isometric, i.e. such that T x 2 = x 1 for all x D The problem is to extend the function T from D to the whole of H THEOREM. The linear isometric function T : D H 2 can be extended in a uniquely determined way to a linear isometric function T : H 1 H 2. PROOF: Let x H 2 be an arbitray element of H 2. A natural idea to define T x is to take some sequence (x n ) H 1 such that x n x and to define T x = lim T x n. To be sure that this procedure works one has to make sure a couple of facts: (1) The definition is a valid equation for x D. (2) For every convergent sequence (x n ) H 1 the sequence (T x n ) is convergent in H 2. (3) If (x n ) and (y n ) are two sequences with the same limit in H 1 then the limits of (T x n ) and T y n ) are equal, too. (4) The function T : H 1 H 2 is linear and isometric. Details of the proof are left as PROBLEM 1.51.

25 Chapter 2 Measures and measurable functions 2.1 Sigma-fields The concept of a sigma-field Let Ω be a (non-empty) set. We are interested in systems of subsets of Ω which are closed under set operations. 2.1 EXAMPLE. In general, a system of subsets need not be closed under set operations. Let Ω = {1, 2, 3}. Consider the system of subsets A = {{1}, {2}, {3}}. This system is not closed under union, intersection or complementation. E.g. the complement of {1} is not in A. It is clear that the power set is closed under any set operations. However, there are smaller systems of sets which are closed under set operations, too. Let Ω = {1, 2, 3}. Consider the system of subsets B = {Ω,, {1}, {2, 3}}. It is easy to see that this system is closed under union, intersection and complementation. Moreover, it follows that these set operations can be repeated in arbitrary order resulting always in sets contained in A. 2.2 DEFINITION. A (non-empty) system F of subsets of Ω is called a σ-field if it closed under union, intersection and complementation als well as under building limits of monotone sequences. The pair (Ω, F) is called a measurable space. There are some obvious necessary properties of a σ-field. PROBLEM 2.1: Show that every σ-field on Ω contains and Ω. PROBLEM 2.2: What is the smallest possible σ-field on Ω? 15

26 16 CHAPTER 2. MEASURES AND MEASURABLE FUNCTIONS If we want to check whether a given system of sets is actually a σ-field then it is sufficient to verify only a minimal set of conditions. The following assertion states such a minimal set of conditions. 2.3 PROPOSITION. A (non-empty) system F of subsets of Ω is a σ-field iff it satisfies the following conditions: (1) Ω F, (2) A F A F, (3) If (A i ) i=1 F then i=1 A i F. The proof is PROBLEM How to construct sigma-fields When one starts to construct a σ-field one usually starts with a family C of sets which in any case should be contained in the σ-field. If this starting family C does not fulfil all conditions of a σ-field then a simple idea could be to add further sets until the family fulfils all required conditions. Actually, this procedure works if the starting family C is a finite system. 2.4 DEFINITION. Let C be any system of subsets on Ω. The σ-field generated by C is the smallest σ-field F which contains C. It is denoted by σ(c). PROBLEM 2.4: Assume that C = {A}. Find σ(c). PROBLEM 2.5: Assume that C = {A, B}. Find σ(c). PROBLEM 2.6: Show by giving an example that the union of two σ-field need not be a σ-field. If the system C is any finite system then σ(c) consists of all sets which can be obtained by finitely many unions, intersections and complementations of sets in C. Although the resulting system σ(c) still is finite a systematic overview over all sets could be rather complicated. Things are much easier if the generating system is a finite partition of Ω. 2.5 PROPOSITION. Assume that C is a finite partition of Ω. Then σ(c) consists of and of all unions of sets in C. The proof is PROBLEM 2.7. PROBLEM 2.8: Let Ω be a finite set. Find the σ-field which is generated by the one-point sets. PROBLEM 2.9: Show that every finite σ-field F is generated by a partition of Ω. Hint: Call a nonempty set A F an atom if it contains no nonempty proper subset in F. Show that the collection of atoms is a partition of Ω and that every set in F is a union of atoms.

27 2.2. MEASURABLE FUNCTIONS Borel sigma-fields Let us discuss σ-fields on R. Clearly, the power set of R is a σ-field. However, the power set is too large. Let us be more modest and start with a system of simple sets and then try to extend the system to a σ-field. The following example shows that such a procedure does not work if we start with one-point sets. PROBLEM 2.1: Let F be the collection of all subsets of R which are countable or are the complement of a countable set. (1) Show that F is a σ-field. (2) Show that F is the smallest σ-field which contains all one-point sets. (3) Does F contain intervals? A reasonable σ-field on R should at least contain all intervals. 2.6 DEFINITION. The smallest σ-field on R which contains all intervals is called the Borel σ-field. It is denoted by B and its elements are called Borel sets. Unfortunately, there is no way of describing all sets in B in a simple manner. All we can say is that any set which can be obtained from intervals by countably many set operations is a Borel set. E.g., every set which is the countable union of intervals is a Borel set. But there are even much more complicated sets in B. On the other hand, however, there are subsets of R which are not in B. The concept of Borel sets is easily extended to R n. 2.7 DEFINITION. The σ-field on R n which is generated by all rectangles R = {I 1 I 2 I n : I k being any interval} is called the Borel σ-field on R n and is denoted by B n. All open and all closed sets in R n are Borel sets since open sets can be represented as a countable union of rectangles and closed sets are the complements of open sets. 2.2 Measurable functions The idea of measurability 2.8 DEFINITION. A function f : (Ω, F) R defined on a measurable space is called F-measurable if the inverse images f 1 (B) = (f B) are in F for all Borel sets B B.

28 18 CHAPTER 2. MEASURES AND MEASURABLE FUNCTIONS To get an idea what measurability means let us consider some simple examples. PROBLEM 2.11: Let (Ω, F, µ) be a measure space and let f = 1 A where A Ω. Show that f is F-measurable iff A F. PROBLEM 2.12: Explain why f = 1 Q is B-measurable. PROBLEM 2.13: Let Ω, F, µ) be a measure space and let f : Ω R be a simple function. Show that f is F-measurable iff all sets of the canonical representation are in F. When we consider functions f : R R then B-measurability is called Borel measurability The basic abstract assertions The notion of measurability is not restricted to real-valued functions. 2.9 DEFINITION. A function f : (Ω, A) (Y, B) is called (A, B)-measurable if f 1 (B) A for all B B. There are two fundamental principles for dealing with measurability. The first principle says that measurability is a property which is preserved under composition of functions. 2.1 THEOREM. Let f : (Ω, A) (Y, B) be (A, B)-measurable, and let g : (Y, B) (Z, C) be (B, C)-measurable. Then g f is (A, C)-measurable. The proof is PROBLEM The second principle is concerned with checking measurability. For checking measurability of f it is sufficient to consider the sets in a generating system of the σ-field in the range of f THEOREM. Let f : (Ω, A) (Y, B) and let C be a generating system of B, i.e. B = σ(c). Then f is (A, B)-measurable iff f 1 (C) A for all C C. PROOF: Let D := {D Y : f 1 (D) A}. It can be shown that D is a σ-field. If f 1 (C) A for all C C then C D. This implies σ(c) D. The details of the proof are PROBLEM The structure of real-valued measurable functions Let (Ω, F) be a measurable space. Let L(F) denote the set of all F-measurable realvalued functions. We start with the most common and most simple criterion for check-

29 2.2. MEASURABLE FUNCTIONS 19 ing measurability of a real-valued function THEOREM. A function f : Ω R is F-measurable iff (f α) F for every α R. The proof is PROBLEM (Hint: Apply 2.11.) This theorem provides us with a lot of examples of Borel-measurable functions. PROBLEM 2.17: Show that every monotone function f : R R is Borelmeasurable. PROBLEM 2.18: Show that every continuous function f : R n R is B n - measurable. Hint: Note that (f α) is a closed set. PROBLEM 2.19: Let f : (Ω, F) R be F-measurable. Show that f +, f, f, and every polynomial a + a 1 f + + a n f n are F-measurable. Even much more is true THEOREM. Let f 1, f 2,..., f n be measurable functions. Then for every continuous function φ : R n R the composition φ(f 1, f 2,..., f n ) is measurable. It follows that applying the usual algebraic operations to measurable functions preserves measurability COROLLARY. Let f 1, f 2 be measurable functions. Then f 1 + f 2, f 1 f 2, f 1 f 2, f 1 f 2 are measurable functions. The proof is PROBLEM 2.2. As a result we see that L(F) is a space of functions where we may perform any algebraic operations without leaving the space. Thus it is a very convenient space for formal manipulations. The next assertion shows that we may even perform all of those operations involving a countable set (e.g. a sequence) of measurable functions! 2.15 THEOREM. Let (f n ) n N be a sequence of measurable functions. Then sup n f n, inf n f n are measurable functions. Let A := ( lim n f n ). Then A F and lim n f n 1 A is measurable. PROOF: Since (sup f n α) = (f n α) n n it follows from 2.12 that sup n f n and inf n f n = sup n ( f n ) are measurable. We have ( ) A := ( lim f n ) = sup inf f n = inf sup f n n k n k k n k This implies A F. The last statement follows from lim n f n = sup inf f n on A. k n k

30 2 CHAPTER 2. MEASURES AND MEASURABLE FUNCTIONS Note that the preceding corollaries are only very special examples of the power of Theorem 2.1. Roughly speaking, any function which can be written as an expression involving countable many operations with countable many measurable functions is measurable. Therefore it is rather difficult to construct non-measurable functions. Let us denote the set of all F-measurable simple functions by S(F). Clearly, all limits of simple measurable functions are measurable. The remarkable fact being fundamental for almost everything in integration theory is the converse of this statement THEOREM. (a) Every measurable function f is the limit of some sequence of simple measurable functions. (b) If f then the approximating sequence can be chosen to be increasing. This theorem is consequence of Theorem Measures The concept of measures Measures are set functions EXAMPLE. Let Ω by an arbitrary set and for any subset A Ω define { k if A contains k elements, µ(a) = A := if A contains infinitely many elements. This set function is called a counting measure. It is defined for all subsets of Ω. Obviously, it is additive, i.e. A B = µ(a B) = µ(a) + µ(b). Measures are set functions which intuitively should be related to the notion of volume. Therefore measures should be nonnegative and additive. In order to apply additivity they should be defined on systems of subsets which are closed under the usual set operations. This leads to the requirement that measures should be defined on σ-fields. Finally, if the underlying σ-field contains infinitely many sets there should be some rule how to handle limits of infinite sequences of sets.

31 2.3. MEASURES 21 Thus, we are ready for the definition of a measure DEFINITION. Let Ω be a non-empty set. A measure µ on Ω is a set function which satisfies the following conditions: (1) µ is defined on a σ-field F on Ω. (2) µ is nonnegative, i.e. µ(a), A F, and µ( ) =. (3) µ is σ-additive, i.e. for every pairwise disjoint sequence (A i ) i=1 F ( ) µ A i = i=1 µ(a i ) i=1 A measure is called finite if µ(ω) <. A measure P is called a probability measure if P (Ω) = 1. If µ F is a measure then (Ω, F, µ) is a measure space. If P F is a probability measure then (Ω, F, P ) is called a probability space. There are some obvious consequences of the preceding definition. Let µ F be a measure. PROBLEM 2.21: Every measure is additive. PROBLEM 2.22: A 1 A 2 implies µ(a 1 ) µ(a 2 ). PROBLEM 2.23: Show the inclusion-exclusion law: µ(a 1 ) + µ(a 2 ) = µ(a 1 A 2 ) + µ(a 1 A 2 ) PROBLEM 2.24: Extend the formula of the preceding problem the union of three sets. PROBLEM 2.25: Any nonnegative linear combination of measures is a measure. The property of being σ-additive both guarantees additivity and implies easy rules for handling infinite sequences of sets. Let µ F be a measure. PROBLEM 2.26: If A i A then µ(a i ) µ(a). PROBLEM 2.27: If A i A and µ(a 1 ) < then µ(a i ) µ(a). PROBLEM 2.28: Every infinite sum of measures is a measure The abstract construction of measures PROBLEM 2.29: Explain the construction of measures on a finite σ-field. Hint: Measures have to be defined for atoms only.