## ABSTRACT

Using an intuitive definition of a 1-0-process, a bijection is established between stopping times and adapted processes that are non-decreasing and takevalues 0 and 1. In the theory of stopping time -algebra and its minimal elements on a filtered probability space, the -algebra of the minimal elements of the stopping times is shown to coincide with the stopping time -algebra. A defined stochastic process relative to a stopping time is proved to be a stopped process.In the belated integral theory, it is established that if two processes are – flat integrable then their product is also -flat integrable and the integral of their product is the product of the integrals.

## TABLE OF CONTENTS

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DECLARATION ………………………………………………………………………………………………. i

CERTIFICATION ……………………………………………………………………………………………. ii

ACKNOWLEDGEMENT………………………………………………………………………………… iii

DEDICATION …………………………………………………………………………………………………. v

TABLE OF CONTENTS …………………………………………………………………………………. vi

LIST OF FIGURES ………………………………………………………………………………………… xii

ABBREVIATIONS DEFINITIONS AND SYMBOLS …………………………………….. xiii

ABSTRACT …………………………………………………………………………………………………… xvi

CHAPTER ONE ………………………………………………………………………………………………. 1

GENERAL BACKGROUND ……………………………………………………………………………. 1

1.1 Preamble …………………………………………………………………………………………………. 1

1.2 Statement of the problem ……………………………………………………………………………. 2

1.3 Justification/ Significance of the Study ……………………………………………………………. 2

1.4 Aim and Objectives ………………………………………………………………………………………. 3

1.5 Research Methodology ………………………………………………………………………………….. 3

1.6 Organization of the Dissertation ……………………………………………………………………… 3

CHAPTER TWO ……………………………………………………………………………………………… 6

LITERATURE REVIEW …………………………………………………………………………………. 6

2.1 Introduction ………………………………………………………………………………………………….. 6

2.1.1 Functional spaces ……………………………………………………………………………………….. 6

2.1.2 Uniform convergence …………………………………………………………………………………. 6

2.1.3 Point-wise convergence ………………………………………………………………………………. 7

2.1.4. Linear operators on a normed space …………………………………………………………….. 7

2.1.5. Norm of a linear operator……………………………………………………………………………. 8

2.1.6 Strong and weak convergence ……………………………………………………………………… 8

2.2 Operator Algebra ………………………………………………………………………………………….. 8

2.2.1 Operators on Hilbert space ………………………………………………………………………….. 9

2.2.2 Adjoint operations………………………………………………………………………………………. 9

2.3 An Algebra …………………………………………………………………………………………………. 10

2.3.1 C*-algebra ……………………………………………………………………………………………….. 10

2.3.2 A concrete C*-algebras ……………………………………………………………………………… 10

2.3.3 Non-degenerate *-algebras ………………………………………………………………………… 10

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2.3.4 * -Homomorphism ……………………………………………………………………………………. 11

2.3.5 Representations ………………………………………………………………………………………… 11

2.3.6 Cyclic representation ………………………………………………………………………………… 12

2.3.7 State ……………………………………………………………………………………………………….. 12

2.3.8 Lemma [Gelfand-Naimark-Segal (GNS)] ……………………………………………………. 13

2.3.9 Remark ……………………………………………………………………………………………………. 14

2.3.10 Weights …………………………………………………………………………………………………. 14

2.3.11 Pullbacks ……………………………………………………………………………………………….. 14

2.3.12 Annihilators …………………………………………………………………………………………… 15

2.3.13 Theorem [Lipschutz, (1974)] ……………………………………………………………………. 15

2.3.14 Trace and trace class: ………………………………………………………………………………. 16

2.3.15 Affiliation………………………………………………………………………………………………. 16

2.3.16 Factor ……………………………………………………………………………………………………. 17

2.4 von Neumann Algebra …………………………………………………………………………………. 18

2.4.1 Projection ………………………………………………………………………………………………… 18

2.4.2 Theorem [Tomiyama (1957)] …………………………………………………………………….. 20

2.4.3 Equivalent projections ………………………………………………………………………………. 20

2.4.4 Proposition [Blackadar (2006)] ………………………………………………………………….. 21

2.4.5 Central Projection and Support …………………………………………………………………… 21

2.4.6 Remark ……………………………………………………………………………………………………. 22

2.4.7 Proposition [SCHRODER-BERNSTEIN] ……………………………………………………. 22

2.4.8 Abelian, finite and infinite projections ………………………………………………………… 22

2.4.9 Type and classification of von Neumann algebra ………………………………………….. 23

2.4.10 Commutative von Neumann algebras ………………………………………………………… 24

2.4.11 Proposition [Blackadar (2006)] ………………………………………………………………… 24

2.4.12 Purely atomic …………………………………………………………………………………………. 25

2.4.13 Gauge space …………………………………………………………………………………………… 25

2.5 Modular Theory ………………………………………………………………………………………….. 26

2.5.1 Standard form representation ……………………………………………………………………… 26

2.5.2 Theorem [Tomita-Takesaki] ………………………………………………………………………. 28

2.5.3 Symmetric and standard forms …………………………………………………………………… 28

2.6 Amplifications and Commutants …………………………………………………………………… 29

2.7 Stochastic Calculus ……………………………………………………………………………………… 30

2.8 Non-commutative Stochastic Integral ……………………………………………………………. 31

2.9 Stopping Time Theory …………………………………………………………………………………. 32

2.10 Martingale Theory …………………………………………………………………………………….. 33

2.11 Measure Theoretic Integration …………………………………………………………………….. 33

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CHAPTER THREE ………………………………………………………………………………………… 35

MEASURES AND STOCHASTIC PROCESSES …………………………………………….. 35

3.1 Introduction ………………………………………………………………………………………………… 35

3.2 Measures ……………………………………………………………………………………………………. 35

3.3 – Algebra ……………………………………………………………………………………………….. 35

3.4 Borel Sets …………………………………………………………………………………………………… 36

3.5 Measure Space ……………………………………………………………………………………………. 37

3.5.1 Lebesgue-Stieltjes measure………………………………………………………………………… 38

3.5.2 Remarks ………………………………………………………………………………………………….. 38

3.5.3 Probability measure ………………………………………………………………………………….. 39

3.5.4 Event ………………………………………………………………………………………………………. 39

3.5.5 Filtered probability space…………………………………………………………………………… 40

3.5.6 Radon-Nykodym theorem………………………………………………………………………….. 40

3.6 Indicator …………………………………………………………………………………………………….. 41

3.7 Simple Function ………………………………………………………………………………………….. 41

3.8 Integrals …………………………………………………………………………………………………….. 41

3.8.1 Square integrability …………………………………………………………………………………… 41

3.8.2 Ito integrals on L2 ……………………………………………………………………………………… 42

3.9 Sample Path ……………………………………………………………………………………………….. 43

3.10 Stochastic Processes ………………………………………………………………………………….. 43

3.10.1 Wiener process ……………………………………………………………………………………….. 44

3.10.2 Adapted process ……………………………………………………………………………………… 46

3.10.3 Separable process ……………………………………………………………………………………. 46

3.10.4 Stochastic equivalence …………………………………………………………………………….. 46

3.10.5 Indistinguishable processes………………………………………………………………………. 47

3.10.6 Continuity in probability processes …………………………………………………………… 47

3.10.7 Stationery and symmetric processes ………………………………………………………….. 47

3.10.8 Periodic stochastic process ………………………………………………………………………. 48

3.10.9 Lévy process ………………………………………………………………………………………….. 48

3.10.10 Step function ………………………………………………………………………………………… 48

3.10.11 Simple process ……………………………………………………………………………………… 48

3.10.12 Additive process …………………………………………………………………………………… 49

3.11 The Clifford Calculus ………………………………………………………………………………… 49

3.12 Quantum Stochastic Process ……………………………………………………………………….. 50

3.13 Symmetric and Anti-symmetric Tensor Products …………………………………………… 51

3.14 Boson and Fermion Fock Spaces …………………………………………………………………. 52

3.15 Clifford Operator Algebra ………………………………………………………………………….. 54

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3.16 The

P L -Martingale …………………………………………………………………………………….. 55

3.17 Lemma [Barnett et al, (1982)] …………………………………………………………………….. 55

3.18 Definite Parity …………………………………………………………………………………………… 56

3.19 Lemma [Barnett et al, (1982)] …………………………………………………………………….. 56

CHAPTER FOUR…………………………………………………………………………………………… 57

STOPPING TIMES ON FILTERED PROBABILITY SPACE ………………………… 57

4.1 Introduction ………………………………………………………………………………………………… 57

4.2 Martingale ………………………………………………………………………………………………….. 57

4.2.1 Conditional expectation. ……………………………………………………………………………. 57

4.2.2 Theorem [Martingale stopping theorem] ……………………………………………………… 60

4.2.3 Examples …………………………………………………………………………………………………. 61

4.2.4 Theorem [Jan, (2013)] ………………………………………………………………………………. 62

4.2.5 Theorem [Doob-Meyer‟s decomposition for discrete sub-martingale] …………….. 62

4.3 Markov Processes ……………………………………………………………………………………….. 62

4.4 Random Walk …………………………………………………………………………………………….. 63

4.4.1Remark …………………………………………………………………………………………………….. 64

4.4.2 Example ………………………………………………………………………………………………….. 64

4.5 Stopping Time…………………………………………………………………………………………….. 66

4.5.1 Properties of stopping times ………………………………………………………………………. 67

4.5.2 Hitting time (First passage)………………………………………………………………………… 69

4.5.3 Hitting times are stopping times …………………………………………………………………. 70

4.5.4 Independent stopping time …………………………………………………………………………. 70

4.5.5 Non-stopping times (Last exit time) ……………………………………………………………. 71

4.5.6 Other stopping times …………………………………………………………………………………. 71

4.6 The 1-0 – Process ………………………………………………………………………………………… 72

4.6.1 Càdlàg …………………………………………………………………………………………………….. 73

4.6.2 Stopping process ………………………………………………………………………………………. 73

4.6.3 Definition ………………………………………………………………………………………………… 74

4.6.4 Stopping time σ-algebra …………………………………………………………………………….. 74

4.6.5 Minimal elements …………………………………………………………………………………….. 74

4.6.6 Definition ………………………………………………………………………………………………… 75

4.6.7 Proposition ………………………………………………………………………………………………. 75

4.7 Stopped Process ………………………………………………………………………………………….. 78

4.8 Stopping Time and Time Projections …………………………………………………………….. 79

4.8.1 Definition ………………………………………………………………………………………………… 79

4.8.2 Quantum stopped process ………………………………………………………………………….. 79

4.8.3 Theorem [Barnett, et al (1996)] ………………………………………………………………….. 80

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4.9 Stopping for L2-Martingale …………………………………………………………………………… 81

4.9.1 Theorem [Barnett, et al (1996)] and [modified by Tijjani, (2001)] …………………. 82

4.9.2 Corollary [Barnett, et al (1996)] …………………………………………………………………. 82

4.9.3 Theorem [Barnett, et al (1996)] ………………………………………………………………….. 82

4.9.4 Theorem [Barnett, et al (1996)] ………………………………………………………………….. 83

4.9.5 Remark ……………………………………………………………………………………………………. 83

4.10 Results and Discussion ………………………………………………………………………………. 84

4.10.1 Theorem [Fulatan, (2015)] ……………………………………………………………………….. 84

4.10.2 Theorem ………………………………………………………………………………………………… 85

4.10.3 Theorem ………………………………………………………………………………………………… 86

CHAPTER FIVE ……………………………………………………………………………………………. 88

THE BELATED INTEGRALS ……………………………………………………………………….. 88

5.1 Introduction ………………………………………………………………………………………………… 88

5.2 Integrals of Ito type ……………………………………………………………………………………… 89

5.3 Local Martingale …………………………………………………………………………………………. 90

5.4 Fundamental Theorem …………………………………………………………………………………. 91

5.5 Integrator ……………………………………………………………………………………………………. 91

5.6 Signed Measure and Bounded Variation ………………………………………………………… 92

5.6.1 Mcshane partitions ……………………………………………………………………………………. 93

5.6.1 Belated semivariation ……………………………………………………………………………….. 94

5.6.2 Right-belated semivariation. ………………………………………………………………………. 95

5.6.3 Lemma [Barnett and Wilde, (1986)]……………………………………………………………. 96

5.6.4 – Flat null. …………………………………………………………………………………………….. 98

5.6.5 Integral of a simple process ……………………………………………………………………….. 98

5.6.6 Lemma ……………………………………………………………………………………………………. 99

5.6.7 Convergence ……………………………………………………………………………………………. 99

5.6.8 Control measure ……………………………………………………………………………………… 100

5.6.9 Proposition [ Barnett and Wilde (1986)] ……………………………………………………. 100

5.7 Outer Set ………………………………………………………………………………………………….. 101

5.8 – Flat Integration …………………………………………………………………………………….. 101

5.8.1 Theorem [ Barnett and Wilde (1986)] ……………………………………………………….. 102

5.8.2 Theorem [Barnett and Wilde (1986)] ………………………………………………………… 103

5.8.3 Essential boundedness …………………………………………………………………………….. 104

5.8.4 Theorem [Barnett and Wilde (1986)] ………………………………………………………… 104

5.8.5 Theorem [Barnett and Wilde (1986)] ………………………………………………………… 104

5.9 Theorem …………………………………………………………………………………………………… 105

5.10 Results and discussions. ……………………………………………………………………………. 106

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CHAPTER SIX …………………………………………………………………………………………….. 107

SUMMARY CONCLUSION AND RECOMMENDATION ……………………………. 107

6.1 Summary ………………………………………………………………………………………………….. 107

(a) Theorem [Stopping Time Process] ……………………………………………………………….. 107

(b)Theorem [Bijection between Stopping Time and Stopping Process] …………………. 107

(c) Theorem [On Minimal Elements of -Algebra] …………………………………………….. 108

(d)Theorem [On the Product of -Flat Integrable Functions] ……………………………….. 108

6.2 Conclusion ……………………………………………………………………………………………….. 108

6.3 Recommendations ……………………………………………………………………………………… 108

REFERENCES……………………………………………………………………………………………… 109

## CHAPTER ONE

GENERAL BACKGROUND

1.1 Preamble

In the seventh century, the theory of probability began in an attempt to calculate the

odds of winning in certain games of chance. Mathematicians, in the middle of twentieth

century, developed general techniques for maximizing the chances of beating a casino

or winning against an intelligent opponent. There is a leavable gambling problems, in

which a player can halt a play at anytime, and unleavable problems, in which a player is

compelled to continue playing forever, Doob, (1971). In a leavable problem, a player

must choose, in addition to a strategy, a rule for stopping. In essence, a decision to stop

at anytime will be allowed to depend on the partial history of states up to that time but

not beyond it. So, a stopping time is thus a mapping from the set of histories. In

probability theory, a stopping time is often defined by a stopping rule, a mechanism for

deciding whether to continue or to stop a process on the basis of the present position and

past events, and which will always lead to a decision to stop at some finite time.

On the other hand, an integration theory relating to the non-commutative stochastic

integral is set up by using a particular field consisting of finite unions of intervals. This

integration called the belated integral will be over all , since the measure involved

will be bounded.

2

1.2 Statement of the problem

In any classical stopping time defined on a filtered probability space, there may be a defined stopping process associated with such time. Here, there is the problem of establishing the one-one correspondence between the stopping time and the associated stopping process if it exists.On the same filtered probability space, there is the problem of construction of minimal -algebra of stopping time and its relation with the -algebra of the stopping time itself.In the algebra of the continuous and bounded linear operators, commutativity is not always guaranteed. Here, there is the problem of construction of two processes whose -flat integrals of product is equal to the product of their -flat integrals. Basically, this study is undertaken with a view to answer certain problems emanating from the literature of stopping times and their algebras. Investigation is carried out in the theories of belated integrals. For the above purposes background surveys of the theories are necessary. Primarily, from the review of the literature in chapter two and the studies of measures and processes in chapter three, results are established in chapters four and five.

1.3 Justification/ Significance of the Study

Scholars such as Sinelnikov, (2012) and other contemporaries approached the theory of 1-0-process in a different construction. It is wished that a different approach can be taken to arrive at another construction. Also,followingBarnett and Wilde, (1986)and the improved McShane‟s division by Toh and Chew, (1999), some analysis and construction are possible.

3

1.4 Aim and Objectives

The main aim of this work is to study existing theories of stopping times, their algebras and contribute to the literatures thereof through the establishment of new results. This can be achieved through the following objectives:

1. To construct a stochastic process relative to a stopping time so that such a stochastic process can be a stopped process.

2. To construct a bijection between a stopping time and a relative stopping process.

3. To analyze the algebra of stopping times,construct the algebra of its minmal elements and compare the collection with the -algebra as a whole.

4. To construct two essentially bounded -flat integrable processes evaluate the integrability of their product.

1.5 Research Methodology

The methodology adopted to realize the above aim and set objectives is such that in the first case the use of characteristic function is carried out to define a stochastic process relative to a stopping time and later it is shown that it is a stopping process. In the second, the intuitive definition of 1-0-process is used to construct the bijective relation between a stopping time and a stopping process. Similar technique is employed with the use of -algebra of stopping times to obtain the desired result. -flat integrability is considered upon an essentially bounded process and a new relationship is established.

1.6 Organization of the Dissertation

This dissertation consists of six chapters with the first concentrating basically on the statements of the problems, aim and objectives and the significance of the studies. The

4

second chapter is primarily the literature review, definitions and other mathematical tools relevant to the research. Measures and stochastic processes are studied in chapter three. Measures of the Riemann Stieltjes type, probability measures and the various stochastic processes are studied. Definition of the stochastic processes culminated into the definition of the stochasticintegrals, sometimes called the Ito integrals are provided. The stochasticity in the Hilbert space together with the concept of Clifford calculus, where some operators are chosen to behave in somewhat the way as in the classical theory has been studied. Stochasticity in quantum sense, that is, the non-commutativity situation which is later used to define integration in the same sense is also attended to. Boson and Fermion Fock spaces whose definitions centered around the concept of direct sum and tensor product has also been discussed.The stochastic processes which are families of random variables on a probability space referred to as commutative processes, that is the classical case, and those which are families of (possibly unbounded) operators on Hilbert space referred to as non-commutative processes, the no-classical case are also studied. The constructions leading to the Ito-Clifford integrals especially by Barnett et al (1982) centered on the square integrable L2 spaces are studied. The generalization of such construction by Carlen and Kree (1998) to L1is also studied. Stopping times are studied in chapter four. Both the classical and the quantum theories are considered. Constructions are put in place in the classical theories. Conditional expectations and the concept of martingales are redefined in quantum sense. In the quantum theory, it is noted that stopping times or random times are projection-valued function adapted to a filtered von Neumann algebra.

5

A study of belated integrals is undertaken in chapter five, where the theoretically admissible integrals that are rather narrow are widened.A consideration of the right belated variation on Borel sets rather than the usual left-handed open intervals is discussed. In this way, the theory is expanded and a result is proved.

Summary of the whole dissertation and its conclusion follow in chapter six together with recommendations. Some contributions are alsohighlighted while some areas of possible research are enlisted.

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