Construction and Analysis of Stopping Times and Belated Integrals on a Filtered Probability Space

 – Construction and Analysis of Stopping Times and Belated Integrals on a Filtered Probability Space –

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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 take values 0 and 1.

In the theory of stopping time s-algebra and its minimal elements on a filtered probability space, the s-algebra of the minimal elements of the stopping times is shown to coincide with the stopping time s-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 m – flat integrable then their product is also m-flat integrable and the integral of their product is the product of the integrals.

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TABLE OF CONTENTS

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

• Preamble………………………………………………………………………………………………………. 1
• Statement of the problem……………………………………………………………………………. 2
• Justification/ Significance of the Study……………………………………………………………. 2
• Aim and Objectives……………………………………………………………………………………….. 3
• Research Methodology…………………………………………………………………………………… 3
• Organization of the Dissertation……………………………………………………………………… 3

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

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

• Introduction………………………………………………………………………………………………….. 6
  • Functional spaces…………………………………………………………………………………………… 6
  • Uniform convergence…………………………………………………………………………………….. 6
  • Point-wise convergence………………………………………………………………………………….. 7
  • Linear operators on a normed space…………………………………………………………………. 7
  • Norm of a linear operator……………………………………………………………………………….. 8

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

• Operator Algebra…………………………………………………………………………………………… 8
  • Operators on Hilbert space……………………………………………………………………………… 9
  • Adjoint operations…………………………………………………………………………………………. 9
• An Algebra…………………………………………………………………………………………………. 10
  • C*-algebra………………………………………………………………………………………………….. 10
  • A concrete C*-algebras………………………………………………………………………………… 10
  • Non-degenerate *-algebras……………………………………………………………………………. 10
  • * -Homomorphism……………………………………………………………………………………………. 11
  • Representations……………………………………………………………………………………………….. 11
  • Cyclic representation………………………………………………………………………………………… 12
  • State……………………………………………………………………………………………………………….. 12
  • Lemma [Gelfand-Naimark-Segal (GNS)]……………………………………………………………. 13
  • Remark…………………………………………………………………………………………………………… 14
  • Weights…………………………………………………………………………………………………………… 14
  • Pullbacks…………………………………………………………………………………………………………. 14
  • Annihilators…………………………………………………………………………………………………….. 15
  • Theorem [Lipschutz, (1974)]……………………………………………………………………………… 15
  • Trace and trace class:………………………………………………………………………………………… 16
  • Affiliation……………………………………………………………………………………………………….. 16
  • Factor……………………………………………………………………………………………………………… 17
• von Neumann Algebra………………………………………………………………………………………. 18
  • Projection………………………………………………………………………………………………………… 18
  • Theorem [Tomiyama (1957)]……………………………………………………………………………… 20
  • Equivalent projections………………………………………………………………………………………. 20
  • Proposition [Blackadar (2006)]………………………………………………………………………….. 21
  • Central Projection and Support………………………………………………………………………….. 21
  • Remark…………………………………………………………………………………………………………… 22
  • Proposition [SCHRODER-BERNSTEIN]………………………………………………………….. 22
  • Abelian, finite and infinite projections………………………………………………………………… 22
  • Type and classification of von Neumann algebra………………………………………………….. 23
  • Commutative von Neumann algebras………………………………………………………………….. 24
  • Proposition [Blackadar (2006)]………………………………………………………………………….. 24
  • Purely atomic…………………………………………………………………………………………………… 25
  • Gauge space…………………………………………………………………………………………………….. 25
• Modular Theory……………………………………………………………………………………………….. 26
  • Standard form representation…………………………………………………………………………….. 26
  • Theorem [Tomita-Takesaki]……………………………………………………………………………….. 28
  • Symmetric and standard forms………………………………………………………………………….. 28
• Amplifications and Commutants………………………………………………………………………… 29
• Stochastic Calculus…………………………………………………………………………………………… 30
• Non-commutative Stochastic Integral…………………………………………………………………. 31
• Stopping Time Theory………………………………………………………………………………………. 32
• Martingale Theory……………………………………………………………………………………………. 33
• Measure Theoretic Integration……………………………………………………………………………. 33

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

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

• Introduction…………………………………………………………………………………………………….. 35
• Measures…………………………………………………………………………………………………………. 35
• s– Algebra…………………………………………………………………………………………………….. 35
• Borel Sets……………………………………………………………………………………………………….. 36
• Measure Space…………………………………………………………………………………………………. 37
  • Lebesgue-Stieltjes measure………………………………………………………………………………… 38
  • Remarks………………………………………………………………………………………………………….. 38
  • Probability measure………………………………………………………………………………………….. 39
  • Event……………………………………………………………………………………………………………… 39
  • Filtered probability space………………………………………………………………………………….. 40
  • Radon-Nykodym theorem…………………………………………………………………………………. 40
• Indicator…………………………………………………………………………………………………………. 41
• Simple Function……………………………………………………………………………………………….. 41
• Integrals………………………………………………………………………………………………………….. 41
  • Square integrability…………………………………………………………………………………………… 41
  • Ito integrals on L²…………………………………………………………………………………………….. 42
• Sample Path…………………………………………………………………………………………………….. 43
• Stochastic Processes…………………………………………………………………………………………. 43
  • Wiener process…………………………………………………………………………………………………. 44
  • Adapted process………………………………………………………………………………………………. 46
  • Separable process……………………………………………………………………………………………… 46
  • Stochastic equivalence………………………………………………………………………………………. 46
  • Indistinguishable processes……………………………………………………………………………….. 47
  • Continuity in probability processes……………………………………………………………………… 47
  • Stationery and symmetric processes……………………………………………………………………. 47
  • Periodic stochastic process………………………………………………………………………………… 48
  • Lévy process……………………………………………………………………………………………………. 48
  • Step function…………………………………………………………………………………………………… 48
  • Simple process…………………………………………………………………………………………………. 48
  • Additive process………………………………………………………………………………………………. 49
• The Clifford Calculus……………………………………………………………………………………….. 49
• Quantum Stochastic Process……………………………………………………………………………… 50
• Symmetric and Anti-symmetric Tensor Products………………………………………………….. 51
• Boson and Fermion Fock Spaces……………………………………………………………………….. 52
• Clifford Operator Algebra…………………………………………………………………………………. 54
• The LP-Martingale…………………………………………………………………………………………….. 55
• Lemma [Barnett et al, (1982)]……………………………………………………………………………. 55
• Definite Parity…………………………………………………………………………………………………. 56
• Lemma [Barnett et al, (1982)]……………………………………………………………………………. 56

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

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

• Introduction…………………………………………………………………………………………………….. 57
• Martingale……………………………………………………………………………………………………….. 57
  • Conditional …………………………………………………………………………………… 57
  • Theorem [Martingale stopping theorem]……………………………………………………………… 60
  • Examples………………………………………………………………………………………………………… 61

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

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

• Markov Processes…………………………………………………………………………………………….. 62
• Random Walk………………………………………………………………………………………………….. 63
  • Remark…………………………………………………………………………………………………………… 64
  • Example………………………………………………………………………………………………………….. 64
• Stopping Time…………………………………………………………………………………………………. 66
  • Properties of stopping times………………………………………………………………………………. 67
  • Hitting time (First passage)……………………………………………………………………………….. 69
  • Hitting times are stopping times…………………………………………………………………………. 70
  • Independent stopping time………………………………………………………………………………… 70
  • Non-stopping times (Last exit time)……………………………………………………………………. 71
  • Other stopping times………………………………………………………………………………………… 71
• The 1-0 – Process……………………………………………………………………………………………… 72
  • Càdlàg……………………………………………………………………………………………………………. 73
  • Stopping process………………………………………………………………………………………………. 73
  • Definition……………………………………………………………………………………………………….. 74
  • Stopping time σ-algebra……………………………………………………………………………………. 74
  • Minimal elements……………………………………………………………………………………………… 74
  • Definition……………………………………………………………………………………………………….. 75
  • Proposition………………………………………………………………………………………………………. 75
• Stopped Process………………………………………………………………………………………………. 78
• Stopping Time and Time Projections…………………………………………………………………… 79
  • Definition……………………………………………………………………………………………………….. 79
  • Quantum stopped process…………………………………………………………………………………. 79
  • Theorem [Barnett, et al (1996)]………………………………………………………………………….. 80
• Stopping for L²-Martingale………………………………………………………………………………… 81
  • Theorem [Barnett, et al (1996)] and [modified by Tijjani, (2001)]…………………………. 82
  • Corollary [Barnett, et al (1996)]…………………………………………………………………………. 82
  • Theorem [Barnett, et al (1996)]………………………………………………………………………….. 82
  • Theorem [Barnett, et al (1996)]………………………………………………………………………….. 83
  • Remark…………………………………………………………………………………………………………… 83
• Results and Discussion……………………………………………………………………………………… 84
  • Theorem [Fulatan, (2015)]…………………………………………………………………………………. 84
  • Theorem………………………………………………………………………………………………………….. 85
  • Theorem………………………………………………………………………………………………………….. 86

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

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

• Introduction…………………………………………………………………………………………………….. 88
• Integrals of Ito type………………………………………………………………………………………….. 89
• Local Martingale………………………………………………………………………………………………. 90
• Fundamental Theorem………………………………………………………………………………………. 91
• Integrator………………………………………………………………………………………………………… 91
• Signed Measure and Bounded Variation…………………………………………………………….. 92
  • Mcshane partitions……………………………………………………………………………………………. 93
  • Belated semivariation……………………………………………………………………………………….. 94
  • Right-belated ………………………………………………………………………………. 95
  • Lemma [Barnett and Wilde, (1986)]…………………………………………………………………… 96
  • m – Flat null……………………………………………………………………………………………………… 98
  • Integral of a simple process……………………………………………………………………………….. 98
  • Lemma……………………………………………………………………………………………………………. 99
  • Convergence……………………………………………………………………………………………………. 99
  • Control measure……………………………………………………………………………………………… 100
  • Proposition [ Barnett and Wilde (1986)]……………………………………………………………. 100
• Outer Set………………………………………………………………………………………………………. 101
• m – Flat Integration…………………………………………………………………………………………. 101
  • Theorem [ Barnett and Wilde (1986)]……………………………………………………………….. 102
  • Theorem [Barnett and Wilde (1986)]………………………………………………………………… 103
  • Essential boundedness…………………………………………………………………………………….. 104
  • Theorem [Barnett and Wilde (1986)]………………………………………………………………… 104
  • Theorem [Barnett and Wilde (1986)]………………………………………………………………… 104
• Theorem………………………………………………………………………………………………………… 105
• Results and ………………………………………………………………………………….. 106

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

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

• Summary………………………………………………………………………………………………….. 107

• Theorem [Stopping Time Process]………………………………………………………………… 107
• Theorem [Bijection between Stopping Time and Stopping Process]…………………… 107
• Theorem [On Minimal Elements of s-Algebra]………………………………………………. 108
• Theorem [On the Product of m-Flat Integrable Functions]………………………………… 108
  • Conclusion………………………………………………………………………………………………… 108
  • Recommendations……………………………………………………………………………………… 108

REFERENCES          109

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GENERAL BACKGROUND

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 t 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.

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REFERENCES

Andrzej, l. and Michael C. M. (2008).Probabilistic Properties of Deterministic Systm,. Cambridge University Press, London, pp. 105-121.

Andreas, E. K. and Juan, C. P. (2012). An Optimal Stopping Problem for Fragmentation                              Processes.       Stochastic       Processes       and       their Applications122 (4): 1210-1225.

.Applebaum, D. (2009). Universal Mallavin Calculus in Fock and Levy-Ito Spaces. Communications in Stochastic Processes, (3): 119-141

Ash, R. B. (1972). Real Analysis and Probability. Academic Press. New York, pp. 201-290

Barnett, C., Streater, R. F. and Wilde, I. F. (1982). The Ito-Clifford Integral.Journal of Functional Analysis, (48): 172-212.

Barnett, C. and Wilde, I. F. (1993). Random time, Predictable processes and Stochastic Integration in Finite von Neumann AlgebraProceedings of London Mathematical Society, (67): 355-383