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Optional Processes: Theory and Applications [Kõva köide]

, (University of Alberta, Edmonton, Canada)
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It is well-known that modern stochastic calculus has been exhaustively developed under usual conditions. Despite such a well-developed theory, there is evidence to suggest that these very convenient technical conditions cannot necessarily be fulfilled in real-world applications.

Optional Processes: Theory and Applications seeks to delve into the existing theory, new developments and applications of optional processes on "unusual" probability spaces. The development of stochastic calculus of optional processes marks the beginning of a new and more general form of stochastic analysis.

This book aims to provide an accessible, comprehensive and up-to-date exposition of optional processes and their numerous properties. Furthermore, the book presents not only current theory of optional processes, but it also contains a spectrum of applications to stochastic differential equations, filtering theory and mathematical finance.

Features

    • Suitable for graduate students and researchers in mathematical finance, actuarial science, applied mathematics and related areas

    • Compiles almost all essential results on the calculus of optional processes in unusual probability spaces

    • Contains many advanced analytical results for stochastic differential equations and statistics pertaining to the calculus of optional processes

    • Develops new methods in finance based on optional processes such as a new portfolio theory, defaultable claim pricing mechanism, etc.

      • Authors

      Mohamed Abdelghani completed his PhD in mathematical finance from the University of Alberta, Edmonton, Canada. He is currently working as a vice president in quantitative finance and machine learning at Morgan Stanley, New York, USA.

      Alexander Melnikov

      is a professor in mathematical finance at the University of Alberta. His research interests belong to the area of contemporary stochastic analysis and its numerous applications in mathematical finance, statistics and actuarial science. He has written six books as well as over 100 research papers in leading academic journals.

    Arvustused

    "Modern stochastics is usually identified with stochastic analysis, a field in mathematics that is well-developed under "usual conditions". Hence, a variety of results of this theory and its applications are also restricted by these technical conditions. Many examples from theory and applications call for further extensions of stochastic analysis. Optional Processes: Theory and Applications is first attempt of such natural extension.

    The authors provide an excellent treatment of papers written in the 1970s and 1980s by Dellacherie, Doob, Galtchouk, Lepingle, and Lenglart among others. Moreover, the authors develop this topic in a comprehensive manner, and while doing so offer beautiful applications to the fields of mathematical finance and filtering theory. This book will be extremely useful for experts in the area of stochastic analysis, mathematical finance, and related fields."

    Svetlozar Rachev, Texas Tech University

    "The usual analysis of stochastic processes in continuous time is developed in a framework of a filtered probability space satisfying the usual conditions. That is, the flow of information modeled by the filtration is assumed to be right continuous. Situations arise where this condition is not satisfied. This important book develops a theory of stochastic processes where unusual conditions are assumed to hold. These have applications in quantitative finance and elsewhere."

    Robert J. Elliott, Faculty Professor and Emeritus Professor at University of Calgary and Research Professor at University of South Australia "Modern stochastics is usually identified with stochastic analysis, a field in mathematics that is well-developed under "usual conditions". Hence, a variety of results of this theory and its applications are also restricted by these technical conditions. Many examples from theory and applications call for further extensions of stochastic analysis. Optional Processes: Theory and Applications is first attempt of such natural extension.

    The authors provide an excellent treatment of papers written in the 1970s and 1980s by Dellacherie, Doob, Galtchouk, Lepingle, and Lenglart among others. Moreover, the authors develop this topic in a comprehensive manner, and while doing so offer beautiful applications to the fields of mathematical finance and filtering theory. This book will be extremely useful for experts in the area of stochastic analysis, mathematical finance, and related fields."

    Svetlozar Rachev, Texas Tech University

    "The usual analysis of stochastic processes in continuous time is developed in a framework of a filtered probability space satisfying the usual conditions. That is, the flow of information modeled by the filtration is assumed to be right continuous. Situations arise where this condition is not satisfied. This important book develops a theory of stochastic processes where unusual conditions are assumed to hold. These have applications in quantitative finance and elsewhere."

    Robert J. Elliott, Faculty Professor and Emeritus Professor at University of Calgary and Research Professor at University of South Australia

    Preface ix
    Introduction xi
    1 Spaces, Laws and Limits
    		1(28)
    1.1 Foundation
    		1(2)
    1.2 Measurable Spaces, Random Variables and Laws
    		3(12)
    1.2.1 Measurable spaces
    		3(1)
    1.2.2 Measurable functions
    		4(1)
    1.2.3 Atoms and separable fields
    		5(1)
    1.2.4 The case of real-valued random variables
    		5(1)
    1.2.5 Monotone class theorem
    		6(1)
    1.2.6 Probability and expectation
    		7(2)
    1.2.6.1 Convergence of random variables
    		9(1)
    1.2.6.2 Fubini's Theorem
    		9(1)
    1.2.6.3 Uniform integrability
    		10(1)
    1.2.6.4 Completion of probability spaces
    		11(1)
    1.2.6.5 Independence
    		12(1)
    1.2.6.6 Conditional expectation
    		12(3)
    1.3 Analytic Set Theory
    		15(14)
    1.3.1 Paving and analytic sets
    		15(2)
    1.3.2 Separable sets
    		17(2)
    1.3.3 Lusin and Souslin spaces
    		19(2)
    1.3.3.1 Souslin-Lusin Theorem
    		21(1)
    1.3.4 Capacities and Choquet's Theorem
    		21(3)
    1.3.4.1 Constructing capacities
    		24(2)
    1.3.5 Theorem of cross-section
    		26(3)
    2 Stochastic Processes
    		29(60)
    2.1 Construction
    		29(5)
    2.1.1 Time law
    		31(1)
    2.1.2 Canonical process
    		32(2)
    2.2 Processes on Filtrations
    		34(2)
    2.2.1 Adapted processes
    		34(1)
    2.2.2 Progressive measurability
    		35(1)
    2.3 Paths Properties
    		36(23)
    2.3.1 Processes on dense sets
    		37(2)
    2.3.2 Upcrossings and downcrossings
    		39(3)
    2.3.3 Separability
    		42(3)
    2.3.3.1 Doob's separability theorems
    		45(2)
    2.3.4 Progressive processes of random sets
    		47(2)
    2.3.5 Almost equivalence
    		49(5)
    2.3.5.1 Pseudo-Paths
    		54(5)
    2.4 Random Times
    		59(27)
    2.4.1 Stopping times
    		60(3)
    2.4.2 Basic properties of stopping times
    		63(3)
    2.4.3 Stochastic intervals
    		66(1)
    2.4.4 Optional and predictable er-fields
    		66(5)
    2.4.5 Predictable stopping times
    		71(6)
    2.4.6 Classification of stopping times
    		77(2)
    2.4.7 Quasi-left-continuous nitrations
    		79(1)
    2.4.8 Optional and predictable cross-sections
    		80(6)
    2.5 Optional and Predictable Processes
    		86(3)
    3 Martingales
    		89(50)
    3.1 Discrete Parameter Martingales
    		89(20)
    3.1.1 Basic properties
    		90(1)
    3.1.2 Right and left closed supermartingales
    		91(1)
    3.1.3 Doob's stopping theorem
    		92(2)
    3.1.3.1 Extension to unbounded stopping times
    		94(3)
    3.1.4 Fundamental inequalities
    		97(1)
    3.1.4.1 Maximal lemma
    		97(2)
    3.1.4.2 Domination in Lp
    		99(1)
    3.1.4.3 Martingales upcrossings and downcrossings
    		100(2)
    3.1.5 Convergence and decomposition theorems
    		102(1)
    3.1.5.1 Almost sure convergence of supermartingales
    		103(1)
    3.1.5.2 Uniform integrability and martingale convergence
    		104(1)
    3.1.5.3 Riesz decompositions of supermartingales
    		105(2)
    3.1.5.4 Krickeberg decomposition of martingales
    		107(1)
    3.1.6 Some applications of convergence theorems
    		108(1)
    3.2 Continuous Parameter Martingales
    		109(30)
    3.2.1 Supermartingales on countable sets
    		109(1)
    3.2.1.1 Fundamental inequalities
    		110(1)
    3.2.1.2 Existence of right and left limits
    		111(3)
    3.2.2 Right-continuous supermartingale
    		114(3)
    3.2.3 Projections theorems
    		117(10)
    3.2.4 Decomposition of supermartingales
    		127(1)
    3.2.4.1 Functional analytic decomposition theorem
    		127(4)
    3.2.4.2 Extension to non-positive functionals
    		131(1)
    3.2.4.3 Decomposition of positive supermartingale of class D
    		132(4)
    3.2.4.4 The general case of Doob decomposition
    		136(3)
    4 Strong Supermartingales
    		139(20)
    4.1 Introduction
    		139(4)
    4.2 Projection Theorems
    		143(6)
    4.3 Special Inequalities
    		149(5)
    4.4 Mertens Decomposition
    		154(1)
    4.5 Snell Envelope
    		155(4)
    5 Optional Martingales
    		159(42)
    5.1 Introduction
    		159(4)
    5.2 Existence and Uniqueness
    		163(3)
    5.3 Increasing and Finite Variation Processes
    		166(5)
    5.3.1 Integration with respect to increasing and finite variation processes
    		167(2)
    5.3.2 Dual projections
    		169(2)
    5.4 Decomposition Results
    		171(13)
    5.4.1 Decomposition of elementary processes
    		172(5)
    5.4.2 Decomposition of optional martingales
    		177(7)
    5.5 Quadratic Variation
    		184(5)
    5.5.1 Predictable and optional
    		184(3)
    5.5.2 Kunita-Watanabe inequalities
    		187(2)
    5.6 Optional Stochastic Integral
    		189(12)
    5.6.1 Integral with respect to square integrable martingales
    		189(3)
    5.6.2 Integral with respect to martingales with integrable variation
    		192(2)
    5.6.3 Integration with respect to local optional martingales
    		194(7)
    6 Optional Supermartingales Decomposition
    		201(14)
    6.1 Introduction
    		201(1)
    6.2 Riesz Decomposition
    		202(2)
    6.3 Doob-Meyer-Galchuk Decomposition
    		204(11)
    6.3.1 Decomposition of DL class
    		214(1)
    7 Calculus of Optional Semimartingales
    		215(28)
    7.1 Integral with Respect to Optional Semimartingales
    		215(2)
    7.2 Formula for Change of Variables
    		217(6)
    7.3 Stochastic Integrals of Random Measures
    		223(6)
    7.4 Semimartingales and Their Characteristics
    		229(7)
    7.4.1 Canonical representation
    		229(3)
    7.4.2 Component representation
    		232(4)
    7.5 Uniform Doob-Meyer Decompositions
    		236(7)
    7.5.1 Supporting lemmas
    		241(2)
    8 Optional Stochastic Equations
    		243(38)
    8.1 Linear Equations, Exponentials and Logarithms
    		243(9)
    8.1.1 Stochastic exponential
    		244(3)
    8.1.2 Stochastic logarithm
    		247(2)
    8.1.3 Nonhomogeneous linear equation
    		249(2)
    8.1.4 Gronwall lemma
    		251(1)
    8.2 Existence and Uniqueness of Solutions of Optional Stochastic Equations
    		252(17)
    8.2.1 Stochastic equation with monotonicity condition
    		253(2)
    8.2.2 Existence and uniqueness results
    		255(2)
    8.2.2.1 Uniqueness
    		257(2)
    8.2.2.2 Existence
    		259(9)
    8.2.3 Remarks and applications
    		268(1)
    8.3 Comparison of Solutions of Optional Stochastic Equations
    		269(12)
    8.3.1 Comparison theorem
    		270(9)
    8.3.2 Remarks and applications
    		279(2)
    9 Optional Financial Markets
    		281(26)
    9.1 Introduction
    		281(1)
    9.2 Market Model
    		282(1)
    9.3 Martingale Deflators
    		283(8)
    9.3.1 The case of stochastic exponentials
    		284(4)
    9.3.2 The case of stochastic logarithms
    		288(3)
    9.4 Pricing and Hedging
    		291(2)
    9.5 Absence of Arbitrage
    		293(4)
    9.6 Examples of Special Cases
    		297(10)
    9.6.1 Ladlag jumps diffusion model
    		297(1)
    9.6.1.1 Computing a local martingale deflator
    		298(1)
    9.6.1.2 Pricing of a European call option
    		299(3)
    9.6.1.3 Hedging of a European call option
    		302(1)
    9.6.2 Basket of stocks
    		303(4)
    9.6.3 Dcfaultable bond and a stock
    		307(1)
    10 Defaultable Markets on t/nusual Space
    		307(1)
    10.1 Introduction
    		307(3)
    10.2 Optional Default
    		310(1)
    10.3 Defaultable Cash-Flow
    		311(4)
    10.3.1 Portfolio with default
    		313(2)
    10.4 Probability of Default
    		315(2)
    10.5 Valuation of Defaultable Cash-Flow and Examples
    		317(8)
    11 Filtering of Optional Semimartingales
    		325(42)
    11.1 The Filtering Problem
    		325(1)
    11.2 The Usual Case of Optimal Filtering
    		326(32)
    11.2.1 Auxiliary results
    		326(9)
    11.2.2 Martingales' integral representation
    		335(13)
    11.2.3 Filtering of cadlag semimartingales
    		348(10)
    11.3 The t/nusual case of Optimal Filtering
    		358(1)
    11.3.1 Filtering on unusual stochastic, basis
    		358(2)
    11.3.2 Filtering on mixed stochastic basis
    		360(2)
    11.4 Filtering in Finance
    		362(5)
    Bibliography 367(10)
    Index 377
    Mohamed Abdelghani completed his PhD in Mathematical Finance from the University of Alberta. He is currently working as a V.P. in quantitative finance and machine learning at Morgan Stanley, New York, USA.

    Alexander Melnikov is a Professor in Mathematical Finance at the University of Alberta, Edmonton, Canada. His research interests belong to the area of contemporary stochastic analysis and its numerous applications in Mathematical Finance, Statistics and Actuarial Science. He has written six books as well as over one hundred research papers in leading academic journals.