Optional Processes ; Theory and Applications (Chapman and Hall/CRC Financial Mathematics Series)

Description

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. Cover Half Title Series Page Title Page Copyright Page Contents Preface Introduction 1. Spaces, Laws and Limits 1.1 Foundation 1.2 Measurable Spaces, Random Variables and Laws 1.2.1 Measurable spaces 1.2.2 Measurable functions 1.2.3 Atoms and separable fields 1.2.4 The case of real-valued random variables 1.2.5 Monotone class theorem 1.2.6 Probability and expectation 1.2.6.1 Convergence of random variables 1.2.6.2 Fubini's Theorem 1.2.6.3 Uniform integrability 1.2.6.4 Completion of probability spaces 1.2.6.5 Independence 1.2.6.6 Conditional expectation 1.3 Analytic Set Theory 1.3.1 Paving and analytic sets 1.3.2 Separable sets 1.3.3 Lusin and Souslin spaces 1.3.3.1 Souslin-Lusin Theorem 1.3.4 Capacities and Choquet's Theorem 1.3.4.1 Constructing capacities 1.3.5 Theorem of cross-section 2. Stochastic Processes 2.1 Construction 2.1.1 Time law 2.1.2 Canonical process 2.2 Processes on Filtrations 2.2.1 Adapted processes 2.2.2 Progressive measurability 2.3 Paths Properties 2.3.1 Processes on dense sets 2.3.2 Upcrossings and downcrossings 2.3.3 Separability 2.3.3.1 Doob's separability theorems 2.3.4 Progressive processes of random sets 2.3.5 Almost equivalence 2.3.5.1 Pseudo-Paths 2.4 Random Times 2.4.1 Stopping times 2.4.2 Basic properties of stopping times 2.4.3 Stochastic intervals 2.4.4 Optional and predictable - fields 2.4.5 Predictable stopping times 2.4.6 Classification of stopping times 2.4.7 Quasi-left-continuous filtrations 2.4.8 Optional and predictable cross-sections 2.5 Optional and Predictable Processes 3. Martingales 3.1 Discrete Parameter Martingales 3.1.1 Basic properties 3.1.2 Right and left closed supermartingales 3.1.3 Doob's stopping theorem 3.1.3.1 Extension to unbounded stopping times 3.1.4 Fundamental inequalities 3.1.4.1 Maximal lemma 3.1.4.2 Domination in Lp 3.1.4.3 Martingales upcrossings and downcrossings 3.1.5 Convergence and decomposition theorems 3.1.5.1 Almost sure convergence of supermartingales 3.1.5.2 Uniform integrability and martingale convergence 3.1.5.3 Riesz decompositions of supermartingales 3.1.5.4 Krickeberg decomposition of martingales 3.1.6 Some applications of convergence theorems 3.2 Continuous Parameter Martingales 3.2.1 Supermartingales on countable sets 3.2.1.1 Fundamental inequalities 3.2.1.2 Existence of right and left limits 3.2.2 Right-continuous supermartingale 3.2.3 Projections theorems 3.2.4 Decomposition of supermartingales 3.2.4.1 Functional analytic decomposition theorem 3.2.4.2 Extension to non-positive functionals 3.2.4.3 Decomposition of positive supermartingale of class D 3.2.4.4 The general case of Doob decomposition 4. Strong Supermartingales 4.1 Introduction 4.2 Projection Theorems 4.3 Special Inequalities 4.4 Mertens Decomposition 4.5 Snell Envelope 5. Optional Martingales 5.1 Introduction 5.2 Existence and Uniqueness 5.3 Increasing and Finite Variation Processes 5.3.1 Integration with respect to increasing and finite variation processes 5.3.2 Dual projections 5.4 Decomposition Results 5.4.1 Decomposition of elementary processes 5.4.2 Decomposition of optional martingales 5.5 Quadratic Variation 5.5.1 Predictable and optional 5.5.2 Kunita-Watanabe inequalities 5.6 Optional Stochastic Integral 5.6.1 Integral with respect to square integrable martingales 5.6.2 Integral with respect to martingales with integrable variation 5.6.3 Integration with respect to local optional martingales 6. Optional Supermartingales Decomposition 6.1 Introduction 6.2 Riesz Decomposition 6.3 Doob-Meyer-Galchuk Decomposition 6.3.1 Decomposition of DL class 7. Calculus of Optional Semimartingales 7.1 Integral with Respect to Optional Semimartingales 7.2 Formula for Change of Variables 7.3 Stochastic Integrals of Random Measures 7.4 Semimartingales and Their Characteristics 7.4.1 Canonical representation 7.4.2 Component representation 7.5 Uniform Doob-Meyer Decompositions 7.5.1 Supporting lemmas 8. Optional Stochastic Equations 8.1 Linear Equations, Exponentials and Logarithms 8.1.1 Stochastic exponential 8.1.2 Stochastic logarithm 8.1.3 Nonhomogeneous linear equation 8.1.4 Gronwall lemma 8.2 Existence and Uniqueness of Solutions of Optional Stochastic Equations 8.2.1 Stochastic equation with monotonicity condition 8.2.2 Existence and uniqueness results 8.2.2.1 Uniqueness 8.2.2.2 Existence 8.2.3 Remarks and applications 8.3 Comparison of Solutions of Optional Stochastic Equations 8.3.1 Comparison theorem 8.3.2 Remarks and applications 9. Optional Financial Markets 9.1 Introduction 9.2 Market Model 9.3 Martingale Deflators 9.3.1 The case of stochastic exponentials 9.3.2 The case of stochastic logarithms 9.4 Pricing and Hedging 9.5 Absence of Arbitrage 9.6 Examples of Special Cases 9.6.1 Ladlag jumps diffusion model 9.6.1.1 Computing a local martingale deflator 9.6.1.2 Pricing of a European call option 9.6.1.3 Hedging of a European call option 9.6.2 Basket of stocks 9.6.3 Defaultable bond and a stock 10. Defaultable Markets on Unusual Space 10.1 Introduction 10.2 Optional Default 10.3 Defaultable Cash-Flow 10.3.1 Portfolio with default 10.4 Probability of Default 10.5 Valuation of Defaultable Cash-Flow and Examples 11. Filtering of Optional Semimartingales 11.1 The Filtering Problem 11.2 The Usual Case of Optimal Filtering 11.2.1 Auxiliary results 11.2.2 Martingales' integral representation 11.2.3 Filtering of cadlag semimartingales 11.3 The Unusual case of Optimal Filtering 11.3.1 Filtering on unusual stochastic basis 11.3.2 Filtering on mixed stochastic basis 11.4 Filtering in Finance Bibliography Index