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

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