Mathematics of the Bond Market: A Lévy Processes Approach (Encyclopedia of Mathematics and its Applications)

Contents
Preface
	The Field
	Levy Modelling
	Aims of the Book
	Structure of the Book
Acknowledgements
Introduction
	I.1 Bonds
	I.2 Models
	I.3 Content of the Book
PART I BOND MARKET IN DISCRETE TIME
	1 Elements of the Bond Market
		1.1 Prices and Rates
		1.2 Models of the Bond Market
		1.3 Portfolios and Strategies
		1.4 Contingent Claims
		1.5 Arbitrage
	2 Arbitrage-Free Bond Markets
		2.1 Martingale Modelling
		2.2 Martingale Measures for HJM Models
			2.2.1 Existence of Martingale Measures
			2.2.2 Uniqueness of the Martingale Measure
		2.3 Martingale Measures and Martingale Representation Property
			2.3.1 Martingale Representation Property
			2.3.2 Generalized Martingale Representation Property
			2.3.3 Girsanov’s Theorems
			2.3.4 Application to HJM Models
		2.4 Markovian Models under the Martingale Measure
			2.4.1 Models with Markovian Trace
			2.4.2 Affine Models
			2.4.3 Dynamics of the Short Rate in Affine Models
			2.4.4 Shape of Forward Curves in Affine Models
			2.4.5 Factor Models
	3 Completeness
		3.1 Concepts of Completeness
		3.2 Necessary Conditions for Completeness
		3.3 Sufficient Conditions for Completeness
		3.4 Approximate Completeness
			3.4.1 General Characterization
			3.4.2 Bond Curves in a Finite Dimensional Space
			3.4.3 Bond Curves in Hilbert Spaces
		3.5 Models with Martingale Prices
			3.5.1 HJM Models
			3.5.2 Multiplicative Factor Model
			3.5.3 Affine Models
		3.6 Replication with Finite Portfolios
		3.7 Completeness and Martingale Measures
PART II FUNDAMENTALS OF STOCHASTIC ANALYSIS
	4 Stochastic Preliminaries
		4.1 Generalities
		4.2 Doob–Meyer Decomposition
			4.2.1 Predictable Quadratic Variation of Square Integrable Martingales
			4.2.2 Compensators of Finite Variation Processes
		4.3 Semimartingales
		4.4 Stochastic Integration
			4.4.1 Bounded Variation Integrators
			4.4.2 Square Integrable Martingales as Integrators
			4.4.3 Integration over Random Measures
			4.4.4 Ito’s Formula
	5 Levy Processes
		5.1 Basics on Levy Processes
		5.2 Levy–Ito Decomposition
		5.3 Special Classes
			5.3.1 Finite Variation Processes
			5.3.2 Subordinators
			5.3.3 Levy Martingales
		5.4 Stochastic Integration
			5.4.1 Square Integrable Integrators
			5.4.2 Integration over Compensated Jump Measures
			5.4.3 Stochastic Fubini’s Theorem
			5.4.4 Ito’s Formula for Levy Processes
	6 Martingale Representation and Girsanov’s Theorems
		6.1 Martingale Representation Theorem
		6.2 Girsanov’s Theorem and Equivalent Measures
PART III BOND MARKET IN CONTINUOUS TIME
	7 Fundamentals
		7.1 Prices and Rates
			7.1.1 Bank Account and Discounted Bond Prices
			7.1.2 Prices and Rates in Function Spaces
		7.2 Portfolios and Strategies
			7.2.1 Portfolios
			7.2.2 Strategies and the Wealth Process
			7.2.3 Wealth Process as Stochastic Integral
		7.3 Non-arbitrage, Claims and Their Prices
		7.4 HJM Modelling
			7.4.1 Bond Prices Formula
			7.4.2 Forward Curves in Function Spaces
		7.5 Factor Models and the Musiela Parametrization
	8 Arbitrage-Free HJM Markets
		8.1 Heath–Jarrow–Morton Conditions
			8.1.1 Proof of Theorem 8.1.1
		8.2 Martingale Measures
			8.2.1 Specification of Drift
			8.2.2 Models with No Martingale Measures
			8.2.3 Invariance of Levy Noise
			8.2.4 Volatility-Based Models
			8.2.5 Uniqueness of the Martingale Measure
	9 Arbitrage-Free Forward Curves Models
		9.1 Term Structure Equation
			9.1.1 Markov Chain and CIR as Factor Processes
			9.1.2 Multiplicative Factor Process
			9.1.3 Affine Term Structure Model
			9.1.4 Ornstein–Uhlenbeck Factors
10 Arbitrage-Free Affine Term Structure
	10.1 Preliminary Model Requirements
	10.2 Jump Diffusion Short Rate
		10.2.1 Analytical HJM Condition
		10.2.2 Generalized CIR Equations
		10.2.3 Exploding Short Rates
		10.2.4 Multidimensional Noise
	10.3 General Markovian Short Rate
		10.3.1 Filipovi´ c’s Theorems
		10.3.2 Comments on Filipovi´ c’s Theorems
		10.3.3 Examples
		10.3.4 Back to Short-Rate Equations
11 Completeness
	11.1 Problem of Completeness
	11.2 Representation of Discounted Bond Prices
	11.3 Admissible Strategies
	11.4 Hedging Equation
	11.5 Completeness for the HJM Model
		11.5.1 Levy Measure with Finite Support
		11.5.2 Proofs of Theorems 11.5.1–11.5.3
		11.5.3 Incomplete Markets
	11.6 Completeness for Affine Models
	11.7 Completeness for Factor Models
	11.8 Approximate Completeness
		11.8.1 HJM Model
		11.8.2 Factor Model
		11.8.3 Affine Model
PART IV STOCHASTIC EQUATIONS IN THE BOND MARKET
	12 Stochastic Equations for Forward Rates
		12.1 Heath–Jarrow–Morton Equation
		12.2 Morton’s Equation
		12.3 The Equations in the Musiela Parametrization
	13 Analysis of the HJMM Equation
		13.1 Existence of Solutions to the HJMM Equation
			13.1.1 Local Solutions
			13.1.2 Global Solutions
			13.1.3 Applications to the Morton–Musiela Equation
	14 Analysis of Morton’s Equation
		14.1 Results
			14.1.1 Comments on Assumptions (A1)–(A3)
		14.2 Applications of the Main Theorems
		14.3 Proof of Theorem 14.1.1
			14.3.1 Outline of the Proof
			14.3.2 Equivalence of Equations (14.1.1) and (14.1.9)
			14.3.3 Auxiliary Results
			14.3.4 Conclusion of the Proof
		14.4 Proof of Theorem 14.1.2
	15 Analysis of the Morton–Musiela Equation
		15.1 Formulation and Comments on the Results
			15.1.1 Comments on the Results
		15.2 Proofs of Theorems 15.1.1 and 15.1.2
			15.2.1 Equivalence Results
			15.2.2 Proof of Theorem 15.1.1
			15.2.3 Proof of Theorem 15.1.2
Appendix A
	A.1 Martingale Representation for Jump Levy Processes
	A.1.1 Multiple Chaos Processes
	A.1.2 Representation of Chaoses
	A.1.3 Chaos Expansion Theorem
	A.1.4 Representation of Square Integrable Martingales
	A.1.5 Representations of Local Martingales
Appendix B
	B.1 Semigroups and Generators
		B.1.1 Generators for Equations with Levy Noise
Appendix C
	C.1 General Evolution Equations
References
Index