Stochastic Methods in Finance: Lectures given at the C.I.M.E.-E.M.S. Summer School held in Bressanone/Brixen, Italy, July 6-12, 2003 (Lecture Notes in Mathematics, 1856)

✍ Scribed by Kerry Back

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 317
• Category: Library

✦ Synopsis

This volume includes the five lecture courses given at the CIME-EMS School on "Stochastic Methods in Finance" held in Bressanone/Brixen, Italy 2003. It deals with innovative methods, mainly from stochastic analysis, that play a fundamental role in the mathematical modelling of finance and insurance: the theory of stochastic processes, optimal and stochastic control, stochastic differential equations, convex analysis and duality theory. Five topics are treated in detail: Utility maximization in incomplete markets; the theory of nonlinear expectations and its relationship with the theory of risk measures in a dynamic setting; credit risk modelling; the interplay between finance and insurance; incomplete information in the context of economic equilibrium and insider trading.

✦ Table of Contents

Title
Preface
Contents
1 Filtering Theory
1.1 Kalman-Bucy Filter
1.2 Two-State Markov Chain
2 Incomplete Information
2.1 Seminal Work
2.2 Markov Chain Models of Production Economies
2.3 Markov Chain Models of Pure Exchange Economies
2.4 Heterogeneous Beliefs
3 Asymmetric Information
3.1 Anticipative Information
3.2 Rational Expectations Models
3.3 Kyle Model
3.4 Continuous-Time Kyle Model
3.5 Multiple Informed Traders in the Kyle Model
References
1 Introduction
2 Structural Approach
2.1 Basic Assumptions
2.2 Classic Structural Models
2.3 Stochastic Interest Rates
2.4 Credit Spreads: A Case Study
2.5 Comments on Structural Models
3 Intensity-Based Approach
3.1 Hazard Function
3.2 Hazard Processes
3.3 Martingale Approach
3.4 Further Developments
3.5 Comments on Intensity-Based Models
4 Dependent Defaults and Credit Migrations
4.1 Basket Credit Derivatives
4.2 Conditionally Independent Defaults
4.3 Copula-Based Approaches
4.4 Jarrow and Yu Model
4.5 Extension of the Jarrow and Yu Model
4.6 Dependent Intensities of Credit Migrations
4.7 Dynamics of Dependent Credit Ratings
4.8 Defaultable Term Structure
4.9 Concluding Remarks
References
1 Preface
2 Introduction Into Insurance Risk
2.1 The Lundberg Risk Model
2.2 Alternatives
2.3 Ruin Probability
2.4 Asymptotic Behavior For Ruin Probabilities
3 Possible Control Variables and Stochastic Control
3.1 Possible Control Variables
3.2 Stochastic Control
4 Optimal Investment for Insurers
4.1 HJB and its Handy Form
4.2 Existence of a Solution
4.3 Exponential Claim Sizes
4.4 Two or More Risky Assets
5 Optimal Reinsurance and Optimal New Business
5.1 Optimal Proportional Reinsurance
5.2 Optimal Unlimited XL Reinsurance
5.3 Optimal XL Reinsurance
5.4 Optimal New Business
6 Asymptotic Behavior for Value Function and Strategies
6.1 Optimal Investment: Exponential Claims
6.2 Optimal Investment: Small Claims
6.3 Optimal Investment: Large Claims
6.4 Optimal Reinsurance
7 A Control Problem with Constraint: Dividends and Ruin
7.1 A Simple Insurance Model with Dividend Payments
7.2 Modified HJB Equation
7.3 Numerical Example and Conjectures
7.4 Earlier and Further Work
8 Conclusions
References
1 Introduction
1.1 Searching the Mechanism of Evaluations of Risky Assets
1.2 Axiomatic Assumptions for Evaluations of Derivatives
1.3 Organization of the Lecture
2 Brownian Filtration Consistent Evaluations and Expectations
2.1 Main Notations and Definitions
2.2 Ft-Consistent Nonlinear Expectations
2.3 Ft-Consistent Nonlinear Evaluations
3 Backward Stochastic Differential Equations: g
–Evaluations and g–Expectations
3.1 BSDE: Existence, Uniqueness and Basic Estimates
3.2 1–Dimensional BSDE
3.3 A Monotonic Limit Theorem of BSDE
3.4 g–Martingales and (Nonlinear) g
–Supermartingale Decomposition Theorem
4 Finding the Mechanism: Is an F–Expectation a g
–Expectation?
4.1 Eμ-Dominated F-Expectations
4.2
Ft-Consistent Martingales
4.3 BSDE under Ft
–Consistent Nonlinear Expectations
4.4 Decomposition Theorem for
-Supermartingales
4.5 Representation Theorem of an F–Expectation by a
g–Expectation
4.6 How to Test and Find
g?
4.7 A General Situation:
Ft–Evaluation Representation Theorem
5 Dynamic Risk Measures
6 Numerical Solution of BSDEs: Euler’s Approximation
7 Appendix
7.1 Martingale Representation Theorem
7.2 A Monotonic Limit Theorem of Itô’s Processes
7.3 Optional Stopping Theorem for Eg–Supermartingale
References
References on BSDE and Nonlinear Expectations
Preface
1 Problem Setting
2 Models on Finite Probability Spaces
2.1 Utility Maximization
3 The General Case
3.1 The Reasonable Asymptotic Elasticity Condition
3.2 Existence Theorems
References
List of Participants
LIST OF C.I.M.E. SEMINARS
2005 COURSES LIST