Lectures given at the C.I.M.E.-E.M.S. Summer School held in Bressanone/Brixen, Italy, July 6--12, 2003
Editors: M. Frittelli, W. Runggaldier

Contents
Incomplete and Asymmetric Information in Asset Pricing
Theory
Kerry Back
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

Modeling and Valuation of Credit Risk 27
Tomasz R. Bielecki, Monique Jeanblanc, Marek Rutkowski
1 Introduction
2 Structural Approach
2.1 Basic Assumptions
Defaultable Claims
Risk-Neutral Valuation Formula
Defaultable Zero-Coupon Bond
2.2 Classic Structural Models
Merton’s Model
  Black and Cox Model
  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
Hazard Function of a Random Time
Associated Martingales
Change of a Probability Measure
Martingale Hazard Function
Defaultable Bonds: Deterministic Intensity
3.2 Hazard Processes
Hazard Process of a Random Time
Valuation of Defaultable Claims
Alternative Recovery Rules
Defaultable Bonds: Stochastic Intensity
Martingale Hazard Process
Martingale Hypothesis
Canonical Construction
Kusuoka’s Counter-Example
Change of a Probability
Statistical Probability
Change of a Numeraire
Preprice of a Defaultable Claim
Credit Default Swaption
A Practical Example
3.3 Martingale Approach
Standing Assumptions
Valuation of Defaultable Claims
Martingale Approach under (H.1)
3.4 Further Developments
Default-Adjusted Martingale Measure
Hybrid Models
Unified Approach
3.5 Comments on Intensity-Based Models

4 Dependent Defaults and Credit Migrations
4.1 Basket Credit Derivatives
The ith -to-Default Contingent Claims
Case of Two Entities
4.2 Conditionally Independent Defaults
Canonical Construction
Independent Default Times
Signed Intensities
Valuation of FDC and LDC
General Valuation Formula
Default Swap of Basket Type
4.3 Copula-Based Approaches
Direct Application
Indirect Application
Simplified Version
4.4 Jarrow and Yu Model
Construction and Properties of the Model
Bond Valuation
4.5 Extension of the Jarrow and Yu Model
Kusuoka’s Construction
Interpretation of Intensities
Bond Valuation
4.6 Dependent Intensities of Credit Migrations
Extension of Kusuoka’s Construction
4.7 Dynamics of Dependent Credit Ratings
4.8 Defaultable Term Structure
Standing Assumptions
Credit Migration Process
Defaultable Term Structure
Premia for Interest Rate and Credit Event Risks
Defaultable Coupon Bond
Examples of Credit Derivatives
4.9 Concluding Remarks
References

Stochastic Control with Application in Insurance
Christian Hipp
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
Investment, One Risky Asset
Investment, Two or More Risky Assets
Proportional Reinsurance
Unlimited XL Reinsurance
XL-Reinsurance
Premium Control
Control of New Business
3.2 Stochastic Control
Objective Functions
Infinitesimal Generators
Hamilton-Jacobi-Bellman Equations
Verification Argument
Steps for Solution
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

Nonlinear Expectations, Nonlinear Evaluations and Risk
Measures
Shige Peng
1 Introduction
1.1 Searching the Mechanism of Evaluations of Risky Assets
1.2 Axiomatic Assumptions for Evaluations of Derivatives
General Situations: Ft –Consistent Nonlinear Evaluations
Ft –Consistent Nonlinear Expectations
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
Comparison Theorem
Backward Stochastic Monotone Semigroups and g–Evaluations
Example: Black–Scholes Evaluations
g–Expectations
Upcrossing Inequality of E g –Supermartingales and Optional
Sampling Inequality
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 E μ -Dominated F -Expectations
4.2 Ft -Consistent Martingales
4.3 BSDE under Ft –Consistent Nonlinear Expectations
4.4 Decomposition Theorem for E-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 Itˆ’s Processes
o
7.3 Optional Stopping Theorem for E g –Supermartingale
References
References on BSDE and Nonlinear Expectations


Utility Maximisation in Incomplete Markets
Walter Schachermayer
1 Problem Setting
2 Models on Finite Probability Spaces
2.1 Utility Maximization
The complete Case (Arrow)
The Incomplete Case
3 The General Case
3.1 The Reasonable Asymptotic Elasticity Condition
3.2 Existence Theorems
References