| Incomplete and Asymmetric Information in Asset Pricing Theory | p. 1 |
| Filtering Theory | p. 1 |
| Kalman-Bucy Filter | p. 3 |
| Two-State Markov Chain | p. 4 |
| Incomplete Information | p. 5 |
| Seminal Work | p. 5 |
| Markov Chain Models of Production Economies | p. 6 |
| Markov Chain Models of Pure Exchange Economies | p. 7 |
| Heterogeneous Beliefs | p. 11 |
| Asymmetric Information | p. 12 |
| Anticipative Information | p. 12 |
| Rational Expectations Models | p. 13 |
| Kyle Model | p. 16 |
| Continuous-Time Kyle Model | p. 18 |
| Multiple Informed Traders in the Kyle Model | p. 20 |
| References | p. 23 |
| Modeling and Valuation of Credit Risk | p. 27 |
| Introduction | p. 27 |
| Structural Approach | p. 29 |
| Basic Assumptions | p. 29 |
| Defaultable Claims | p. 29 |
| Risk-Neutral Valuation Formula | p. 31 |
| Defaultable Zero-Coupon Bond | p. 32 |
| Classic Structural Models | p. 34 |
| Merlon's Model | p. 34 |
| Black and Cox Model | p. 37 |
| Stochastic Interest Rates | p. 43 |
| Credit Spreads: A Case Study | p. 45 |
| Comments on Structural Models | p. 46 |
| Intensity-Based Approach | p. 47 |
| Hazard Function | p. 47 |
| Hazard Function of a Random Time | p. 48 |
| Associated Martingales | p. 49 |
| Change of a Probability Measure | p. 50 |
| Martingale Hazard Function | p. 53 |
| Defaultable Bonds: Deterministic Intensity | p. 53 |
| Hazard Processes | p. 55 |
| Hazard Process of a Random Time | p. 56 |
| Valuation of Defaultable Claims | p. 57 |
| Alternative Recovery Rules | p. 59 |
| Defaultable Bonds: Stochastic Intensity | p. 63 |
| Martingale Hazard Process | p. 64 |
| Martingale Hypothesis | p. 65 |
| Canonical Construction | p. 67 |
| Kusuoka's Counter-Example | p. 69 |
| Change of a Probability | p. 70 |
| Statistical Probability | p. 72 |
| Change of a Numeraire | p. 74 |
| Preprice ofa Defaultable Claim | p. 77 |
| Credit Default Swaption | p. 79 |
| A Practical Example | p. 82 |
| Martingale Approach | p. 84 |
| Standing Assumptions | p. 85 |
| Valuation of Defaultable Claims | p. 85 |
| Martingale Approach under (H.1) | p. 87 |
| Further Developments | p. 88 |
| Default-Adjusted Martingale Measure | p. 88 |
| Hybrid Models | p. 89 |
| Unified Approach | p. 90 |
| Comments on Intensity-Based Models | p. 90 |
| Dependent Defaults and Credit Migrations | p. 91 |
| Basket Credit Derivatives | p. 92 |
| The ith-to-Default Contingent Claims | p. 92 |
| Case of Two Entities | p. 93 |
| Conditionally Independent Defaults | p. 94 |
| Canonical Construction | p. 94 |
| Independent Default Times | p. 95 |
| Signed Intensities | p. 96 |
| Valuation of FDC and LDC | p. 96 |
| General Valuation Formula | p. 97 |
| Default Swap of Basket Type | p. 98 |
| Copula-Based Approaches | p. 99 |
| Direct Application | p. 100 |
| Indirect Application | p. 100 |
| Simplified Version | p. 102 |
| Jarrow and Yu Model | p. 103 |
| Construction and Properties of the Model | p. 103 |
| Bond Valuation | p. 105 |
| Extension of the Jarrow and Yu Model | p. 106 |
| Kusuoka's Construction | p. 107 |
| Interpretation of Intensities | p. 108 |
| Bond Valuation | p. 108 |
| Dependent Intensities of Credit Migrations | p. 109 |
| Extension of Kusuoka's Construction | p. 109 |
| Dynamics of Dependent Credit Ratings | p. 112 |
| Defaultable Term Structure | p. 113 |
| Standing Assumptions | p. 113 |
| Credit Migration Process | p. 116 |
| Defaultable Term Structure | p. 117 |
| Premia for Interest Rate and Credit Event Risks | p. 119 |
| Defaultable Coupon Bond | p. 120 |
| Examples of Credit Derivatives | p. 121 |
| Concluding Remarks | p. 122 |
| References | p. 123 |
| Stochastic Control with Application in Insurance | p. 127 |
| Preface | p. 127 |
| Introduction Into Insurance Risk | p. 128 |
| The Lundberg Risk Model | p. 128 |
| Alternatives | p. 129 |
| Ruin Probability | p. 129 |
| Asymptotic Behavior For Ruin Probabilities | p. 131 |
| Possible Control Variables and Stochastic Control | p. 132 |
| Possible Control Variables | p. 132 |
| Investment, One Risky Asset | p. 132 |
| Investment Two or More Risky Assets | p. 133 |
| Proportional Reinsurance | p. 134 |
| Unlimited XL Reinsurance | p. 134 |
| XL-Reinsurance | p. 135 |
| Premium Control | p. 135 |
| Control of New Business | p. 135 |
| Stochastic Control | p. 136 |
| Objective Functions | p. 136 |
| Infinitesimal Generators | p. 137 |
| Hamilton-Jacobi-Bellman Equations | p. 139 |
| Verification Argument | p. 141 |
| Steps for Solution | p. 143 |
| Optimal Investment for Insurers | p. 143 |
| HJB and its Handy Form | p. 143 |
| Existence of a Solution | p. 145 |
| Exponential Claim Sizes | p. 145 |
| Two or More Risky Assets | p. 147 |
| Optimal Reinsurance and Optimal New Business | p. 148 |
| Optimal Proportional Reinsurance | p. 150 |
| Optimal Unlimited XL Reinsurance | p. 151 |
| Optimal XL Reinsurance | p. 152 |
| Optimal New Business | p. 153 |
| Asymptotic Behavior for Value Function and Strategies | p. 154 |
| Optimal Investment: Exponential Claims | p. 154 |
| Optimal Investment: Small Claims | p. 154 |
| Optimal Investment: Large Claims | p. 155 |
| Optimal Reinsurance | p. 156 |
| A Control Problem with Constraint: Dividends and Ruin | p. 157 |
| A Simple Insurance Model with Dividend Payments | p. 157 |
| Modified HJB Equation | p. 158 |
| Numerical Example and Conjectures | p. 159 |
| Earlier and Further Work | p. 161 |
| Conclusions | p. 162 |
| References | p. 163 |
| Nonlinear Expectations, Nonlinear Evaluations and Risk Measures | p. 165 |
| Introduction | p. 165 |
| Searching the Mechanism of Evaluations of Risky Assets | p. 165 |
| Axiomatic Assumptions for Evaluations of Derivatives | p. 166 |
| General Situations: <$>{cal F}_t^X<$>-Consistent Nonlinear Evaluations | p. 166 |
| <$>{cal F}_t^X<$>-Consistent Nonlinear Expectations | p. 167 |
| Organization of the Lecture | p. 168 |
| Brownian Filtration Consistent Evaluations and Expectations | p. 169 |
| Main Notations and Definitions | p. 169 |
| <$>{cal F}_t<$>-Consistent Nonlinear Expectations | p. 171 |
| <$>{cal F}_t<$>-Consistent Nonlinear Evaluations | p. 173 |
| Backward Stochastic Differential Equations: g-Evaluations and g-Expectations | p. 176 |
| BSDE: Existence, Uniqueness and Basic Estimates | p. 176 |
| 1-Dimensional BSDE | p. 182 |
| Comparison Theorem | p. 183 |
| Backward Stochastic Monotone Semigroups and g-Evaluations | p. 186 |
| Example: Black-Scholes Evaluations | p. 188 |
| Expectations | p. 189 |
| Upcrossing Inequality of g-Supermartingales and Optional Sampling Inequality | p. 193 |
| A Monotonie Limit Theorem of BSDE | p. 199 |
| g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem | p. 201 |
| Finding the Mechanism: Is an <$>{cal F}<$>-Expectation a g-Expectation? | p. 204 |
| -Dominated <$>{cal F}<$>-Expectations | p. 204 |
| <$>{cal F}_t<$>-Consistent Martingales | p. 207 |
| BSDE under <$>{cal F}_t<$>-Consistent Nonlinear Expectations | p. 210 |
| Decomposition Theorem for <$>cal E}<$>-Supermartingales | p. 213 |
| Representation Theorem of an <$>{cal F}<$>-Expectation by a g-Expectation | p. 216 |
| How to Test and Find g? | p. 219 |
| A General Situation: <$>{cal F}_t<$>-Evaluation Representation Theorem | p. 220 |
| Dynamic Risk Measures | p. 221 |
| Numerical Solution of BSDEs: Euler's Approximation | p. 222 |
| Appendix | p. 224 |
| Martingale Representation Theorem | p. 224 |
| A Monotonic Limit Theorem of Itô's Processes | p. 226 |
| Optional Stopping Theorem for <$>{cal E}^g<$>-Supermartingale | p. 232 |
| References | p. 238 |
| References on BSDE and Nonlinear Expectations | p. 240 |
| Utility Maximisation in Incomplete Markets | p. 255 |
| Problem Setting | p. 255 |
| Models on Finite Probability Spaces | p. 259 |
| Utility Maximization | p. 266 |
| The complete Case (Arrow) | p. 266 |
| The Incomplete Case | p. 272 |
| The General Case | p. 277 |
| The Reasonable Asymptotic Elasticity Condition | p. 277 |
| Existence Theorems | p. 281 |
| References | p. 289 |
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