Author | Steele, J. Michael. author |
---|---|
Title | Stochastic Calculus and Financial Applications [electronic resource] / by J. Michael Steele |
Imprint | New York, NY : Springer New York, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9305-4 |
Descript | X, 302 p. online resource |
1. Random Walk and First Step Analysis -- 1.1. First Step Analysis -- 1.2. Time and Infinity -- 1.3. Tossing an Unfair Coin -- 1.4. Numerical Calculation and Intuition -- 1.5. First Steps with Generating Functions -- 1.6. Exercises -- 2. First Martingale Steps -- 2.1. Classic Examples -- 2.2. New Martingales from Old -- 2.3. Revisiting the Old Ruins -- 2.4. Submartingales -- 2.5. Doobโs Inequalities -- 2.6. Martingale Convergence -- 2.7. Exercises -- 3. Brownian Motion -- 3.1. Covariances and Characteristic Functions -- 3.2. Visions of a Series Approximation -- 3.3. Two Wavelets -- 3.4. Wavelet Representation of Brownian Motion -- 3.5. Scaling and Inverting Brownian Motion -- 3.6. Exercises -- 4. Martingales: The Next Steps -- 4.1. Foundation Stones -- 4.2. Conditional Expectations -- 4.3. Uniform Integrability -- 4.4. Martingales in Continuous Time -- 4.5. Classic Brownian Motion Martingales -- 4.6. Exercises -- 5. Richness of Paths -- 5.1. Quantitative Smoothness -- 5.2. Not Too Smooth -- 5.3. Two Reflection Principles -- 5.4. The Invariance Principle and Donskerโs Theorem -- 5.5. Random Walks Inside Brownian Motion -- 5.6. Exercises -- 6. Itรด Integration -- 6.1. Definition of the Ito Integral: First Two Steps -- 6.2. Third Step: Itรดโs Integral as a Process -- 6.3. The Integral Sign: Benefits and Costs -- 6.4. An Explicit Calculation -- 6.5. Pathwise Interpretation of Ito Integrals -- 6.6. Approximation in H2 -- 6.7. Exercises -- 7. Localization and Itรดโs Integral -- 7.1. Itรดโs Integral on L2LOC -- 7.2. An Intuitive Representation -- 7.3. Why Just L2LOC? -- 7.4. Local Martingales and Honest Ones -- 7.5. Alternative Fields and Changes of Time -- 7.6. Exercises -- 8. Itรดโs Formula -- 8.1. Analysis and Synthesis -- 8.2. First Consequences and Enhancements -- 8.3. Vector Extension and Harmonic Functions -- 8.4. Functions of Processes -- 8.5. The General Ito Formula -- 8.6. Quadratic Variation -- 8.7. Exercises -- 9. Stochastic Differential Equations -- 9.1. Matching Itรดโs Coefficients -- 9.2. Ornstein-Uhlenbeck Processes -- 9.3. Matching Product Process Coefficients -- 9.4. Existence and Uniqueness Theorems -- 9.5. Systems of SDEs -- 9.6. Exercises -- 10. Arbitrage and SDEs -- 10.1. Replication and Three Examples of Arbitrage -- 10.2. The Black-Scholes Model -- 10.3. The Black-Scholes Formula -- 10.4. Two Original Derivations -- 10.5. The Perplexing Power of a Formula -- 10.6. Exercises -- 11. The Diffusion Equation -- 11.1. The Diffusion of Mice -- 11.2. Solutions of the Diffusion Equation -- 11.3. Uniqueness of Solutions -- 11.4. How to Solve the Black-Scholes PDE -- 11.5. Uniqueness and the Black-Scholes PDE -- 11.6. Exercises -- 12. Representation Theorems -- 12.1. Stochastic Integral Representation Theorem -- 12.2. The Martingale Representation Theorem -- 12.3. Continuity of Conditional Expectations -- 12.4. Lรฉvyโs Representation Theorem -- 12.5. Two Consequences of Lรฉvyโs Representation -- 12.6. Bedrock Approximation Techniques -- 12.7. Exercises -- 13. Girsanov Theory -- 13.1. Importance Sampling -- 13.2. Tilting a Process -- 13.3. Simplest Girsanov Theorem -- 13.4. Creation of Martingales -- 13.5. Shifting the General Drift -- 13.6. Exponential Martingales and Novikovโs Condition -- 13.7. Exercises -- 14. Arbitrage and Martingales -- 14.1. Reexamination of the Binomial Arbitrage -- 14.2. The Valuation Formula in Continuous Time -- 14.3. The Black-Scholes Formula via Martingales -- 14.4. American Options -- 14.5. Self-Financing and Self-Doubt -- 14.6. Admissible Strategies and Completeness -- 14.7. Perspective on Theory and Practice -- 14.8. Exercises -- 15. The Feynman-Kac Connection -- 15.1. First Links -- 15.2. The Feynman-Kac Connection for Brownian Motion -- 15.3. Lรฉvyโs Arcsin Law -- 15.4. The Feynman-Kac Connection for Diffusions -- 15.5. Feynman-Kac and the Black-Scholes PDEs -- 15.6. Exercises -- Appendix I. Mathematical Tools -- Appendix II. Comments and Credits