Stochastic Calculus and Financial Applications / by J. Michael Steele.

Format:

Book

Author/Creator:

Steele, J. Michael, author.

Contributor:

SpringerLink (Online service)

Series:

Stochastic modelling and applied probability 0172-4568 ; 45.

Stochastic Modelling and Applied Probability, 0172-4568 ; 45

Language:

English

Subjects (All):

Probabilities.

Economics, Mathematical.

Statistics.

Probability Theory and Stochastic Processes.

Quantitative Finance.

Statistical Theory and Methods.

Local Subjects:

Probability Theory and Stochastic Processes.

Quantitative Finance.

Statistical Theory and Methods.

Physical Description:

1 online resource.

Edition:

First edition 2001.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 2001.

System Details:

text file PDF

Summary:

This book is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance. The Wharton School course that forms the basis for this book is designed for energetic students who have had some experience with probability and statistics but have not had ad- vanced courses in stochastic processes. Although the course assumes only a modest background, it moves quickly, and in the end, students can expect to have tools that are deep enough and rich enough to be relied on throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more de- manding development of continuous-time stochastic processes, especially Brownian motion. The construction of Brownian motion is given in detail, and enough mate- rial on the subtle nature of Brownian paths is developed for the student to evolve a good sense of when intuition can be trusted and when it cannot. The course then takes up the Ito integral in earnest. The development of stochastic integration aims to be careful and complete without being pedantic.

Contents:

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. Doob's 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 Donsker's Theorem

5.5. Random Walks Inside Brownian Motion

5.6. Exercises

6. Itô Integration

6.1. Definition of the Ito Integral: First Two Steps

6.2. Third Step: Itô's 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 Itô's Integral

7.1. Itô's 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. Itô's 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 Itô's 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. Lévy's Representation Theorem

12.5. Two Consequences of Lévy's 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 Novikov's 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. Lévy's 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.

Other Format:

Printed edition:

ISBN:

9781468493054

Access Restriction:

Restricted for use by site license.