Ubbo F.Wiersema – Brownian Motion Calculus

Preface.

1 Brownian Motion.

1.1 Origins.

1.2 Brownian Motion Specification.

1.3 Use Brownian Motion Stock Price Dynamics

1.4 Construction Brownian Motion A Symmetric Random Walk.

1.5 Covariance Brownian Motion.

1.6 Correlated Brownian Motions.

1.7 Successive Brownian Motion Increments.

1.8 Features Brownian Motion Path.

1.9 Exercises.

1.10 Summary.

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2 Martingales.

2.1 Simple example

2.2 Filtration.

2.3 Conditional Expectation.

2.4 Martingale Description.

2.5 Martingale Analysis Steps.

Martingale Analysis: 2.6 examples

2.7 Process of Independent Increments

2.8 Exercises.

2.9 Summary.

3 Itō Stochastic Integral.

3.1 How a Stochastic Integral Arises.

3.2 Stochastic integral for Non-Random Step-Functions.

3.3 Stochastic Integrational for Non-Anticipating Random Step-Functions.

3.4 Extension to Not-Anticipating General Random Integrands.

3.5 Properties of an Itō Stochastic Integral.

3.6 Significance Integrand Position

3.7 Itō integral of Non-Random Integrand.

3.8 Area under a Brownian Motion Path.

3.9 Exercises.

3.10 Summary.

3.11 A Tribute to Kiyosi Itō.

Acknowledgment.

4 Itō Calculus.

4.1 Stochastic Differential Notion.

4.2 Taylor Expansion for Ordinary Calculus.

4.3 Itō’s Formula as a Set of Rules.

4.4 Illustrations of Itō’s Formula.

4.5 Lévy Characterization of Brownian Motion.

4.6 Combinations Brownian Motions.

Multiple Correlated Brownian Motions.

4.8 Area under a Brownian Motion Path – Revisited.

4.9 Justification of Itō’s Formula.

4.10 Exercises.

4.11 Summary.

5 Stochastic Differential Equilibrium Equations

5.1 Structure and Function of a Stochastic Differential equation.

5.2 Arithmetic Brownian Motion SDE.

5.3 Geometric Brownian Motion SDE.

5.4 Ornstein–Uhlenbeck SDE.

5.5-Reversion SDE.

5.6-Square Reversion-Root Diffusion SDE

5.7 Square Foot Expected Value-Root Diffusion Process.

5.8 Coupled SDDEs

5.9 Checking the Solution to a SDE.

5.10 General Solutions Methods for Linear SDEs

5.11 Martingale Representation.

5.12 Exercises.

5.13 Summary.

6 Option Valuation

6.1 Partial Differential Equation Method.

6.2 Martingale method in One-Period Binomial Framework.

6.3 Martingale Method in Continuous-Time Framework.

6.4 A Review of Risk-Neutral Method.

6.5 Martingale Valuation Method for Some European Options

6.6 Links Between Methods

6.6.1 Feynman-Kač Link between PDE Method and Martingale Method.

6.6.2 Multi-Period Binomial Link to Constant

6.7 Exercise.

6.8 Summary.

7 Changes in Probability

7.1 Change in Discrete Probability Mass

7.2 Changes in Normal Density

7.3 Change in Brownian Motion.

7.4 Girsanov Transformation.

Stock Price Dynamics Revisited – 7.5

7.6 General Drift Change

7.7 Importance Sampling

7.8 Use in Deriving Conditionsal Expectations.

7.9 The Concept of Probability Change

7.10 Exercises.

7.11 Summary.

8 Numeraire.

8.1 Change of Numeraire

8.2 Forward Price Dynamics

8.3 Option Value under the most Suitable Numeraire

8.4 Relating the Change of Numeraire to the Change of Probability

8.5 Geometric Change of Numeraire Brownian Motion.

8.6. Numeraire Change in LIBOR Market Model

8.7 Credit Risk Modelling Application

8.8 Exercises.

8.9 Summary.

ANNEXES.

A Annexe A: Computations Brownian Motion.

A.1 Moment Generating function and Moments Brownian Motion.

A.2 Probability Brownian Motion Position.

A.3 Brownian Motion Reflected at Origin

A.4 First Barrier Passage.

Alternative: A.5 Brownian Motion Specification.

B Annex B: Ordinary Incorporation.

B.1 Riemann Integral.

B.2 Riemann–Stieltjes Integral.

B.3 Other useful properties

B.4 References.

C Annexe C: Brownian Motion Variability.

C.1 Quadratic Variation.

C.2 First Variation.

D Annex D : Norms.

Distance between points: D.1.

D.2 Norm of Function.

D.3 Norm for a Random Variable.

D.4 Norm for a Random Process

D.5 Reference.

E Annex E Convergence Concepts

E.1 Central Limit Theorem.

E.2 =-Square Convergence.

E.3 Almost Sure Convergence.

E.4 Convergence of Probability.

E.5 Summary.

Answers to Exercises

References.

Index.

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