金融数学中的带跳随机微分方程数值解

Preface
Suggestions for the Reader
Basic Notation
Motivation and Brief Survey
1 Stochastic Differential Equations with Jumps
1.1 Stochastic Processes
1.2 Supermartingales and Martingajes
1.3 Quadratic Variation and Covariation
1.4 Ito Integral
1.5 Ito Formula
1.6 Stochastic Differential Equations
1.7 Linear SDEs
1.8 SDEs with Jumps
1.9 Existence and Uniqueness of Solutions of SDEs
1.10 Exercises
2 Exact Simulation of Solutions of SDEs
2.1 Motivation of Exact Simulation
2.2 Sampling from Transition Distributions
2.3 Exact Solutions of Multi—dimensional SDEs
24 Functions of Exact Solutions
2.5 Almost Exact Solutions by Conditioning
2.6 Almost Exact Simulation by Time Change
2.7 Functionals of Solutions of SDEs
2.8 Exercises
3 Benchmark Approach to Finance and Insurance
3.1 Market Model
3.2 Best Performing Portfolio
3.3 Supermartingale Property and Pricing
3.4 Diversification
3.5 Real World Pricing Under Some Models
3.6 Real World Pricing Under the MMM
3.7 Binomial Option Pricing
3.8 Exercises
4 Stochastic Expansions
4.1 Introduction to Wagner—Platen Expansions
4.2 Multiple Stochastic Integrals
4.3 Coefficient Functions
4.4 Wagner—Platen Expansions
4.5 Moments of Multiple Stochastic Integrals
4.6 Exercises
5 Introduction to Scenario Simulation
5.1 Approximating Solutions of ODEs
5.2 Scenario Simulation
5.3 Strong Taylor Schemes
5.4 Derivative—Free Strong Schemes
5.5 Exercises
6 Regular Strong Taylor Approximations with Jumps
6.1 Discrete—Time Approximation
6.2 Strong Order 1.0 Taylor Scheme
6.3 Conunutativity Conditions
6.4 Convergence Results
6.5 Lemma on Multiple Ito Integrals
6.6 Proof of the Convergence Theorem
6.7 Exercises
7 Regular Strong Ito Approximations
7.1 Explicit Regular Strong Schemes
7.2 Drift—Implicit Schemes
7.3 Balanced Implicit Methods
7.4 Predictor—Corrector Schemes
7.5 Convergence Results
7.6 Exercises
8 Jump—Adapted Strong Approximations
8.1 Introduction to Jump—Adapted Approximations
8.2 Jump—Adapted Strong Taylor Schemes
8.3 Jump—Adapted Derivative—Free Strong Schemes
8.4 Jump—Adapted Drift—Implicit Schemes
8.5 Predictor—Corrector Strong Schemes
8.6 Jump—Adapted Exact Simulation
8.7 Convergence Results
8.8 Numerical Results on Strong Schemes
8.9 Approximation of Pure Jump Processes
8.10 Exercises
9 Estimating Discretely Observed Diffusions
9.1 Maximum Likelihood Estimation
9.2 Discretization of Estimators
9.3 Transform Functions for Diffusions
9.4 Estimation of Affine Diffusions
9.5 Asymptotics of Estimating Functions
9.6 Estimating Jump Diffusions
9.7 Exercises
10 Filtering
10.1 Kalman—Bucy Filter
10.2 Hidden Markov Chain Filters
10.3 Filtering a Mean Reverting Process
10.4 Balanced Method in Filtering
10.5 A Benchmark Approach to Filtering in Finance
10.6 Exercises
11 Monte Carlo Simulation of SDEs
11.1 Introduction to Monte Carlo Simulation
11.2 Weak Taylor Schemes
11.3 Derivative—Free Weak Approximations
11.4 Extrapolation Methods
11.5 Implicit and Predictor—Corrector Methods
11.6 Exercises
12 Regular Weak Taylor Approximations
12.1 Weak Taylor Schemes
12.2 Commutativity Conditions
12.3 Convergence Results
12.4 Exercises
13 Jump—Adapted Weak Approximations
13.1 Jump—Adapted Weak Schemes
13.2 Derivative—Free Schemes
13.3 Predictor—Corrector Schemes
13.4 Some Jump—Adapted Exact Weak Schemes
13.5 Convergence of Jump—Adapted Weak Taylor Schemes
13.6 Convergence of Jump—Adapted Weak Schemes
13.7 Numerical Results on Weak Schemes
13.8 Exercises
14 Numerical Stability
14.1 Asymptotic p—Stability
14.2 Stability of Predictor—Corrector Methods
14.3 Stability of Some Implicit Methods
14.4 Stability of Simplified Schemes
14.5 Exercises
15 Martingale Representations and Hedge Ratios
15.1 General Contingent Claim Pricing
15.2 Hedge Ratios for One—dimensional Processes
15.3 Explicit Hedge Ratios
15.4 Martingale R，epresentation for Non—Smooth Payoffs
15.5 Absolutely Continuous Payoff Functions
15.6 Maximum of Several Assets
15.7 Hedge Ratios for Lookback Options
15.8 Exercises
16 Variance Reduction Techniques
16.1 Various Variance Reduction Methods
16.2 Measure Transformation Techniques
16.3 Discrete—Time Variance Reduced Estimators
16.4 Control Variates
16.5 HP Variance Reduction
16.6 Exercises
17 Trees and Markov Chain Approxirnations
17.1 Numerical Effects of Tree Methods
17.2 Efficiency of Simplified Schemes
17.3 Higher Order Markov Chain Approximations
17.4 Finite Difference Methods
17.5 ConvergenCP， Theorem for Markov Chains
17.6 Exercises
18 Solutions for Exercises
Acknowledgements
Bibliographical Notes
References
Author Index
Index