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 Martingales 1.3 Quadratic Variation and Covariation 1.4 Ito Integral 1.5 It5 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 E.xercises 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 2.4 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 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 Commutativity Conditions
6.4 Convergence Results 6.5 Lemma on Multiple It5 Integrals 6.6 Proof of the Convergence Theorem 6.7 Exercises 7 Regular Strong It6 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 Representation for Non-Smooth Payoffs .. 15.5 Absolutely Continuous Payoff ~nctions 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 Approximations 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 Convergence Theorem for Markov Chains 17.6 Exercises 18 Solutions for Exercises Acknowledgements Bibliographical Notes References Author Index Index
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