Mehr zum Titel: | ""Preface""; ""Contents""; ""Suggestions for the Reader""; ""Basic Notation""; ""Motivation and Brief Survey""; ""SDEs with Jumps""; ""Stochastic Processes""; ""Supermartingales and Martingales""; ""Quadratic Variation and Covariation""; ""ItÃ? Integral""; ""ItÃ? Formula""; ""Stochastic Differential Equations""; ""Linear SDEs""; ""SDEs with Jumps""; ""Existence and Uniqueness of Solutions of SDEs""; ""Exercises""; ""Exact Simulation of Solutions of SDEs""; ""Motivation of Exact Simulation""; ""Sampling from Transition Distributions""; ""Exact Solutions of Multi-dimensional SDEs"" ""Functions of Exact Solutions""""Almost Exact Solutions by Conditioning""; ""Almost Exact Simulation by Time Change""; ""Functionals of Solutions of SDEs""; ""Exercises""; ""Benchmark Approach to Finance""; ""Market Model""; ""Best Performing Portfolio""; ""Supermartingale Property and Pricing""; ""Diversification""; ""Real World Pricing Under Some Models""; ""Real World Pricing Under the MMM""; ""Binomial Option Pricing""; ""Exercises""; ""Stochastic Expansions""; ""Introduction to Wagner-Platen Expansions""; ""Multiple Stochastic Integrals""; ""Coefficient Functions"" ""Wagner-Platen Expansions""""Moments of Multiple Stochastic Integrals""; ""Exercises""; ""Introduction to Scenario Simulation""; ""Approximating Solutions of ODEs""; ""Scenario Simulation""; ""Strong Taylor Schemes""; ""Derivative-Free Strong Schemes""; ""Exercises""; ""Regular Strong Taylor Approximations""; ""Discrete-Time Approximation""; ""Strong Order 1.0 Taylor Scheme""; ""Commutativity Conditions""; ""Convergence Results""; ""Lemma on Multiple ItÃ? Integrals""; ""Proof of the Convergence Theorem""; ""Exercises""; ""Regular Strong ItÃ? Approximations"" ""Explicit Regular Strong Schemes""""Drift-Implicit Schemes""; ""Balanced Implicit Methods""; ""Predictor-Corrector Schemes""; ""Convergence Results""; ""Exercises""; ""Jump-Adapted Strong Approximations""; ""Introduction to Jump-Adapted Approximations""; ""Jump-Adapted Strong Taylor Schemes""; ""Jump-Adapted Derivative-Free Strong Schemes""; ""Jump-Adapted Drift-Implicit Schemes""; ""Predictor-Corrector Strong Schemes""; ""Jump-Adapted Exact Simulation""; ""Convergence Results""; ""Numerical Results on Strong Schemes""; ""Approximation of Pure Jump Processes""; ""Exercises"" ""Estimating Discretely Observed Diffusions""""Maximum Likelihood Estimation""; ""Discretization of Estimators""; ""Transform Functions for Diffusions""; ""Estimation of Affine Diffusions""; ""Asymptotics of Estimating Functions""; ""Estimating Jump Diffusions""; ""Exercises""; ""Filtering""; ""Kalman-Bucy Filter""; ""Hidden Markov Chain Filters""; ""Filtering a Mean Reverting Process""; ""Balanced Method in Filtering""; ""A Benchmark Approach to Filtering in Finance""; ""Exercises""; ""Monte Carlo Simulation of SDEs""; ""Introduction to Monte Carlo Simulation""; ""Weak Taylor Schemes"" ""Derivative-Free Weak Approximations"" |
Inhalt: | Stochastic Differential Equations with Jumps -- Exact Simulation of Solutions of SDEs -- Benchmark Approach to Finance and Insurance -- Stochastic Expansions -- to Scenario Simulation -- Regular Strong Taylor Approximations with Jumps -- Regular Strong Itô Approximations -- Jump-Adapted Strong Approximations -- Estimating Discretely Observed Diffusions -- Filtering -- Monte Carlo Simulation of SDEs -- Regular Weak Taylor Approximations -- Jump-Adapted Weak Approximations -- Numerical Stability -- Martingale Representations and Hedge Ratios -- Variance Reduction Techniques -- Trees and Markov Chain Approximations -- Solutions for Exercises. In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics. |
