Inhalt

•  
   Abstract
•  
   Acknowledgements
•  
   Contents
•  
   Notation
•  
   Abbreviations
•  
   1 Introduction
•  
   1.1 Related scientific works
•  
   1.2 Outline of the thesis
•  
  2 Stochastic control problems and Hamilton-Jacobi-Bellman equations
•  
   2.1 General stochastic control problem
•  
   2.1.1 One-dimensional stochastic control problem and HJB equation
•  
   2.1.2 Two-dimensional stochastic control problem and HJB equation
•  
   2.2 Viscosity solutions and convergence theory
•  
  3 Finite difference numerical methods
•  
  3.1 Standard finite difference methods
•  
   3.1.1 Discretization of the Hamilton-Jacobi-Bellman equation
•  
   3.1.2 Classical implicit FDM with policy iteration
•  
   3.1.3 Piecewise constant policy timestepping method
•  
  3.2 Non-Existence of higher order monotone approximation schemes
•  
   3.2.1 Main Results
•  
   3.2.2 Application of the results to the HJB equation
•  
  4 Piecewise predicted policy timestepping method
•  
   4.1 Main idea and algorithm
•  
   4.2 Numerical example: mean-variance optimal investment problem
•  
   4.3 Numerical example: passport option pricing problem
•  
  5 One-dimensional Tree-Grid method
•  
   5.1 Recapitulation: problem formulation
•  
  5.2 Construction of the Tree-Grid method
•  
   5.2.1 The basic idea
•  
   5.2.2 Excursion: FSG method
•  
   5.2.3 The basic Tree-Grid method
•  
   5.2.4 The Tree-Grid method with artificial diffusion
•  
   5.2.5 The final Tree-Grid method algorithm
•  
   5.2.6 Relationship to other numerical methods
•  
   5.3 Convergence of the Tree-Grid method
•  
   5.3.1 Consistency of the scheme
•  
   5.3.2 Monotonicity, stability, convergence
•  
   5.4 Numerical example: uncertain volatility model
•  
   5.5 Numerical example: passport option pricing problem
•  
  6 Tree-Grid method with control independent stencil
•  
   6.1 Tree-Grid method revisited
•  
  6.2 Modification: control-independent stencil
•  
   6.2.1 Derivation of the modified scheme
•  
   6.2.2 Analytical solution of the control problem in the modified scheme
•  
   6.2.3 The Fibonacci algorithm for finding the optimal control
•  
   6.3 Numerical example: passport option pricing problem
•  
  7 Two-dimensional Tree-Grid method
•  
   7.1 Recapitulation: problem formulation
•  
   7.2 Construction of 2D Tree-Grid method
•  
   7.2.1 Notation
•  
   7.2.2 Choosing the stencil nodes
•  
   7.2.3 Choosing the stencil weights (probabilities)
•  
   7.2.4 Artificial diffusion and covariance adjustment
•  
   7.2.5 Setting parameter K and stencil size reduction
•  
   7.2.6 The final 2D Tree-Grid method algorithm
•  
   7.2.7 Comparison to other wide stencil methods
•  
   7.3 Convergence of the 2D Tree-Grid method
•  
   7.4 Numerical example: two-factor uncertain volatility model
•  
  8 Restrictions for the higher dimensional generalization of the Tree-Grid method
•  
   8.1 P-dimensional stochastic control problem
•  
   8.2 Construction of the P-dimensional Tree-Grid scheme
•  
   8.2.1 Notation
•  
   8.2.2 Choosing the stencil nodes
•  
   8.2.3 Choosing the stencil weights (probabilities)
•  
   8.3 Appearance of possibly negative weights
•  
   8.4 Ideas from Tree-Grid schemes applicable to other methods
•  
  9 Conclusion and outlook
•  
   9.1 Outlook of the future research
•  
   References