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LABEL="Seite 103" ORDER="103"><mets:fptr FILEID="IMG_DEFAULT_4132428"/><mets:fptr FILEID="IMG_THUMBS_4132428"/><mets:fptr FILEID="IMG_MIN_4132428"/><mets:fptr FILEID="IMG_MAX_4132428"/><mets:fptr FILEID="ALTO4132428"/></mets:div><mets:div ID="phys4132429" TYPE="page" LABEL="Seite 104" ORDER="104"><mets:fptr FILEID="IMG_DEFAULT_4132429"/><mets:fptr FILEID="IMG_THUMBS_4132429"/><mets:fptr FILEID="IMG_MIN_4132429"/><mets:fptr FILEID="IMG_MAX_4132429"/><mets:fptr FILEID="ALTO4132429"/></mets:div></mets:div></mets:structMap><mets:structMap TYPE="LOGICAL"><mets:div ID="log3555940" DMDID="md3555940" ADMID="amd3555940" TYPE="document" LABEL="The tree-grid method" ORDER="1" CONTENTIDS="urn:nbn:de:hbz:468-20181029-153921-8"><mets:fptr FILEID="PDF_3555940"/><mets:fptr FILEID="IMG_TEASER_4132257"/><mets:div ID="log4132259" TYPE="section" LABEL="Abstract" ORDER="1"/><mets:div ID="log4132262" TYPE="section" LABEL="Acknowledgements" ORDER="2"/><mets:div ID="log4132265" TYPE="section" LABEL="Contents" ORDER="3"/><mets:div ID="log4132268" TYPE="section" LABEL="Notation" ORDER="4"><mets:div ID="log4132270" TYPE="section" LABEL="Abbreviations" ORDER="1"/></mets:div><mets:div ID="log4132272" TYPE="section" LABEL="1 Introduction" ORDER="5"><mets:div ID="log4132274" TYPE="section" LABEL="1.1 Related scientific works" ORDER="1"/><mets:div ID="log4132277" TYPE="section" LABEL="1.2 Outline of the thesis" ORDER="2"/></mets:div><mets:div ID="log4132281" TYPE="section" LABEL="2 Stochastic control problems and Hamilton-Jacobi-Bellman equations" ORDER="6"><mets:div ID="log4132282" TYPE="section" LABEL="2.1 General stochastic control problem" ORDER="1"><mets:div ID="log4132286" TYPE="section" LABEL="2.1.1 One-dimensional stochastic control problem and HJB equation" ORDER="1"/><mets:div ID="log4132288" TYPE="section" LABEL="2.1.2 Two-dimensional stochastic control problem and HJB equation" ORDER="2"/></mets:div><mets:div ID="log4132290" TYPE="section" LABEL="2.2 Viscosity solutions and convergence theory" ORDER="2"/></mets:div><mets:div ID="log4132294" TYPE="section" LABEL="3 Finite difference numerical methods" ORDER="7"><mets:div ID="log4132295" TYPE="section" LABEL="3.1 Standard finite difference methods" ORDER="1"><mets:div ID="log4132296" TYPE="section" LABEL="3.1.1 Discretization of the Hamilton-Jacobi-Bellman equation" ORDER="1"/><mets:div ID="log4132298" TYPE="section" LABEL="3.1.2 Classical implicit FDM with policy iteration" ORDER="2"/><mets:div ID="log4132300" TYPE="section" LABEL="3.1.3 Piecewise constant policy timestepping method" ORDER="3"/></mets:div><mets:div ID="log4132302" TYPE="section" LABEL="3.2 Non-Existence of higher order monotone approximation schemes" ORDER="2"><mets:div ID="log4132303" TYPE="section" LABEL="3.2.1 Main Results" ORDER="1"/><mets:div ID="log4132306" TYPE="section" LABEL="3.2.2 Application of the results to the HJB equation" ORDER="2"/></mets:div></mets:div><mets:div ID="log4132310" TYPE="section" LABEL="4 Piecewise predicted policy timestepping method" ORDER="8"><mets:div ID="log4132311" TYPE="section" LABEL="4.1 Main idea and algorithm" ORDER="1"/><mets:div ID="log4132315" TYPE="section" LABEL="4.2 Numerical example: mean-variance optimal investment problem" ORDER="2"/><mets:div ID="log4132319" TYPE="section" LABEL="4.3 Numerical example: passport option pricing problem" ORDER="3"/></mets:div><mets:div ID="log4132324" TYPE="section" LABEL="5 One-dimensional Tree-Grid method" ORDER="9"><mets:div ID="log4132325" TYPE="section" LABEL="5.1 Recapitulation: problem formulation" ORDER="1"/><mets:div ID="log4132327" TYPE="section" LABEL="5.2 Construction of the Tree-Grid method" ORDER="2"><mets:div ID="log4132328" TYPE="section" LABEL="5.2.1 The basic idea" ORDER="1"/><mets:div ID="log4132330" TYPE="section" LABEL="5.2.2 Excursion: FSG method" ORDER="2"/><mets:div ID="log4132332" TYPE="section" LABEL="5.2.3 The basic Tree-Grid method" ORDER="3"/><mets:div ID="log4132335" TYPE="section" LABEL="5.2.4 The Tree-Grid method with artificial diffusion" ORDER="4"/><mets:div ID="log4132338" TYPE="section" LABEL="5.2.5 The final Tree-Grid method algorithm" ORDER="5"/><mets:div ID="log4132340" TYPE="section" LABEL="5.2.6 Relationship to other numerical methods" ORDER="6"/></mets:div><mets:div ID="log4132343" TYPE="section" LABEL="5.3 Convergence of the Tree-Grid method" ORDER="3"><mets:div ID="log4132345" TYPE="section" LABEL="5.3.1 Consistency of the scheme" ORDER="1"/><mets:div ID="log4132350" TYPE="section" LABEL="5.3.2 Monotonicity, stability, convergence" ORDER="2"/></mets:div><mets:div ID="log4132353" TYPE="section" LABEL="5.4 Numerical example: uncertain volatility model" ORDER="4"/><mets:div ID="log4132356" TYPE="section" LABEL="5.5 Numerical example: passport option pricing problem" ORDER="5"/></mets:div><mets:div ID="log4132360" TYPE="section" LABEL="6 Tree-Grid method with control independent stencil" ORDER="10"><mets:div ID="log4132361" TYPE="section" LABEL="6.1 Tree-Grid method revisited" ORDER="1"/><mets:div ID="log4132364" TYPE="section" LABEL="6.2 Modification: control-independent stencil" ORDER="2"><mets:div ID="log4132365" TYPE="section" LABEL="6.2.1 Derivation of the modified scheme" ORDER="1"/><mets:div ID="log4132367" TYPE="section" LABEL="6.2.2 Analytical solution of the control problem in the modified scheme" ORDER="2"/><mets:div ID="log4132369" TYPE="section" LABEL="6.2.3 The Fibonacci algorithm for finding the optimal control" ORDER="3"/></mets:div><mets:div ID="log4132371" TYPE="section" LABEL="6.3 Numerical example: passport option pricing problem" ORDER="3"/></mets:div><mets:div ID="log4132375" TYPE="section" LABEL="7 Two-dimensional Tree-Grid method" ORDER="11"><mets:div ID="log4132376" TYPE="section" LABEL="7.1 Recapitulation: problem formulation" ORDER="1"/><mets:div ID="log4132378" TYPE="section" LABEL="7.2 Construction of 2D Tree-Grid method" ORDER="2"><mets:div ID="log4132380" TYPE="section" LABEL="7.2.1 Notation" ORDER="1"/><mets:div ID="log4132382" TYPE="section" LABEL="7.2.2 Choosing the stencil nodes" ORDER="2"/><mets:div ID="log4132383" TYPE="section" LABEL="7.2.3 Choosing the stencil weights (probabilities)" ORDER="3"/><mets:div ID="log4132385" TYPE="section" LABEL="7.2.4 Artificial diffusion and covariance adjustment" ORDER="4"/><mets:div ID="log4132388" TYPE="section" LABEL="7.2.5 Setting parameter K and stencil size reduction" ORDER="5"/><mets:div ID="log4132390" TYPE="section" LABEL="7.2.6 The final 2D Tree-Grid method algorithm" ORDER="6"/><mets:div ID="log4132392" TYPE="section" LABEL="7.2.7 Comparison to other wide stencil methods" ORDER="7"/></mets:div><mets:div ID="log4132395" TYPE="section" LABEL="7.3 Convergence of the 2D Tree-Grid method" ORDER="3"/><mets:div ID="log4132399" TYPE="section" LABEL="7.4 Numerical example: two-factor uncertain volatility model" ORDER="4"/></mets:div><mets:div ID="log4132405" TYPE="section" LABEL="8 Restrictions for the higher dimensional generalization of the Tree-Grid method" ORDER="12"><mets:div ID="log4132406" TYPE="section" LABEL="8.1 P-dimensional stochastic control problem" ORDER="1"/><mets:div ID="log4132408" TYPE="section" LABEL="8.2 Construction of the P-dimensional Tree-Grid scheme" ORDER="2"><mets:div ID="log4132409" TYPE="section" LABEL="8.2.1 Notation" ORDER="1"/><mets:div ID="log4132411" TYPE="section" LABEL="8.2.2 Choosing the stencil nodes" ORDER="2"/><mets:div ID="log4132412" TYPE="section" LABEL="8.2.3 Choosing the stencil weights (probabilities)" ORDER="3"/></mets:div><mets:div ID="log4132414" TYPE="section" LABEL="8.3 Appearance of possibly negative weights" ORDER="3"/><mets:div ID="log4132416" TYPE="section" LABEL="8.4 Ideas from Tree-Grid schemes applicable to other methods" ORDER="4"/></mets:div><mets:div ID="log4132419" TYPE="section" LABEL="9 Conclusion and outlook" ORDER="13"><mets:div ID="log4132420" TYPE="section" LABEL="9.1 Outlook of the future research" ORDER="1"/></mets:div><mets:div ID="log4132425" TYPE="section" 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