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Viscosity Solutions and Optimization in Mathematical Finance

Tesi di Dottorato

Dipartimento: Matematica

Autore: Marco Papi Contatta »

Composta da 226 pagine.

 

Questa tesi ha raggiunto 473 click dal 20/03/2004.

 

Consultata integralmente una volta.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.

 

 

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<br><b>Acknowledgments </b><br/> <br><b>Introduction </b><br/> Notations & Definitions <br/> <br><b>Part I </b><br/> <br><b>1 Financial Markets Modeling </b><br/> 1.1 FinancialMarkets <br/> 1.1.1 Trading Strat&egrave;gies and Arbitrage <br/> 1.1.2 Equivalent Martingale Measures <br/> 1.2 Completeness of Financial Markets <br/> <br><b>2 Pricing Mortgage-Backed Securities </b><br/> 2.1 Cash-FlowsModeling <br/> 2.1.1 TheMBSMarket <br/> 2.2 Pricing M BSs: A PDE Approach <br/> 2.3 Appendix <br/> <br><b>3 Viscosity Solutions for Degenerate Nonlinear Equations </b><br/> 3.1 Definitions and Preliminaries <br/> 3.2 Well Posedness of the Cauchy Problem <br/> <br><b>4 A Generalized Osgood Condition for Viscosity Solutions to Fully Nonlinear PDEs </b><br/> 4.1 Comparison Principle and Existence <br/> 4.2 Application to the Financial Model <br/> <br><b>5 Regularity Results for a Class of Semilinear PDEs and Applications </b><br/> 5.1 Financial motivations: the MBS model <br/> 5.2 MainResults <br/> 5.3 Proof of the Results <br/> 5.3.1 Time regularity <br/> 5.4 Regularity and the Ito's formula. <br/> 5.5 Conclusions <br/> <br><b>Part II </b><br/> <br><b>6 A Model for the Optimal Asset-Liability Management for Insurance Companies </b><br/> 6.1 Type of Contract <br/> 6.2 HedgingApproach <br/> 6.3 Stochastic Optimization <br/> 6.4 The Model <br/> 6.4.1 The evolution of the assets <br/> 6.4.2 The evolution of the fund <br/> 6.4.3 The Reserves and Solvency Margin <br/> 6.4.4 The Performance Index <br/> 6.4.5 Additional Constraints <br/> 6.5 Numericaltests <br/> 6.6 Conclusions and Further Developments <br/> <br><b>7 Optimal ALM with Constraints: A Dynamic Programming Approach </b><br/> 7.1 The Model <br/> 7.2 The Dynamic Programming Algorithm <br/> 7.3 The Numerical Algorithm <br/> 7.3.1 Probabilistic State Constraints <br/> 7.3.2 Projection Property <br/> 7.3.3 The Numerical Approximation of the DP Algorithm ..<br/> 7.4 A Priori Error Estimates <br/> 7.4.1 The Regularity of the Value Function <br/> Appendix A <br/> 7.4.2 A Convergence Result <br/> 7.5 NumericalResults <br/> 7.5.1 Stochastic Tests <br/> 7.5.2 Deterministic Tests <br/> 7.6 Conclusions <br/> 7.7 Appendix: Proof of Lemma 7.11 <br/> <br><b>8 Lipschitzian Estimates in Discrete-Time Constrained Optimal Control </b><br/> 8.1 The Optimization Problem <br/> 8.2 Main Notations and Definitions <br/> 8.3 The Lipschitz Regularity <br/> 8.4 The dH-Regularity of the Control Set <br/> 8.5 Proof of the Results <br/> <br><b>9 Regularity Properties of Constrained Set- Valued Mappings </b><br/> 9.1 Generalized Jacobians <br/> 9.2 The Lipschitz Regularity <br/> 9.3 The Implicit Function Theorem <br/> <br><b>References </b><br/>

Indice della Tesi di Marco Papi

Indice della tesi: Viscosity Solutions and Optimization in Mathematical Finance, Pagina 2