Logarithmic utility function

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Table of contents

INTRODUCTION GÉNÉRALE 
0.1 Maximisation d’utilité dans un modèle avec défauts
0.1.1 Maximisation de la fonction d’utilité exponentielle et prix d’indifférence dans un marché avec défaut
0.1.2 Optimisation de portefeuille dans un marché avec défaut sous information totale/partielle
0.2 Grossissement progressif de filtrations et EDSR à sauts
0.3 Modélisation du spread bid-ask
I MAXIMIZATION OF UTILITY IN AN INCOMPLETE MARKET WITH DEFAULTS AND TOTAL/PARTIAL INFORMATION 
1 Exponential utility maximization 
1.1 Introduction
1.2 The market model
1.3 Strategies valued in a compact set
1.4 The non constrained case
1.4.1 The set of admissible strategies
1.4.2 Characterization of the dynamic value function as the maximal subsolution of a BSDE
1.5 Approximation of the value function
1.6 Case of bounded coefficients
1.7 Coefficients satisfying some integrability conditions
1.7.1 Case of strategies valued in a convex-compact set
1.7.2 The non constrained case
1.8 Indifference pricing
1.9 Generalizations
1.9.1 Several default times and several stocks
1.9.2 Poisson jumps
1.10 Appendix
1.10.1 Essential supremum
1.10.2 A classical lemma of analysis
1.10.3 Proof of the closedness by binding of A′
1.10.4 Proof of the existence of a càd-làg modification of (Jt)
1.10.5 Proof of equality (1.5.2)
1.10.6 Proof of optimality criterion (Proposition 1.7.2)
1.10.7 Characterization of the value function as the maximum solution of BSDE (1.3.3)
2 Optimization under Full/Partial Information 
2.1 Introduction
2.2 The model
2.3 Logarithmic utility function
2.4 Power utility
2.4.1 Optimization over bounded strategies
2.4.2 General case
2.4.3 Several default times and several assets
2.5 The partial information case
2.5.1 Filtering
2.5.2 Optimization problem for the logarithmic and power utility functions
2.5.3 Optimization problem for the exponential utility function and indifference pricing
2.6 Appendix
2.6.1 Proof of Propositions 2.4.2 and 2.4.3
2.6.2 Proof of Theorem 2.4.1
2.6.3 Proof of Theorem 2.4.2
2.6.4 Proof of Lemma 2.5.3
II PROGRESSIVE ENLARGEMENT OF FILTRATIONS AND BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 
3 BSDEs with jumps 
3.1 Introduction
3.2 Progressive enlargement of filtrations
3.3 Decomposition of BSDEs with jumps
3.3.1 Existence of a solution
3.3.2 Application to the pricing of a European option in a complete market with default
3.3.3 Uniqueness
3.4 Decomposition of Feynman-Kac formula for IPDE
3.5 Utility maximization in a jump market model
III BID-ASK SPREAD MODELING 
4 Bid-Ask spread modeling 
4.1 Introduction
4.2 The model
4.2.1 Theoretical analysis of path sensitivity and approximation
4.2.2 Bid-Ask model
4.3 Optimal liquidation portfolio problem
4.3.1 The economic motivations and the objective functions
4.3.2 Theoretical solution of the optimization problem
4.3.3 Log-Normal and Constant Elasticity of Variance Diffusions
4.4 Numerical results
4.4.1 Black-Scholes case
4.4.2 CEV case
4.4.3 Comparison on scenarios
4.5 Appendix
4.5.1 Proofs of Lemmas 4.2.1 and 4.2.2
Bibliography