Market weights equation and its first properties

For more info about our services contact : help@bestpfe.com

Table of contents

1 Introduction – Français 
1.1 Modélisation des capitalisations relatives des actifs
1.1.1 Processus et distribution des poids du marché
1.1.2 Structure du marché et arbitrage relatif
1.1.3 Applications et perspectives
1.2 Un schéma probabiliste pour les équations de McKean-Vlasov
1.2.1 Itération de Picard et méthode de continuation
1.2.2 Représentation probabiliste de la loi marginale
1.2.3 Analyse de la convergence et de l’erreur et perspectives
1.3 Représentation probabiliste et formules d’intégration par parties pour certains modèles à volatilité stochastique avec drift non-borné
1.3.1 Représentation probabiliste du couple spot-volatilité
1.3.2 Formules d’intégration par parties
1.4 Méthodes numériques pour les EDSPRs issues des jeux à champ moyen .
1.4.1 Jeux à champ moyen et approche probabiliste
1.4.2 Algorithmes d’arbre et de grille pour les EDSPRs
2 Introduction – English 
2.1 Modeling of relative capitalizations of assets
2.1.1 Process and distribution of market weights
2.1.2 Structure of the market and relative arbitrage
2.1.3 Applications and perspectives
2.2 A probabilistic scheme for McKean-Vlasov equations
2.2.1 Picard iteration and method of continuation
2.2.2 Probabilistic representation of marginal law
2.2.3 Convergence and error analysis and perspectives
2.3 Probabilistic representation and integration by parts forumulae for some stochastic volatility model with unbounded drift
2.3.1 Probabilistic representation of spot-volatility couple
2.3.2 Integration by parts formulae for sensitivities
2.4 Numerical methods for FBSDEs arising from mean-field games
2.4.1 Mean-field games and probabilistic approach
2.4.2 Tree and grid algorithms for FBSDEs
I Modeling the market by capital distribution 
3 Modeling the market by capital distribution 
3.1 Introduction
3.2 Equation and process of market weights
3.2.1 Market weights equation and its first properties
3.2.2 Distribution and transition density of market weights
3.2.3 Stationary distribution of market weights
3.3 Structure of market and relative arbitrage
3.3.1 Market portfolio and growth rate
3.3.2 Trading strategy and relative arbitrage
3.3.3 Conditions of arbitrage relative to the market
3.4 Future works and conclusion
3.4.1 Functionally generated portfolio
3.4.2 Estimation of model parameters with application to stock market indices
3.4.3 Long-term portfolio optimization under market weights equation
3.4.4 Conclusion and perspectives
3.5 Appendix
3.5.1 Attainment of boundary for Theorem 3.2.1
3.5.2 Hörmander’s condition and theorem for Proposition 3.2.2
3.5.3 Choice of Lyapunov function V for Theorem 3.2.2
3.5.4 Generalized local martingale problem and candidate measure for
II A Probabilistic Scheme for McKean-Vlasov Equations 
4 A Probabilistic Scheme for McKean-Vlasov Equations 
4.1 Introduction
4.2 Description of the numerical probabilistic scheme
4.2.1 Assumptions and well-posedness of the McKean-Vlasov SDE
4.2.2 Construction of the Picard iteration scheme
4.2.3 Probabilistic representation of the marginal law of the Picard iteration scheme
4.2.4 Construction of the Monte Carlo estimator
4.3 Convergence Analysis
4.3.1 Decomposition of the global error and complexity of the algorithm
4.3.2 Convergence analysis of the local Picard iteration schemes .
4.3.3 Convergence analysis of the global Picard iteration scheme on [0; T]
4.4 Numerical results
4.4.1 Standard linear model
4.4.2 Kuramoto model
4.4.3 Polynomial drift model
4.5 Appendix
4.5.1 Proof of Theorem 4.2.1
4.5.2 Proof of Proposition 4.3.1
III Probabilistic Representation and Integration by Parts Formulae for some Stochastic Volatility Models 
5 Probabilistic Representation and Integration by Parts Formulae for some Stochastic Volatility Models
5.1 Introduction
5.2 Preliminaries: assumptions, definition of the underlying Markov chain and related Malliavin calculus
5.2.1 Assumptions
5.2.2 Choice of the approximation process
5.2.3 Markov chain on random time grid
5.2.4 Tailor-made Malliavin calculus for the Markov chain
5.3 Probabilistic representation for the couple (ST ; YT )
5.4 Integration by parts formulae
5.4.1 The transfer of derivative formula
5.4.2 The integration by parts formulae
5.5 Numerical Results
5.5.1 Black-Scholes Model
5.5.2 A Stein-Stein type model
5.5.3 A model with a periodic diffusion coefficient function
5.6 Proof of Theorem 5.3.1 and Lemma 5.4.1
5.6.1 Proof of Theorem 5.3.1
5.6.2 Proof of Lemma 5.4.1
5.7 Some technical results
5.7.1 Emergence of jumps in the renewal process N
5.7.2 Formulae for the computations of price, Delta and Vega
IV Probabilistic Numerical Methods for Mean-Field Games 
6 Probabilistic Numerical Methods for Mean-Field Games 
6.1 Introduction
6.2 Overview of Mean Field Games and FBSDEs
6.2.1 N Player Stochastic Differential Games
6.2.2 Mean Field Games
6.2.3 General System
6.3 Algorithms
6.3.1 Global Picard Iteration on a Small Time Interval
6.3.2 Continuation in Time of the Global Method for Arbitrary Interval/ Coupling
6.3.3 Tree Algorithm for the Global Method
6.3.4 Picard Iteration on the Marginal Laws: a Grid Algorithm
6.4 Examples
6.4.1 Linear Example
6.4.2 Trigonometric Drivers Example
6.4.3 Mixed Model
6.4.4 Examples: Linear Quadratic Mean Field Games
6.5 Conclusion
Bibliography