Continuous-time Stochastic Control and Optimization with Financial Applications

Transcription

1 Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer

2 Some elements of stochastic analysis Stochastic processes Filtration and processes Stopping times Brownian motion Martingales, semimartingales Stochastic integral and applications.. : Stochastic integral with respect to a continuous semimartingale Ito process ' Ito's formula Martingale representation theorem Girsanov's theorem Stochastic differential equations Strong solutions of SDE Estimates on the moments of solutions to SDE Feynman-Kac formula ( 25 Stochastic optimization problems. Examples in finance Introduction Examples Portfolio allocation Production-consumption model Irreversible investment model Quadratic hedging of options Superreplication cost in uncertain volatility Optimal selling of an asset Valuation of natural resources Other optimization problems in finance Ergodic and risk-sensitive control problems Superreplication under gamma constraints 33

3 XII Robust utility maximization problem and risk measures Forward performance criterion ' : Bibliographical remarks 34 3 The classical PDE approach to dynamic programming Introduction Controlled diffusion processes Dynamic programming principle Hamilton-Jacobi-Bellman equation Formal derivation of H.IB Remarks and extensions Verification theorem Applications Merton portfolio allocation problem in finite horizon Investment-consumption problem with random time horizon A model of production-consumption on infinite horizon Example of singular stochastic control problem Bibliographical remarks 59 4 The viscosity solutions approach to stochastic control problems Introduction Definition of viscosity solutions From dynamic programming to viscosity solutions of HJB equations Viscosity properties inside the domain Terminal condition Comparison principles and uniqueness results Classical comparison principle Strong comparison principle An irreversible investment model Problem Regularity and construction of the value function Optimal strategy Superreplication cost in uncertain volatility model Bounded volatility Unbounded volatility Bibliographical remarks 94 5 Optimal switching and free boundary problems Introduction Optimal stopping Dynamic programming and viscosity property Smooth-fit principle Optimal strategy Methods of solution in the one-dimensional case Examples of applications 104

4 XIII 5.3 Optimal switching Problem formulation Dynamic programming and system of variational inequalities Switching regions The one-dimensional case Explicit solution in the two-regime case Bibliographical remarks Backward stochastic differential equations and optimal control Introduction General properties Existence and uniqueness results Linear BSDE Comparison principles BSDE. PDE and nonlinear Feynman-Kac formulae Control and BSDE Optimization of a family of BSDEs Stochastic maximum principle Reflected BSDEs and optimal stopping problems Existence and approximation via penalization Connection with variational inequalities Applications Exponential utility maximization with option payoff Mean-variance criterion for portfolio selection Bibliographical remarks Martingale and convex duality methods Introduction Dual representation for the superreplication cost Formulation of the superreplication problem Martingale probability measures and no arbitrage Optional decomposition theorem and dual representation for the superreplication cost Ito processes and Brownian filtration framework Duality for the utility maximization problem Formulation of the portfolio optimization problem General existence result Resolution via the dual formulation The case of complete markets Examples in incomplete markets Quadratic hedging problem Problem formulation The martingale case Variance optimal martingale measure and quadratic hedging numeraire 201

5 XIV Problem resolution by change of numeraire Example 210' 7.5 Bibliographical remarks 212 A Complements of integration 213 A.I Uniform integrability 213 A.2 Essential supremum of a family of random variables 215 A.3 Some compactness theorems in probability 215 B Convex analysis considerations 217 B.I Semicontinuous, convex functions '. 217 B.2 Fenchel-Legendre transform 218 B.3 Example in R 219 References 223 Index 231