| Some elements of stochastic analysis | p. 1 |
| Stochastic processes | p. 1 |
| Filtration and processes | p. 1 |
| Stopping times | p. 3 |
| Brownian motion | p. 5 |
| Martingales, semimartingales | p. 6 |
| Stochastic integral and applications | p. 12 |
| Stochastic integral with respect to a continuous semimartingale | p. 12 |
| It&ocap; process | p. 16 |
| It&ocap;'s formula | p. 17 |
| Martingale representation theorem | p. 18 |
| Girsanov's theorem | p. 18 |
| Stochastic differential equations | p. 22 |
| Strong solutions of SDE | p. 22 |
| Estimates on the moments of solutions to SDE | p. 24 |
| Feynman-Kac formula | p. 25 |
| Stochastic optimization problems. Examples in finance | p. 27 |
| Introduction | p. 27 |
| Examples | p. 28 |
| Portfolio allocation | p. 28 |
| Production-consumption model | p. 29 |
| Irreversible investment model | p. 30 |
| Quadratic hedging of options | p. 31 |
| Superreplication cost in uncertain volatility | p. 31 |
| Optimal selling of an asset | p. 32 |
| Valuation of natural resources | p. 32 |
| Other optimization problems in finance | p. 32 |
| Ergodic and risk-sensitive control problems | p. 32 |
| Superreplication under gamma constraints | p. 33 |
| Robust utility maximization problem and risk measures | p. 33 |
| Forward performance criterion | p. 34 |
| Bibliographical remarks | p. 34 |
| The Classical PDE approach to dynamic programming | p. 37 |
| Introduction | p. 37 |
| Controlled diffusion processes | p. 37 |
| Dynamic programming principle | p. 40 |
| Hamilton-Jacobi-Bellman equation | p. 42 |
| Formal derivation of HJB | p. 42 |
| Remarks and extensions | p. 45 |
| Verification theorem | p. 47 |
| Applications | p. 51 |
| Merton portfolio allocation problem in finite horizon | p. 51 |
| Investment-consumption problem with random time horizon | p. 53 |
| A model of production-consumption on infinite horizon | p. 55 |
| Example of singular stochastic control problem | p. 58 |
| Bibliographical remarks | p. 59 |
| The viscosity solutions approach to stochastic control problems | p. 61 |
| Introduction | p. 61 |
| Definition of viscosity solutions | p. 61 |
| From dynamic programming to viscosity solutions of HJB equations | p. 64 |
| Viscosity properties inside the domain | p. 64 |
| Terminal condition | p. 69 |
| Comparison principles and uniqueness results | p. 75 |
| Classical comparison principle | p. 76 |
| Strong comparison principle | p. 77 |
| An irreversible investment model | p. 82 |
| Problem | p. 82 |
| Regularity and construction of the value function | p. 83 |
| Optimal strategy | p. 88 |
| Superreplication cost in uncertain volatility model | p. 89 |
| Bounded volatility | p. 90 |
| Unbounded volatility | p. 91 |
| Bibliographical remarks | p. 94 |
| Optimal switching and free boundary problems | p. 95 |
| Introduction | p. 95 |
| Optimal stopping | p. 95 |
| Dynamic programming and viscosity property | p. 96 |
| Smooth-fit principle | p. 99 |
| Optimal strategy | p. 101 |
| Methods of solution in the one-dimensional case | p. 103 |
| Examples of applications | p. 104 |
| Optimal switching | p. 107 |
| Problem formulation | p. 108 |
| Dynamic programming and system of variational inequalities | p. 109 |
| Switching regions | p. 114 |
| The one-dimensional case | p. 116 |
| Explicit solution in the two-regime case | p. 119 |
| Bibliographical remarks | p. 137 |
| Backward stochastic differential equations and optimal control | p. 139 |
| Introduction | p. 139 |
| General properties | p. 139 |
| Existence and uniqueness results | p. 139 |
| Linear BSDE | p. 141 |
| Comparison principles | p. 142 |
| BSDE, PDE and nonlinear Feynman-Kac formulae | p. 143 |
| Control and BSDE | p. 147 |
| Optimization of a family of BSDEs | p. 147 |
| Stochastic maximum principle | p. 149 |
| Reflected BSDEs and optimal stopping problems | p. 152 |
| Existence and approximation via penalization | p. 154 |
| Connection with variational inequalities | p. 159 |
| Applications | p. 162 |
| Exponential utility maximization with option payoff | p. 162 |
| Mean-variance criterion for portfolio selection | p. 162 |
| Bibliographical remarks | p. 169 |
| Martingale and convex duality methods | p. 171 |
| Introduction | p. 171 |
| Dual representation for the superreplication cost | p. 172 |
| Formulation of the superreplication problem | p. 172 |
| Martingale probability measures and no arbitrage | p. 173 |
| Optional decomposition theorem and dual representation for the superreplication cost | p. 174 |
| It&ocap; processes and Brownian filtration framework | p. 177 |
| Duality for the utility maximization problem | p. 181 |
| Formulation of the portfolio optimization problem | p. 181 |
| General existence result | p. 181 |
| Resolution via the dual formulation | p. 183 |
| The case of complete markets | p. 195 |
| Examples in incomplete markets | p. 197 |
| Quadratic hedging problem | p. 199 |
| Problem formulation | p. 199 |
| The martingale case | p. 200 |
| Variance optimal martingale measure and quadratic hedging numéraire | p. 201 |
| Problem resolution by change of numéraire | p. 206 |
| Example | p. 210 |
| Bibliographical remarks | p. 212 |
| Complements of integration | p. 213 |
| Uniform integrability | p. 213 |
| Essential supremum of a family of random variables | p. 215 |
| Some compactness theorems in probability | p. 215 |
| Convex analysis considerations | p. 217 |
| Semicontinuous, convex functions | p. 217 |
| Fenchel-Legendre transform | p. 218 |
| Example in R | p. 219 |
| References | p. 223 |
| Index | p. 231 |
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