| |
| |
| Preface | |
| |
| |
| |
| Stochastic processes | |
| |
| |
| |
| Random functions | |
| |
| |
| |
| Trajectories of stochastic processes | |
| |
| |
| |
| Jumps of stochastic processes | |
| |
| |
| |
| When are stochastic processes equal? | |
| |
| |
| |
| Measurability of Stochastic Processes | |
| |
| |
| |
| Filtration, adapted, and progressively measurable processes | |
| |
| |
| |
| Stopping times | |
| |
| |
| |
| Stopped variables, [sigma]-algebras, and truncated processes | |
| |
| |
| |
| Predictable processes | |
| |
| |
| |
| Martingales | |
| |
| |
| |
| Doob's inequalities | |
| |
| |
| |
| The energy equality | |
| |
| |
| |
| The quadratic variation of discrete time martingales | |
| |
| |
| |
| The downcrossings inequality | |
| |
| |
| |
| Regularization of martingales | |
| |
| |
| |
| The Optional Sampling Theorem | |
| |
| |
| |
| Application: elementary properties of Levy processes | |
| |
| |
| |
| Application: the first passage times of the Wiener processes | |
| |
| |
| |
| Some remarks on the usual assumptions | |
| |
| |
| |
| Localization | |
| |
| |
| |
| Stability under truncation | |
| |
| |
| |
| Local martingales | |
| |
| |
| |
| Convergence of local martingales: uniform convergence on compacts in probability | |
| |
| |
| |
| Locally bounded processes | |
| |
| |
| |
| Stochastic Integration with Locally Square-Integrable Martingales | |
| |
| |
| |
| The Ito-Stieltjes Integrals | |
| |
| |
| |
| Ito-Stieltjes integrals when the integrators have finite variation | |
| |
| |
| |
| Ito-Stieltjes integrals when the integrators are locally square-integrable martingales | |
| |
| |
| |
| Ito-Stieltjes integrals when the integrators are semimartingales | |
| |
| |
| |
| Properties of the Ito-Stieltjes integral | |
| |
| |
| |
| The integral process | |
| |
| |
| |
| Integration by parts and the existence of the quadratic variation | |
| |
| |
| |
| The Kunita-Watanabe inequality | |
| |
| |
| |
| The Quadratic Variation of Continuous Local Martingales | |
| |
| |
| |
| Integration when Integrators are Continuous Semimartingales | |
| |
| |
| |
| The space of square-integrable continuous local martingales | |
| |
| |
| |
| Integration with respect to continuous local martingales | |
| |
| |
| |
| Integration with respect to semimartingales | |
| |
| |
| |
| The Dominated Convergence Theorem for stochastic integrals | |
| |
| |
| |
| Stochastic integration and the Ito-Stieltjes integral | |
| |
| |
| |
| Integration when Integrators are Locally Square-Integrable Martingales | |
| |
| |
| |
| The quadratic variation of locally square-integrable martingales | |
| |
| |
| |
| Integration when the integrators are locally square-integrable martingales | |
| |
| |
| |
| Stochastic integration when the integrators are semimartingales | |
| |
| |
| |
| The Structure of Local Martingales | |
| |
| |
| |
| Predictable Projection | |
| |
| |
| |
| Predictable stopping times | |
| |
| |
| |
| Decomposition of thin sets | |
| |
| |
| |
| The extended conditional expectation | |
| |
| |
| |
| Definition of the predictable projection | |
| |
| |
| |
| The uniqueness of the predictable projection, the predictable section theorem | |
| |
| |
| |
| Properties of the predictable projection | |
| |
| |
| |
| Predictable projection of local martingales | |
| |
| |
| |
| Existence of the predictable projection | |
| |
| |
| |
| Predictable Compensators | |
| |
| |
| |
| Predictable Radon-Nikodym Theorem | |
| |
| |
| |
| Predictable Compensator of locally integrable processes | |
| |
| |
| |
| Properties of the Predictable Compensator | |
| |
| |
| |
| The Fundamental Theorem of Local Martingales | |
| |
| |
| |
| Quadratic Variation | |
| |
| |
| |
| General Theory of Stochastic Integration | |
| |
| |
| |
| Purely Discontinuous Local Martingales | |
| |
| |
| |
| Orthogonality of local martingales | |
| |
| |
| |
| Decomposition of local martingales | |
| |
| |
| |
| Decomposition of semimartingales | |
| |
| |
| |
| Purely Discontinuous Local Martingales and Compensated Jumps | |
| |
| |
| |
| Construction of purely discontinuous local martingales | |
| |
| |
| |
| Quadratic variation of purely discontinuous local martingales | |
| |
| |
| |
| Stochastic Integration With Respect To Local Martingales | |
| |
| |
| |
| Definition of stochastic integration | |
| |
| |
| |
| Properties of stochastic integration | |
| |
| |
| |
| Stochastic Integration With Respect To Semimartingales | |
| |
| |
| |
| Integration with respect to special semimartingales | |
| |
| |
| |
| Linearity of the stochastic integral | |
| |
| |
| |
| The associativity rule | |
| |
| |
| |
| Change of measure | |
| |
| |
| |
| The Proof of Davis' Inequality | |
| |
| |
| |
| Discrete-time Davis' inequality | |
| |
| |
| |
| Burkholder's inequality | |
| |
| |
| |
| Some Other Theorems | |
| |
| |
| |
| The Doob-Meyer Decomposition | |
| |
| |
| |
| The proof of the theorem | |
| |
| |
| |
| Dellacherie's formulas and the natural processes | |
| |
| |
| |
| The sub- super- and the quasi-martingales are semimartingales | |
| |
| |
| |
| Semimartingales as Good Integrators | |
| |
| |
| |
| Integration of Adapted Product Measurable Processes | |
| |
| |
| |
| Theorem of Fubini for Stochastic Integrals | |
| |
| |
| |
| Martingale Representation | |
| |
| |
| |
| Ito's Formula | |
| |
| |
| |
| Ito's Formula for Continuous Semimartingales | |
| |
| |
| |
| Some Applications of the Formula | |
| |
| |
| |
| Zeros of Wiener processes | |
| |
| |
| |
| Continuous Levy processes | |
| |
| |
| |
| Levy's characterization of Wiener processes | |
| |
| |
| |
| Integral representation theorems for Wiener processes | |
| |
| |
| |
| Bessel processes | |
| |
| |
| |
| Change of Measure for Continuous Semimartingales | |
| |
| |
| |
| Locally absolutely continuous change of measure | |
| |
| |
| |
| Semimartingales and change of measure | |
| |
| |
| |
| Change of measure for continuous semimartingales | |
| |
| |
| |
| Girsanov's formula for Wiener processes | |
| |
| |
| |
| Kazamaki-Novikov criteria | |
| |
| |
| |
| Ito's Formula for Non-Continuous Semimartingales | |
| |
| |
| |
| Ito's formula for processes with finite variation | |
| |
| |
| |
| The proof of Ito's formula | |
| |
| |
| |
| Exponential semimartingales | |
| |
| |
| |
| Ito's Formula For Convex Functions | |
| |
| |
| |
| Derivative of convex functions | |
| |
| |
| |
| Definition of local times | |
| |
| |
| |
| Meyer-Ito formula | |
| |
| |
| |
| Local times of continuous semimartingales | |
| |
| |
| |
| Local time of Wiener processes | |
| |
| |
| |
| Ray-Knight theorem | |
| |
| |
| |
| Theorem of Dvoretzky Erdos and Kakutani | |
| |
| |
| |
| Processes with Independent Increments | |
| |
| |
| |
| Levy processes | |
| |
| |
| |
| Poisson processes | |
| |
| |
| |
| Compound Poisson processes generated by the jumps | |
| |
| |
| |
| Spectral measure of Levy processes | |
| |
| |
| |
| Decomposition of Levy processes | |
| |
| |
| |
| Levy-Khintchine formula for Levy processes | |
| |
| |
| |
| Construction of Levy processes | |
| |
| |
| |
| Uniqueness of the representation | |
| |
| |
| |
| Predictable Compensators of Random Measures | |
| |
| |
| |
| Measurable random measures | |
| |
| |
| |
| Existence of predictable compensator | |
| |
| |
| |
| Characteristics of Semimartingales | |
| |
| |
| |
| Levy-Khintchine Formula for Semimartingales with Independent Increments | |
| |
| |
| |
| Examples: probability of jumps of processes with independent increments | |
| |
| |
| |
| Predictable cumulants | |
| |
| |
| |
| Semimartingales with independent increments | |
| |
| |
| |
| Characteristics of semimartingales with independent increments | |
| |
| |
| |
| The proof of the formula | |
| |
| |
| |
| Decomposition of Processes with Independent Increments | |
| |
| |
| Appendix | |
| |
| |
| |
| Results from Measure Theory | |
| |
| |
| |
| The Monotone Class Theorem | |
| |
| |
| |
| Projection and the Measurable Selection Theorems | |
| |
| |
| |
| Cramer's Theorem | |
| |
| |
| |
| Interpretation of Stopped [sigma]-algebras | |
| |
| |
| |
| Wiener Processes | |
| |
| |
| |
| Basic Properties | |
| |
| |
| |
| Existence of Wiener Processes | |
| |
| |
| |
| Quadratic Variation of Wiener Processes | |
| |
| |
| |
| Poisson processes | |
| |
| |
| Notes and Comments | |
| |
| |
| References | |
| |
| |
| Index | |