Contents -- Preface -- 1 Stochastic processes -- 1.1 Random functions -- 1.1.1 Trajectories of stochastic processes -- 1.1.2 Jumps of stochastic processes -- 1.1.3 When are stochastic processes equal? -- 1.2 Measurability of Stochastic Processes -- 1.2.1 Filtration, adapted, and progressively measurable processes -- 1.2.2 Stopping times -- 1.2.3 Stopped variables, σ-algebras, and truncated processes -- 1.2.4 Predictable processes -- 1.3 Martingales -- 1.3.1 Doob's inequalities -- 1.3.2 The energy equality -- 1.3.3 The quadratic variation of discrete time martingales -- 1.3.4 The downcrossings inequality -- 1.3.5 Regularization of martingales -- 1.3.6 The Optional Sampling Theorem -- 1.3.7 Application: elementary properties of Lévy processes -- 1.3.8 Application: the first passage times of the Wiener processes -- 1.3.9 Some remarks on the usual assumptions -- 1.4 Localization -- 1.4.1 Stability under truncation -- 1.4.2 Local martingales -- 1.4.3 Convergence of local martingales: uniform convergence on compacts in probability -- 1.4.4 Locally bounded processes -- 2 Stochastic Integration with Locally Square-Integrable Martingales -- 2.1 The Itô-Stieltjes Integrals -- 2.1.1 Itô-Stieltjes integrals when the integrators have finite variation -- 2.1.2 Itô-Stieltjes integrals when the integrators are locally square-integrable martingales -- 2.1.3 Itô-Stieltjes integrals when the integrators are semimartingales -- 2.1.4 Properties of the Itô-Stieltjes integral -- 2.1.5 The integral process -- 2.1.6 Integration by parts and the existence of the quadratic variation -- 2.1.7 The Kunita-Watanabe inequality -- 2.2 The Quadratic Variation of Continuous Local Martingales -- 2.3 Integration when Integrators are Continuous Semimartingales -- 2.3.1 The space of square-integrable continuous local martingales.

2.3.2 Integration with respect to continuous local martingales -- 2.3.3 Integration with respect to semimartingales -- 2.3.4 The Dominated Convergence Theorem for stochastic integrals -- 2.3.5 Stochastic integration and the Itô-Stieltjes integral -- 2.4 Integration when Integrators are Locally Square-Integrable Martingales -- 2.4.1 The quadratic variation of locally square-integrable martingales -- 2.4.2 Integration when the integrators are locally square-integrable martingales -- 2.4.3 Stochastic integration when the integrators are semimartingales -- 3 The Structure of Local Martingales -- 3.1 Predictable Projection -- 3.1.1 Predictable stopping times -- 3.1.2 Decomposition of thin sets -- 3.1.3 The extended conditional expectation -- 3.1.4 Definition of the predictable projection -- 3.1.5 The uniqueness of the predictable projection, the predictable section theorem -- 3.1.6 Properties of the predictable projection -- 3.1.7 Predictable projection of local martingales -- 3.1.8 Existence of the predictable projection -- 3.2 Predictable Compensators -- 3.2.1 Predictable Radon-Nikodym Theorem -- 3.2.2 Predictable Compensator of locally integrable processes -- 3.2.3 Properties of the Predictable Compensator -- 3.3 The Fundamental Theorem of Local Martingales -- 3.4 Quadratic Variation -- 4 General Theory of Stochastic Integration -- 4.1 Purely Discontinuous Local Martingales -- 4.1.1 Orthogonality of local martingales -- 4.1.2 Decomposition of local martingales -- 4.1.3 Decomposition of semimartingales -- 4.2 Purely Discontinuous Local Martingales and Compensated Jumps -- 4.2.1 Construction of purely discontinuous local martingales -- 4.2.2 Quadratic variation of purely discontinuous local martingales -- 4.3 Stochastic Integration With Respect To Local Martingales -- 4.3.1 Definition of stochastic integration.

4.3.2 Properties of stochastic integration -- 4.4 Stochastic Integration With Respect To Semimartingales -- 4.4.1 Integration with respect to special semimartingales -- 4.4.2 Linearity of the stochastic integral -- 4.4.3 The associativity rule -- 4.4.4 Change of measure -- 4.5 The Proof of Davis' Inequality -- 4.5.1 Discrete-time Davis' inequality -- 4.5.2 Burkholder's inequality -- 5 Some Other Theorems -- 5.1 The Doob-Meyer Decomposition -- 5.1.1 The proof of the theorem -- 5.1.2 Dellacherie's formulas and the natural processes -- 5.1.3 The sub- super- and the quasi-martingales are semimartingales -- 5.2 Semimartingales as Good Integrators -- 5.3 Integration of Adapted Product Measurable Processes -- 5.4 Theorem of Fubini for Stochastic Integrals -- 5.5 Martingale Representation -- 6 Itô's Formula -- 6.1 Itô's Formula for Continuous Semimartingales -- 6.2 Some Applications of the Formula -- 6.2.1 Zeros of Wiener processes -- 6.2.2 Continuous Lévy processes -- 6.2.3 Lévy's characterization of Wiener processes -- 6.2.4 Integral representation theorems for Wiener processes -- 6.2.5 Bessel processes -- 6.3 Change of Measure for Continuous Semimartingales -- 6.3.1 Locally absolutely continuous change of measure -- 6.3.2 Semimartingales and change of measure -- 6.3.3 Change of measure for continuous semimartingales -- 6.3.4 Girsanov's formula for Wiener processes -- 6.3.5 Kazamaki-Novikov criteria -- 6.4 Itô's Formula for Non-Continuous Semimartingales -- 6.4.1 Itô's formula for processes with finite variation -- 6.4.2 The proof of Itô's formula -- 6.4.3 Exponential semimartingales -- 6.5 Itô's Formula For Convex Functions -- 6.5.1 Derivative of convex functions -- 6.5.2 Definition of local times -- 6.5.3 Meyer-Itô formula -- 6.5.4 Local times of continuous semimartingales -- 6.5.5 Local time of Wiener processes -- 6.5.6 Ray-Knight theorem.

6.5.7 Theorem of Dvoretzky Erdos and Kakutani -- 7 Processes with Independent Increments -- 7.1 Lévy processes -- 7.1.1 Poisson processes -- 7.1.2 Compound Poisson processes generated by the jumps -- 7.1.3 Spectral measure of Lévy processes -- 7.1.4 Decomposition of Lévy processes -- 7.1.5 Lévy-Khintchine formula for Lévy processes -- 7.1.6 Construction of Lévy processes -- 7.1.7 Uniqueness of the representation -- 7.2 Predictable Compensators of Random Measures -- 7.2.1 Measurable random measures -- 7.2.2 Existence of predictable compensator -- 7.3 Characteristics of Semimartingales -- 7.4 Lévy-Khintchine Formula for Semimartingales with Independent Increments -- 7.4.1 Examples: probability of jumps of processes with independent increments -- 7.4.2 Predictable cumulants -- 7.4.3 Semimartingales with independent increments -- 7.4.4 Characteristics of semimartingales with independent increments -- 7.4.5 The proof of the formula -- 7.5 Decomposition of Processes with Independent Increments -- Appendix -- A: Results from Measure Theory -- A.1 The Monotone Class Theorem -- A.2 Projection and the Measurable Selection Theorems -- A.3 Cramér's Theorem -- A.4 Interpretation of Stopped σ-algebras -- B: Wiener Processes -- B.1 Basic Properties -- B.2 Existence of Wiener Processes -- B.3 Quadratic Variation of Wiener Processes -- C: Poisson processes -- Notes and Comments -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Y.