Probability Theory II - Stochastic Calculus

Preface
Frequently Used Symbols and Notations
Contents
Abbreviations
1 Stochastic Processes
	1.1 Stochastic Processes: Law and Finite-Dimensional Distributions
		1.1.1 Measurable Processes
	1.2 Uniqueness
	1.3 Existence
	1.4 Filtrations and Martingales
	1.5 Proof of Kolmogorov's Extension Theorem
	1.6 Key Ideas to Remember
2 Markov Processes
	2.1 Transition Law and Feller Processes
	2.2 Markov Property
	2.3 Processes with Independent Increments and Martingales
	2.4 Finite-Dimensional Laws and Chapman-Kolmogorov Equation
	2.5 Characteristic Operator and Kolmogorov Equations
		2.5.1 The Local Case
		2.5.2 Backward Kolmogorov Equation
		2.5.3 Forward Kolmogorov (or Fokker-Planck) Equation
	2.6 Markov Processes and Diffusions
	2.7 Key Ideas to Remember
3 Continuous Processes
	3.1 Continuity and a.s. Continuity
	3.2 Canonical Version of a Continuous Process
	3.3 Kolmogorov's Continuity Theorem
	3.4 Proof of Kolmogorov's Continuity Theorem
	3.5 Key Ideas to Remember
4 Brownian Motion
	4.1 Definition
	4.2 Markov and Feller Properties
	4.3 Wiener Space
	4.4 Brownian Martingales
	4.5 Key Ideas to Remember
5 Poisson Process
	5.1 Definition
	5.2 Markov and Feller Properties
	5.3 Martingale Properties
	5.4 Proof of Theorem 5.2.1
	5.5 Key Ideas to Remember
6 Stopping Times
	6.1 The Discrete Case
		6.1.1 Optional Sampling, Maximal Inequalities, and Upcrossing Lemma
	6.2 The Continuous Case
		6.2.1 Usual Conditions and Stopping Times
		6.2.2 Filtration Enlargement and Markov Processes
		6.2.3 Filtration Enlargement and Lévy Processes
		6.2.4 General Results on Stopping Times
	6.3 Key Ideas to Remember
7 Strong Markov Property
	7.1 Feller and Strong Markov Properties
	7.2 Reflection Principle
	7.3 The Homogeneous Case
8 Continuous Martingales
	8.1 Optional Sampling and Maximal Inequalities
	8.2 Càdlàg Martingales
	8.3 The Space Mc,2 of Square-Integrable Continuous Martingales
	8.4 The Space Mc,loc of Continuous Local Martingales
	8.5 Uniformly Square-Integrable Martingales
	8.6 Key Ideas to Remember
9 Theory of Variation
	9.1 Riemann-Stieltjes Integral
	9.2 Lebesgue-Stieltjes Integral
	9.3 Semimartingales
		9.3.1 Brownian Motion as a Semimartingale
		9.3.2 Semimartingales of Bounded Variation
	9.4 Doob's Decomposition and Quadratic Variation Process
	9.5 Covariation Matrix
	9.6 Proof of Doob's Decomposition Theorem
	9.7 Key Ideas to Remember
10 Stochastic Integral
	10.1 Integral with Respect to a Brownian Motion
		10.1.1 Proof of Lemma 10.1.7
	10.2 Integral with Respect to Continuous Square-Integrable Martingales
		10.2.1 Integral of Indicator Processes
		10.2.2 Integral of Simple Processes
		10.2.3 Integral in L2
		10.2.4 Integral in Lloc2
		10.2.5 Stochastic Integral as a Riemann-Stieltjes Integral
	10.3 Integral with Respect to Continuous Semimartingales
	10.4 Scalar Itô Processes
	10.5 Key Ideas to Remember
11 Itô's Formula
	11.1 Itô's Formula for Continuous Semimartingales
		11.1.1 Itô's Formula for Brownian Motion
		11.1.2 Itô's Formula for Itô Processes
	11.2 Some Consequences of Itô's Formula
		11.2.1 Burkholder-Davis-Gundy Inequalities
		11.2.2 Quadratic Variation Process
	11.3 Proof of Itô's Formula
	11.4 Key Ideas to Remember
12 Multidimensional Stochastic Calculus
	12.1 Multidimensional Brownian Motion
	12.2 Multidimensional Itô Processes
	12.3 Multidimensional Itô's Formula
	12.4 Lévy's Characterization and Correlated Brownian Motion
	12.5 Key Ideas to Remember
13 Changes of Measure and Martingale Representation
	13.1 Change of Measure and Itô Processes
		13.1.1 An Application: Risk-Neutral Valuation of Financial Derivatives
	13.2 Integrability of Exponential Martingales
	13.3 Girsanov Theorem
	13.4 Approximation by Exponential Martingales
	13.5 Representation of Brownian Martingales
		13.5.1 Proof of Theorem 13.1.1
	13.6 Key Ideas to Remember
14 Stochastic Differential Equations
	14.1 Solving SDEs: Concepts of Existence and Uniqueness
	14.2 Weak Existence and Uniqueness via Girsanov Theorem
	14.3 Weak vs Strong Solutions: The Yamada-Watanabe Theorem
	14.4 Standard Assumptions and Preliminary Estimates
	14.5 Some A Priori Estimates
	14.6 Key Ideas to Remember
15 Feynman-Kac Formulas
	15.1 Characteristic Operator of an SDE
	15.2 Exit Time from a Bounded Domain
	15.3 The Autonomous Case: The Dirichlet Problem
	15.4 The Evolutionary Case: The Cauchy Problem
	15.5 Key Ideas to Remember
16 Linear Equations
	16.1 Solution and Transition Law of a Linear SDE
	16.2 Controllability of Linear Systems and Absolute Continuity
	16.3 Kalman Rank Condition
	16.4 Hörmander's Condition
	16.5 Examples and Applications
	16.6 Key Ideas to Remember
17 Strong Solutions
	17.1 Uniqueness
	17.2 Existence
	17.3 Markov Property
		17.3.1 Forward Kolmogorov Equation
	17.4 Continuous Dependence on Parameters
18 Weak Solutions
	18.1 The Stroock-Varadhan Martingale Problem
	18.2 Equations with Hölder Coefficients
	18.3 Other Results for the Martingale Problem
	18.4 Strong Uniqueness Through Regularization by Noise
	18.5 Key Ideas to Remember
19 Complements
	19.1 Markovian Projection and Gyöngy's Lemma
	19.2 Backward Stochastic Differential Equations
	19.3 Filtering and Stochastic Heat Equation
	19.4 Backward Stochastic Integral and Krylov's SPDE
20 A Primer on Parabolic PDEs
	20.1 Uniqueness: The Maximum Principle
		20.1.1 Cauchy-Dirichlet Problem
		20.1.2 Cauchy Problem
	20.2 Existence: The Fundamental Solution
	20.3 The Parametrix Method
		20.3.1 Gaussian Estimates
		20.3.2 Proof of Proposition 20.3.2
		20.3.3 Potential Estimates
		20.3.4 Proof of Theorem 20.2.5
		20.3.5 Proof of Proposition 18.4.3
	20.4 Key Ideas to Remember
References
Index