Stochastic Differential Equations, Backward SDEs, Partial Differential Equations (Stochastic Modelling and Applied Probability, 69)

✍ Scribed by Etienne Pardoux, Aurel Rӑşcanu

• Publisher: Springer
• Year: 2014
• Tongue: English
• Leaves: 680
• Edition: 2014
• Category: Library

✦ Synopsis

This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter.

Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has become an important subject of Mathematics and Applied Mathematics, because of its mathematical richness and its importance for applications in many areas of Physics, Biology, Economics and Finance, where random processes play an increasingly important role. One important aspect is the connection between diffusion processes and linear partial differential equations of second order, which is in particular the basis for Monte Carlo numerical methods for linear PDEs. Since the pioneering work of Peng and Pardoux in the early 1990s, a new type of SDEs called backward stochastic differential equations (BSDEs) has emerged. The two main reasons why this new class of equations is important are the connection between BSDEs and semilinear PDEs, and the fact that BSDEs constitute a natural generalization of the famous Black and Scholes model from Mathematical Finance, and thus offer a natural mathematical framework for the formulation of many new models in Finance.

✦ Table of Contents

Contents
Introduction
Notations
Chapter 1:
Background of Stochastic Analysis
1.1 Preliminaries
1.1.1 Preliminaries of Probability Theory
1.1.2 Filtrations
1.1.3 Conditional Expectation
1.1.4 Stochastic Processes
1.1.5 Complements on Tightness
1.1.6 Stopping Times
1.1.7 Fundamental Inequalities
1.2 Continuous Martingales
1.2.1 Basic Results
1.2.2 Martingales and Bounded Variation Processes
1.3 Brownian Motion
1.3.1 Gaussian Spaces
1.3.2 Definition and Main Properties
1.3.3 Ft-Brownian Motion
1.4 Exercises
Chapter 2:
Itô's Stochastic Calculus
2.1 Notations: Preliminaries
2.2 Definition of Itô's Stochastic Integral
2.3 Itô's Formula
2.3.1 Applications of Itô's Formula
2.3.2 A Stochastic Subdifferential Inequality
2.4 Martingale Representation Theorems
2.5 Girsanov's Theorem
2.6 Exercises
Chapter 3:
Stochastic Differential Equations
3.1 Introduction
3.2 A Basic Inequality
3.3 Estimates, Uniqueness and Comparison Results
3.3.1 Classical SDE
3.3.2 SDEs with Stieltjes Integrals
3.3.3 Stochastic Linear Equations
3.3.4 Comparison Results
3.3.4.1 Lipschitz Case
3.3.4.2 Monotone Case
3.4 Lipschitz Coefficients
3.4.1 Classical SDEs
3.4.2 SDEs with Stieltjes Integrals
3.5 Global Monotonicity
3.5.1 A Deterministic Problem
3.5.2 Main Result
3.5.3 SDEs with Deterministic Initial Condition
3.5.4 SDEs with Stieltjes Integrals
3.6 Local Monotonicity
3.6.1 Locally Monotone Drift
3.6.2 Locally Lipschitz Coefficients
3.7 Markov Solutions of SDEs
3.7.1 Markov Processes
3.7.2 The Markov Property of Solutions of SDEs
3.8 The Feynman–Kac Formula
3.8.1 Backward Parabolic PDEs
3.8.2 Forward Parabolic PDEs
3.8.3 Parabolic PDEs with Dirichlet Boundary Conditions
3.8.4 Elliptic Equations with Dirichlet Boundary Condition
3.8.5 Elliptic PDEs in Rd
3.9 Remarks on Weak and Strong Solutions
3.10 Exercises
Chapter 4:
SDEs with Multivalued Drift
4.1 Introduction
4.2 SDEs with a Maximal Monotone Operator in the Drift
4.2.1 Assumptions: Definitions
4.2.2 A Priori Estimates: Uniqueness
4.2.3 The Generalized Convex Skorohod Problem
4.2.4 Main Result: Existence
4.2.5 SDEs with a Subdifferential Operator in the Drift
4.3 Reflected SDEs
4.3.1 The Generalized Skorohod Problem
4.3.1.1 Preliminaries
4.3.1.2 The Generalized Skorohod Problem
4.3.2 The Classical Skorohod Problem
4.3.3 Skorohod Equations
4.3.4 Markov Solutions of Reflected SDEs
4.3.5 SDEs with Oblique Reflection
4.4 The Feynman–Kac Formula
4.4.1 Parabolic PDEs with Neumann Boundary Conditions
4.4.2 Elliptic Equations with Neumann Boundary Conditions
4.5 Invariant Sets of SDEs
4.6 Exercises
Chapter 5:
Backward Stochastic Differential Equations
5.1 Introduction
5.2 Basic Inequalities
5.2.1 Backward Itô's Formula
5.2.2 A Fundamental Inequality
5.3 BSDEs with Deterministic Final Time
5.3.1 A Priori Estimates and Uniqueness
5.3.2 Complementary Results
5.3.3 BSDEs with Lipschitz Coefficients
5.3.3.1 BSDEs with Deterministic Lipschitz Conditions
5.3.3.2 BSDEs with Random Lipschitz Conditions
5.3.3.3 BSDEs with Locally Lipschitz Coefficients
5.3.4 BSDEs with Monotone Coefficients
5.3.4.1 The First BSDE: Monotone Coefficient (s,Ys) dQs
5.3.4.2 The Second BSDE: Monotone Coefficient F(t,Yt,Zt) dt
5.3.4.3 The Third BSDE: Monotone Coefficient (s,Ys,Zs) dQs
5.3.5 Linear BSDEs
5.3.6 Comparison Results
5.3.6.1 Lipschitz Case
5.3.6.2 Monotone Case
5.4 Semilinear Parabolic PDEs
5.4.1 Parabolic Systems in the Whole Space
5.4.2 Parabolic Dirichlet Problem
5.4.3 Parabolic Neumann Problem
5.5 BSDEs with a Subdifferential Coefficient
5.5.1 Uniqueness
5.5.2 Existence
5.6 BSDEs with Random Final Time
5.6.1 BSDEs with a Monotone Coefficient
5.6.2 BSVIs with Random Final Time
5.6.3 Weak Variational Solutions
5.7 Semilinear Elliptic PDEs
5.7.1 Elliptic Equations in the Whole Space
5.7.2 Elliptic Dirichlet Problem
5.7.3 Elliptic Equations with Neumann Boundary Conditions
5.8 Parabolic Variational Inequality
5.9 Invariant Sets of BSDEs
5.10 Exercises
Chapter 6:
Annexes
6.1 Introduction
6.2 Annex A: Vectors and Matrices
6.3 Annex B: Elements of Nonlinear Analysis
6.3.1 Notations
6.3.2 Maximal Monotone Operators
6.3.3 Stochastic Monotone Functions
6.3.4 Compactness Results
6.3.5 Bounded Variation Functions
6.3.6 Semicontinuity
6.3.7 Convex Functions
6.3.7.1 Definitions: Properties
6.3.7.2 Regularization of Convex Functions
6.3.7.3 Convex Functions on C([0,T];Rd)
6.3.8 Semiconvex Functions
6.3.9 Differential Equations
6.3.10 Auxiliary Results
6.4 Annex C: Deterministic and Stochastic Inequalities
6.4.1 Deterministic Inequalities
6.4.2 Stochastic Inequalities
6.4.3 Forward Stochastic Inequalities
6.4.4 Backward Stochastic Inequalities
6.5 Annex D: Viscosity Solutions
6.5.1 Definitions
6.5.1.1 Elliptic PDE
6.5.1.2 Systems of PDEs
6.5.1.3 Boundary Conditions
6.5.1.4 Parabolic PDEs
6.5.2 A First Uniqueness Result
6.5.3 A Second Uniqueness Result
6.5.4 A Third Uniqueness Result
6.6 Annex E: Hints for Some Exercises
References
Index