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E-grāmata: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

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  • Formāts: PDF+DRM
  • Sērija : Fields Institute Monographs 29
  • Izdošanas datums: 25-Sep-2012
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461442868
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Optimal Stochastic Control, Stochastic Target Problems, and Backward SDEPalielināt
  • Formāts: PDF+DRM
  • Sērija : Fields Institute Monographs 29
  • Izdošanas datums: 25-Sep-2012
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781461442868
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?This book collects some recent developments in stochasticcontrol theory with applications to financial mathematics. We first addressstandard stochastic control problems from the viewpoint of the recentlydeveloped weak dynamic programming principle. A special emphasis is put on theregularity issues and, in particular, on the behavior of the value functionnear the boundary. We then provide a quick review of the main tools fromviscosity solutions which allow to overcome all regularity problems.We next address the class of stochastic target problemswhich extends in a nontrivial way the standard stochastic control problems. Herethe theory of viscosity solutions plays a crucial role in the derivation of thedynamic programming equation as the infinitesimal counterpart of thecorresponding geometric dynamic programming equation. The various developmentsof this theory have been stimulated by applications in finance and by relevantconnections with geometric flows. Namely, the second order extension wasmotivated by illiquidity modeling, and the controlled loss version wasintroduced following the problem of quantile hedging.The third part specializes to an overview of Backwardstochastic differential equations, and their extensions to the quadratic case.?

This book collects recent developments in stochastic control theory with applications to financial mathematics. It approaches quadratic backward stochastic differential equations following the point of view of Tevzadze.

Recenzijas

This is an excellent book on the topic of Stochastic Control Problems (SCP). The author transformed his notes for a graduate course at the Field Institute into a volume that will serve also as a good reference in the area. The author has chosen the framework of diffusions, which makes the exposition more friendly and accessible to a larger audience, in particular for those who want to learn this topic. (Jaime San Martķn, Bulletin of the American Mathematical Society, Vol. 54 (2), April, 2017)

1 Introduction
1(4)
2 Conditional Expectation and Linear Parabolic PDEs
5(16)
2.1 Stochastic Differential Equations
5(5)
2.2 Markovian Solutions of SDEs
10(1)
2.3 Connection with Linear Partial Differential Equations
11(4)
2.3.1 Generator
11(1)
2.3.2 Cauchy Problem and the Feynman-Kac Representation
12(2)
2.3.3 Representation of the Dirichlet Problem
14(1)
2.4 The Black-Scholes Model
15(6)
2.4.1 The Continuous-Time Financial Market
15(1)
2.4.2 Portfolio and Wealth Process
16(2)
2.4.3 Admissible Portfolios and No-Arbitrage
18(1)
2.4.4 Super-Hedging and No-Arbitrage Bounds
18(1)
2.4.5 The No-Arbitrage Valuation Formula
19(1)
2.4.6 PDE Characterization of the Black-Scholes Price
20(1)
3 Stochastic Control and Dynamic Programming
21(18)
3.1 Stochastic Control Problems in Standard Form
21(4)
3.2 The Dynamic Programming Principle
25(5)
3.2.1 A Weak Dynamic Programming Principle
25(2)
3.2.2 Dynamic Programming Without Measurable Selection
27(3)
3.3 The Dynamic Programming Equation
30(3)
3.4 On the Regularity of the Value Function
33(6)
3.4.1 Continuity of the Value Function for Bounded Controls
33(3)
3.4.2 A Deterministic Control Problem with Non-smooth Value Function
36(1)
3.4.3 A Stochastic Control Problem with Non-smooth Value Function
36(3)
4 Optimal Stopping and Dynamic Programming
39(14)
4.1 Optimal Stopping Problems
39(2)
4.2 The Dynamic Programming Principle
41(2)
4.3 The Dynamic Programming Equation
43(2)
4.4 Regularity of the Value Function
45(8)
4.4.1 Finite Horizon Optimal Stopping
45(2)
4.4.2 Infinite Horizon Optimal Stopping
47(3)
4.4.3 An Optimal Stopping Problem with Nonsmooth Value
50(3)
5 Solving Control Problems by Verification
53(14)
5.1 The Verification Argument for Stochastic Control Problems
53(4)
5.2 Examples of Control Problems with Explicit Solutions
57(5)
5.2.1 Optimal Portfolio Allocation
57(1)
5.2.2 Law of Iterated Logarithm for Double Stochastic Integrals
58(4)
5.3 The Verification Argument for Optimal Stopping Problems
62(2)
5.4 Examples of Optimal Stopping Problems with Explicit Solutions
64(3)
5.4.1 Perpetual American Options
64(2)
5.4.2 Finite Horizon American Options
66(1)
6 Introduction to Viscosity Solutions
67(22)
6.1 Intuition Behind Viscosity Solutions
67(1)
6.2 Definition of Viscosity Solutions
68(1)
6.3 First Properties
69(4)
6.4 Comparison Result and Uniqueness
73(7)
6.4.1 Comparison of Classical Solutions in a Bounded Domain
73(1)
6.4.2 Semijets Definition of Viscosity Solutions
74(1)
6.4.3 The Crandall-Ishii's Lemma
75(1)
6.4.4 Comparison of Viscosity Solutions in a Bounded Domain
76(4)
6.5 Comparison in Unbounded Domains
80(3)
6.6 Useful Applications
83(1)
6.7 Proof of the Crandall-Ishii's Lemma
84(5)
7 Dynamic Programming Equation in the Viscosity Sense
89(12)
7.1 DPE for Stochastic Control Problems
89(6)
7.2 DPE for Optimal Stopping Problems
95(3)
7.3 A Comparison Result for Obstacle Problems
98(3)
8 Stochastic Target Problems
101(22)
8.1 Stochastic Target Problems
101(11)
8.1.1 Formulation
101(1)
8.1.2 Geometric Dynamic Programming Principle
102(2)
8.1.3 The Dynamic Programming Equation
104(6)
8.1.4 Application: Hedging Under Portfolio Constraints
110(2)
8.2 Stochastic Target Problem with Controlled Probability of Success
112(11)
8.2.1 Reduction to a Stochastic Target Problem
113(1)
8.2.2 The Dynamic Programming Equation
114(1)
8.2.3 Application: Quantile Hedging in the Black-Scholes Model
115(8)
9 Second Order Stochastic Target Problems
123(26)
9.1 Superhedging Under Gamma Constraints
123(11)
9.1.1 Problem Formulation
124(2)
9.1.2 Hedging Under Upper Gamma Constraint
126(6)
9.1.3 Including the Lower Bound on the Gamma
132(2)
9.2 Second Order Target Problem
134(11)
9.2.1 Problem Formulation
134(2)
9.2.2 The Geometric Dynamic Programming
136(1)
9.2.3 The Dynamic Programming Equation
137(8)
9.3 Superhedging Under Illiquidity Cost
145(4)
10 Backward SDEs and Stochastic Control
149(16)
10.1 Motivation and Examples
149(5)
10.1.1 The Stochastic Pontryagin Maximum Principle
150(2)
10.1.2 BSDEs and Stochastic Target Problems
152(1)
10.1.3 BSDEs and Finance
153(1)
10.2 Wellposedness of BSDEs
154(5)
10.2.1 Martingale Representation for Zero Generator
154(1)
10.2.2 BSDEs with Affine Generator
155(1)
10.2.3 The Main Existence and Uniqueness Result
156(3)
10.3 Comparison and Stability
159(1)
10.4 BSDEs and Stochastic Control
160(2)
10.5 BSDEs and Semilinear PDEs
162(2)
10.6 Appendix: Essential Supremum
164(1)
11 Quadratic Backward SDEs
165(24)
11.1 A Priori Estimates and Uniqueness
166(3)
11.1.1 A Priori Estimates for Bounded Y
166(1)
11.1.2 Some Properties of BMO Martingales
167(1)
11.1.3 Uniqueness
168(1)
11.2 Existence
169(6)
11.2.1 Existence for Small Final Condition
169(3)
11.2.2 Existence for Bounded Final Condition
172(3)
11.3 Portfolio Optimization Under Constraints
175(6)
11.3.1 Problem Formulation
175(2)
11.3.2 BSDE Characterization
177(4)
11.4 Interacting Investors with Performance Concern
181(8)
11.4.1 The Nash Equilibrium Problem
181(1)
11.4.2 The Individual Optimization Problem
182(1)
11.4.3 The Case of Linear Constraints
183(3)
11.4.4 Nash Equilibrium Under Deterministic Coefficients
186(3)
12 Probabilistic Numerical Methods for Nonlinear PDEs
189(12)
12.1 Discretization
190(3)
12.2 Convergence of the Discrete-Time Approximation
193(2)
12.3 Consistency, Monotonicity and Stability
195(2)
12.4 The Barles-Souganidis Monotone Scheme
197(4)
13 Introduction to Finite Differences Methods
201(12)
13.1 Overview of the Barles-Souganidis Framework
201(2)
13.2 First Examples
203(3)
13.2.1 The Heat Equation: The Classic Explicit and Implicit Schemes
203(3)
13.2.2 The Black-Scholes-Merton PDE
206(1)
13.3 A Nonlinear Example: The Passport Option
206(3)
13.3.1 Problem Formulation
206(1)
13.3.2 Finite Difference Approximation
207(2)
13.3.3 Howard Algorithm
209(1)
13.4 The Bonnans-Zidani [ 7] Approximation
209(2)
13.5 Working in a Finite Domain
211(1)
13.6 Variational Inequalities and Splitting Methods
211(2)
13.6.1 The American Option
211(2)
References 213
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