Stochastic Modelling and Applied Probability

By: N. V. Krylov, A. B. Aries (Translator)

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Hardcover | 15 October 2008

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Stochastic control theory is a relatively young branch of mathematics. The beginning of its intensive development falls in the late 1950s and early 1960s. ~urin~ that period an extensive literature appeared on optimal stochastic control using the quadratic performance criterion (see references in Wonham [76]). At the same time, Girsanov [25] and Howard [26] made the first steps in constructing a general theory, based on Bellman's technique of dynamic programming, developed by him somewhat earlier [4]. Two types of engineering problems engendered two different parts of stochastic control theory. Problems of the first type are associated with multistep decision making in discrete time, and are treated in the theory of discrete stochastic dynamic programming. For more on this theory, we note in addition to the work of Howard and Bellman, mentioned above, the books by Derman [8], Mine and Osaki [55], and Dynkin and Yushkevich [12]. Another class of engineering problems which encouraged the development of the theory of stochastic control involves time continuous control of a dynamic system in the presence of random noise. The case where the system is described by a differential equation and the noise is modeled as a time continuous random process is the core of the optimal control theory of diffusion processes. This book deals with this latter theory.

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"The book treats a large class of fully nonlinear parabolic PDEs via probabilistic methods. ... The monograph may be strongly recommended as an excellent reading to PhD students, postdocs et al working in the area of controlled stochastic processes and/or nonlinear partial differential equations of the second order. ... recommended to a wider audience of all students specializing in stochastic analysis or stochastic finance starting from MSc level." (Alexander Yu Veretennikov, Zentralblatt MATH, Vol. 1171, 2009)

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Non-Fiction Engineering & Technology Electronics & Communications Engineering Electronics Engineering Automatic Control Engineering Robotics Mathematics Probability & Statistics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Applied Mathematics Stochastics

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Published: 7th December 1980

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Published: 12th October 2011

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Notation	p. xi
Introduction to the Theory of Controlled Diffusion Processes	p. 1
The Statement of Problems-Bellman's Principle-Bellman's Equation	p. 2
Examples of the Bellman Equations-The Normed Bellman Equation	p. 7
Application of Optimal Control Theory-Techniques for Obtaining Some Estimates	p. 16
One-Dimensional Controlled Processes	p. 22
Optimal Stopping of a One-Dimensional Controlled Process	p. 35
Notes	p. 42
Auxiliary Propositions	p. 45
Notation and Definitions	p. 45
Estimates of the Distribution of a Stochastic Integral in a Bounded Region	p. 51
Estimates of the Distribution of a Stochastic Integral in the Whole Space	p. 61
Limit Behavior of Some Functions	p. 67
Solutions of Stochastic Integral Equations and Estimates of the Moments	p. 77
Existence of a Solution of a Stochastic Equation with Measurable Coefficients	p. 86
Some Properties of a Random Process Depending on a Parameter	p. 91
The Dependence of Solutions of a Stochastic Equation on a Parameter	p. 102
The Markov Property of Solutions of Stochastic Equations	p. 110
Ito's Formula with Generalized Derivatives	p. 121
Notes	p. 128
General Properties of a Payoff Function	p. 129
Basic Results	p. 129
Some Preliminary Considerations	p. 140
The Proof of Theorems 1.5-1.7	p. 147
The Proof of Theorems 1.8-1.11 for the Optimal Stopping Problem	p. 152
Notes	p. 161
The Bellman Equation	p. 163
Estimation of First Derivatives of Payoff Functions	p. 165
Estimation from Below of Second Derivatives of a Payoff Function	p. 173
Estimation from Above of Second Derivatives of a Payoff Function	p. 181
Estimation of a Derivative of a Payoff Function with Respect to t	p. 188
Passage to the Limit in the Bellman Equation	p. 193
The Approximation of Degenerate Controlled Processes by Nondegenerate Ones	p. 200
The Bellman Equation	p. 203
Notes	p. 211
The Construction of [epsilon]-Optimal Strategies	p. 213
[epsilon]-Optimal Markov Strategies and the Bellman Equation	p. 213
[epilson]-Optimal Markov Strategies. The Bellman Equation in the Presence of Degeneracy	p. 218
The Payoff Function and Solution of the Bellman Equation: The Uniqueness of the Solution of the Bellman Equation	p. 228
Notes	p. 243
Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation	p. 245
Generalizations of the Results Obtained in Section 3.1	p. 245
General Methods for Estimating Derivatives of Payoff Functions	p. 254
The Normed Bellman Equation	p. 266
The Optimal Stopping of a Controlled Process on an Infinite Interval of Time	p. 275
Control on an Infinite Interval of Time	p. 285
Notes	p. 291
Appendices
Some Properties of Stochastic Integrals	p. 293
Some Properties of Submartingales	p. 299
Bibliography	p. 303
Index	p. 307
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Stochastic Modelling and Applied Probability

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