Reflecting Stochastic Differential Equations with Jumps and Applications
1st Edition
Published on 5. August 1999
Book
Paperback/Softback
224 pages
978-1-58488-125-4 (ISBN)
Description
Many important physical variables satisfy certain dynamic evolution systems and can take only non-negative values. Therefore, one can study such variables by studying these dynamic systems. One can put some conditions on the coefficients to ensure non-negative values in deterministic cases. However, as a random process disturbs the system, the components of solutions to stochastic differential equations (SDE) can keep changing between arbitrary large positive and negative values-even in the simplest case. To overcome this difficulty, the author examines the reflecting stochastic differential equation (RSDE) with the coordinate planes as its boundary-or with a more general boundary.
Reflecting Stochastic Differential Equations with Jumps and Applications systematically studies the general theory and applications of these equations. In particular, the author examines the existence, uniqueness, comparison, convergence, and stability of strong solutions to cases where the RSDE has discontinuous coefficients-with greater than linear growth-that may include jump reflection. He derives the nonlinear filtering and Zakai equations, the Maximum Principle for stochastic optimal control, and the necessary and sufficient conditions for the existence of optimal control.
Most of the material presented in this book is new, including much new work by the author concerning SDEs both with and without reflection. Much of it appears here for the first time. With the application of RSDEs to various real-life problems, such as the stochastic population and neurophysiological control problems-both addressed in the text-scientists dealing with stochastic dynamic systems will find this an interesting and useful work.
Reflecting Stochastic Differential Equations with Jumps and Applications systematically studies the general theory and applications of these equations. In particular, the author examines the existence, uniqueness, comparison, convergence, and stability of strong solutions to cases where the RSDE has discontinuous coefficients-with greater than linear growth-that may include jump reflection. He derives the nonlinear filtering and Zakai equations, the Maximum Principle for stochastic optimal control, and the necessary and sufficient conditions for the existence of optimal control.
Most of the material presented in this book is new, including much new work by the author concerning SDEs both with and without reflection. Much of it appears here for the first time. With the application of RSDEs to various real-life problems, such as the stochastic population and neurophysiological control problems-both addressed in the text-scientists dealing with stochastic dynamic systems will find this an interesting and useful work.
Reviews / Votes
"The author tries to obtain the weakest assumptions in his theorems, so this should be of interest to specialists who have to deal with systems with unusual conditions."-J. Picard, Zentralblatt MATH Vol. 942
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More details
Series
Language
English
Place of publication
Oxford
United States
Publishing group
Taylor & Francis Inc
Target group
College/higher education
Professional and scholarly
Dimensions
Height: 229 mm
Width: 152 mm
Weight
334 gr
ISBN-13
978-1-58488-125-4 (9781584881254)
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Schweitzer Classification
Persons
Content
SOME RECENT RESULTS ON SDE WITH JUMPS IN 1-DIMENSIONAL SPACE
Local Time and Occupation Density Formula
A Generalization of Ito's Formula
The Continuity of Local Time
Krylov Estimation
Tanaka Formula
Uniqueness of Solutions to Stochastic Differential Equations
Comparison for Solutions of Stochastic Differential Equations
Convergence of Solutions to Stochastic differential Equations
Existence of Solutions to Stochastic Differential Equations
Tanaka formula for SDE with Poisson Jumps in n-Dimensional Space
SKOROHOD PROBLEMS WITH GIVEN CADLAG FUNCTIONS
The Space D and Skorohod's Topology
Skorohod's Problem in a General Domain. Solution with Jumps
Skorohod Problem with Jump Reflection in a Half Space
REFLECTING STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS
Yamada-Watanabe Theorem, Tanaka Formula and Krylov Estimate
Moment Estimates and Existence of Solutions for Random Coefficients
Existence of Solutions for RSDE with Jumps
Existence of Solutions with Jump Reflection in a Half Space
PROPERTIES OF SOLUTIONS TO RSDE WITH JUMPS
Convergence Theorems for Solutions
Stability of Solutions
Comparison of Solutions
Applications of Comparison Theorem to 1-Dimensional RSDE
Uniqueness of Solutions
Convergence of Solutions in Half Space
NONLINEAR FILTERING OF RSDE
Representation of Martingales (Functional Coefficient Case)
Non-Linear Filtering Equation
Zakai Equation
STOCHASTIC CONTROL
Girsanov Theorem with Weak Conditions
Martingale Method, Necessary and Sufficient Conditions for Optimal Control
STOCHASTIC POPULATION CONTROL
Stochastic Population Control Model and Maximum Principle
Pathwise Stochastic Population Control and Stability of Population
Applications to Neurophysiological Control and Others
BIBLIOGRAPHY
Local Time and Occupation Density Formula
A Generalization of Ito's Formula
The Continuity of Local Time
Krylov Estimation
Tanaka Formula
Uniqueness of Solutions to Stochastic Differential Equations
Comparison for Solutions of Stochastic Differential Equations
Convergence of Solutions to Stochastic differential Equations
Existence of Solutions to Stochastic Differential Equations
Tanaka formula for SDE with Poisson Jumps in n-Dimensional Space
SKOROHOD PROBLEMS WITH GIVEN CADLAG FUNCTIONS
The Space D and Skorohod's Topology
Skorohod's Problem in a General Domain. Solution with Jumps
Skorohod Problem with Jump Reflection in a Half Space
REFLECTING STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS
Yamada-Watanabe Theorem, Tanaka Formula and Krylov Estimate
Moment Estimates and Existence of Solutions for Random Coefficients
Existence of Solutions for RSDE with Jumps
Existence of Solutions with Jump Reflection in a Half Space
PROPERTIES OF SOLUTIONS TO RSDE WITH JUMPS
Convergence Theorems for Solutions
Stability of Solutions
Comparison of Solutions
Applications of Comparison Theorem to 1-Dimensional RSDE
Uniqueness of Solutions
Convergence of Solutions in Half Space
NONLINEAR FILTERING OF RSDE
Representation of Martingales (Functional Coefficient Case)
Non-Linear Filtering Equation
Zakai Equation
STOCHASTIC CONTROL
Girsanov Theorem with Weak Conditions
Martingale Method, Necessary and Sufficient Conditions for Optimal Control
STOCHASTIC POPULATION CONTROL
Stochastic Population Control Model and Maximum Principle
Pathwise Stochastic Population Control and Stability of Population
Applications to Neurophysiological Control and Others
BIBLIOGRAPHY