Reflecting Stochastic Differential Equations with Jumps and Applications

Reflecting Stochastic Differential Equations with Jumps and Applications

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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 more

Product details

  • Paperback | 224 pages
  • 155.7 x 232.7 x 13mm | 358.31g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2003.
  • 1584881259
  • 9781584881254

Review quote

"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 Promo Copyshow more

Table of contents

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 BIBLIOGRAPHYshow more