Stochastic Flows and Jump-Diffusions

This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps.In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heatequations are constructed independently of the theory of partial differential equations.Researchers and graduate student in probability theory will find this book very useful.

List of contents

Preface.- Introduction.- 1.Probability distributions and stochastic processes.- 2.Stochastic integrals based on Wiener processes and Poisson random measures.- 3.Stochastic differential equations and stochastic flows.- 4.Diffusions, jump-diffusions and heat equations.- 5.Malliavin calculus for Wiener processes and Poisson random measures.- 6.Smooth densities and heat kernels.- 7.Jump-diffusions on manifolds and smooth densities.- Bibliography.- Index.

About the author

Kunita was an invited speaker at the ICM 1986.

Summary

Report

"The presentation is self-contained, clear and precise. The book is definitely a must-read for researchers in the field of stochastic flows and stochastic differential equations." (G. V. Riabov, Mathematical Reviews, October, 2020)

Product details

• Authors: Hiroshi Kunita
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.01.2019
• EAN: 9789811338007
• ISBN: 978-981-1338-00-7
• No. of pages: 352
• Dimensions: 159 mm x 241 mm x 239 mm
• Weight: 716 g
• Illustrations: XVII, 352 p. 145 illus.
• Series: Probability Theory and Stochastic Modelling; Probability Theory and Stochas; Probability Theory and Stochastic Modelling
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Analysis, Stochastik, B, Angewandte Mathematik, Differentialrechnung und -gleichungen, Mathematics and Statistics, Applications of Mathematics, Finance & accounting, Probability Theory and Stochastic Processes, Partial Differential Equations, Differential calculus & equations, Differential equations, Probabilities, Stochastics, Probability Theory, Economics, Mathematical, Quantitative Finance, Mathematics in Business, Economics and Finance