Stochastic Flows and Jump-Diffusions: 92 (Probability Theory and Stochastic Modelling, 92) - Hardcover

Kunita, Hiroshi

Hardcover

ISBN 10: 9811338000 ISBN 13: 9789811338007

Publisher: Springer, 2019

Synopsis

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.

About this title

About the Author

Kunita was an invited speaker at the ICM 1986. 

From the Back Cover

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.

Edition details

• Publisher: Springer
• Publication date: 2019
• Language: English
• ISBN 10: 9811338000
• ISBN 13: 9789811338007
• Binding: Hardcover
• Edition number: 1
• Number of pages: 369