Stochastic Flows and Jump-Diffusions

Nonfiction, Science & Nature, Mathematics, Applied, Statistics
Cover of the book Stochastic Flows and Jump-Diffusions by Hiroshi Kunita, Springer Singapore
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Author: Hiroshi Kunita ISBN: 9789811338014
Publisher: Springer Singapore Publication: March 26, 2019
Imprint: Springer Language: English
Author: Hiroshi Kunita
ISBN: 9789811338014
Publisher: Springer Singapore
Publication: March 26, 2019
Imprint: Springer
Language: English

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 heat equations are constructed independently of the theory of partial differential equations.

Researchers and graduate student in probability theory will find this book very useful.

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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 heat equations are constructed independently of the theory of partial differential equations.

Researchers and graduate student in probability theory will find this book very useful.

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