Authors:
- Provides systematic treatment of the Malliavin calculus on the Wiener–Poisson space, introducing Sobolev norms
- Uses the flow property of the solution of stochastic differential equations and application to dual jump-diffusions
- Is a study of fundamental solutions through stochastic analysis without the aid of partial differential equations
Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 92)
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Table of contents (7 chapters)
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Front Matter
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Back Matter
About this book
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.
Keywords
- stochastic differential equation with jumps
- jump-diffusion process
- Malliavin calculus
- Wiener space
- fundamental solution
- asymptotic short time estimate
- smooth density
- stochastic flow
- diffeomorphism
- diffusion and jump-diffusion processes
- heat equations
- backward heat equations
- 60H05, 60H07, 60H30
- 35K08, 35K10, 58J05
- quantitative finance
- partial differential equations
Reviews
“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)
Authors and Affiliations
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Kyushu University (emeritus), Fukuoka, Japan
Hiroshi Kunita
Bibliographic Information
Book Title: Stochastic Flows and Jump-Diffusions
Authors: Hiroshi Kunita
Series Title: Probability Theory and Stochastic Modelling
DOI: https://doi.org/10.1007/978-981-13-3801-4
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2019
Hardcover ISBN: 978-981-13-3800-7Published: 09 April 2019
eBook ISBN: 978-981-13-3801-4Published: 26 March 2019
Series ISSN: 2199-3130
Series E-ISSN: 2199-3149
Edition Number: 1
Number of Pages: XVII, 352
Number of Illustrations: 145 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Quantitative Finance, Partial Differential Equations