Stochastic Flows and Jump-Diffusions
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Beschreibung
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.
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
Autor*in
Hiroshi Kunita
Themen in »Stochastic Flows and Jump-Diffusions«
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
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“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)
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