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Stochastic Flows and Jump-Diffusions (Hardcover, 1st ed. 2019)
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Stochastic Flows and Jump-Diffusions (Hardcover, 1st ed. 2019)
Series: Probability Theory and Stochastic Modelling, 92
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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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