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Kunita, H.: Stochastic differential equations and stochastic flows
of diffeomorphisms.-Elworthy, D.: Geometric aspects of diffusions
on manifolds.-Ancona, A.: Theorie du potential sur les graphs et
les varieties.-Emery, M.: Continuous martingales in differentiable
manifolds.
During the last fifty years, Gopinath Kallianpur has made extensive
and significant contributions to diverse areas of probability and
statistics, including stochastic finance, Fisher consistent
estimation, non-linear prediction and filtering problems, zero-one
laws for Gaussian processes and reproducing kernel Hilbert space
theory, and stochastic differential equations in infinite
dimensions. To honor Kallianpur's pioneering work and scholarly
achievements, a number of leading experts have written research
articles highlighting progress and new directions of research in
these and related areas. This commemorative volume, dedicated to
Kallianpur on the occasion of his seventy-fifth birthday, will pay
tribute to his multi-faceted achievements and to the deep insight
and inspiration he has so graciously offered his students and
colleagues throughout his career. Contributors to the volume: S.
Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P.
S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami,
Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov,
I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B.
Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R.
Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii,
I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C.
Tudor, W. A. Woycynski, J. Xiong
Professor Kiyosi Ito is well known as the creator of the modern
theory of stochastic analysis. Although Ito first proposed his
theory, now known as Ito's stochastic analysis or Ito's stochastic
calculus, about fifty years ago, its value in both pure and applied
mathematics is becoming greater and greater. For almost all modern
theories at the forefront of probability and related fields, Ito's
analysis is indispensable as an essential instrument, and it will
remain so in the future. For example, a basic formula, called the
Ito formula, is well known and widely used in fields as diverse as
physics and economics.
This volume contains 27 papers written by world-renowned
probability theorists. Their subjects vary widely and they present
new results and ideas in the fields where stochastic analysis plays
an important role. Also included are several expository articles by
well-known experts surveying recent developments. Not only
mathematicians but also physicists, biologists, economists and
researchers in other fields who are interested in the effectiveness
of stochastic theory will find valuable suggestions for their
research. In addition, students who are beginning their study and
research in stochastic analysis and related fields will find
instructive and useful guidance here.
This volume is dedicated to Professor Ito on the occasion of his
eightieth birthday as a token of deep appreciation for his great
achievements and contributions. An introduction to and commentary
on the scientific works of Professor Ito are also included.
During the last fifty years, Gopinath Kallianpur has made extensive
and significant contributions to diverse areas of probability and
statistics, including stochastic finance, Fisher consistent
estimation, non-linear prediction and filtering problems, zero-one
laws for Gaussian processes and reproducing kernel Hilbert space
theory, and stochastic differential equations in infinite
dimensions. To honor Kallianpur's pioneering work and scholarly
achievements, a number of leading experts have written research
articles highlighting progress and new directions of research in
these and related areas. This commemorative volume, dedicated to
Kallianpur on the occasion of his seventy-fifth birthday, will pay
tribute to his multi-faceted achievements and to the deep insight
and inspiration he has so graciously offered his students and
colleagues throughout his career. Contributors to the volume: S.
Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P.
S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami,
Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov,
I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B.
Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R.
Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii,
I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C.
Tudor, W. A. Woycynski, J. Xiong
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.
Stochastic analysis and stochastic differential equations are rapidly developing fields in probability theory and its applications. This book provides a systematic treatment of stochastic differential equations and stochastic flow of diffeomorphisms and describes the properties of stochastic flows. Professor Kunita's approach regards the stochastic differential equation as a dynamical system driven by a random vector field, including K. Itô's classical theory. Beginning with a discussion of Markov processes, martingales and Brownian motion, Kunita reviews Itô's stochastic analysis. He places emphasis on establishing that the solution defines a flow of diffeomorphisms. This flow property is basic in the modern and comprehensive analysis of the solution and will be applied to solve the first and second order stochastic partial differential equations. This book will be valued by graduate students and researchers in probability. It can also be used as a textbook for advanced probability courses.
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