In probability theory and related fields,a stochastic or random process is a mathematical object usually defined as a sequence of random variables in a probability space,where the index of the sequence often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population,an electrical current fluctuating due to thermal noise,or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology,chemistry,ecology,neuroscience,physics,image processing,signal processing,control theory,information theory,computer science,and telecommunications. Furthermore,seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
Analysis is the branch of mathematics dealing with continuous functions,limits,and related theories,such as differentiation,integration,measure,infinite sequences,series,and analytic functions.
Edward Nelson was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical logic,he was noted especially for his internal set theory,and views on ultrafinitism and the consistency of arithmetic. In philosophy of mathematics he advocated the view of formalism rather than platonism or intuitionism. He also wrote on the relationship between religion and mathematics.
In probability theory and related fields,Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular,it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then,the significant contributors such as S. Kusuoka,D. Stroock,J-M. Bismut,Shinzo Watanabe,I. Shigekawa,and so on finally completed the foundations.
Kiyosi Itô was a Japanese mathematician who made fundamental contributions to probability theory,in particular,the theory of stochastic processes. He invented the concept of stochastic integral and stochastic differential equation,and is known as the founder of so-called Itôcalculus. He also pioneered the world connections between stochastic calculus and differential geometry,known as stochastic differential geometry,invited for the ICM in Stockholm.
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process,resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices,random growth models or physical systems that are subjected to thermal fluctuations.
Anatoliy Volodymyrovych Skorokhod was a Ukrainian mathematician.
Paul Malliavin was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus,an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem. He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences from 1979 to 2010.
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients,in the same way ordinary stochastic differential equations generalize ordinary differential equations.
In mathematics,the Skorokhod integral,also named Hitsuda–Skorokhod integral,often denoted ,is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and Japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts:
Jean-Michel Bismut is a French mathematician who has been a professor at the UniversitéParis-Sud since 1981. His mathematical career covers two apparently different branches of mathematics:probability theory and differential geometry. Ideas from probability play an important role in his works on geometry.
Sergio Albeverio is a Swiss mathematician and mathematical physicist working in numerous fields of mathematics and its applications. In particular he is known for his work in probability theory,analysis,mathematical physics,and in the areas algebra,geometry,number theory,as well as in applications,from natural to social-economic sciences.
Kalyan Bidhan Sinha is an Indian mathematician. He is a professor at the Jawaharlal Nehru Centre for Advanced Scientific Research,and Professor Emeritus for life of the Indian Statistical Institute.
ShinzōWatanabe is a Japanese mathematician,who has made fundamental contributions to probability theory,stochastic processes and stochastic differential equations. He is regarded and revered as one of the fundamental contributors to the modern probability theory and Stochastic calculus. The pioneering book “Stochastic Differential Equations and Diffusion Processes”he wrote with Nobuyuki Ikeda has attracted a lot of researchers into the area and is known as the “Ikeda-Watanabe”for researchers in the field of stochastic analysis. He had been served as the editor of Springer Mathematics.
In probability theory,a branch of mathematics,white noise analysis,otherwise known as Hida calculus,is a framework for infinite-dimensional and stochastic calculus,based on the Gaussian white noise probability space,to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes.
Igor Dmitrievich Chueshov was a Ukrainian mathematician. He was both a correspondent member of the Mathematics section of the National Academy of Sciences of Ukraine and a professor in the Department of Mathematical Physics and Computational Mathematics at the National University of Kharkiv.
Tetsuji Miwa is a Japanese mathematician,specializing in mathematical physics.
Leonard Gross is an American mathematician and Professor Emeritus of Mathematics at Cornell University.
Ambar Niel Sengupta is an Indian-American mathematician. He is a professor of mathematics at the University of Connecticut.
In quantum mechanics,a quantum Markov semigroup describes the dynamics in a Markovian open quantum system. The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski in 1972,and then developed by V. Gorini,A. M. Kossakowski,E. C. G. Sudarshan and Göran Lindblad in 1976.
