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Russo–Vallois Integral
In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral :\int f \, dg=\int fg' \, ds for suitable functions f and g. The idea is to replace the derivative g' by the difference quotient :g(s+\varepsilon)-g(s)\over\varepsilon and to pull the limit out of the integral. In addition one changes the type of convergence. Definitions Definition: A sequence H_n of stochastic processes Convergence of random variables, converges uniformly on compact sets in probability to a process H, :H=\text\lim_H_n, if, for every \varepsilon>0 and T>0, :\lim_\mathbb(\sup_, H_n(t)-H(t), >\varepsilon)=0. One sets: :I^-(\varepsilon,t,f,dg)=\int_0^tf(s)(g(s+\varepsilon)-g(s))\,ds :I^+(\varepsilon,t,f,dg)=\int_0^t f(s)(g(s)-g(s-\varepsilon)) \, ds and :[f,g]_\varepsilon (t)=\int_0^t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s))\,ds. Definition: The forward integral is defined as the ucp-limit of :I^-: \int_0^t fd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |