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Semimartingale
In probability theory, a real-valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a cà dlà g adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales. Definition A real-valued process ''X'' defined on the filtered probability space (Ω,''F'',(''F''''t'')''t'' ≥ 0,P) is called a semimartingale if it can be decomposed as :X_t = M_t + A_t where ''M'' is a local martingale and ''A'' is a cà dlà g adapted process of locally bounded variation. This means that for almost all \omega \in \Omega and all compact intervals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Itô Calculus
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: Y_t = \int_0^t H_s\,dX_s, where is a locally square-integrable process adapted to the filtration generated by , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stochastic Calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese people, Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates. The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Wiener Process
In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best known Lévy processes (cà dlà g stochastic processes with stationary increments, stationary independent increments). It occurs frequently in pure and applied mathematics, economy, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingale (probability theory), martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quadratic Variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued stochastic process defined on a probability space (\Omega,\mathcal,\mathbb) and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as t, defined as : t=\lim_\sum_^n(X_-X_)^2 where P ranges over partitions of the interval ,t/math> and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian motion. More generally, the covariation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Martingale (probability Theory)
In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given the past, is equal to the present value. Martingales are used to model fair games, where future expected winnings are equal to the current amount regardless of past outcomes. History Originally, ''martingale (betting system), martingale'' referred to a class of betting strategy, betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Girsanov Theorem
In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it explains how to convert from the physical measure, which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values, to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a measure that is absolutely continuous with respect to ''P'' then every ''P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov Chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Émery Topology
In martingale theory, Émery topology is a topology on the space of semimartingales. The topology is used in financial mathematics. The class of stochastic integrals with general predictable integrands coincides with the closure of the set of all simple integrals. The topology was introduced in 1979 by the French mathematician Michel Émery. Definition Let (\Omega,\mathcal,\,P) be a filtered probability space, where the filtration satisfies the usual conditions and T\in (0,\infty). Let \mathcal(P) be the space of real semimartingales and \mathcal(1) the space of simple predictable processes H with , H, =1. We define :\, X\, _:=\sup\limits_\mathbb\left (H\cdot X)_t, \right)\right Then (\mathcal(P),d) with the metric d(X,Y):=\, X-Y\, _ is a complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points miss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Special Semimartingale
Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer * Special police forces Military * Special forces * Special operations Literature * ''Specials'' (novel), a novel by Scott Westerfeld * ''Specials'', the comic book heroes, see ''Rising Stars'' (comic) Film and television * Special (lighting), a stage light that is used for a single, specific purpose * ''Special'' (film), a 2006 scifi dramedy * ''The Specials'' (2000 film), a comedy film about a group of superheroes * Special 26, a 2013 Indian Hindi-language period heist thriller film * ''The Specials'' (2019 film), a film by Olivier Nakache and Éric Toledano * Television special, television programming that temporarily replaces scheduled programming * ''Special'' (TV series), a 2019 Netflix Original TV series * ''Specials'' (TV series), a 1991 TV series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Doléans-Dade Exponential
In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation dY_t = Y_\,dX_t,\quad\quad Y_0=1,where Y_ denotes the process of left limits, i.e., Y_=\lim_Y_s. The concept is named after Catherine Doléans-Dade. Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since X measures the cumulative percentage change in Y. Notation and terminology Process Y obtained above is commonly denoted by \mathcal(X). The terminology "stochastic exponential" arises from the similarity of \mathcal(X)=Y to the natural exponential of X: If ''X'' is absolutely continuous with respect to time, then ''Y'' solves, path-by-path, the differential equation dY_t/\mathrmt = Y_tdX_t/dt, whose solution is Y=\exp(X-X_0). General formula and special cases * Without any assumptions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
