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Stochastic Differential Equation
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,Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the distributional derivative of a Brownian motion or more generally a semimartingale. However, other types of random behaviour are possible, such as jump processes like Lévy processes or semimartingales with jumps. Stochastic differential equations are in general neither differential equations nor random differential equations. Random differential equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Louis Bachelier
Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part of his doctoral thesis ''The Theory of Speculation'' (''Théorie de la spéculation'', defended in 1900). Bachelier's doctoral thesis, which introduced the first mathematical model of Brownian motion and its use for valuing stock options, was the first paper to use advanced mathematics in the study of finance. His Bachelier model has been influential in the development of other widely used models, including the Black-Scholes model. Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes. Early years Bachelier was born in Le Havre, in Seine-Maritime. His father was a wine merchant and amateur scientist, and the vice-consul of Venezuela at Le Havre. His mother was the da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Smoluchowski Equation
Marian Smoluchowski (; 28 May 1872 – 5 September 1917) was a Polish physicist who worked in the territories of the Austro-Hungarian Empire. He was a pioneer of statistical physics and made significant contributions to the theory of Brownian motion and stochastic processes. Smoluchowski graduated in physics from the University of Vienna in 1895 before becoming a ''privatdozent'' at the University of Lemberg three years later. In 1913, he was appointed the chair of the Faculty of Experimental Physics at the Jagellonian University in Kraków. He is known for the Smoluchowski equation, Einstein–Smoluchowski relation and Feynman–Smoluchowski ratchet. Life He was born in 1872 into an upper-class family in Vorder-Brühl, near Vienna, to father Wilhelm and mother Teofila (née Szczepanowska). He attended the prestigious Collegium Theresianum and subsequently studied physics at the University of Vienna (1890-95). In 1895, he obtained his doctorate based on his dissertati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Itô Integral
Ito, Itō or Itoh may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan * Itō, Shizuoka People * Itō (surname), for people with the Japanese surname Itō * , Japanese voice actor * Kiyosi Itô (1915–2008), Japanese mathematician * Princess Ito (died 861), Japanese imperial princess * Ito Giani (1941–2018), Italian sprinter * Ito (footballer, born 1961), full name Andrés Alonso García, Spanish footballer * Ito (footballer, born 1975), full name Antonio Álvarez Pérez, Spanish footballer * Ito (footballer, born 1992), full name Jorge Delgado Fidalgo, Spanish footballer * Ito (footballer, born 1994), full name Mario Manuel de Oliveira, Angolan footballer * , Japanese fashion model and actress *Ito Smith (born 1995), American football player * Ito Curata (1959–2020), Filipino fashion designe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Stratonovich Integral
In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics. In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike the Itô calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds. Perhaps the most common situation in which these are encountered is as the solution to Stratonovich stochastic differential equations (SDEs). These are equivalent to Itô SDEs and it is possible to convert between the two whenever one definition is more convenient. Definition The Stratonovich integral can be defined in a manner similar to the Riemann integral, that is as a limit of Riemann sums. Suppose that W : , T\times \Omega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Stochastic Difference Equation
Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a ''stochastic process'' is also referred to as a ''random process''. Stochasticity is used in many different fields, including image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance (e.g., stochastic oscillator), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. Etymology The word ''stochastic'' in English was originally used as an adjective with the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ruslan L
Ruslan may refer to: * ''Ruslan'' (film), a 2009 film starring Steven Segal * '' Ruslaan'', a 2024 Indian film * Ruslan (given name), masculine given name (contains list of people) * Antonov An-124 ''Ruslan'', large Soviet cargo aircraft, later built in Ukraine and Russia * SS ''Ruslan'', a Russian cargo ship in the Third Aliyah in 1919 See also * Aslan (other) Aslan is the fictional lion in C. S. Lewis's ''Chronicles of Narnia''. Aslan or ''Arslan'' (both spellings of a Turkic word meaning "fearless", "warrior", "lion") may also refer to: People Given name Arsalan * Arsalan Anwar (born 1986), P ..., cognate * Rusian (other) * Ruslan and Ludmila (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Stochastic Integral
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 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 integral is the mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kiyosi Itô
was a Japanese people, 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 Itô calculus. He also pioneered the world connections between stochastic calculus and differential geometry, known as stochastic differential geometry. He was invited for the International Congress of Mathematicians in Stockholm in 1962. So much were Itô's results useful to financial mathematics that he was sometimes called "the most famous Japanese in Wall Street". Itô was a member of the faculty at University of Kyoto for most of his career and eventually became the director of their Research Institute for Mathematical Sciences. But he also spent multi-year stints at several foreign institutions, the longest of which took place at Cornell University. Overview Itô pioneered the theory of stoch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |