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Jump Diffusion
Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and pattern theory and computational vision. In physics In crystals, atomic diffusion typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular diffusion. Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering and by Mößbauer spectroscopy. Closed expressions for the autocorrelation function have been derived for several jump(-diffusion) models: * Singwi, Sjölander 1960: alternation between oscillatory motion and directed motion * Chudley, Elliott 1961: jumps on a lattice * Sears 1966, 1967: jump diffusion of rotational degrees of freedom * Hall, Ross 1981: jump diffusion within a restricted volume In economics and finance In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. 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, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jump Model
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process. In finance, various stochastic models are used to model the price movements of financial instruments; for example the Black–Scholes model for pricing options assumes that the underlying instrument follows a traditional diffusion process, with continuous, random movements at all scales, no matter how small. John Carrington Cox and Stephen Ross proposed that prices actually follow a 'jump process'. Robert C. Merton extended this approach to a hybrid model known as jump diffusion, which states that the prices have large jumps interspersed with small continuous movements. See also *Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid System
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena. In general, the ''state'' of a hybrid system is defined by the values of the ''continuous variables'' and a discrete ''mode''. The state changes either continuously, according to a flow condition, or discretely according to a ''control graph''. Continuous flow is permitted as long as so-called ''invariants'' hold, while discrete transitions can occur as soon as given ''jump conditions'' are satisfied. Discrete transi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise-deterministic Markov Process
In probability theory, a piecewise-deterministic Markov process (PDMP) is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those times. The class of models is "wide enough to include as special cases virtually all the non-diffusion models of applied probability." The process is defined by three quantities: the flow, the jump rate, and the transition measure. The model was first introduced in a paper by Mark H. A. Davis in 1984. Examples Piecewise linear models such as Markov chains, continuous-time Markov chains, the M/G/1 queue, the GI/G/1 queue and the fluid queue can be encapsulated as PDMPs with simple differential equations. Applications PDMPs have been shown useful in ruin theory, queueing theory, for modelling biochemical processes such as DNA replication in eukaryotes and subtilin production by the organism B. subtilis, and for modelling earthquakes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jump Process
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process. In finance, various stochastic models are used to model the price movements of financial instruments; for example the Black–Scholes model for pricing options assumes that the underlying instrument follows a traditional diffusion process, with continuous, random movements at all scales, no matter how small. John Carrington Cox and Stephen Ross proposed that prices actually follow a 'jump process'. Robert C. Merton extended this approach to a hybrid model known as jump diffusion, which states that the prices have large jumps interspersed with small continuous movements. See also *Poisson process, an example of a jump process *Continuous-time Markov chain A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid System
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena. In general, the ''state'' of a hybrid system is defined by the values of the ''continuous variables'' and a discrete ''mode''. The state changes either continuously, according to a flow condition, or discretely according to a ''control graph''. Continuous flow is permitted as long as so-called ''invariants'' hold, while discrete transitions can occur as soon as given ''jump conditions'' are satisfied. Discrete transi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saccade
A saccade ( , French for ''jerk'') is a quick, simultaneous movement of both eyes between two or more phases of fixation in the same direction.Cassin, B. and Solomon, S. ''Dictionary of Eye Terminology''. Gainesville, Florida: Triad Publishing Company, 1990. In contrast, in smooth pursuit movements, the eyes move smoothly instead of in jumps. The phenomenon can be associated with a shift in frequency of an emitted signal or a movement of a body part or device. Controlled cortically by the frontal eye fields (FEF), or subcortically by the superior colliculus, saccades serve as a mechanism for fixation, rapid eye movement, and the fast phase of optokinetic nystagmus. The word appears to have been coined in the 1880s by French ophthalmologist Émile Javal, who used a mirror on one side of a page to observe eye movement in silent reading, and found that it involves a succession of discontinuous individual movements. Function Humans and many animals do not look at a scene in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Sampling
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population in question. Sampling has lower costs and faster data collection than measuring the entire population and can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medical Imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. Medical imaging also establishes a database of normal anatomy and physiology to make it possible to identify abnormalities. Although imaging of removed organs and tissues can be performed for medical reasons, such procedures are usually considered part of pathology instead of medical imaging. Measurement and recording techniques that are not primarily designed to produce images, such as electroencephalography (EEG), magnetoencephalography (MEG), electrocardiography (ECG), and others, represent other technologies that produce data susceptible to representation as a parameter graph versus time or maps that contain data about the measureme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short-rate Model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate, r_t \,, then, is the ( continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire yield curve. However, no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of r_t \, as a stochastic process under a risk-neutral measure Q, then the price at time t of a zero-coupon bond maturing at time T with a payoff of 1 is given by : P(t,T) = \operatorname^Q\left \mathcal_t \right where \mathcal is the natural filtration for the process. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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