Jump diffusion is a
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 appea ...
that involves
jumps
Jumping or leaping is a form of locomotion or movement in which an organism or non-living (e.g., robotic) mechanical system propels itself through the air along a ballistic trajectory. Jumping can be distinguished from running, galloping and o ...
and
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. It has important applications in
magnetic reconnection
Magnetic reconnection is a physical process occurring in highly conducting plasmas in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle acceleration. Magnetic reconnectio ...
,
coronal mass ejection
A coronal mass ejection (CME) is a significant release of plasma and accompanying magnetic field from the Sun's corona into the heliosphere. CMEs are often associated with solar flares and other forms of solar activity, but a broadly accepted ...
s,
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
,
option pricing
In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the impl ...
, and
pattern theory and
computational vision.
In physics
In crystals,
atomic diffusion
Atomic may refer to:
* Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties
* Atomic physics, the study of the atom
* Atomic Age, also known as the "Atomic Era"
* Atomic scale, distances comp ...
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
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
.
Jump diffusion can be studied on a microscopic scale by
inelastic neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
and by
Mößbauer spectroscopy. Closed expressions for the
autocorrelation function
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
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
option pricing
In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the impl ...
, a jump-diffusion model is a form of
mixture model
In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation ...
, mixing a
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 mod ...
and a
diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
. Jump-diffusion models have been introduced by
Robert C. Merton
Robert Cox Merton (born July 31, 1944) is an American economist, Nobel Memorial Prize in Economic Sciences laureate, and professor at the MIT Sloan School of Management, known for his pioneering contributions to continuous-time finance, especia ...
as an extension of
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 mod ...
s. Due to their computational tractability, the special case of a
basic affine jump diffusion is popular for some
credit risk
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
and
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 sh ...
s.
In pattern theory, computer vision, and medical imaging
In
pattern theory and
computational vision in
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 rev ...
, jump-diffusion processes were first introduced by Grenander and Miller
as a form of
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 attemp ...
algorithm that mixes "focus"-like motions, the
diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
es, with
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 Publishi ...
-like motions, via
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 mod ...
es.
The approach modelled sciences of electron-micrographs as containing multiple shapes, each having some fixed dimensional representation, with the collection of micrographs filling out the sample space corresponding to the unions of multiple finite-dimensional spaces.
Using techniques from
pattern theory, a posterior probability model was constructed over the countable union of sample space; this is therefore a
hybrid system model, containing the discrete notions of object number along with the continuum notions of shape.
The jump-diffusion process was constructed to have
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 tha ...
properties so that after initially flowing away from its initial condition it would generate samples from the posterior probability model.
See also
*
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 mod ...
, an example of jump diffusion
*
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 ...
(PDMP), an example of jump diffusion and a generalization of the jump process
*
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 te ...
(in the context of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s), a generalization of jump diffusion
References
{{Stochastic processes
Stochastic processes
Options (finance)