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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
that involves
jumps 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 electrically conducting Plasma (physics), plasmas, in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle accelerati ...
,
coronal mass ejection
A coronal mass ejection (CME) is a significant ejection of plasma mass 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 theoretical understandin ...
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 State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, and
pattern theory
Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machin ...
and
computational vision.
In physics
In crystals,
atomic diffusion
In chemical physics, atomic diffusion is a diffusion process whereby the random, thermally-activated movement of atoms in a solid results in the net transport of atoms. For example, helium atoms inside a balloon can diffuse through the wall of ...
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, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
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
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 observati ...
, 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 mo ...
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. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic sy ...
. In finance, jump-diffusion models were first 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 ...
. Such models have a range of financial applications from
option pricing
In finance, a price (premium) is paid or received for purchasing or selling options.
The calculation of this premium will require sophisticated mathematics.
Premium components
This price can be split into two components: intrinsic value, and ...
, to
credit risk
Credit risk is the chance that a borrower does not repay a loan
In finance, a loan is the tender of money by one party to another with an agreement to pay it back. The recipient, or borrower, incurs a debt and is usually required to pay ...
, to
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
forecasting
Forecasting is the process of making predictions based on past and present data. Later these can be compared with what actually happens. For example, a company might Estimation, estimate their revenue in the next year, then compare it against the ...
.
In pattern theory, computer vision, and medical imaging
In
pattern theory
Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machin ...
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 revea ...
, jump-diffusion processes were first introduced by Grenander and Miller
as a form of
random sampling
In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the who ...
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. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic sy ...
es, with
saccade
In vision science, a saccade ( ; ; ) is a quick, simultaneous movement of both Eye movement (sensory), eyes between two or more phases of focal points in the same direction. In contrast, in Smooth pursuit, smooth-pursuit movements, the eyes mov ...
-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 mo ...
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
Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machin ...
, 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 th ...
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 mo ...
, 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, automaton, or a differ ...
(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, such as in a parametric curve. Examples include the mathematical models ...
s), a generalization of jump diffusion
References
{{Stochastic processes
Stochastic processes
Options (finance)