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. It has important applications in
magnetic reconnection,
coronal mass ejections,
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, and
pattern theory and
computational vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
.
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
option pricing, a jump-diffusion model is a form of
mixture model, mixing a
jump process and a
diffusion process. Jump-diffusion models have been introduced by
Robert C. Merton as an extension of
jump models. Due to their computational tractability, the special case of a
basic affine jump diffusion
In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form
: dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0,
Z_\geq 0,
where B is a standard Brownian motion, and ...
is popular for some
credit risk and
short-rate models.
In pattern theory, computer vision, and medical imaging
In
pattern theory and
computational vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
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 algorithm that mixes "focus"-like motions, the
diffusion processes, with
saccade-like motions, via
jump processes.
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, 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 (in the context of
dynamical systems), a generalization of jump diffusion
References
{{Stochastic processes
Stochastic processes
Options (finance)