A jump process is a type of
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
process that has discrete movements, called
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 ...
, with random arrival times, rather than continuous movement, typically modelled as a
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
or
compound Poisson process.
In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, various stochastic models are used to model the price movements of
financial instrument
Financial instruments are monetary contracts between parties. They can be created, traded, modified and settled. They can be cash (currency), evidence of an ownership interest in an entity or a contractual right to receive or deliver in the form ...
s; 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
John Carrington Cox is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world's leading experts on options theory and one of the inventors of the Cox–Ross–Rubinstein model for option pricing, as well as ...
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 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 p ...
, 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 (CTMC), an example of a jump process and a generalization of the Poisson process
*
Counting process A counting process is a stochastic process with values that are non-negative, integer, and non-decreasing:
# ''N''(''t'') ≥ 0.
# ''N''(''t'') is an integer.
# If ''s'' ≤ ''t'' then ''N''(''s'') ≤ ''N''(''t'').
If ''s'' < ''t'', then ''N''(' ...
, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs
*
Interacting particle system, an example of a jump process
*
Kolmogorov equations (continuous-time Markov chains)
In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evolut ...
References
Stochastic processes
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