A jump process is a type of
stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...
process that has discrete movements, called
jumps, 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 John ...
or
compound Poisson process.
In
finance
Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
, 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
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 ...
, with continuous, random movements at all scales, no matter how small.
John Carrington Cox
John Carrington Cox is the Nomura Professor of Finance Emeritus 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, ...
and
Stephen Ross proposed that prices actually follow a 'jump process'.
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 ...
extended this approach to a hybrid model known as
jump diffusion
Jump diffusion is a stochastic process that involves jump process, jumps and diffusion process, diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, and pattern theory and computationa ...
, which states that the prices have large jumps interspersed with small continuous movements.
See also
*
Poisson process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...
, 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 will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a ...
(CTMC), an example of a jump process and a generalization of the Poisson process
*
Counting process A counting process 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 famil ...
, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs
*
Interacting particle system
In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countably-infinite-order graph G and a local state space, a compact metric space S ...
, an example of a jump process
*
Kolmogorov equations (continuous-time Markov chains)
In probability theory, Kolmogorov equations characterize Markov process, continuous-time Markov processes. In particular, they describe how the probability of a continuous-time Markov process in a certain state changes over time. There are four ...
References
Stochastic processes
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