An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process (a Lévy process is an additive process with identically distributed increments). An example of an additive process that is not a Lévy process is a Brownian motion with a time-dependent drift. The additive process was introduced by Paul Lévy in 1937.
There are applications of the additive process in quantitative finance (this family of processes can capture important features of the implied volatility) and in digital image processing.
Definition
An additive process is a generalization of a Lévy process obtained relaxing the hypothesis of identically distributed increments. Thanks to this feature an additive process can describe more complex phenomenons than a Lévy process.
A stochastic process on such that almost surely is an additive process if it satisfy the following hypothesis: It has independent increments. It is continuous in probability.
Main properties
Independent increments
A stochastic process has independent increments if and only if for any the random variable is independent from the random variable .
Continuity in probability
A stochastic process is continuous in probability if, and only if, for any
Lévy–Khintchine representation
There is a strong link between additive process and infinitely divisible distributions. An additive process at time has an infinitely divisible distribution characterized by the generating triplet . is a vector in , is a matrix in and is a measure on such that and .
is called drift term, covariance matrix and Lévy measure. It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula:
where is a vector in and is the indicator function of the set .
A Lèvy process characteristic function has the same structure but with and with a vector in , a positive definite matrix in and is a measure on .
Existence and uniqueness in law of additive process
The following result together with the Lévy–Khintchine formula characterizes the additive process.
Let be an additive process on . Then, its infinitely divisible distribution is such that: For all , is a positive definite matrix. and for all is such that , is a positive definite matrix and for every in . If and every in , .
Conversely for family of infinitely divisible distributions characterized by a generating triplet that satisfies 1, 2 and 3, it exists an additive process with this distribution.
Subclass of additive process
Additive Logistic Process
Family of additive processes with generalized logistic distribution. Their 5 parameters characteristic function is
Two subcases of additive logistic process are the symmetric logistic additive process with standard logistic distribution (, , ) and the conjugate-power Dagum additive process with Dagum distribution (, , ).
The function can always be chosen s.t. the additive process is a martingale.
Additive Normal Tempered Stable Process
Extension of the Lévy normal tempered stable processes; some well-known Lévy normal tempered stable processes have normal-inverse Gaussian distribution and the variance-gamma distribution. Additive normal tempered stable processes have the same characteristic function of Lévy normal tempered stable processes but with time dependent parameters (the level of the volatility), (the variance of jumps) and (linked to the skew):
where
The function can always be chosen s.t. the additive process is a martingale.
Additive Subordinator
A positive non decreasing additive process with values in is an additive subordinator. An additive subordinator is a semimartingale (thanks to the fact that it is not decreasing) and it is always possible to rewrite its Laplace transform as
It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes.
Sato Process
An additive self-similar process is called Sato process. It is possible to construct a Sato process from a Lévy process such that has the same law of .
An example is the variance gamma SSD, the Sato process obtained starting from the variance gamma process.
The characteristic function of the Variance gamma at time is
where and are positive constant.