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 stationary 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.

An additive process is a generalization of a Lévy process obtained relaxing the hypothesis of stationary increments. Thanks to this feature an additive process can describe more complex phenomenons than a Lévy process.

A stochastic process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ on ${\displaystyle \mathbb {R} ^{d}}$ such that ${\displaystyle X_{0}=0}$ almost surely is an additive process if it satisfy the following hypothesis:

A stochastic process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ has independent increments if and only if for any ${\displaystyle 0\leq p<r\leq s<t}$ the random variable ${\displaystyle X_{t}-X_{s}}$ is independent from the random variable ${\displaystyle X_{r}-X_{p}}$.^{[clarification needed]}

A stochastic process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ is continuous in probability if, and only if, for any ${\displaystyle t>0}$

There is a strong link between additive process and infinitely divisible distributions. An additive process at time ${\displaystyle t}$ has an infinitely divisible distribution characterized by the generating triplet ${\displaystyle (\gamma _{t},A_{t},\nu _{t})}$. ${\displaystyle \gamma _{t}}$ is a vector in ${\displaystyle \mathbb {R} ^{d}}$, ${\displaystyle A_{t}}$ is a matrix in ${\displaystyle \mathbb {R} ^{d\times d}}$ and ${\displaystyle \nu _{t}}$ is a measure on ${\displaystyle \mathbb {R} ^{d}}$ such that ${\displaystyle \nu _{t}(\{0\})=0}$ and ${\displaystyle \int _{\mathbb {R} ^{d}}(1\wedge x^{2})\nu _{t}(dx)<\infty }$.

${\displaystyle \gamma _{t}}$ is called drift term, ${\displaystyle A_{t}}$ covariance matrix and ${\displaystyle \nu _{t}}$ Lévy measure. It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula: