Additive process
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 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 formula_1 on formula_2 such that formula_3 almost surely is an additive process if it satisfy the following hypothesis: Main properties. Independent increments. A stochastic process formula_1 has independent increments if and only if for any formula_5 is independent from the random variable formula_6. Continuity in probability. A stochastic process formula_1 is continuous in probability if, and only if, for any formula_8 Lévy–Khintchine representation. There is a strong link between  additive process and infinitely divisible distributions. An additive process at time formula_10 has an infinitely divisible distribution characterized by the generating triplet formula_11. formula_12 is a vector in formula_2, formula_14 is a matrix in formula_15 and formula_16 is a measure on formula_2 such that formula_18 and formula_19. formula_12 is called drift term, formula_14 covariance matrix and formula_16 Lévy measure. It is possible to write explicitly the additive process characteristic function using the "Lévy–Khintchine formula": where formula_24 is a vector in formula_2 and formula_26 is the indicator function of the set formula_27. A Lèvy process characteristic function has the same  structure but with formula_28 and formula_29 with formula_30 a vector in formula_2, formula_32 a positive definite matrix in formula_33 and formula_34 is a measure on formula_2. Existence and uniqueness in law of additive process. The following result together with the "Lévy–Khintchine formula" characterizes the additive process. Let formula_1 be an additive process on formula_2. Then, its infinitely divisible distribution is such that: Conversely for family of infinitely divisible distributions characterized by a generating triplet formula_11 that satisfies 1, 2 and 3, it exists an additive process formula_1 with this distribution. Subclass of additive process. Additive subordinator. A positive non  decreasing additive process formula_55 with values in formula_56 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 formula_58 is called Sato process. It is possible to construct a Sato process from a Lévy process formula_1 such that formula_60 has the same law of formula_61. 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 formula_62 is where formula_64 and formula_65 are positive constant. The characteristic function of the variance gamma SSD is Applications. Quantitative finance. Lévy process is used to model the log-returns of market prices. Unfortunately, the "stationarity " of the increments does not reproduce correctly market data. A Lévy process fit well call option and put option prices (implied volatility) for a single expiration date but is unable to fit options prices with different maturities (volatility surface). The additive process introduces a "deterministic" non -stationarity that allows it to fit all expiration dates. A four-parameters Sato process (self-similar additive process) can reproduce correctly the volatility surface (3% error on the S&P 500 equity market). This order of magnitude of error is usually obtained using models with 6-10 parameters to fit market data. A self-similar process correctly describes market data because of its flat skewness and excess kurtosis; empirical studies had observed this behavior in market skewness and excess kurtosis. Some of the processes that fit option prices with a 3% error are VGSSD, NIGSSD, MXNRSSD obtained from variance gamma process , normal inverse Gaussian process and Meixner process. Lévy subordination is used to construct new Lévy processes (for example variance gamma process and normal inverse Gaussian process). There is a large number of financial applications of processes constructed by Lévy subordination. An additive process built via additive subordination maintains the analytical tractability of a process built via Lévy subordination but it reflects better the time-inhomogeneus structure of market data. Additive subordination is applied to the commodity market and to VIX options. Digital image processing. An estimator based on the minimum of an additive process can be applied to image processing . Such estimator aims to distinguish between real signal and noise in the picture pixels.