There is a strong link between additive process and infinitely divisible distributions.
has an infinitely divisible distribution characterized by the generating triplet
It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula: where
[7] A Lèvy process characteristic function has the same structure but with
[8] The following result together with the Lévy–Khintchine formula characterizes the additive process.
[9][10] Family of additive processes with generalized logistic distribution.
[11] 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[12] have the same characteristic function of Lévy normal tempered stable processes but with time dependent parameters
The characteristic function of the Variance gamma at time
The characteristic function of the variance gamma SSD is Simulation of Additive process is computationally efficient thanks to the independence of increments.
[17] Jump simulation is a generalization to the class of additive processes of the jump simulation technique developed for Lévy processes.
Moreover, Gaussian approximation can be applied to replace small jumps with a diffusive term.
It is also possible to use the Ziggurat algorithm to speed up the simulation of jumps.
[18] Simulation of Lévy process via characteristic function inversion is a well established technique in the literature.
The inversion speed is enhanced by the use of the Fast Fourier transform.
The method has similar computational cost as simulating a standard geometric Brownian motion.
[20] 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.
[3] 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.
[21] 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.
[23] Additive normal tempered stable processes fit accurately equity market data ( error below 0.8% on the S&P 500 equity market) specifically for short maturities.
These family of processes reproduces very well also the equity market implied volatility skew.
Moreover, an interesting power scaling characteristic arises in calibrated parameters
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.
[25] Additive subordination is applied to the commodity market[26] and to VIX options.
Such estimator aims to distinguish between real signal and noise in the picture pixels.