additive process

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 $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ on $\backslash mathbb\; R^d$ such that $X\_0=0$ almost surely is an additive process if it satisfy the following hypothesis:

1. It has independent increments.
2. It is continuous in probability.{{sfn|Tankov|Cont|2003|p=455}}

A stochastic process $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ has independent increments if and only if for any

A stochastic process $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ is continuous in probability if, and only if, for any $t>0$

:$\backslash lim\_\{s\; \backslash to\; t^-\}\; \backslash Pr\; \backslash left(\backslash big|\; X\_s-\; X\_t\; \backslash big|\; \backslash geq\; \backslash varepsilon\; \backslash right)\; =\; 0.${{sfn|Tankov|Cont|2003|p=80}}

There is a strong link between additive process and infinitely divisible distributions. An additive process at time $t$ has an infinitely divisible distribution characterized by the generating triplet $(\backslash gamma\_t,\; A\_t,\; \backslash nu\_t)$. $\backslash gamma\_t$ is a vector in $\backslash mathbb\; R^d$, $A\_t$ is a matrix in $\backslash mathbb\; R^\{d\backslash times\; d\}$ and $\backslash nu\_t$ is a measure on $\backslash mathbb\; R^d$ such that

$\backslash nu\_t(\backslash \{0\backslash \})=0$ and . {{sfn|Sato|1999|p=47}}

$\backslash gamma\_t$ is called drift term, $A\_t$ covariance matrix and $\backslash nu\_t$ Lévy measure.

It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula:

:

where $u$ is a vector in $\backslash mathbb\; R^d$ and $\backslash mathbf\{I\_C\}$ is the indicator function of the set $C$.{{sfn|Sato|1999|pp=37–38}}

A Lèvy process characteristic function has the same structure but with $\backslash gamma\_t\; =t\backslash gamma,\; \backslash nu\_t\; =\; t\backslash nu$ and $A\_t\; =\; At$ with $\backslash gamma$ a vector in $\backslash mathbb\; R^d$, $A$ a positive definite matrix in $\backslash mathbb\; R^\{d\; \backslash times\; d\}$ and $\backslash nu$ is a measure on $\backslash mathbb\; R^d$.{{sfn|Tankov|Cont|2003|p=95}}

The following result together with the Lévy–Khintchine formula characterizes the additive process.

Let $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ be an additive process on $\backslash mathbb\; R^d$. Then, its infinitely divisible distribution is such that:

1. For all $t$, $A\_t$ is a positive definite matrix.
2. $\backslash gamma\_0=0,\; A\_0=0,\; \backslash nu\_0=0$ and for all $s,\; t$ is such that
3. If $s\backslash to\; t$ $\backslash gamma\_s\backslash to\; \backslash gamma\_t,\; A\_s\; \backslash to\; A\_t$ and $\backslash nu\_s(B)\backslash to\; \backslash nu\_t(B)$ every $B$ in $\backslash mathbf\{B\}(\backslash mathbb\; R^d)$, $0\backslash not\backslash in\; B$.

Conversely for family of infinitely divisible distributions characterized by a generating triplet $(\backslash gamma\_t,\; A\_t,\; \backslash nu\_t)$ that satisfies 1, 2 and 3, it exists an additive process $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ with this distribution.{{sfn|Tankov|Cont|2003|p=458}}{{sfn|Sato|1999|p=63}}

Family of additive processes with generalized logistic distribution. Their 5 parameters characteristic function is

: $\backslash operatorname\; E\; \backslash left[e^\{iuX\_t\}\backslash right]\; =\; \backslash left(\backslash frac\{B(\backslash alpha\_t+i\; \backslash sigma\_t\; u,\; \backslash beta\_t-\; i\; \backslash sigma\; u\; )\}\{B(\backslash alpha\_t,\; \backslash beta\_t)\}\; \backslash right)^\{\backslash delta\_t\}\; e^\{\; i\; \backslash mu\_t\; u\}\backslash ;\backslash ;.$

Two subcases of additive logistic process are the symmetric logistic additive process with standard logistic distribution ($\backslash alpha\_t=1$, $\backslash beta\_t=1$, $\backslash delta\_t=1$) and the conjugate-power Dagum additive process with Dagum distribution ($\backslash alpha\_t=1$, $\backslash beta\_t=1-\backslash sigma(t)$, $\backslash alpha\_t=1$).

The function $\backslash mu\_t$ can always be chosen s.t. the additive process is a martingale.{{sfn|Carr|Torricelli|2021|p= 698}}

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{{sfn|Azzone|Baviera|2022|p= 503}} have the same characteristic function of Lévy normal tempered stable processes but with time dependent parameters $\backslash sigma\_t$ (the level of the volatility), $k\_t$ (the variance of jumps) and $\backslash eta\_t$ (linked to the skew):

: $\backslash operatorname\; E\; \backslash left[e^\{iuX\_t\}\backslash right]\; =\; \{\backslash cal\; L\}\_t\; \backslash left(iu\; \backslash left(\backslash frac\{1\}\{2\}+\backslash eta\_t\; \backslash right)\backslash sigma\_t^2+\backslash frac\{u^2\backslash sigma^2\_t\}\{2\};\backslash ;k\_t,\backslash ;\backslash alpha\; \backslash right)e^\{iu\backslash varphi\_tt\},$

where

: $\backslash ln\; \{\backslash cal\; L\}\_t\; \backslash left(u;\backslash ;k\_t,\backslash ;\backslash alpha\backslash right)\; :=$

\begin{cases}

\displaystyle \frac{t}{k_t}

\displaystyle \frac{1-\alpha}{\alpha}

\left \{1- \left(1+\frac{u \; k_t}{1-\alpha}\right)^\alpha \right \} & \mbox{if } \; 0< \alpha < 1 \\[4mm]

\displaystyle -\frac{t}{k_t}

\ln \left(1+u \; k_t\right) & \mbox{if } \; \alpha = 0 \end{cases}

The function $\backslash varphi\_t$ can always be chosen s.t. the additive process is a martingale.{{sfn|Azzone|Baviera|2022|p= 503}}

A positive non decreasing additive process $\backslash \{S\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ with values in $\backslash mathbb\; R$ 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

: $\backslash operatorname\; E\backslash left[\; e^\{-u\; S\_t\}\; \backslash right]\; =\; \backslash exp\backslash left(u\; b\_t\; +\; \backslash int\_\{\backslash mathbb\; R^d\}\; (e^\{i\; u\; x\}-1)\; \backslash nu\_t(dx)\backslash right).${{sfn|Li|Li|Mendoza-Arriaga|2016|pp= 5–6}}

It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes.{{sfn|Li|Li|Mendoza-Arriaga|2016|p= 1}}

An additive self-similar process $\backslash \{Z\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ is called Sato process.{{sfn|Eberlein|Madan|2009|p= 5}}

It is possible to construct a Sato process from a Lévy process $\backslash \{X\_t\backslash \}\_\{t\; \backslash geq\; 0\}$ such that $Z\_t$ has the same law of $t^hX\_1$.

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 $t=1$ is

: $\backslash operatorname\; E\; \backslash left[e^\{iuX\_1\}\backslash right]\; =\; \backslash left(\backslash frac\{1\}\{1-iu\backslash theta\backslash nu+0.5\backslash sigma^2\backslash nu\; u^2\}\backslash right)^\{1/\backslash nu\},$

where $\backslash theta,\; \backslash nu$ and $\backslash sigma$ are positive constant.

The characteristic function of the variance gamma SSD is

: $\backslash operatorname\; E\; \backslash left[e^\{iuZ\_t\}\backslash right]\; =\; \backslash left(\backslash frac\{1\}\{1-iut^h\backslash theta\backslash nu+0.5\backslash sigma^2\backslash nu\; u^2t^2h\}\backslash right)^\{1/\backslash nu\}${{sfn|Carr|Geman|Madan|Yor|2007|p= 39}}

Simulation of Additive process is computationally efficient thanks to the independence of increments. The additive process increments can be simulated separately and simulation can also be parallelized.{{sfn|Eberlein|Madan|2009|}}

Jump simulation is a generalization to the class of additive processes of the jump simulation technique developed for Lévy processes.

The method is based on truncating small jumps below a certain threshold and simulating the finite number of independent jumps.

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.{{sfn|Eberlein|Madan|2009|p= 19}}

Simulation of Lévy process via characteristic function inversion is a well established technique in the literature.{{sfn|Ballotta|Kyriakou|2014|p= 1}}

This technique can be extended to additive processes. The key idea is obtaining an approximation of the cumulative distribution function (CDF) by inverting the characteristic function. The inversion speed is enhanced by the use of the Fast Fourier transform. Once the approximation of the CDF is available is it possible to simulate an additive process increment just by simulating a uniform random variable. The method has similar computational cost as simulating a standard geometric Brownian motion.{{sfn|Azzone|Baviera|2023|pp=1-5}}

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.{{sfn|Tankov|Cont|2003|p=454}}

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.{{sfn|Carr|Geman|Madan|Yor|2007|p= 32}} 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.{{sfn|Carr|Geman|Madan|Yor|2007|p= 37}}

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.{{sfn|Carr|Geman|Madan|Yor|2007|pp= 39–42}}

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 $k\_t=\backslash bar\{k\}t^\backslash beta$ and $\backslash eta\_t=\backslash bar\{\backslash eta\}t^\backslash delta$. There is statistical evidence that $\backslash beta=1$

and $\backslash delta=-1/2$.{{sfn|Azzone|Baviera|2022|pp= 506-508}}

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.{{sfn|Li|Li|Mendoza-Arriaga|2016|pp= 3}} Additive subordination is applied to the commodity market{{sfn|Li|Li|Mendoza-Arriaga|2016|p= 17}} and to VIX options.{{sfn|Li|Li|Zhang|2017|p= 1}}

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.{{sfn|Bhattacharya|Brockwell|1976|p= 71}}

{{Reflist}}

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{{Stochastic processes}}

Category:Probability theory