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.^[1] The additive process was introduced by Paul Lévy in 1937.^[2]

There are applications of the additive process in quantitative finance^[3] (this family of processes can capture important features of the implied volatility) and in digital image processing.^[4]

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

1. It has independent increments.
2. It is continuous in probability.^[1]

Main properties[]

Independent increments[]

A stochastic process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ has independent increments if and only if for any ${\displaystyle 0\leq p<r<s<t}$ the random variable ${\displaystyle X_{t}-X_{s}}$ is independent from the random variable ${\displaystyle X_{r}-X_{p}}$.^[5]

Continuity in probability[]

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

${\displaystyle \lim _{s\to t}\Pr \left({\big |}X_{s}-X_{t}{\big |}\geq \varepsilon \right)=0.}$^[5]

Lévy–Khintchine representation[]

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 ${\displaystyle \int _{\mathbb {R} ^{d}}(1\wedge x^{2})\nu _{t}(dx)<\infty }$. ^[6]

${\displaystyle \gamma _{t}}$ 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:

${\displaystyle \varphi _{X}(u)(t):=\operatorname {E} \left[e^{iu'X_{t}}\right]=\exp \left(u'\gamma _{t}i-{\frac {1}{2}}u'A_{t}u+\int _{\mathbb {R} ^{d}}\left(e^{iu'x}-1-iu'x\mathbf {I} _{|x|<1}\right)\,\nu _{t}(dx)\right),}$

where ${\displaystyle u}$ is a vector in ${\displaystyle \mathbb {R} ^{d}}$ and ${\displaystyle \mathbf {I_{C}} }$ is the indicator function of the set ${\displaystyle C}$.^[7]

A Lèvy process characteristic function has the same structure but with ${\displaystyle \gamma _{t}=t\gamma ,\nu _{t}=t\nu }$ and ${\displaystyle A_{t}=At}$ with ${\displaystyle \gamma }$ a vector in ${\displaystyle \mathbb {R} ^{d}}$, ${\displaystyle A}$ a positive definite matrix in ${\displaystyle \mathbb {R} ^{d\times d}}$ and ${\displaystyle \nu }$ is a measure on ${\displaystyle \mathbb {R} ^{d}}$.^[8]

Existence and uniqueness in law of additive process[]

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

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

1. For all ${\displaystyle t}$, ${\displaystyle A_{t}}$ is a positive definite matrix.
2. ${\displaystyle \gamma _{0}=0,A_{0}=0,\nu _{0}=0}$ and for all ${\displaystyle s,t}$ is such that ${\displaystyle s<t}$, ${\displaystyle A_{t}-A_{s}}$ is a positive definite matrix and ${\displaystyle \nu _{t}(B)\geq \nu _{s}(B)}$ for every ${\displaystyle B}$ in ${\displaystyle \mathbf {B} (\mathbb {R} ^{d})}$.
3. If ${\displaystyle s\to t}$ ${\displaystyle \gamma _{s}\to \gamma _{t},A_{s}\to A_{t}}$ and ${\displaystyle \nu _{s}(B)\to \nu _{t}(B)}$ every ${\displaystyle B}$ in ${\displaystyle \mathbf {B} (\mathbb {R} ^{d})}$, ${\displaystyle 0\not \in B}$.

Conversely for family of infinitely divisible distributions characterized by a generating triplet ${\displaystyle (\gamma _{t},A_{t},\nu _{t})}$ that satisfies 1, 2 and 3, it exists an additive process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ with this distribution.^[9][10]

Subclass of additive process[]

Additive subordinator[]

A positive non decreasing additive process ${\displaystyle \{S_{t}\}_{t\geq 0}}$ with values in ${\displaystyle \mathbb {R} }$ 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

${\displaystyle \operatorname {E} \left[e^{-uS_{t}}\right]=\exp \left(ub_{t}+\int _{\mathbb {R} ^{d}}(e^{iux}-1)\nu _{t}(dx)\right).}$^[11]

It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes.^[12]

Sato process[]

An additive self-similar process ${\displaystyle \{Z_{t}\}_{t\geq 0}}$ is called Sato process.^[13] It is possible to construct a Sato process from a Lévy process ${\displaystyle \{X_{t}\}_{t\geq 0}}$ such that ${\displaystyle Z_{t}}$ has the same law of ${\displaystyle t^{h}X_{1}}$.

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 ${\displaystyle t=1}$ is

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

where ${\displaystyle \theta ,\nu }$ and ${\displaystyle \sigma }$ are positive constant.

The characteristic function of the variance gamma SSD is

${\displaystyle \operatorname {E} \left[e^{iuZ_{t}}\right]=\left({\frac {1}{1-iut^{h}\theta \nu +0.5\sigma ^{2}\nu u^{2}t^{2}h}}\right)^{1/\nu }}$^[14]

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.^[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.^[15] 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.^[16] 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.^[17]

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.^[18] Additive subordination is applied to the commodity market^[19] and to VIX options.^[20]

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.^[4]

References[]

Sources[]

• Tankov, Peter; Cont, Rama (2003). Financial modelling with jump processes. Chapman and Hall. ISBN 1584884134.
• Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. ISBN 9780521553025.
• Li, Jing; Li, Lingfei; Mendoza-Arriaga, Rafael (2016). "Additive subordination and its applications in finance". Finance and Stochastics. 20 (3): 2–6. doi:10.1007/s00780-016-0300-8.
• Eberlein, Ernst; Madan, Dilip B. (2009). "Sato processes and the valuation of structured products". Quantitative Finance. 9 (1). doi:10.1080/14697680701861419.
• Carr, Peter; Geman, Hélyette; Madan, Dilip B.; Yor, Marc (2007). "SELF-DECOMPOSABILITY AND OPTION PRICING". Mathematical Finance. 17 (1). doi:10.1111/j.1467-9965.2007.00293.x.
• Li, Jing; Li, Lingfei; Zhang, Gongqiu (2017). "Pure jump models for pricing and hedging VIX derivatives". Journal of Economic Dynamics and Control. 74. doi:10.1016/j.jedc.2016.11.001.
• Bhattacharya, P. K.; Brockwell, P. J. (1976). "The minimum of an additive process with applications to signal estimation and storage theory". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 37 (1). doi:10.1007/BF00536298.