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 is 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 increasing 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 smile) 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

1. Tankov & Cont 2003, p. 455.
2. Tankov & Cont 2003, p. 468.
3. Tankov & Cont 2003, p. 454.
4. Bhattacharya & Brockwell 1976, p. 71.
5. Tankov & Cont 2003, p. 80.
6. Sato 1999, p. 47.
7. Sato 1999, pp. 37–38.
8. Tankov & Cont 2003, p. 95.
9. Tankov & Cont 2003, p. 458.
10. Sato 1999, p. 63.
11. Li, Li & Mendoza-Arriaga 2016, pp. 5–6.
12. Li, Li & Mendoza-Arriaga 2016, p. 1.
13. Eberlein & Madan 2009, p. 5.
14. Carr et al. 2007, p. 39.
15. Carr et al. 2007, p. 32.
16. Carr et al. 2007, pp. 37.
17. Carr et al. 2007, pp. 39–42.
18. Li, Li & Mendoza-Arriaga 2016, pp. 3.
19. Li, Li & Mendoza-Arriaga 2016, p. 17.
20. Li, Li & Zhang 2017, p. 1.

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 for Wahrscheinlichkeitstheorie und Verwandte Gebiete. 37 (1). doi:10.1007/BF00536298.