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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
[edit]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 on such that almost surely is an additive process if it satisfy the following:
- It has independent increments.
- It is continuous in probability.[1]
Main Properties
[edit]Independent Increments
[edit]A stochastic process has independent increments if and only if for any the random variable is independent from the random variable .[5]
Continuity in Probability
[edit]A stochastic process is continuous in probability if and only if for any
Lévy–Khintchine representation
[edit]There is a strong link between additive process and infinitely divisible distributions. An additive process at time has an infinitely divisible distribution characterized by the generating triplet . is a vector in , is a matrix in and is a measure on such that and . [6]
is called drift term, covariance matrix and Lévy measure. It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula:
,
where is a vector in and is the indicator function of the set .[7]
A Lèvy process characteristic function has the same structure but with and with a vector in , a positive definite matrix in and is a measure on .[8]
Existence and uniqueness in law of additive Process
[edit]The following result together with the Lévy–Khintchine formula characterizes the additive process.
Let be an additive process on . Then, its infinitely divisible distribution is such that:
- For all , is a positive definite matrix
- and for all such that is a positive definite matrix and for every in .
- If and every in , .
Conversely for family of infinitely divisible distribution characterized by a generating triplet that satisfies 1, 2 and 3 it exists an additive process with this distribution.[9][10]
Subclass of additive process
[edit]Additive subordinator
[edit]A positive non increasing additive process with values in 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
.[11]
It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes.[12]
Sato process
[edit]An additive self-similar process is called Sato process.[13] It is possible to construct a Sato process from a Lévy process such that has the same law of .
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 is
,
where and are positive constant.
The characteristic function of the variance gamma SSD is
Applications
[edit]Quantitative Finance
[edit]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
[edit]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
[edit]- ^ a b Tankov & Cont 2003, p. 455.
- ^ Tankov & Cont 2003, p. 468.
- ^ a b Tankov & Cont 2003, p. 454.
- ^ a b Bhattacharya & Brockwell 1976, p. 71.
- ^ a b Tankov & Cont 2003, p. 80.
- ^ Sato 1999, p. 47.
- ^ Sato 1999, pp. 37–38.
- ^ Tankov & Cont 2003, p. 95.
- ^ Tankov & Cont 2003, p. 458.
- ^ Sato 1999, p. 63.
- ^ Li, Li & Mendoza-Arriaga 2016, pp. 5–6.
- ^ Li, Li & Mendoza-Arriaga 2016, p. 1.
- ^ Eberlein & Madan 2009, p. 5.
- ^ Carr et al. 2007, p. 39.
- ^ Carr et al. 2007, p. 32.
- ^ Carr et al. 2007, pp. 37.
- ^ Carr et al. 2007, pp. 39–42.
- ^ Li, Li & Mendoza-Arriaga 2016, pp. 3.
- ^ Li, Li & Mendoza-Arriaga 2016, p. 17.
- ^ Li, Li & Zhang 2017, p. 1.
Sources
[edit]- 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.