Additive process

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]

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]

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]

• 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.