Levy process

Lévyprozesse , named after the French mathematician Paul Lévy (1886–1971), are stochastic processes with stationary, independent increments . They describe the development of variables over time, which are random but are exposed to influences that remain the same and independent of one another over time (in distribution). Many important processes, such as the Wiener process or the Poisson process , are Lévy processes.

definition

Be a stochastic process over the index set (mostly or ). It is said to have independent increases if the random variables (the increases of ) for all are independent . ${\displaystyle (X_{t}),\;t\in T}$ ${\displaystyle T}$ ${\displaystyle T=\mathbb {R} _{+}}$ ${\displaystyle T=\mathbb {N} _{0}}$ ${\displaystyle X_{t}}$ ${\displaystyle t_{1}<t_{2}<\dotsb <t_{n}\in T}$ ${\displaystyle X_{t_{2}}-X_{t_{1}},X_{t_{3}}-X_{t_{2}},\dotsc ,X_{t_{n}}-X_{t_{n-1}}}$ ${\displaystyle X_{t}}$

Is the distribution of the increases over equally long time intervals the same, i. H. applies

${\displaystyle X_{t_{1}+h}-X_{t_{1}}\sim X_{t_{2}+h}-X_{t_{2}}\;\forall t_{1},t_{2}\in T,\;h>0,}$

this is what one calls a process with steady growth . ${\displaystyle X_{t}}$

Levy processes are precisely those processes that have independent and steady growth. Often it is also required that ( almost certainly ) applies. Is a general Lévyprozess, then by a Lévyprozess with defined. The following is always assumed. ${\displaystyle (X_{t})}$ ${\displaystyle X_{0}=0}$ ${\displaystyle (X_{t})}$ ${\displaystyle Y_{t}=X_{t}-X_{0}}$ ${\displaystyle (Y_{t})}$ ${\displaystyle Y_{0}=0}$ ${\displaystyle X_{0}=0}$

Discrete-time levy processes

If it applies specifically , the class of Levy processes can be characterized very easily: there is a representation for all such processes ${\displaystyle T=\mathbb {N} _{0}}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} _{0}}}$

${\displaystyle X_{n}=\sum _{i=1}^{n}Z_{i},}$

where are independent and identically distributed random variables. On the other hand, for each sequence of independent random variables , which all have the same arbitrarily predetermined distribution, a Levy process X is defined by and . In the time-discrete case, a Levy process is basically nothing more than a random walk with any but constant jump distribution. The simplest example of a discrete-time Lévy process is therefore also the simple, symmetrical random walk , in which Bernoulli is symmetrically distributed . Here the process X, starting at , moves in each step with a probability ½ by one up, otherwise by one down. ${\displaystyle Z_{1},Z_{2},\ldots }$ ${\displaystyle (Z_{i})_{i\in \mathbb {N} }}$ ${\displaystyle X_{0}=0}$ ${\displaystyle \textstyle X_{n}=\sum _{i=1}^{n}Z_{i}}$ ${\displaystyle 2X_{1}-1}$ ${\displaystyle X_{0}=0}$

Continuous Levy processes

A gamma process is a Levy process in which the gains are independent and gamma distributed . This is possible because the gamma distribution is infinitely divisible. The process is almost certainly monotonously growing, so it is a subordinator . The process has infinite activity and no diffusion component. The two random paths are from trajectories of gamma processes, with the shape parameters 0.7 (red) and 0.25 (blue)

In this case , characterization is no longer so easy: For example, there is no time-continuous Lévy process in which Bernoulli is distributed as above. ${\displaystyle T=[0,\infty )}$ ${\displaystyle X_{1}}$

However, continuous Lévy processes are closely related to the concept of infinite divisibility : if a Lévy process is, it is infinitely divisible. On the other hand, an infinitely divisible random variable already clearly defines the distribution of the entire Lévi process. Each Levy process corresponds to an infinitely divisible distribution function and vice versa. ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{1}}$ ${\displaystyle (X_{t})_{t\geq 1}}$

Three trajectories of Levy processes of the Variance-Gamma type

Important examples of continuous Lévy processes are the Wiener process (also called Brownian motion), in which the infinitely divisible distribution of is a normal distribution , or the Poisson process , in which the Poisson distribution is. But also many other distributions, for example the gamma distribution or the Cauchy distribution , can be used for the construction of Levy processes. Besides the deterministic process , the Wiener process with constant drift and constant volatility is the only continuous Lévy process, i.e. H. the normal distribution of its increases already follows from the continuity of a Levy process. However, there is, for example, no Lévy process with uniformly distributed states. ${\displaystyle X_{1}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{t}=\sigma t}$

The concept of finite and infinite activity is also important : Are there infinitely many (and thus infinitely small) jumps in an interval or not? Information about this is also provided by the Lévymass .

Furthermore, subordinators are of importance, these are Levy processes with almost certainly monotonically growing paths. One example of this is the gamma process . The difference between two gamma processes is known as the variance-gamma-process .

Further definition

A stochastic process over a probability space is called a Levyprocess , if ${\displaystyle X_{t},\,t\geq 0}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$

• ${\displaystyle X_{0}=0}$,
• ${\displaystyle X_{t}}$ has independent and stationary gains and
• ${\displaystyle X_{t}}$is stochastically continuous, d. H. for any and holds ${\displaystyle \varepsilon >0}$ ${\displaystyle t_{0}\geq 0}$

${\displaystyle \lim _{t\to t_{0}}P(\vert X_{t}-X_{t_{0}}\vert >\varepsilon )=0}$.

Lévy-Chinchin formula

→ Main article : Lévy Khinchin formula

For every valued Lévy process its characteristic function can be written in the form: ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle (X_{t})_{t\geq 0}}$

${\displaystyle \operatorname {E} (e^{izX_{t}})=e^{t\psi (z)}}$

with the characteristic exponent

${\displaystyle \psi (z)=-{\frac {1}{2}}\langle z,Az\rangle +i\langle \gamma ,z\rangle +\int _{\mathbb {R} ^{d}}(e^{i\langle z,x\rangle }-1-i\langle z,x\rangle 1_{|x|\leq 1})\nu (dx)}$

and the characteristic triple . There is a symmetric positive definite matrix, a vector and a measure on with ${\displaystyle (A,\nu ,\gamma )}$ ${\displaystyle A\in \mathbb {R} ^{d\times d}}$ ${\displaystyle \gamma \in \mathbb {R} ^{d}}$ ${\displaystyle \nu }$ ${\displaystyle \mathbb {R} ^{d}}$

${\displaystyle \nu (\{0\})=0}$ and ${\displaystyle \int _{\mathbb {R} ^{d}}\min(|x|^{2},1)\,\mathrm {d} \nu (x)<\infty .}$

The characteristic triple is clearly determined by the process.

This representation of the characteristic function of a Lévy process is named after Paul Lévy and Alexandr Chintschin .

Lévy-Itō decomposition

Each Levy process can be represented as a sum of a Brownian movement , a linear drift process and a pure jump process , which contains all the jumps of the original Lévy process. This statement is known as the Lévy-Itō decomposition.

Be a Levyprocess in with a characteristic triple . Then there are three independent Lévyprozesse, all of which are defined on the same probability space , , such that: ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle (A,\nu ,\gamma )}$ ${\displaystyle X^{(1)}}$ ${\displaystyle X^{(2)}}$ ${\displaystyle X^{(3)}}$

• ${\displaystyle X^{(1)}}$is a Brownian movement with drift, ie a Levy process with a characteristic triple ; ${\displaystyle (A,0,\gamma )}$
• ${\displaystyle X^{(2)}}$is a Levy process with a characteristic triple ( i.e. a compound Poisson process); ${\displaystyle (0,\nu |_{\mathbb {R} ^{d}\backslash \{x;|x|\geq 1\}},0)}$
• ${\displaystyle X^{(3)}}$is a square integrable martingale and a pure jumping process with the characteristic triple . ${\displaystyle (0,\nu |_{\{x;|x|<1\}},0)}$

Important properties

• The expectation function of a Levy process is linear in t , i.e. H. ${\displaystyle (X_{t})}$

${\displaystyle \operatorname {E} (X_{t})=t\operatorname {E} (X_{1})}$. The same applies to the variance

${\displaystyle \operatorname {Var} (X_{t})=t\operatorname {Var} (X_{1})}$(provided the corresponding moments exist at time 1). The following applies to the covariance function

${\displaystyle \operatorname {Cov} (X_{s},X_{t})=\operatorname {Var} (X_{\min(s,t)})=\min(s,t)\operatorname {Var} (X_{1})}$.

• If true, then it is a martingale . ${\displaystyle E(X_{1})=0}$ ${\displaystyle (X_{t})}$

literature

• J. Bertoin: Lévy Processes . Cambridge Tracts in Mathematics, Vol. 121, Cambridge University Press 2002, ISBN 0-521-64632-4
• AE Kyprianou: Introductory Lectures on fluctuations of Lévy process with applications . University text, Springer.
• Philip E. Protter: Stochastic Integration and Differential Equations . Springer, Berlin 2003, ISBN 3-540-00313-4
• Rama Cont, Peter Tankov: Financial Modeling with Jump Processes . Chapman & Hall, 2003, ISBN 1-58488-413-4
• Ken-iti Sato: Lévy Processes and Infinitely Divisible Distributions . Cambridge studies in advanced mathematics, 1999, ISBN 0-521-55302-4

Web links

• University script about Lévy processes
• Jan Kallsen: Lévy processes clearly explained. (PDF)