Contributed reagents/materials/analysis tools: BB. Wrote the paper: BB.
The author has declared that no competing interests exist.
We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Lévy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Lévy processes, of which Brownian Motion is a special case, have become increasingly popular. Lévy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Lévy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.
The paper is written especially for practitioners and applied scientists. It is based on two recent papers in stochastic analysis
The main part of the paper contains a simple existence result for Feller processes and a description of the general simulation scheme. These results will be followed by several examples.
The source code for the simulations can be found as supporting information (
Brownian motion and more general Lévy processes are used as models in many areas: For example in medicine to model the spreading of diseases
In hydrology stable processes are used as models for the movement of particles in contaminated ground water. It has been shown that state space dependent models provide a better fit to empirical data
In geology also stable processes are used in models for the temperature change. Based on ice-core data the temperatures in the last-glacial and Holocene periods are recorded. Statistical analysis showed that the temperature change in the last-glacial periods is stable with index 1.75 and in the Holocene periods it is Gaussian, i.e. stable with index 2 (see Fig. 4 in
For a technical example from physics note that the fluctuations of the ion saturation current measured by Langmuir probes in the edge plasma of the Uragan-3M stellarator-torsatron are alpha-stable and the alpha depends on the distance from the plasma boundary
Anomalous diffusive behavior has been observed in various physical systems and a standard model for this behavior are continuous time random walks (CTRWs)
In mathematical finance the idea of extending Lévy processes to Lévy-like Feller processes was first introduced in
Thus there is plenty of evidence that Feller processes can be used as suitable models for real-world phenomena.
Up to now general Feller processes were not very popular in applications. This might be due to the fact that the existence and construction of Feller processes is a major problem. There are many approaches: Using the Hille-Yosida theorem and Kolmogorov's construction
Our construction will not yield processes as general as the previous ones, but it will still provide a rich class of examples. In fact the presented method is just a simple consequence of a recent result on the solutions to certain stochastic differential equations
Furthermore each of the above mentioned methods also provides an approximation to the constructed Feller process. Most of them are not usable for simulations or work only under technical conditions. Also further general approximation schemes exist, for example the Markov chain approximation in
In contrast to these we derived in
Within different fields the terms
A stochastic process is a family of random variables indexed by a time parameter
Although this will not appear explicitly in the sequel, a process will always be equipped with its so-called natural filtration, which is a formal way of taking into account all the information related to the history of the process. Technically the filtration, which is an increasing family of sigma fields indexed by time, is important since a change from the natural filtration to another filtration might alter the properties of the process dramatically.
A
- independent increments: The random variables
- stationary increments:
- càdlàg paths: Almost every sample path is a right continuous function with left limits.
For equivalent definitions and a comprehensive mathematical treatment of Lévy processes and their properties see
Note that the term
A Lévy process
The most popular Lévy process is Brownian motion (
Classes of Lévy exponents depend, especially in modeling, on some parameters. Thus one can easily construct a family of Lévy processes by replacing these parameters by state space dependent functions. Another approach to construct families of Lévy processes is to introduce a state space dependent mixing of some given Lévy processes. We will elaborate this in the next section.
Given a family of Lévy processes
The chain starts at time 0 in
The first step is at time
The second step is at time
The third step is at time
etc. until time
This Markov chain is spatially inhomogeneous since the distribution of the next step always depends on the current position. If the chain converges (in distribution for
satisfy
and
for all
A Feller processes is sometimes also called: Lévy-type process, jump-diffusion, process generated by a pseudo-differential operator, process with a Lévy generator or process with a Lévy-type operator as generator. Note that in mathematical finance often the Cox-Ingersoll-Ross process
The generator
If the corresponding family of Lévy processes is a subset of a
In general, as mentioned in the previous section, the construction of a Feller process corresponding to a given family of Lévy processes is very complicated. It even might be impossible as the following examples show: Let
However, we will present in the next section a very simple method to construct Feller processes.
Suppose we know (for example based on an empirical study) that the process we want to model behaves like a Lévy process
If the sets
Let
Now set for
Note that the theorem extends to any finite number of Lévy processes
To avoid pathological cases one should assume
Thus if one knows how to simulate increments of the
Given a Feller process
Select a starting point
The first point of the sample path is
Draw a random number
The next point of the sample path is
Repeat 3. and 4. until
The simulated path is an approximation of the sample path of the Feller process, in the sense that for
The reader familiar with the Euler scheme for Brownian or Lévy-driven stochastic differential equations (SDEs) will note that the approximation looks like an Euler scheme for an SDE. In fact it is an Euler scheme, but the corresponding SDE does not have such a nice form as for example the Lévy-driven SDEs discussed in
We will now present some examples of Feller processes together with simulations of their sample paths. The first example will show the generality of the mixture approach, the remaining examples are special cases for which the existence has been shown by different techniques.
All simulations are done with the software package R
To show the range of possibilities which are covered by the mixture approach we construct a process which behaves like
For this we just define a family of Lévy processes by the family of characteristic exponents
These functions are Lipschitz continuous and thus a corresponding Feller process exists.
Around the origin it behaves like a Poisson process, for smaller values like Brownian motion and a for larger values like a Cauchy process.
A Lévy process
If we now define a function
A corresponding Feller process exists and is unique if the function
Each position is color coded by the corresponding value of the exponent. In the yellow regions it behaves almost like Brownian motion, in the red regions it behaves almost like a Cauchy process.
The characteristic function of a normal inverse Gaussian process
Note that the mean reversion is not introduced by using simply a drift which drags the process back to the origin. It is the choice of
The process features mean reversion to 0.
The characteristic function of a Meixner process
A family of Meixner processes which corresponds to a Feller process can be constructed by substituting the parameters
The process moves
The chosen functions satisfy the existence conditions from above. Furthermore the function
Using the presented mixture approach one can easily construct Feller models based on given Lévy models. In these cases the existence of the process is granted.
Furthermore the presented approximation is a very intuitive way to generate the sample path of a Feller processes. Obviously the method requires that one can simulate the increments of the corresponding Lévy processes. But for Lévy processes used in applications, especially together with Monte Carlo techniques, this poses no new restriction.
Thus all necessary tools are available to use Feller processes as models for a wide range of applications.
The simulations where done in R
Source code of the figures.
(TXT)