Tempered multistable processes
In real world phenomena, one always needs to use mathematical processes to study finance or economics models. Sometimes, we need the processes to be non-continuous, have finite expectations and moments, for example in studies carried out in stochastic optimal control area. An extension of multistable processes is tempered multistable processes. The intuitive idea of tempered multistable processes involves removing the big jumps from multistable processes and replacing them with reasonable small jumps instead. The tempered multistable processes are "locally stable", but they tend to behave like Brownian motion over long time scales. An advantage of such processes is that they possess moments of all orders.
We define the multistable stochastic integral of a indicator function on by specifying the finite-dimensional distributions of as a stochastic process on the space of functions
. Let and , , then for , and
is the characteristic function of a probability distribution on the random vector . We define the -multistable random measure by
The tempered multistable measure has independently scattered property.
We call tempered multistable Lévy motion. The tempered multistable Lévy motion has finite expectation and moments.
An equivalent way of defining the tempered multistable Lévy motion is series representation of tempered multistable Lévy Motion.
Let be a sequence of arrival times of a Poisson process with unit arrival time, be a sequence of i.i.d random variables with uniform distribution on , where is a fixed number. Let be a sequence of i.i.d random variables with uniform distribution on , be a sequence of i.i.d exponential random variables with parameter 1, and be a sequence of i.i.d random variables with distribution . Assume that the five sequences , , , , and are independent. We denote that .
Under the above assumption,
with is the series representation of tempered multistable Lévy motion.