In probability theory, a Lévy process is a stochastic process (taking values in some Euclidean space) which has stationary independent increments. The purpose of this article is to explore how to define Lévy processes on an arbitrary first-order structure \({\mathfrak A}\). One needs to first figure out the correct notion of random variable. The approach taken in this paper is to take a nonstandard extension of \({\mathfrak A}\), namely \({\mathfrak A}^*\), and look at all measures obtained by restricting the Loeb measures of internal measures to the \(\sigma\)-algebra generated by the internal sets; call this set \({\mathcal M}\). Such measures provide a source of random variables with values in \({\mathfrak A}\), for one views such a measure as a generalization of an element of \({\mathfrak A}\), which corresponds to a deterministic random variable via Dirac measures.
For the purposes of constructing Lévy processes, a certain subclass of \({\mathcal M}\) is distiguished, namely the set \({\mathcal D}({\mathfrak A})\) of definable probabilities on \({\mathfrak A}\). (For example, if \({\mathfrak A}\) is stable, then \({\mathcal M}={\mathcal D}({\mathfrak A})\).) Assuming that a commutative semigroup is definable in the structure, one can define a convolution product \(\star\) on \({\mathcal D}({\mathfrak A})\) and a convolution exponential on \({\mathcal D}({\mathfrak A})\).
As in classical stochastic analysis, an element \(\mu\) of \({\mathcal D}({\mathfrak A})\) is called infinitely divisible if it has \(n^{\text{th}}\) roots (with respect to convolution) for every \(n\). For an interval \(I\) with endpoints (called 0 and 1) in a linearly ordered semigroup and an infinitely divisible \(\mu\), a Lévy process corresponding to \(\mu\) with respect to \(I\) is a mapping \(X:I\to{\mathcal D}({\mathfrak A})\) such that \(X(0)\) is the Dirac measure corresponding to \(0\), \(X(1)=\mu\), and \(X(s+t)=X(s)\star X(t)\) for all \(s,t\). The main question this paper examines is: when does an infinitely divisible element have a Lévy process corresponding to it with respect to intervals \(I\) of the form \([0,1]\), \([0,1]\cap\mathbb Q\), or any hyperfinite interval. It is shown that if \(\mu\) is a convolution exponential, then such Lévy processes exist. Furthermore, using an analogous construction, one can define random variables with values in \({\mathfrak A}^*\) and one can prove that Lévy processes exist for convolution exponentials in this larger structure.