The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach

In this paper we study the effect that mean-reverting components in the arithmetic dynamics of electricity spot price have on the price of a call option on a swap. Our model allows for seasonal effects, spikes, and negative values of the price of electricity. We show that for sufficiently large delivery periods of the swap contract, the error that one makes by neglecting some of the mean-reverting processes affecting the spot price evolution converges to zero. The decay rate is explicitly calculated. This is achieved by exploiting the additive structure of the electricity price process in order to determine an explicit closed-form formula for the price of the call on a swap. The theoretical analysis is then illustrated via a numerical example.

A question for iterated Galois groups in arithmetic dynamics

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

Zsigmondy theorem for arithmetic dynamics induced by a drinfeld module

Let [Formula: see text] be a Drinfeld [Formula: see text]-module defined over a global function field [Formula: see text] Let [Formula: see text] be a non-torsion point of [Formula: see text] with infinite [Formula: see text]-orbit. For each [Formula: see text] write the ideal [Formula: see text] as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence [Formula: see text] i.e. for all but finitely many [Formula: see text] there exists a prime ideal [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]

On the arithmetic dynamics of monomial maps

We prove several results for the arithmetic dynamics of monomial maps, including Silverman’s conjectures on height growth, dynamical Mordell–Lang conjecture, and dynamical Manin–Mumford conjecture. These results were originally known for monomial maps on algebraic tori. We extend them to arbitrary toric varieties.

Bounded Height in Families of Dynamical Systems

Let $a,b\in\overline{\mathbb{Q}}$ be such that exactly one of $a$ and $b$ is an algebraic integer, and let $f_t(z):=z^2+t$ be a family of polynomials parameterized by $t\in\overline{\mathbb{Q}}$. We prove that the set of all $t\in\overline{\mathbb{Q}}$ for which there exist $m,n\geq 0$ such that $f_t^m(a)=f_t^n(b)$ has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.

Discrete resonances

The concept of resonance has been instrumental to the study of Hamiltonian systems with divided phase space. One can also define such systems over discrete spaces, which have a finite or countable number of points, but in this new setting the notion of resonance must be re-considered from scratch. I review some recent developments in the area of arithmetic dynamics which outline some salient features of linear and nonlinear stable (elliptic) orbits over a discrete space, and also underline the difficulties that emerge in their analysis.