Measure-valued affine and polynomial diffusions and applications to energy modeling

Venue: The event will take in presence only for the Phd student and part of the commission and online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)
Time: 9.30

Francesco Guida - PhD in Mathematics, University of Trento

Abstract:
The central theme of this thesis is the study of stochastic processes in the infinite dimensional setup of (non-negative) measures. We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on R^m_+, in cases where their domain is large enough. In general, the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson-Watanabe superprocesses. The polynomial framework is especially attractive from a mathematical finance point of view. Indeed, it allows to transfer some of the most famous finite dimensional models, such as the Black-Scholes one, to an infinite dimensional measure-valued setting. 
We outline the applicability of our approach to energy markets term structure modeling by introducing a framework allowing to employ (non-negative) measure-valued processes to consider electricity and gas futures. Interpreting the process' spatial structure as time to maturity, we show how the Heath-Jarrow-Morton (HJM) approach can be translated to such framework, thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an analogue to the HJM-drift condition, then considering existence of (non-negative) measure-valued diffusions satisfying this condition in a Markovian setting. To analyze mathematically convenient classes of models, we also consider measure-valued polynomial and affine diffusions allowing for tractable pricing procedures via the moment formula and Fourier approaches.

Supervisors : Luca Di Persio - Christa Cuchiero