A random journey through dynamics and finance: pullback attractors, price impact, nonlinear valuation and FX market.

Abstract

The main objective of this thesis is to explore several areas of Random Dynamical Systems and Mathematical Finance. We start by considering random dynamical systems with two different sources of noise, which we call common and intrinsic. We study the interplay between these two sources of randomness from a novel point of view, going beyond the usual statistical approach. We determine the stochastic Fokker-Planck equation describing the system and prove that such equation has a pullback attractor for almost all realizations of the common noise. On the mathematical finance side, we start by discussing consistency properties of jump-diffusion models with respect to inversion, with applications to the Foreign Exchange market. We first solve the constant jump size case, and then analyze the more involved case of the compound Poisson process. We determine a fairly general class of admissible densities for the jump size in the domestic measure. Then, we delve into the nonlinear valuation framework under credit risk, collateral and funding costs, generalizing the mathematical framework of \cite{brigo2019nonlinear} for what concerns in particular the filtrations and the default times. Finally, we propose a first theory of price impact in presence of an interest-rates term structure. We formulate an instantaneous and transient price impact model for zero-coupon bond, defining a cross price impact that is endogenous to the term structure. We extend this setup to coupon-bearing bonds, HJM framework and conclude by solving an optimal execution problem in interest rates market.