Interacting particle systems in varying environment, stochastic domination in statistical mechanics and optimal pairs trading in finance

Doktorsavhandling, 2010

In this thesis we first consider the contact process in a randomly evolving environment, introduced by Erik Broman. This process is a generalization of the contact process where the recovery rate can vary between two values. The rate which it chooses is determined by a background process, which evolves independently at different sites. We prove that survival of the process is independent of how we start the background process, that finite and infinite survival are equivalent and finally that the process dies out at criticality. Second, we consider spin systems on $\Z$ whose rates are again determined by a background process, which is more general than that considered above. We prove that, if the background process has a unique stationary distribution and if the rates satisfy a certain positivity condition, then there are at most two extremal stationary distributions. Third, we discuss various aspects concerning stochastic domination for the Ising and fuzzy Potts models. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\RR$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the parameters $J_1$, $J_2$, $h_1$ and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on $\Zd$ the fuzzy Potts measures dominate the same set of product measures while on $\Td$, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. Finally, we study the problem of optimally closing a pair trading strategy when the difference of the underlying assets is assumed to be an Ornstein-Uhlenbeck type process driven by a jump-diffusion process. We prove a verification theorem and analyze a numerical method for the associated free boundary problem. We prove rigorous error estimates, which are used to draw some conclusions from numerical simulations.