Stochastic domination for the Ising and fuzzy Potts models

We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. 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 three parameters $J_1$, $J_2$, $h$ 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. For the Ising model, Liggett and Steif proved that on \Zd the plus measures dominate the same set of product measures while on \T^2 that statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure

Optimal closing of a pair trade with a model containing jumps

A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely applied investment strategy in the financial industry. Recently, Ekstr\"om, Lindberg and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In this paper we study the same problem, but the model is generalized to also include jumps. More precisely we assume that the above difference is an Ornstein-Uhlenbeck type process, driven by a L\'evy process of finite activity. 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.Comment: 17 pages, 4 figures

Interacting particle systems in a randomly evolving environment

This thesis concerns interacting particle systems in a randomly evolving environment. In the first paper, we consider the so called contact process in a randomly evolving environment (CPREE), 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. As for the contact process, we can similarly define a critical value in terms of survival for this process. We prove that this definition is independent of how we start the background process, that finite and infinite survival (meaning nontriviality of the upper invariant measure) are equivalent and finally that the process dies out at criticality. In the second paper, we consider spin systems on the integers (i.e. interacting particle systems on the integers in which each coordinate has only two possible values and only one coordinate changes in each transition) whose rates are determined by a background process, which is more general than in the first paper. We prove a generalization of a result by Liggett, that under certain conditions on the rates there are only two extremal invariant distributions

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

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

Interacting particle systems in a randomly evolving environment

This thesis concerns interacting particle systems in a randomly evolving environment. In the first paper, we consider the so called contact process in a randomly evolving environment (CPREE), 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. As for the contact process, we can similarly define a critical value in terms of survival for this process. We prove that this definition is independent of how we start the background process, that finite and infinite survival (meaning nontriviality of the upper invariant measure) are equivalent and finally that the process dies out at criticality. In the second paper, we consider spin systems on the integers (i.e. interacting particle systems on the integers in which each coordinate has only two possible values and only one coordinate changes in each transition) whose rates are determined by a background process, which is more general than in the first paper. We prove a generalization of a result by Liggett, that under certain conditions on the rates there are only two extremal invariant distributions

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

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