A Reflected Moving Boundary Problem Driven by Space-Time White Noise

We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and show that the solution is global when the boundary speed is bounded. We also derive the expected H\"older continuity for the process and hence for the derivative of the moving boundary. Both the case when the spatial domains are given by fixed finite distances from the shared boundary, and when the spatial domains are the semi-infinite intervals on either side of the shared boundary are considered. In the second case, our results require us to further develop the known theory for reflected SPDEs on infinite spatial domains by extending the uniqueness theory and establishing the local H\"older continuity of the solutions

Optimal execution with rough path signatures

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimisation problem, we are able to calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where exact solutions are not known. In the latter case, we obtain favourable comparisons with standard execution strategies

Stefan Problems for Reflected SPDEs Driven by Space-Time White Noise

We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space-time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary

Moving boundary problems for reflected SPDEs

The work in this thesis is based on the study of reflected SPDE (stochastic partial differential equation) moving boundary problems. These are systems consisting of two competing profiles which each evolve according to a reflected stochastic heat equation in one spatial dimension, and share a common boundary point. The reflection here minimally pushes the profiles upwards in order to maintain positivity. The evolution of the shared boundary depends on the state of the profiles, and so is coupled with the dynamics of the two competing sides. Such equations are suited to modelling competition between two types. An example of this is the limit order book. We can think of the competing profiles as being order volumes to buy/sell an asset at different prices, with positivity of these ensured by the reflection terms. The shared boundary then represents the current midprice. Key results include scaling laws, which show how we can connect particle systems to particular reflected SPDE moving boundary problems, and existence and uniqueness results for different classes of reflected SPDE moving boundary models. In the latter case, we first study the situation where the boundary mechanism is driven by the competing profiles considered as continuous functions. Following this we impose stronger assumptions and examine the case where the boundary mechanism is driven by the derivatives of the competing profiles at the shared interface, as in the classical Stefan problem. We also study other properties of our systems, including their Hölder regularity and the existence of invariant measures.</p

Moving boundary problems for reflected SPDEs

The work in this thesis is based on the study of reflected SPDE (stochastic partial differential equation) moving boundary problems. These are systems consisting of two competing profiles which each evolve according to a reflected stochastic heat equation in one spatial dimension, and share a common boundary point. The reflection here minimally pushes the profiles upwards in order to maintain positivity. The evolution of the shared boundary depends on the state of the profiles, and so is coupled with the dynamics of the two competing sides. Such equations are suited to modelling competition between two types. An example of this is the limit order book. We can think of the competing profiles as being order volumes to buy/sell an asset at different prices, with positivity of these ensured by the reflection terms. The shared boundary then represents the current midprice. Key results include scaling laws, which show how we can connect particle systems to particular reflected SPDE moving boundary problems, and existence and uniqueness results for different classes of reflected SPDE moving boundary models. In the latter case, we first study the situation where the boundary mechanism is driven by the competing profiles considered as continuous functions. Following this we impose stronger assumptions and examine the case where the boundary mechanism is driven by the derivatives of the competing profiles at the shared interface, as in the classical Stefan problem. We also study other properties of our systems, including their Hölder regularity and the existence of invariant measures.</p