Discretionary stopping of stochastic differential equations with generalised drift

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of vari- ational inequalities. In particular, we prove that the problem’s value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem’s data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take sev- eral qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case shows that the so-called “principle of smooth fit” does not hold in general for optimal stopping problems involving solutions to SDEs with generalised drift

The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions

Testing for fractional cointegration: the relationship between government popularity and economic performance in the UK

Author's draft (March 2003) of chapter published in New Trends in Macroeconomics, Claude Diebolt, Catherine Kyrtsou, editors, Springer Verlag (2005

The stochastic elementary formula method and approximate travelling waves for semi-linear reaction diffusion equations

In this thesis we consider approximate travelling wave solutions for stochastic and generalised KPP equations and systems by using the stochastic elementary formula method of Elworthy and Truman. We begin with the semi-classical analysis for generalised KPP equations. With a so-called "late caustic" assumption we prove that the global wave front is given by the Hamilton Jacobi function. We prove a Huygens principle on complete Riemannian manifolds without cut locus, with some bounds on their volume elements, in particular Cartan-Hadamard manifolds. Based on the semiclassical analysis we then consider the propagation of approximate travelling waves for stochastic generalised KPP equations. Three regimes of perturbation are considered: weak, mild, and strong. We show that weak perturbations have little effect on the wave like solutions of the unperturbed equations while strong perturbations essentially destroy the wave and force the solutions to decay rapidly. In the more difficult mild case we show the existence of a 'wave front', in front of which the solution is close to zero (of order exp(-c1μ-2) as μ~0 for c1 random) and behind which it has at least order exp(-c2μ-1) for some random c2 depending on the increment of the noise. For an alternative stochastic equation we classify the effect of the noise by the Lyapunov exponent of a corresponding stochastic ODE. Finally we study the asymptotic behaviour of reaction-diffusion systems with a small parameter by using the n-dimensional Feynman-Kac formula and Freidlin's large deviation theory. We obtain the travelling wave with nonlinear ergodic interactions and a special case with nonlinear reducible interactions

Online change-point detection for a transient change

We consider a popular online change-point problem of detecting a transient change in distributions of independent random variables. For this change-point problem, several change-point procedures are formulated and some advanced results for a particular procedure are surveyed. Some new approximations for the average run length to false alarm are offered and the power of these procedures for detecting a transient change in mean of a sequence of normal random variables is compared