Some Minorants and Majorants of Random Walks and Lévy Processes

This thesis consists of four chapters, all relating to some sort of minorant or majorant of random walks or Lévy processes.In Chapter 1 we provide an overview of recent work on descriptions and properties of the convex minorant of random walks and Lévy processes as detailed in Chapter 2, [72] and [73]. This work rejuvenated the field of minorants, and led to the work in all the subsequent chapters. The results surveyed include point process descriptions of the convex minorant of random walks and Lévy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.Chapter 1 serves as a long introduction to Chapter 2, in which we offer a unified approach to the theory of concave majorants of random walks. The reasons for the switch from convex minorants to concave majorants are discussed in Section 1.1, but the results are all equivalent. This unified theory is arrived at by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant - the path transformation is different from the one discussed in Chapter 1, but this is necessary to deal with a more general case than the standard one as done in Section 2.6. The path transformation of Chapter 1, which is discussed in detail in Section 2.8, is more relevant to the limiting results for Lévy processes that are of interest in Chapter 1. Our results lead to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean (the more general case mentioned above), we investigate three nested compositions that naturally arise from our construction of the concave majorant.In Chapter 3, we study the Lipschitz minorant of a Lévy process. For α > 0, the α-Lipschitz minorant of a function f: R → R is the greatest function m : R → R such that m ≤ f and |m(s)-m(t)| ≤ α |s-t| for all s,t in R, should such a function exist. If X=(X_t){t in R} is a real-valued Lévy process that is not pure linear drift with slope ± α, then the sample paths of X have an α-Lipschitz minorant almost surely if and only if | E[X_1] | < α. Denoting the minorant by M, we investigate properties of the random closed set Z := { t in R : Mt = max( Xt, Xt-), which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator "made stationary" in a suitable sense. We give conditions for the contact set Z to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the Lévy measure of the associated subordinator. We study the limit of Z as α → ∞ and find for the so-called abrupt Lévy processes introduced by Vigon that this limit is the set of local infima of X. When X is a Brownian motion with drift β such that |β| < α, we calculate explicitly the densities of various random variables related to the minorant.Finally, in Chapter 4 we study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by Lévy noise, or equivalently when the initial potential is a two-sided Lévy process X. This shock structure turns out to give rise to a parabolic minorant of the Lévy process - see Section 4.2 for details. The main results are that when X is abrupt in the sense of Vigon or has bounded variation with limsup|h| → 0 h-2 X(h) = ∞, the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When X is abrupt the shock structure is discrete and when X is eroded there are no rarefaction intervals