In memory of Larry Shepp: An editorial

Abstract

This special issue of High Frequency is dedicated to the memory of Professor Larry Shepp, who passed on April 23, 2013. We are humbled to honor his immense scientific impact through this issue. The authors who have written for this special issue contributed their work from a call which we circulated at a conference organized in Prof. Shepp's honor, at Rice University, in June 2018. The conference spanned a number of topics in the areas of applied probability, as influenced by Shepp, including optimal stopping, stochastic control, financial mathematics, Gaussian, and Lévy processes.

While Prof. Shepp himself did not explicitly describe any of his work as part of high-frequency (HF) data analysis, any methodology which follows the stopping and control rules which he pioneered, would necessarily run into the use of high-frequency data to ensure a close match with continuous-time modeling, and avoid overly sub-optimal approximations. In this issue, we see examples of this interpretation through four thought-provoking papers, some where the data analysis is implicit in the continuous-time framework of stochastic processes, some which invoke HF data analysis explicitly, all which honor the influence of Larry Shepp by proposing problems in optimal stopping, mathematical finance, and the theory of stochastic processes.

This issue's first paper, by Pavel Gapeev, revisits Larry Shepp's famous Russian option (see below) in a situation where the continuation region for optimal stopping has at least two disconnected components, specifically the dual Russian option pricing problem he proposed with Albert Shiryaev in 1993, now with a positive exponential discount rate. The implementation of the corresponding stopping scheme would easily miss this feature of a so-called double-continuation region unless decisions are made with sufficiently high frequency. This issue's paper by Paolo Guasoni and Ali Sanjari also treats an optimal trading strategy problem, but one of a very different nature: the liquidation of a large position in the financial market. Assuming high-frequency data and trading is available, the liquidator and the other market actors engage in competing objectives, which can lead to severely reduced liquidity in the late stages of liquidation. The paper explores this and other sources of nonlinear price impact, in the context of stochastic control and utility maximization. Switching gears, this issue's paper by Guodong Pang and Murad Taqqu honors the themes of Gaussian and Poisson stochastic processes, including the study of path regularity features, another area influenced by Larry Shepp. They study how the assumption of a power law in the shot distribution of a Poisson-based shot noise process leads to self-similar limiting Gaussian processes which extend fractional Brownian motion by dissociating the high-frequency regularity behavior from the self-similarity property. Finally, keeping within the study of stochastic processes where the continuous-time framework implies access to ultra-high-frequency data, the last article in this issue, by Paavo Salminen and Ernesto Mordecki, echoes the influence of Shepp's Russian option, while studying an optimal stopping problem with discounting: for a Brownian motion with a drift discontinuity, a threshold exists beyond which the stopping region has two disconnected components. It is easy to see how all these topics are influenced by the work of Larry Shepp.

We take this opportunity to describe Prof. Shepp's most notable areas of contribution, placing this issue and its articles in the context of his broader scientific legacy.

Within optimal stopping, Shepp is best known for his breakthrough work on foundational Chow-Robbins problems of optimal stopping (Shepp, 1969). Shepp invented the “Russian option” (Shepp & Shiryaev, 1993a) which is now widely used on Wall Street for the lookback hedge option guaranteeing the discounted maximum price to the buyer. A dual short-sell version of the Russian option would subsequently be developed in (Shepp & Shiryaev, 1996). Shepp also made fundamental contributions to the study of optimal stopping for Bessel processes and maximal inequalities for Bessel processes (Dubins, Shepp, & Shiryaev, 1994).

Within stochastic control, Shepp is widely acclaimed for his pioneering work on solvable singular stochastic control problems (Beneš, Shepp, & Witsenhausen, 1980) as well as for a series of seminal papers on optimal corporate planning (Radner & Shepp, 1996; Shepp & Shiryaev, 1993b) for purposes of profit-taking and for hiring/firing of company personnel. Shepp also made fundamental contributions to gambling theory; in (Chen, Shepp, Yao, & Zhang, 2005), Shepp reexamines the theorem of Dubins and Savage stating that the gambler wishing to maximize the probability of obtaining a fortune 1 (starting with fortune 0 < f < 1) in any sub-fair casino cannot do better than to follow the modified bold-play strategy.

Within probability theory, Shepp is best known for his work on random covers, in particular for his paper which obtained the exact condition on a sequence of numbers, 1 > l₁, l₂, … such that if arcs of these lengths are centered uniformly at random on a circumference of unit length then every point is covered with probability 1 (Shepp, 1972). He is also well known for his pioneering work on connectedness of random graphs (Shepp, 1989), which provided impetus for future contributions by Durrett, Kesten, and others.

Last but not least, Shepp is undoubtedly most famous for his paper (Shepp & Logan, 1974), which built the mathematical foundation for the Nobel Prize winning invention of the Computational Tomography (CT) scanner by G. Hounsfield. Shepp was also widely recognized within radiology for his patents on CT scanning designs. He was an essential contributor to emission tomography; his work (Shepp & Vardi, 1982) has continued to be widely implemented by practitioners. While not initially described as an HF device at the time of its computational inception and medical implementation, the CT scanner is a prime example where mathematics helped solve a question of HF data analysis, in terms of both spatial and time resolutions, with great societal impact.