Recovering Brownian and jump parts from high-frequency observations of a L\'evy process

We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a L\'evy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a carefully chosen threshold, in which case both methods yield the same rate of convergence. This rate depends on the small-jump activity and is given in terms of the Blumenthal-Getoor index. Finally, we discuss possible extensions, including the multidimensional case, and provide numerical illustrations

Implementable coupling of L\'evy process and Brownian motion

We provide a simple algorithm for construction of Brownian paths approximating those of a L\'evy process on a finite time interval. It requires knowledge of the L\'evy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of increments. Applications to model risk and multilevel Monte Carlo are discussed in detail, and numerical examples are provided.Comment: 21 pages, 6 figure

Joint density of a stable process and its supremum: regularity and upper bounds

This article develops integration-by-parts formulae for the joint law of a stable process and its supremum at a fixed time. The argument rests on a multilevel representation for the joint law and uses ideas from Malliavin calculus, the theory of convex majorants for stable processes and the Chambers-Mallows-Stuck representation for stable laws. As our main application, we prove that an infinitely differentiable joint density exists and establish upper bounds (on the entire support of the joint law) for this density and its partial derivatives of any order.Comment: 33 pages, 1 figur

Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes

Exact Simulation of the Extrema of Stable Processes

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of L\'evy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).Comment: 26 pages, 3 figures, Julia implementation of the exact simulation algorithm is in the GitHub repository: https://github.com/jorgeignaciogc/StableSupremum.j

Physical activity and risk of Metabolic Syndrome in an urban Mexican cohort

Abstract Background In the Mexican population metabolic syndrome (MS) is highly prevalent. It is well documented that regular physical activity (PA) prevents coronary diseases, type 2 diabetes and MS. Most studies of PA have focused on moderate-vigorous leisure-time activity, because it involves higher energy expenditures, increase physical fitness, and decrease the risk of MS. However, for most people it is difficult to get a significant amount of PA from only moderately-vigorous leisure activity, so workplace activity may be an option for working populations, because, although may not be as vigorous in terms of cardio-respiratory efforts, it comprises a considerable proportion of the total daily activity with important energy expenditure. Since studies have also documented that different types and intensity of daily PA, including low-intensity, seem to confer important health benefits such as prevent MS, we sought to assess the impact of different amounts of leisure-time and workplace activities, including low-intensity level on MS prevention, in a sample of urban Mexican adults. Methods The study population consisted of 5118 employees and their relatives, aged 20 to 70 years, who were enrolled in the baseline evaluation of a cohort study. MS was assessed according to the criteria of the National Cholesterol Education Program, ATP III and physical activity with a validated self-administered questionnaire. Associations between physical activity and MS risk were assessed with multivariate logistic regression models. Results The prevalence of the components of MS in the study population were: high glucose levels 14.2%, high triglycerides 40.9%, high blood pressure 20.4%, greater than healthful waist circumference 43.2% and low-high density lipoprotein 76.9%. The prevalence of MS was 24.4%; 25.3% in men and 21.8% in women. MS risk was reduced among men (OR 0.72; 95%CI 0.57–0.95) and women (OR 0.78; 95%CI 0.64–0.94) who reported an amount of ≥30 minutes/day of leisure-time activity, and among women who reported an amount of ≥3 hours/day of workplace activity (OR 0.75; 95%CI 0.59–0.96). Conclusion Our results indicate that both leisure-time and workplace activity at different intensity levels, including low-intensity significantly reduce the risk of MS. This finding highlights the need for more recommendations regarding the specific amount and intensity of leisure-time and workplace activity needed to prevent MS

Convex minorants and the simulation of the extrema of Lévy processes

In this thesis we will establish the stick-breaking representation of the convex minorant and the extrema of an arbitrary Levy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of Levy processes. We then use the stick-breaking representation to create geometrically convergent simulation algorithm for the extrema of a Levy process whose increments can be sampled. For processes whose increments cannot be sampled we develop a multilevel Monte Carlo algorithm using the stick-breaking representation. In all cases, the algorithms present in this thesis outperform the existing algorithms in the literature