Tightened Exponential Bounds for Discrete Time, Conditionally Symmetric Martingales with Bounded Jumps

This letter derives some new exponential bounds for discrete time, real valued, conditionally symmetric martingales with bounded jumps. The new bounds are extended to conditionally symmetric sub/ supermartingales, and they are compared to some existing bounds.Comment: To appear in the Statistics and Probability Letters, final version of the manuscript (dated May 1, 2013). Presented in part at the 2012 International Workshop on Applied Probability (IWAP), Jerusalem, Israel, June 201

Exponential inequalities for martingales with applications

The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pe\~{n}a, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of {de la Pe\~{n}a's} inequality to self-normalized deviations is also provided.Comment: 22 page

Sequential and Adaptive Inference Based on Martingale Concentration

Randomized experiments hold a well-deserved place at the top of the hierarchy of scientific evidence, and as such have received a great deal of attention from the statistical research community. In the simplest setting, a fixed group of subjects is available to the experimenter, who assigns one of two treatments to each subject via randomization, then observes corresponding outcomes. The goal is to draw inference about the effect of the experimental treatment on the observed outcome.Classical, frequentist statistical inference provides a powerful set of tools for this fixed-sample setting. We begin with an observed sample of some deterministic size and seek procedures which yield valid hypothesis tests, p-values, and confidence intervals---for example, a t-test of the null hypothesis that the experimental treatment has no effect, on average, or a corresponding confidence interval for the average treatment effect. The fixed-sample paradigm demands that we plan the experiment ahead of time, including the size of the experimental sample and the exact hypotheses to be tested, and that we adhere rigidly to this plan.In contrast, modern data analysis demands adaptivity. In particular, often the sample we choose to analyze is itself selected on the basis of observed data. For example, in an online A/B test, we may observe an ongoing stream of visitors enrolled into an experiment, so that the experimental sample is growing over time. The final experimental sample will include all of the visitors observed up to the time we decide to stop the experiment. The decision to stop could be made adaptively, by monitoring observed results and stopping early if a strong effect is observed, later if not. This is the realm of sequential, as opposed to fixed-sample, analysis.There are many other kinds of adaptivity that arise in practice. A second example is in the analysis of nonrandomized, or observational, studies of causal effects. In testing for statistical evidence of an effect, we may choose to focus on a subpopulation which we believe to be highly affected by the treatment of interest. For example, in studying the effect of fish consumption on mercury levels in the blood, we may focus on individuals whose diets are especially high in fish. Classical statistics requires that we define precisely which diets will be classified as "especially high in fish" before we analyze outcomes, but experimenters may prefer for this choice to be guided by the observed outcomes themselves.In both of the above examples---the sequential stopping of a randomized experiment and the adaptive choice of subgroup in an observational study---the use of fixed-sample methods, which do not account for adaptivity, will lead to violations of statistical guarantees such as false positive control. These violations are commonly included under the label "p-hacking" and have received much blame for the lack of reproducibility in various fields of scientific research. Fortunately, alternative statistical methods are available, methods that explicitly account for adaptivity to yield robust inference while placing fewer restrictions on the researcher. Such methods are the ultimate aim of the present work.This thesis develops a framework for constructing sequential and adaptive statistical procedures by taking advantage of the time-uniform concentration properties of certain martingales. Chapter 1 begins by laying out a mathematical framework for the derivation of time-uniform concentration inequalities for various classes of martingales. This framework unifies and strengthens a plethora of results from the exponential concentration literature and provides a toolbox for developing sequential and adaptive statistical procedures. The remaining three chapters develop such procedures.Chapter 2 builds upon the techniques of Chapter 1 to develop uniform concentration bounds which are somewhat more analytically and computationally complex but are much more useful for statistical applications. We frame these methods in terms of confidence sequences, that is, sequences of confidence intervals that are uniformly valid over an unbounded time horizon. One of the key results of this work is an empirical-Bernstein confidence sequence which provides a time-uniform, nonparametric, and non-asymptotic analogue of the t-test applicable to any distribution with bounded support. We explore applications to sequential estimation of average treatment effects in a randomized experiment, our first example above, as well as sequential estimation of a covariance matrix.Chapter 3 applies ideas from Chapters 1 and 2 to develop methods for the two related problems of estimating quantiles and estimating the entire cumulative distribution function, based on i.i.d. samples. We present confidence sequences for these estimands which are valid uniformly over time for any distribution, and we explore applications to A/B testing and best-arm identification when objectives are based on quantiles rather than means. Finally, Chapter 4 explores an application of uniform martingale concentration to the second example given above, the adaptive choice of subgroup within the analysis of an observational study. We introduce Rosenbaum's sensitivity analysis framework for observational studies, and show how our procedure yields qualitative improvements over existing methods within this framework.The martingale-based inferential methods we explore in this work trace their origins to Abraham Wald's work on the sequential probability ratio test during the 1940s, as well as to pioneering extensions developed in the late 1960s and early 1970s by Herbert Robbins, Donald Darling, David Siegmund, and Tze Leung Lai, not to mention many others. However, despite the decades of relevant literature, we believe most of the potential of the core ideas has yet to be realized. The key to unlocking this potential, we hope, is a fuller understanding of the nonparametric applicability of these methods, a detailed study of their implementation and tuning in practice, and an exploration of their utility beyond the sequential setting. While we propose several procedures that have immediate practical utility, we hope the larger contribution of the work will be as a first step towards a deeper appreciation of the power of martingale-based methods for adaptive inference, and ultimately to the development of a new class of statistical procedures which permit the kinds of adaptivity contemporary data analysts desire

Pathwise functional calculus and applications to continuous-time finance

This thesis develops a mathematical framework for the analysis of continuous- time trading strategies which, in contrast to the classical setting of continuous-time finance, does not rely on stochastic integrals or other probabilistic notions. Using the recently developed `non-anticipative functional calculus', we first develop a pathwise definition of the gain process for a large class of continuous-time trading strategies which include the important class of delta-hedging strategies, as well as a pathwise definition of the self-financing condition. Using these concepts, we propose a framework for analyzing the performance and robustness of delta-hedging strategies for path-dependent derivatives across a given set of scenarios. Our setting allows for general path-dependent payoffs and does not require any probabilistic assumption on the dynamics of the underlying asset, thereby extending previous results on robustness of hedging strategies in the setting of diffusion models. We obtain a pathwise formula for the hedging error for a general path-dependent derivative and provide sufficient conditions ensuring the robustness of the delta hedge. We show in particular that robust hedges may be obtained in a large class of continuous exponential martingale models under a vertical convexity condition on the payoffs functional. Under the same conditions, we show that discontinuities in the underlying asset always deteriorate the hedging performance. These results are applied to the case of Asian options and barrier options. The last chapter, independent of the rest of the thesis, proposes a novel method, jointly developed with Andrea Pascucci and Stefano Pagliarani, for analytical approximations in local volatility models with L\ue9vy jumps. The main result is an expansion of the characteristic function in a local L\ue9vy model, which is worked out in the Fourier space by considering the adjoint formulation of the pricing problem. Combined with standard Fourier methods, our result provides effcient and accurate pricing formulae. In the case of Gaussian jumps, we also derive an explicit approximation of the transition density of the underlying process by a heat kernel expansion; the approximation is obtained in two ways: using PIDE techniques and working in the Fourier space. Numerical tests confirm the effectiveness of the method

Scalable Performance Analysis of Massively Parallel Stochastic Systems

The accurate performance analysis of large-scale computer and communication systems is directly inhibited by an exponential growth in the state-space of the underlying Markovian performance model. This is particularly true when considering massively-parallel architectures such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative performance measures such as passage-time distributions from performance models of these systems is critical for providers of these services. Indeed, without such an ability, they remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have any confidence of honouring. Additionally, this must be possible in a short enough period of time to allow many different parameter combinations in a complex system to be tested. If we can achieve this rapid performance analysis goal, it will enable service providers and engineers to determine the cost-optimal behaviour which satisfies the SLAs. In this thesis, we develop a scalable performance analysis framework for the grouped PEPA stochastic process algebra. Our approach is based on the approximation of key model quantities such as means and variances by tractable systems of ordinary differential equations (ODEs). Crucially, the size of these systems of ODEs is independent of the number of interacting entities within the model, making these analysis techniques extremely scalable. The reliability of our approach is directly supported by convergence results and, in some cases, explicit error bounds. We focus on extracting passage-time measures from performance models since these are very commonly the language in which a service level agreement is phrased. We design scalable analysis techniques which can handle passages defined both in terms of entire component populations as well as individual or tagged members of a large population. A precise and straightforward specification of a passage-time service level agreement is as important to the performance engineering process as its evaluation. This is especially true of large and complex models of industrial-scale systems. To address this, we introduce the unified stochastic probe framework. Unified stochastic probes are used to generate a model augmentation which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis, we deploy these probes to define many detailed and derived performance measures that can be automatically and directly analysed using rapid ODE techniques. In this way, we tackle applicable problems at many levels of the performance engineering process: from specification and model representation to efficient and scalable analysis

Applied financial econometric analysis: The dynamics of swap spreads and the estimation of volatility.

This Thesis contains an examination of the time-series properties of swap spreads, their relation with credit spreads and an estimation of the risk premium embedded in the swap spread curve. Chapter 2 introduces the main institutional aspects of swap markets, and studies the time-series properties of swap spreads. These are shown to be non-stationary and display a time-varying conditional volatility. Chapter 3 provides evidence of cointegration between corporate bond spreads and swap spreads. We estimate an error-correction model, including additional variables such as the level and slope of the yield curve, taking into account the exogenous structural break due to the crisis of August 1998. We find evidence that the relation between swap and credit spreads arises from the swap cash flows being indexed to Libor rates. Chapter 4 studies the risk premium in the term structure of the swap spreads, obtaining evidence that it is time-varying. The slope of the swap spread curve is shown to predict the changes in swap spreads. These results are relevant for the study of the risk premium in credit markets, and extend the existing literature on riskless Treasury securities. Chapter 6 develops the asymptotic properties of the quadratic variation estimator of the volatility of a continuous time diffusion process. We explore the case in which the number of observations tends to infinity, while the time between them remains fixed. For the case of a geometric Brownian motion, we show that the estimator is asymptotically biased, but the bias is a random variable that converges. We study the behaviour of this random variable via a simulation study, that shows that it typically has a "small" effect. We conclude by exploring some practical applications related the specification of the volatility for financial time series