Quasi-sure essential supremum and applications to finance
A notion of essential supremum is developed when the uncertainty is measured by a family of non- dominated and non-compact probability measures. It provides new perspectives on super-replication and allows the Absence of Instantaneous Profit (AIP) to be characterized.
Computable method for pricing European options with general transaction costs.
In the literature, the only known result to evaluate super-hedging prices of a European claim is a dual characterization and this one is only valid for linear transaction costs. Here, we consider general transaction costs that may be not convex and using an approach based on random set theory, we show that, in discrete time, the super-hedging prices satisfy a dynamic programming principle so that it is possible to compute them backwardly.
Exploring Fixed-Accuracy Estimation of Population Gini Income Inequality Index Under Big Data: Practical Sequential Strategies with Significant Complexities
We will begin with sequential minimum risk point estimation (MRPE) theory in a normal distribution by briefly visiting (i) what has been customarily known and (ii) what is new in the field generating a sense of vigorous research as of now especially by young researchers. This part will share classical formulations and some of the customary first-order and second-order asymptotics as cost per unit observation goes down.
In the same vein,
Chattopadhyay, B. and De, S. K. (2016, 4:30, DOI 10.3390/econometrics4030030
as well as De, S. K. and Chattopadhyay, B. (2017, Sankhya, Series B 79: 247-277)
respectively developed elegant sequential fixed-width confidence interval (FWCI) and MRPE methodologies for GF , the celebrated Gini income inequality index in a population associated with an unknown distribution function F having its support on (0; ∞). We will mainly focus on the sequential MRPE strategies. We revisit such problems from the vantage point of big data science by proposing newly designed easyto-implement sequential estimation strategies with nearly minimal computational complexities and technical difficulties. The second part of this presentation will emphasize inference techniques introduced in
Mukhopadhyay, N. (2021, in Gini Inequality Index: Methods and Applications, Chapter 11, N. Mukhopadhyay and P. P. Sengupta, eds., pp. 217-241, CRC Press/Chapman & Hall, Boca Raton, ISBN: 978-0-367-68835-6).
We show that these new sequential estimation strategies have a wide range of appealing asymptotic properties including both first-order and second-order approximations. The proposed approaches under discussion will be flexible enough to embrace other non-standard inference problems in the future. A part of this has been borrowed from my joint recent research with Professor Yan Zhuang (Connecticut College, New London, Connecticut, USA) and with my PhD student, Mr. Boyi Zhang (Department of Statistics, University of Connecticut-Storrs, Connecticut, USA).
Model-based Adaptive Designs for Dose-ranging Studies
In this presentation, I will focus on adaptive designs in dose-ranging studies -- the exploratory phase of the drug development process designed and carried out to establish drug efficacy and dose response relationships. A failure to identify the correct dose or detect important dose- limiting toxicities, or a reliance on a surrogate endpoint that ultimately behaves differently than the endpoint required for a Phase III confirmatory trial, can lead to a failure in Phase III. In part based on these concerns, there has been increasing interest in, and utilization of, adaptive approaches for the design and analysis of Phase II trials. These designs directly address the goals of the exploratory phase trial with respect to identification of dose to carry forward in the confirmatory phase, estimation of likelihood of success in confirmatory trial, and efficient early stopping for efficacy or for futility. A critical component of these designs is the dose-response model for efficacy and/or safety endpoints that captures prior information about the form and location of the clinically important dose response relationship. The optimal experimental design framework, with available doses as design region and response variables following a mathematical model of dose-response relationship, provides enough structure to address the objectives of the dose-ranging studies. The focus is on choosing the dose levels and the patient allocation per dose in some optimal way to enhance the process of estimating the unknown parameters of the model. The challenge is that the optimal design for non-linear models depends on unknown parameters. The solution is adaptive design – running the experiment sequentially. Initial design is chosen, and preliminary parameter estimates are obtained. Then, the next stage doses are selected from the available range of doses that satisfy the efficacy and toxicity constraints and provide the maximal improvement of the design with respect to the selected criterion of optimality and current parameter estimates. The next available cohort of patients is allocated to these doses. The estimates of unknown parameters are refined given these additional observations. These design-estimation steps are repeated until an early stopping decision is achieved, or the maximum number of patients is enrolled. Adaptive designs employ frequent interim analyses of all accumulated data (and, possibly, external trial data) to determine whether pre-planned design modifications will be ‘triggered’. Interim analyses partition the trial into multiple stages, each trial stage’s characteristics (number of treatment arms, number of patients to be enrolled, their allocation between arms, stage duration, etc.) defined by the preceding interim analysis results. The ability to sequentially examine available data to determine whether trial modifications are necessary and, when indicated, implement pre-defined design changes gives adaptive design its strength and flexibility. I will also provide an overview of available adaptive designs appropriate for drug development at different level: trial, program, and portfolio, including platform, umbrella, and basket designs.