On Completely Mixed Stochastic Games

In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, we assume the transition probability of the undiscounted stochastic game is controlled by one player and all the optimal strategies of the game are strictly positive. Under all the above assumptions, we show that the $\beta$-discounted stochastic games with the same payoff matrices and $\beta$ sufficiently close to 1 are also completely mixed. We give a counterexample to show that the converse of the above result in not true. We also show that, if we have non-zero value in some state for the undiscounted stochastic game then for $\beta$ sufficiently close to 1 the $\beta$-discounted stochastic game also possess nonzero value in the same state

Racial Indianization An analysis of civic discourses of racism denial

This dissertation seeks to understand the civic discourses of racism denial that represent the views of the government officials, Dalits, academicians in the mainstream media, and how the logics of the discourses might potentially re-constitute or re-define the Indian nation. Specifically, this project aims to understand the changing dynamics of the constitution of the Indian nation vis- -vis the internal Other through the discourses about race, caste and racism denial in the civic mainstream domain. This project identified two discourses of denial of race and racism in the media representation of the government officials, Dalits, and academicians responses to the Dalit accusation that casteism is racism. The dominant discourse Caste is Not Race denied racism directly. The other discourse, Caste is Race only apparently supported the charge that casteism is racism, albeit with deep ambivalence, and circumvented the racism charge and shifted the focus to broader caste issues. Based on the analyses, I argue that the two discourses of denial and apparent support of the Dalits interacted with each other in the same discursive domain, and reworked, reconstituted and reified the extant cultural logics of race, caste and nation to deny racism in the postcolonial Indian context

Quadratic variation and quadratic roughness

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the quadratic roughness of a path along a partition sequence and show that, for Holder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely. Using these results we derive a formulation of Foellmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence

Level crossings of fractional Brownian motion

Since the classical work of L\'evy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or martingale settings, this work presents a highly anticipated extension to fractional Brownian motion -- a prominent non-Markovian and non-martingale process. Our result is viewed as a fractional analogue of Chacon et al. (1981). Consequently, it provides a global path-by-path construction of fractional Brownian local time. Due to the absence of conventional probabilistic tools in the fractional setting, our approach utilizes completely different argument with a flavor of the subadditive ergodic theorem, combined with the shifted stochastic sewing lemma recently obtained in Matsuda and Perkowski (22, arXiv:2206.01686). Furthermore, we prove an almost-sure convergence of the (1/H)-th variation of fractional Brownian motion with the Hurst parameter H, along random partitions defined by level crossings, called Lebesgue partitions. This result raises an interesting conjecture on the limit, which seems to capture non-Markovian nature of fractional Brownian motion.Comment: 37 pages, 6 figures. Simulation python code for Figure 3 can be found in the sourc

Determination of charge spread, position resolution, energy resolution and gain uniformity of Gas Electron Multipliers (GEM)

Gas electron multipliers (GEM) detectors are gaseous detectors widely used for tracking and imaging applications due to their good position resolution, high efficiency at high irradiation rates, among other factors. In the present work, position resolution, charge spread, energy resolution and gain uniformity have been investigated experimentally for single and double GEM geometries using an Fe-55 source. The position resolution measurements have been performed by a novel method, using a high precision instrument for source movement and is found to be highly successful. The result shows that the double GEM can resolve positions with sigma values up to 36.8 micron and 54.6 micron in x and y directions, respectively. To validate the experimental results, a Garfield simulation work has been carried out on charge spread

Roughness properties of paths and signals

Functions and processes with irregular behaviour in time are ubiquitous in physics, engineering, and finance and have been the focus of various pathwise theories of integration in stochastic analysis, in which the degree of 'roughness' of the function plays an important role. This thesis focuses on various concepts of 'roughness' for continuous functions and processes and their interplay with pathwise integration. We first explore these issues using the concept of pathwise quadratic variation, then expand results to the more general setting of p-th order variation. The first chapter discusses some motivations and background for the questions explored in the thesis and provides an overview of the results. In the second chapter, we study quadratic variation along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We introduce a property which we call quadratic roughness, and show that for H ̈older-continuous paths satisfying this roughness condition, the quadratic variation along ‘balanced’ partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion satisfy this quadratic roughness property almost-surely along partitions with fine enough mesh. Using these results we derive a formulation of the pathwise F ̈ollmer-Itˆo calculus which is invariant with respect to the partition sequences. Furthermore, we provide an invariance result for local time under quadratic roughness. In the third chapter, instead of balanced partition sequences (which is a key condition in Chapter 2) we consider (finitely) refining partition sequences, without any bound on mesh size. We construct a generalized Haar basis along any such finite refining sequence of partitions. We provide a closed-form representation of quadratic variation in terms of Faber-Schauder coefficients along this basis. Further, we construct a class of continuous processes with linear and prescribed quadratic variations along any given finitely refining partition sequence. We provide an example of a rough class of continuous processes with invariant quadratic variations along finitely refining sequences of partitions. Brownian motion belongs to this ‘rough’ class, but we also give examples of processes with 1/2 -H ̈older continuity in this class. Finally, we extend these constructions to higher dimensions. In the fourth chapter of the thesis, we consider a more general concept of roughness based on p-th variation and the associated notions of variation and roughness index of a continuous function. We define the normalized p-th variation of a path and use it to introduce a pathwise estimator to estimate the order of roughness of a signal. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and Takagi-Landsberg functions. In the final chapter we use our ‘roughness’ estimator (discussed in Chapter 4) to investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H < 0.5 for the modelling of volatility of financial assets, using a non-parametric, model-free approach. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5, which suggests that the origin of the roughness observed in realized volatility time-series lies in the estimation error rather than the volatility process itself. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the value of H, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index ˆH < 0.5 but this is not necessarily indicative of a similar rough behaviour of the spot volatility process which may have H ≥ 1/2

Rough volatility: fact or artefact?

We investigate the statistical evidence for the use of `rough' fractional processes with Hurst exponent $H< 0.5$ for the modeling of volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized $p$-th variation along a sequence of partitions. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than $0.5$. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits `rough' behaviour with an apparent Hurst index $\hat{H}<0.5$. These results suggest that the origin of the roughness observed in realized volatility time-series lies in the microstructure noise rather than the volatility process itself

Quadratic variation along refining partitions: Constructions and examples

We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose quadratic variation along a partition sequence is invariant under coarsening. This class is shown to include typical sample paths of Brownian motion, but also paths which are 1/2 -Hölder continuous. Finally, we show how to extend these constructions to higher dimensions

Quadratic variation and quadratic roughness

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the quadratic roughness of a path along a partition sequence and show that, for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely. Using these results we derive a formulation of Föllmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence