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In the first six months of its deadly spread across the world, the Covid-19 incidence has exhibited interesting dichotomy between the rich and the poor countries. Surprisingly, the incidence and the Case Fatality Rate has been much higher in the richer countries compared with the poorer countries. However, the reasons behind this dichotomy have not been explained based on data or evidence, although some of the factors for the susceptibility of populations to SARS-CoV-2 infections have been proposed. We have taken into consideration all publicly available data and mined for the possible explanations in order to understand the reasons for this phenomenon. The data included many parameters including demography of nations, prevalence of communicable and non-communicable diseases, sanitation parameters etc. Results of our analyses suggest that demography, improved sanitation and hygiene, and higher incidence of autoimmune disorders as the most plausible factors to explain higher death rates in the richer countries Thus, the much debated “hygiene hypothesis” appears to lend credence to the Case Fatality Rate dichotomy between the rich and the poor countries.SignificanceThe current COVID-19 epidemic has emerged as one of the deadliest of all infectious diseases in recent times and has affected all nations, especially the developed ones. In such times it is imperative to understand the most significant factor contributing towards higher mortality. Our analysis shows a higher association of demography, sanitation & autoimmunity to COVID-19 mortality as compared to the developmental parameters such as the GDP and the HDI globally. The dependence of sanitation parameters as well as autoimmunity upon the mortality gives direct evidences in support of the lower deaths in nations whose population do not confer to higher standards of hygiene practices and have lower prevalence of autoimmune diseases. This study calls attention to immune training and strengthening through various therapeutic interventions across populations.

A standard global protocol for the treatment of COVID-19 patients is absent. Many clinicians have used the SOFA score to decide the use of treatments. This has created a problem to correctly interpret the results of clinical and epidemiological research around the globe. We show, with a practical example, what a critical role the asymptomatic patients of COVID-19 can play in determining the efficacy of hydroxychloroquine as a treatment. Our critique applies to any treatment for COVID-19.

Statistics evolved as a science in an era when the amount of data available was small and efforts were on to extract maximum information from them. Are the techniques developed during those times relevant anymore in the era of data science? We will illustrate using examples that several statistical concepts developed over the last 150 years are as relevant in this era as they were then

When a test is conducted in several sessions using distinct question papers, normalization of scores is required to have a fair assessment of the candidates. Several selection tests nowadays are conducted in multiple sessions (using multiple choice questions). In this article we discuss various normalization schemes used in India when an examination involving multiple choice questions is conducted across various sessions. We illustrate through simulation, that the percentile-based normalization scheme outperforms all the other schemes.

In this chapter, we will give definitions, set up notations that will be used in the rest of the book and give some basic results. While some proofs are included, several results are stated without proof. The proofs of these results can be found in standard books on stochastic processes.

We begin this chapter with the quadratic variation and Levy’s characterization of the Brownian motion. Later, we will outline the basic development of the Ito’s Integral w.r.t. Brownian motion. We also discuss existence and uniqueness of solutions to the classical stochastic differential equations driven by Brownian motion.

We have discussed predictable \(\sigma \)-field and seen the crucial role played by predictable integrands in the theory of stochastic integration. In our treatment of the integration, we have so far suppressed another role played by predictable processes. In the decomposition of semimartingales, Theorem 5.55, the process A with finite variation paths turns out to be a predictable process. Indeed, this identification played a major part in the development of the theory of stochastic integration.

In this chapter, we will consider stochastic differential equations as in Sect. 7.3 where the driving semimartingale need not be continuous.

In this chapter we will consider the question as to when do all martingales adapted to a filtration \(({\mathcal F}_\centerdot )\) admit a representation as a stochastic integral with respect to a given local martingale M. This result was proved by Ito’s when the underlying filtration is the filtration generated by a multidimensional Wiener process. Ito’s had proven the integral representation property for square integrable martingales and this was extended to all martingales by Clark.

In this chapter, we would give the continuous-time version of the Burkholder–Davis–Gundy inequality \(-p=1\) case. This is due to Davis. This plays an important role in answering various questions on the stochastic integral w.r.t. a martingale M—including condition on \(f\in {\mathbb L}(M)\) under which \(\int f\,d\, M\) is a local martingale. This naturally leads us to the notion of a sigma-martingale which we discuss. We will begin with a result on martingales obtained from process with a single jump.

In this chapter, we will discuss martingales indexed by integers (mostly positive integers) and obtain basic inequalities on martingales and other results which are the basis of most of the developments in later chapters on stochastic integration. We will begin with a discussion on conditional expectation and then on filtration—two notions central to martingales.

In Chap. 7, we saw that using random time change, any continuous semimartingale can be transformed into a amenable semimartingale, and then one can have a growth estimate on the stochastic integral similar to the one satisfied by integrals w.r.t. Brownian motion.

In Karandikar-Rao [11], the quadratic variation [M, M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using [M, M], avoiding using the predictable quadratic variation 〈M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new. We then discuss the vector stochastic integral ∫ 〈f, dY〉 where f is ℝ d valued predictable process, Y is ℝ d valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M1, … M d : If N n are martingales such that N → N t for every t and if ∃f n such that N = ∫ 〈f n , dM〉, then ∃f such that N = ∫ 〈f, dM〉. Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above. This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.

We give a construction of an explicit mapping $\Psi :\mathsf {D}([0,\infty ), \mathbb {R} )\rightarrow \mathsf {D}([0,\infty ), \mathbb {R} ),$ where $\mathsf {D}([0,\infty ), \mathbb {R} )$ denotes the class of real valued r.c.l.l. functions on $[0,\infty )$ such that for a locally square integrable martingale (M t ) with r.c.l.l. paths, Ψ(M.(ω)) = A.(ω) gives the quadratic variation process (written usually as [M, M] t ) of (M t ). We also show that this process (A t ) is the unique increasing process (B t ) such that $M^2_t-B_t$ is a local martingale, B 0 = 0 and $\mathbb {P}((\Delta B)_t=[(\Delta M)_t]^2, \;0

An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, ||| G ||| is not bigger than the geometric mean of |||A1|||, . . . , |||Am|||.

In this chapter, we present an approach to system identification based on viewing identification as a problem in statistical learning theory. Apparently, this approach was first mooted in [396]. The main motivation for initiating such a program is that traditionally system identification theory provide asymptotic results. In contrast, statistical learning theory is devoted to the derivation of finite time estimates. If system identification is to be combined with robust control theory to develop a sound theory of indirect adaptive control, it is essential to have finite time estimates of the sort provided by statistical learning theory. As an illustration of the approach, a result is derived showing that in the case of systems with fading memory, it is possible to combine standard results in statistical learning theory (suitably modified to the present situation) with some fading memory arguments to obtain finite time estimates of the desired kind. It is also shown that the time series generated by a large class of BIBO stable nonlinear systems has a property known as β-mixing. As a result, earlier results of [394] can be applied to many more situations than shown in that paper.

Markov Chain Monte Carlo (MCMC) is a popular method used to generate samples from arbitrary distributions, which may be specified indirectly. In this article, we give an introduction to this method along with some examples.

I analyse various schemes of negative marking for tests consisting of multiple-choice questions and propose a scheme that reduces the impact of random guessing. I also propose an alternate style of multiple-choice questions, where each question may have several correct answers and the candidate is required to tick all correct answers in order to get credit.

We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.

It has been shown previously by Nobel and Dembo (Stat. Probab. Lett. 17 (1993) 169) that, if a family of functions F has the property that empirical means based on an i.i.d. process converge uniformly to their values as the number of samples approaches infinity, then F continues to have the same property if the i.i.d. process is replaced by a β-mixing process. In this note, this result is extended to the case where the underlying probability is itself not fixed, but varies over a family of measures. Further, explicit upper bounds are derived on the rate at which the empirical means converge to their true values, when the underlying process is β-mixing. These bounds are less conservative than those derived by Yu (Ann. Probab. 22 (1994) 94).

We consider the question of robustness of the optimal nonlinear filter when the signal process X and the observation noise are possibly correlated. The signal X and observations Y are given by a SDE where the coefficients can depend on the entire past. Using results on pathwise solutions of stochastic differential equations we express X as a functional of two independent Brownian motions under the reference probability measure P0. This allows us to write the filter π as a ratio of two expectations. This is the main step in proving robustness. In this framework we show that when (Xn,Yn) converge to (X,Y) in law, then the corresponding filters also converge in law. Moreover, when the signal and observation processes converge in probability, so do the filters. We also prove that the paths of the filter are continuous in this framework.

We consider the nonlinear filtering model with signal and observation noise independent, and show that in case the signal is continuous in probability, the filter admits a version whose paths are continuous. The analysis is based on expressing the nonlinear filter as a Wiener functional via the Kallianpur-Striebel Bayes formula.

In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair (signal, filter) are Feller--Markov processes. The state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded. Our proofs, in contrast to those of Kunita [ J. Multivariate Anal., 1 (1971), pp. 365--393; Spatial Stochastic Processes, Birkhauser, 1991, pp. 233--256] and Stettner [ Stochastic Differential Equations, Springer-Verlag, 1989, pp. 279--292], do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of Ocone and Pardoux [ SIAM J. Control Optim., 34 (1996), pp. 226--243] on asymptotic stability of the nonlinear filter, which use the Kunita--Stettner setup, hold for the general situation considered in this paper.

In the nonlinear filtering model with signal and observation noise independent, we show that the filter depends continuously on the law of the signal. We do not assume that the signal process is Markov and prove the result under minimal integrability conditions. The analysis is based on expressing the nonlinear filter as a Wiener functional via the Kallianpur–Striebel Bayes formula.

We characterize the optimal filter in the nonlinear filtering theory as the unique solution to the Zakai equation. The results are very general as we allow the function h appearing in the filtering model to be discontinuous and unbounded. We consider the standard Markov model as well as the case when the signal X and the observation Y are solutions of a stochastic differential equation where the coefficients are allowed to depend on the past of Y. This is done via some results on existence of stationary solutions and solutions corresponding to certain measure valued evolution equations for controlled (and uncontrolled) martingale problems

In this article, several general questions regarding opinion polls and seat prediction in the context of Indian parliamentary democracy are diacussed and the methodology used by the author to tackle these is described. Also included are the results of the opinion polls leading to the 1998 parliamentary polls.

A 2×2 game is played repeatedly by two satisficing players. The game considered includes the Prisoner's Dilemma, as well as games of coordination and common interest. Each player has anaspirationat each date, and takes an action. The action is switched at the subsequent period only if the achieved payoff falls below aspirations; the switching probability depends on the shortfall. Aspirations are periodically updated according to payoff experience, but are occasionally subject to trembles. For sufficiently slow updating of aspirations and small tremble probability, it is shown that both players must ultimately cooperate most of the time.Journal of Economic LiteratureClassification Numbers C72, D83. © 1998 Academic Press., Ray gratefully acknowledges support under National Science Foundation Grant SBR- 9414114. Vega-Redondo acknowledges support from the Spanish Ministry of Education, CICYT Project PB 94-1504.

A nonlinear Hilbert-space-valued stochastic differential equation where L -1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L -1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L -1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ 0 of the martingale problem posed by the corresponding McKean—Vlasov equation.

Whether random phenomena exist in nature or not, it is useful to think of the notion of randomness as a mathematical model for a phenomenon whose outcome is uncertain. Such a model can be obtained by exploiting the observation that, in many phenomena, even though the outcome in any given instance is uncertain, collectively there is a pattern. An axiomatic development of such a model is given below. It is also shown that in such a set-up an interpretation of the probability of an event can be provided using the ‘Law of Large Numbers’.

In this article, we construct a mapping I : D[0, ∞)×D[0,∞)→D[0,∞) such that if (Xt) is a semimartingale on a probability space (Ω, F , P) with respect to a filtration ( F t) and if (ft) is an r.c.l.l. ( F t) adapted process, then I (ƒ.(ω), X. (ω))= ∫ 0 . ƒ−dX(ω) a.s. This is of significance when using stochastic integrals in statistical inference problems. Similar results on solutions to SDEs are also given.

LetX be a Markov process taking values in a complete, separable metric spaceE and characterized via a martingale problem for an operatorA. We develop a criterion for invariant measures when rangeA is a subset of continuous functions onE. Using this, uniqueness in the class of all positive finite measures of solutions to a (perturbed) measure-valued evolution equation is proved when the test functions are taken from the domain ofA. As a consequence, it is shown that in the characterization of the optimal filter (in the white-noise theory of filtering) as the unique solution to an analogue of Zakai (as well as Fujisaki-Kallianpur-Kunita) equation, it suffices to take domainA as the class of test functions where the signal process is the solution to the martingale problem forA.

An upper bound is given for the mean square Wasserstein distance between the empirical measure of a sequence of i.i.d. random vectors and the common probability law of the sequence. The same result holds for an infinite exchangeable sequence and its directing measure. Similarly, for an i.i.d. sequence of stochastic processes, an upper bound is obtained for the mean square of the maximum, over 0 ⩽ t ⩽ T, of the Wasserstein distance between the empirical measure of the sequence at time t and the common marginal law at t. These upper bounds are derived under weak assumptions and are not very far from the known rate of convergence pertaining to an i.i.d. sequence of uniform random vectors on the unit cube. Our approach, however, allows us to get results for arbitrary distributions under moment conditions and also gives results for processes. An application is given to so-called diffusions with jumps. Moment estimates for these processes are derived which may be of independent interest.

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(γ,H,B) is absolutely continuous with respect to the abstract Wiener measureμ. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Ito integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.

In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.

We consider a signal process X taking values in a complete, sep- arable metric space E. X is assumed to be a Markov process charachterized via the martingale problem for an operator A. In the context of the finitely additive white noise theory of filtering, we show that the optimal filter it(y) is the unique solution of the analogue of the Zakai equation for every y, not necessarily continuous. This is done by first proving uniqueness of solution to a (perturbed) measure valued evolution equation associated with A. An additional assumption of uniqueness of the local martingale problem for A is imposed.