Arbitrage Bounds for Prices of Weighted Variance Swaps

We develop robust pricing and hedging of a weighted variance swap when market prices for a finite number of co--maturing put options are given. We assume the given prices do not admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super- and sub- replicating strategies which enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup. In particular we use and extend F\"ollmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the 'log contract' and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs.Comment: 25 pages, 4 figure

Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups; Part 1: Two Killing fields spanned on u^{\alpha} and w^{\alpha }

For a rotating dust with a 3-dimensional symmetry group all possible metric forms can be classified and, within each class, explicitly written out. This is made possible by the formalism of Pleba\'nski based on the Darboux theorem. In the resulting coordinates, the Killing vector fields (if any exist) assume a special form. Each Killing vector field may be either spanned on the fields of velocity and rotation or linearly independent of them. By considering all such cases one arrives at the classification. With respect to the structures of the groups, this is just the Bianchi classification, but with all possible orientations of the orbits taken into account. In this paper, which is part 1 of a 3-part series, all solutions are considered for which two Killing fields are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del are identified as special cases, and their new invariant definitions are provided. In addition, a new invariant definition is given of the Ozsvath class III solution.Comment: 23 pages, LaTe

A Non - Singular Cosmological Model with Shear and Rotation

We have investigated a non-static and rotating model of the universe with an imperfect fluid distribution. It is found that the model is free from singularity and represents an ever expanding universe with shear and rotation vanishing for large value of time.Comment: 10 pages, late

Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups, Part 3: All Killing fields linearly independent of u^{\alpha} and w^{\alpha}

This is the third and last part of a series of 3 papers. Using the same method and the same coordinates as in parts 1 and 2, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under the assumption that each of the Killing vectors is linearly independent of velocity $u^{\alpha}$ and rotation $w^{\alpha}$ at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations are solved for the components of the metric tensor in every case that arises. No progress was made with the Einstein equations in any of the cases, and no previously known solutions were identified. A brief overview of literature on solutions with rotating sources is given.Comment: One missing piece, signaled after eq. (10.7), is added after (10.21). List of corrections: In (3.7) wrong subscript in vorticity; In (3.10) wrong subscript in last term of g_{23}; In (4.23) wrong formulae for g_{12} and g_{22}; In (7.17) missing factor in velocity; In (7.18) one wrong factor in g_{22}; In (10.9) factor in vorticity; In (10.15) - (10.20) y_0 = 0; In (10.20) wrong second term in y. The rewriting typos did not influence result

Measurement of cervical length using transvaginal sonography for prediction of preterm labour

Background: Preterm labour and delivery cause major health burden to the society due to high perinatal morbidities and mortality and long-term health implications and also affects maternal. An effective and objective way for predicting preterm delivery is measurement of cervical length by transvaginal sonography as it allows better quality and accurate visualization of uterine cervix. Cervical length (<25 mm) is good and accurate cervical biometry for prediction of preterm birth. The objective of this study was to measure cervical length by transvaginal sonography for predicting preterm labour and fetal outcome.Methods: This prospective observational study was conducted in department of obstetrics and gynaecology, at SVPIMSR hospital, Ahmedabad from July 2018 to December 2019 in 150 antenatal women to assess cervical changes (cervical length, dilatation of internal OS, funnelling etc.) between 16 to 24 weeks of gestation and these cases followed till delivery and results were analysed.Results:150 antenatal women who fulfilled the selection criteria were studied using transvaginal ultrasound between 16-24 weeks of gestation, out of them 36 (24%) women delivered preterm babies. The sensitivity, specificity, positive predictive value (PPV) and negative predictive value (NPV) recorded in this study were 80.5%, 94.73%, 82.85% and 93.91% respectively.Conclusion: Transvaginal sonography is the most useful and better, safe, accurate, most effective, less expensive, objective and acceptable technique for assessing cervical length in all antenatal women and predicting the preterm labour when assessed between 16 -24 weeks of gestational age

Pregnancy outcome in patients with fibroid: a retrospective study

Background: Fibroids are benign smooth muscle cell tumour of the uterus. In some patients of pregnancy associated with fibroid, it does not affect the outcome of pregnancy. On the other hand, various complications have been reported. The aim of our study was to evaluate the outcome in antenatal women with fibroids.Methods: This retrospective study was conducted at tertiary care center, obstetrics & gynecology department over a period of eighteen months between June 2018 to November 2019. Total 26 pregnant patients with >3 cm fibroid was included in the study. They were followed during antenatal period. Maternal age, parity, size of fibroid, complications during pregnancy, labour, and delivery, mode of delivery and indications of cesarean section were noted.Results: Out of 26 patients, 15 (57.6%) were between 26-30 years of age group and 16 (61.53%) were multigravidas. Normal vaginal delivery occurred in 8 (33.33%), while 16 (66.66%) delivered by caesarean section. There were 9 (34.61%) patients who had no complication whereas 17 (65.38%) had some complication. Pain was present in 8 (30.76%). PROM and preterm labour were present in 3 (18.75%) and 5 (19.23%) respectively. PPH was present in 2 (7.69%).Conclusions: Pregnant patients who have fibroids are to be carefully screened in the antenatal period, so as to have a regular follow up. The widespread use of ultrasonography has facilitated diagnosis and management of fibroids in pregnancy. The site and size of fibroid is very important to predict its effect on pregnancy

Quantum Theory of Non-Relativistic Particles Interacting with Gravity

We investigate the effects of the gravitational field on the quantum dynamics of non-relativistic particles. We consider N non-relativistic particles, interacting with the linearized gravitational field. Using the Feynman - Vernon influence functional technique, we trace out the graviton field, to obtain a master equation for the system of particles to first order in $G$. The effective interaction between the particles, as well as the self-interaction is non-local in time and in general non-markovian. We show that the gravitational self-interaction cannot be held responsible for decoherence of microscopic particles due to the fast vanishing of the diffusion function. For macroscopic particles though, it leads to diagonalization to the energy eigenstate basis, a desirable feature in gravity induced collapse models. We finally comment on possible applications.Comment: Latex,14 pages, replaced to correct the titl

Relation Between Einstein And Quantum Field Equations

We show that there exists a choice of scalar field modes, such that the evolution of the quantum field in the zero-mass and large-mass limits is consistent with the Einstein equations for the background geometry. This choice of modes is also consistent with zero production of these particles and thus corresponds to a preferred vacuum state preserved by the evolution. In the zero-mass limit, we find that the quantum field equation implies the Einstein equation for the scale factor of a radiation-dominated universe; in the large-mass case, it implies the corresponding Einstein equation for a matter-dominated universe. Conversely, if the classical radiation-dominated or matter-dominated Einstein equations hold, there is no production of scalar particles in the zero and large mass limits, respectively. The suppression of particle production in the large mass limit is over and above the expected suppression at large mass. Our results hold for a certain class of conformally ultrastatic background geometries and therefore generalize previous results by one of us for spatially flat Robertson-Walker background geometries. In these geometries, we find that the temporal part of the graviton equations reduces to the temporal equation for a massless minimally coupled scalar field, and therefore the results for massless particle production hold also for gravitons. Within the class of modes we study, we also find that the requirement of zero production of massless scalar particles is not consistent with a non-zero cosmological constant. Possible implications are discussed.Comment: Latex, 24 pages. Minor changes in text from original versio