Author

# Arindam Sengupta

Other affiliations: Indian Statistical Institute, Indian Institute of Technology Guwahati

Bio: Arindam Sengupta is an academic researcher from University of Calcutta. The author has contributed to research in topics: Random variable & Natural filtration. The author has an hindex of 5, co-authored 15 publications receiving 56 citations. Previous affiliations of Arindam Sengupta include Indian Statistical Institute & Indian Institute of Technology Guwahati.

##### Papers

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TL;DR: In this paper, conditions for the existence of time-space harmonic polynomials of each degree in the second, "space", argument were investigated and various properties a sequence of time space harmonic poynomials may possess and the interaction of these properties with distributional properties of the underlying process.

Abstract: A time-space harmonic polynomial for a stochastic process M=(M
t) is a polynomial P in two variables such that P(t, M
t) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, “space,” argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,(2) where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a “measure” proposed by Hochberg(3) on path space is obtained.

18 citations

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TL;DR: In this article, the authors provide sufficient conditions on the distribution for the properly normalized partial sums to converge to a standard normal distribution, and show that their conditions are general enough so that the examples provided by Arnold and Villasenor (1999) are covered by their results.

Abstract: Arnold and Villasenor (1999) raised several questions for upper records, including characterizing all limit distributions of normalized partial sums of upper records. We provide some answers in the case when the distribution from which the samples are drawn is bounded above. When the distribution is not bounded above, we give sufficient conditions on the distribution for the properly normalized partial sums to converge to a standard normal distribution. We show that our conditions are general enough so that the examples provided by Arnold and Villasenor (1999) are covered by our results.

9 citations

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TL;DR: In this article, the conditions for the existence of polynomialsP(·,·) of two variables, "time" and "space", and of arbitrary degree in the latter, such that P(n, Mn) is a martingale for the natural filtration of M. The necessary and sufficient conditions for general martingales are given.

Abstract: We investigate, for a given martingaleM={M
n: n≥0}, the conditions for the existence of polynomialsP(·,·) of two variables, “time” and “space,” and of arbitrary degree in the latter, such that{P(n, M
n)} is a martingale for the natural filtration ofM. Denoting by ℘ the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span ℘. A complete necessary and sufficient condition is obtained in the case whenM has independent increments. For generalM, we obtain a necessary condition which entails, under mild additional hypotheses, thatM is necessarily Markovian. Considering a slightly more general class of polynomials than ℘ we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set ℘ determines the law of the martingale in a certain sense.

8 citations

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TL;DR: In this article, the authors studied the properties of sums of lower records from a distribution on [0, oo] which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers.

Abstract: We study the properties of sums of lower records from a distribution on [0, oo) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Levy measure on [0, oo) originate as infinite sums of lower records.

7 citations

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TL;DR: In this paper, the authors consider stochastic processes (M t ) t ≥ 0 for which the class V of time-space harmonic functions is rich enough to yield the Markov property for the process.

5 citations

##### Cited by

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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

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TL;DR: A class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials, which contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields.

Abstract: We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Levy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Levy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

97 citations

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TL;DR: In this paper, the authors introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order$m$ only requires the computation of matrix exponentials.

Abstract: We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as L\'evy-driven SDEs with affine vector fields. Thus, many popular models such as exponential L\'evy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

85 citations

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TL;DR: This book can be used as a textbook for first-year or second-year graduate students and the mathematical presentation is thorough and the exercises are quite useful in understanding the concepts of linear models.

Abstract: An elementary introduction to mathematical finance, third edition, by Sheldon M. Ross, Cambridge, Cambridge University Press, 2011, xv+305 pp., £35.00 or US$60.00 (hardback), ISBN 978-0-521-19253-8...

25 citations

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TL;DR: In this article, a closed form and a recurrence relation for a family of time-space harmonic polynomials relative to a Levy process is given, and the relationship with the Kailath-Segall (orthogonal) polynomial associated to the process is also discussed.

Abstract: In this work, we give a closed form and a recurrence relation for a family of time--space harmonic polynomials relative to a Levy process. We also state the relationship with the Kailath--Segall (orthogonal) polynomials associated to the process.

21 citations