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

Convergence of Probability Measures

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.

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Polynomial processes and their applications to mathematical Finance

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.

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...

An elementary introduction to mathematical finance

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.

Time--space harmonic polynomials relative to a L\'{e}vy process

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.

Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

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.

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Asymptotic properties of sums of upper records

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.

Time-space polynomial martingales cenerated by a discrete-time martingale

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.

/pdf/convergence-of-lower-records-and-infinite-divisibility-52c2lqrlht.pdf

Arindam Sengupta

Papers

Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

Asymptotic properties of sums of upper records

Time-space polynomial martingales cenerated by a discrete-time martingale

Convergence of lower records and infinite divisibility

Markov processes, time–space harmonic functions and polynomials