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.

/pdf/polynomial-processes-and-their-applications-to-mathematical-2qy6ahc5tb.pdf

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 First Course in Machine Learning by Simon Rogers and Mark Girolami

Introduction to Probability

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.

Convergence of lower records

Extending an idea of Spitzer [2], a way to compute the moments of the time of escape from (−N,L) by a symmetric simple random walk is exhibited. It is shown that all these moments depend polynomially onL andN.

Moments of escape times of random walk

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Social Networks: Modelling and AnalysisNiyatiAggrawal and AdarshAnandCRC Press, 2022, xvii + 235 pages, hardcover ($170); e‐book ($44.21). ISBN: 78‐0‐367‐54139‐2 (hardcover), 978‐0‐367‐54173‐6 (paperback), 978‐1‐003‐08806‐6 (e‐book)

Arindam Sengupta

Papers

A First Course in Machine Learning by Simon Rogers and Mark Girolami

Introduction to Probability

Convergence of lower records

Moments of escape times of random walk

Social Networks: Modelling and AnalysisNiyatiAggrawal and AdarshAnandCRC Press, 2022, xvii + 235 pages, hardcover ($170); e‐book ($44.21). ISBN: 78‐0‐367‐54139‐2 (hardcover), 978‐0‐367‐54173‐6 (paperback), 978‐1‐003‐08806‐6 (e‐book)