Mathematics

# Mathematical and Statistical Methods in Insurance and by Cira Perna, Marilena Sibillo

By Cira Perna, Marilena Sibillo

The interplay among mathematicians and statisticians unearths to be a good method of the research of assurance and monetary difficulties, particularly in an operative point of view.

The Maf2006 convention, held on the collage of Salerno in 2006, had accurately this goal and the gathering released the following gathers a few of the papers provided on the convention and successively labored out to this objective. They hide a wide selection of matters in assurance and monetary fields.

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Ciurlia In more detail, in the following we consider the Kantorivich functional defined as ⎧ ⎫ ⎨ N M ⎬ (4) µˆ c (P, Q) := min cT (ξi , ξ j )ηi j ηi j ⎩ ⎭ i=1 j =1 N i=1 ηi j = q j ∀ j (5) ηi j = pi ∀i (6) M j =1 ηi j ≥ 0 ∀i, j , (7) where ct (·) measures the distance among two scenarios ξi and ξ j and is defined as T cT (ξi , ξ j ) := t =1 |ξi,t − ξ j,t | , (8) Rn . with | · | a suitable norm in Given that the original probability measure P and the approximating one Q are discrete with finitely many scenarios, the Kantorovich distance represents the optimal value of a linear transportation problem where the transport metric measures the cost of moving a scenario ξi , or a set of scenarios, of the original tree to the nearest scenario in the new reduced tree.

The interest towards the traded volumes is suggested by the Multifractal Model of Asset Returns (MMAR) which, introduced by [CF02], is the focus of a wide debate. The basic idea beyond the MMAR is to compound a Brownian motion (eventually a fractional one) with a multifractal trading time, defined as the cumulative distribution function of a self-similar multifractal measure which deforms the physical time in order to take into account the different number of transactions per unit of time. According to this model, we expect to find self-similarity in the traded volume.

As basis functions we have employed either powers or Laguerre polynomials. 24 A. R. Bacinello Observe that in each iteration (h), once the stochastic date T at which the benefit is due has been simulated, the simulated path of the reference portfolio can be generated • Forwards, from time t1 to T (h) , by using the conditional distribution above recalled. • Backwards, by simulating first the value of the reference portfolio at time T (h) from a binomial distribution, and after its values between times T (h) and t1 from the corresponding conditional distributions.