Download Semi-Markov Risk Models for Finance, Insurance and by Jacques Janssen PDF

By Jacques Janssen

Every person operating in comparable fields from utilized mathematicians to statisticians to actuaries and operations researchers will locate this a brilliantly worthwhile useful textual content. The ebook provides functions of semi-Markov methods in finance, assurance and reliability, utilizing real-life difficulties as examples. After a presentation of the most probabilistic instruments important for knowing of the ebook, the authors convey tips to practice semi-Markov methods in finance, ranging from the axiomatic definition and carrying on with finally to the main complicated monetary instruments.

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Example text

14) The classification of a renewal process is based on three concepts: recurrence, transience and periodicity. 1 (i) A renewal process ( Tn , n ≥ 1 ) is recurrent if X n < ∞ for all n; otherwise it is called transient. (ii) A renewal process ( Tn , n ≥ 1 ) is periodic with period δ if the possible values of the random variables X n , n ≥ 1 form the denumerable set { 0, δ , 2δ ,…} , and δ is the largest such number. Otherwise (that is, if there is no such strictly positive δ ), the renewal process is aperiodic.

54) E ( S N2 ) = E E S N2 N . 56) = n var( X ) + ( nE ( X )) 2 . 53), we get: var( S N ) = E ( N ) var( X ) + E ( N 2 )( E ( X 2 )) − ( E ( X )) 2 ( E ( N ))2 and finally we obtain the second Wald’s identity in the form var( S N ) = E ( N ) var( X ) + var( N )( E ( X )) 2 . v. Y given ℑ1 by an implicit relation. Now the question is: can we define the conditional expectation with an explicit relation? s. s. 62) (iii) n =1 ⎝ n =1 ⎠ ∀i, ∀j , i ≠ j. It is important to note here that the null events N1 , N 2 , N3 , on which respectively these last three properties are not true, are generally not identical, so that for each ω , the random set function P (.

13) is called the hitting time of Λ by the process X. Probability tools 37 It is easily shown that the properties of stopping and hitting times are (see Protter (1990)): (i) If X is càdlàg, adapted and Λ ∈ β , then the hitting time related to Λ is a stopping time. v. 14) are also stopping times. 15) is called the stopping time σ -algebra. In fact, the σ -algebra ℑT represents the information of all observable sets up to the stopping time T. v. v. s. 17) (ii) ℑS ∩ ℑT , = ℑS ∧T . 8 MARTINGALES In this section, we shall briefly present some topics related to the most wellknown category of stochastic processes called martingales.

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