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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**Additional info for Semi-Markov Risk Models for Finance, Insurance and Reliability**

**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.