By Jacques Janssen, Raimondo Manca (auth.)

ISBN-10: 038729547X

ISBN-13: 9780387295473

ISBN-10: 0387295488

ISBN-13: 9780387295480

**Applied Semi-Markov procedures **aims to offer to the reader the instruments essential to follow semi-Markov techniques in real-life difficulties. The booklet is self-contained and, ranging from a low point of chance options, steadily brings the reader to a deep wisdom of semi-Markov approaches. The publication offers homogeneous and non-homogeneous semi-Markov techniques, in addition to Markov and semi-Markov rewards tactics. those strategies are basic for lots of functions, yet they don't seem to be as completely provided in different books at the topic as they're right here.

*Audience*

This ebook is meant for graduate scholars and researchers in arithmetic, operations learn and engineering; it could actually additionally entice actuaries and monetary managers, and an individual attracted to its functions for banks, mechanical industries for reliability elements, and insurance firms.

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**Additional info for Applied Semi-Markov Processes**

**Example text**

S. s. (resp. equality. 2), the martingale equality means that the best predicted value simply is the observed value of the process at the time of predicting s. The use of martingale in finance is frequently (see Janssen and Skiadas (1995)) to model the concept of an efficient financial market. s.. It is possible to prove the following result (see for example Protter (1990)). (X^) is right continuous iff there exists a unique modification Y ofX such that Y is cddldg. (ii) IfXis a martingale then, up to a modification, the function ^h->£'(X^) is right continuous.

Also, we define a new random variable A'^ giving the total number of renewals on [0,L). >0}. 19) 50 Chapter 2 In reliability theory, the event {N = k} would mean that the (A:+l)th component introduced in the system would have an infinite lifetime! 21) and in general, for A: G N : P{N = k) = (F(+oo))' (1 -F(+oo)). , A^ = +oo. 26): So, it is possible to compute the mean of the total number of renewals very easily in the transient case. We can also give the distribution function of L. Indeed, we have: P{L

The reply is given by the so-called optional sampling theorem also called Doob 's theorem. s. v. s. s. This result is interesting for the concept of stopped process. 5 Let X be a stochastic process and T a stopping time. 15) where: Probability Tools withtAT 43 = mf{t,T]. From this definition, it follows that if the process X is adapted and cadlag, then so is the stopped process X^. This is due to the fact that / A T is also a stopping time and moreover: This leads to the last result we want to mention.

### Applied Semi-Markov Processes by Jacques Janssen, Raimondo Manca (auth.)

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