By Jacques Janssen, Raimondo Manca

ISBN-10: 038729547X

ISBN-13: 9780387295473

ISBN-10: 0387295488

ISBN-13: 9780387295480

Aims to offer to the reader the instruments essential to observe semi-Markov approaches in real-life problems.

The booklet is self-contained and, ranging from a low point of likelihood recommendations, progressively brings the reader to a deep wisdom of semi-Markov processes.

Presents homogeneous and non-homogeneous semi-Markov techniques, in addition to Markov and semi-Markov rewards processes.

The thoughts are primary for plenty of purposes, yet they don't seem to be as completely offered in different books at the topic as they're here.

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

**Example text**

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.

X„yE[Y%). 49) This type of property is very useful for computing probabilities using conditioning and will often be used in the following chapters. d. real random variables and IS! v. with integer values independent of the given sequence. v. X^ have a variance. 52) we also have: E(S,) = E(NE(X)l and finally the so-called the first Wald's identity. 54) it suffices to evaluate E(Sl) as we did above. 28), we can write that: E(SI)=E(E(SI\N)). 56) we obtain on the set [co: N{co) = n): E{sl\N = n) = E(Y,X, = var Z^,- +k Z^.

21) converges to O . This theorem was used by the Nobel Prize winner H. Markowitz (1959) to justify that the return of a diversified portfolio of assets has a normal distribution. As a particular case of the Central limit Theorem, let us mention the de Moivre 's theorem starting with f 1, with prob. 25) [0, with prob. v. 22) has a binomial distribution with parameters {n,p). By applying now the Central Limit Theorem, w e get the following result: S,-np la^ >iV(0,l), ^np(\-p) called de Moivre's result.

### Applied Semi-Markov Processes by Jacques Janssen, Raimondo Manca

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