By Jacques Janssen, Raimondo Manca

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

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

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

The options are primary for plenty of functions, yet they aren't as completely offered in different books at the topic as they're here.

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Extra resources for Applied Semi-Markov Processes

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14), let us choose X(t) = R(t)--^, m We shall compute G such that this integral equation is valid. We get: G(t) = X(t)-X^F(t) = R(t) - -^ - /? 33) H(t) + -i- \F(t-x)dx. 35) And thus: G(t) = l _1 \{l-F(x))dx.

In fact, the a -algebra 3^^ represents the information of all observable sets up to the stopping time T. W e can also say that S^. v. v. s,^ 3s ^^r^ C7-21) ( i i ) 3 , n 3 , , = 3,,,. 22) 8 MARTINGALES In this paragraph, we shall briefly present some topics related to the most wellknown category of stochastic processes called martingales. Let Jf be a real stochastic process defined on the filtered complete probability space ( Q , 5 , P , ( 3 , , / e r ) ) . 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.

V^G5. 11) and so P^is called the conditional probability measure given B. v. 11) P\B) Q For our next step, we shall now consider a countable event partition {B^,n>\) of the sample space Q. That is: Q = Q 5 „ 5 , n 5 , = 0 , V/,7:/^7. 10): PiA) = XPiB„)P(A\B„). v. 11): E{Y) = Y,P{B„)E,^{Y). 16) as E^ (7) called the conditional expectation of Y given 3 , . 11) with B=Bn. 16), we get: E(E^^(Y)(co)) = E(Y). 17): Probability Tools 27 1^3, {Y){co)dP =X K (^)^«. 11), we get: K ( r ) ( . , . P . 21) = JY((o)dP.

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