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 techniques in real-life problems.

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

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

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

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

**Sample text**

X,^Y„\ft^T)^\. ,for all t eT: X^=Y^, as. yteT) for alU e r . = l. 16) definedfor each coeQ, is called a trajectory or sample path of the process. It must be clearly understood that the so-called "modem" study of stochastic processes is concerned with the study of the properties of these trajectories. For example, we can affirm that if two processes X and Y are indistinguishable, then there exists a set A/^ belonging to 3 of probability 0 such that: '^co^N:XXco) = Y^(cD)yteT. 17) In other words, for each co element of the set Cl-N, the two functions t h^ X^ (co) and / h-> }^ (o)) are equal.

10) belongs to the a -algebra 3 . 1 If(£',X) = (M,y5), the process is called a real stochastic process with values i n R ; if(£,K) = (M",y5"), it is called a real multidimensional process with values in R''. }, the process X is called a discrete time stochastic process or a random sequence; if T is the positive half real line [0,oo), the process X is called a continuous time stochastic process. v. Xt is 3^ -measurable. This means that, for all t ^T: x;'{B) = {(D\XXco)eB]e:5^yB^^. ,for all t GT : X,=Y,.

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. It follows that every martingale such that the function t\-^E[X^) is right continuous is cadlag.

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

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