Convexity by Klee V. (ed.)

By Klee V. (ed.)

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Then we can write the generator of the Markov chain as 0: Q PA -(A +a) (3 0 -A Q= ( o o pa p(3 qA o -(A + (3) qa q(3 00) A 0 o A o -(3 -a . 0 Assuming a > 0 and (3 > 0, then Q is irreducible. Therefore there exists a unique stationary distribution that can be determined by solving the system of equations: vQ = 0 and vi = 1. ) competing for the lives of individuals. For each of the individuals, one of these risks will "win," and this individual will die. The competing risk theory aims to study these risks and their prevention.

In either of the models above, 0:( t) E M = {1, ... , m}, t 2:: 0, is a finite-state Markov chain characterizing the piecewise-deterministic behavior of the parameter process. ), and aims at deriving the optimality. Recent interest for such jump linear systems stems from the fact they can be used to describe unpredictable structural changes. Owing to the various source of uncertainties, in many real-world applications, the parameter process is of very high dimension. This brings about much of the difficulty in analyzing such systems.

Then aCt) = i on the interval [TO, Tl). The first jump time Tl has the probability distribution Ph E B) = L exp {lot qii(S)ds } (-qii(t)) dt, where B C [0,00) is a Borel set. \. qi j -qii Tl Note that qii(Tl) may equal o. In this case, define P(a(Tl) = jh) = 0, j -=I i. We claim P(qii(Tt} = 0) = O. In fact, if we let Bi = {t : qii(t) = O}, then P(qii(Tl) = 0) = P(TI E B i ) = Li exp {lot qii(S)ds } (-qii(t)) dt = O. In general, aCt) = a(Td on the interval [Tl' Tl+t}. The jump time Tl+l has the conditional probability distribution P( Tl+l - Tl E BdT1, ...

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