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By Jacob Kogan

ISBN-10: 3540168184

ISBN-13: 9783540168188

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Consequently, with i 0 as the initial state for the chain, which itself is the step process X ◦ = X t◦ given by X t◦ = η t , where η0 ≡ i 0 , ◦ ◦ t Nt = ληn ((τn+1 ∧ t) − τn ) = n=0 writing ◦i = ◦ ({i}) ◦i t for i ∈ G so that t = 0 ◦ (A) = ◦ =i λ X ◦ π X ◦ i ds = 1( X s− ) s− s− t 0 i∈A ◦ ds; λ X s− ◦i , we have t 0 ◦ =i q X ◦ i ds. 1( X s− ) s− We come now to a key lemma that will prove immensely useful in the future. For s ≥ 0, define the shift θs mapping W, respectively M, into itself, by (θs N ◦ )t = Nt◦ − Ns◦ 0 if t ≥ s if t < s, θs µ◦ = µ◦ (· ∩ (]s, ∞[ × E)) .

Furthermore, let PB|x be the probability on (G, G) such that under PB|x the process πt◦ t≥0 is a standard Brownian motion on R, starting from x: PB|x (X 0 = x) = 1; and for all n ∈ N, 0 = t0 < t1 < · · · < tn , the increments πt◦k − πt◦k−1 1≤k≤n are independent with πt◦ − πs◦ for s < t Gaussian with mean 0 and variance t − s. For r ≥ 1, let {G}r denote the set of all sets {w1 , . . , wr } consisting of r elements of G. As a mark space we shall use E = {0} ∪ ∞ {G}r r =1 with the understanding that if the mark 0 occurs, the branching diffusion has become extinct, while if the mark {w1 , .

31) for n = 0 ◦ is that H ∈ H0 iff H ∩ Nt◦ = 0 , resp. H ∩ N t = 0 , equals either the empty set (n) or the set (τ1 > t) . Also, for n = 0 the function f z n (t) is a function f (0) (t) of t only. 2 The description of adapted processes merely states (apart from measurability properties) that a process is adapted iff its value at t can be computed from the number of jumps on [0, t] and the timepoints and marks for these jumps. , (M, H) is piecewise deterministic, (cf. 12)). For a process to be predictable, to find its value at t it suffices to know the number of jumps on [0, t[, their location in time and the marks.

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Bifurcation of Extremals in Optimal Control by Jacob Kogan

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