Let N(t), t 0 be a Poisson process with parameter 0 Moreover, define the stochastic process K(t), t 0 , where K(t) N(t) t for every t 0 a) Let 0 s t Determine E(N(t) N(s)) and E((K(t) K(s))2 ) b) Show that E(K(t) 2 K(u), 0 u s) K(s) 2 (t s) ...

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Let {N(t), t 0} be a Poisson process with parameter > 0. Moreover, define the

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Let {N(t), t ≥ 0} be a Poisson process with parameter λ > 0. Moreover, define the stochastic process {K(t), t ≥ 0}, where K(t) := N(t) − λt for every t ≥ 0. a) Let 0 ≤ s < t. Determine E(N(t) − N(s)) and E((K(t) − K(s))2 ). b) Show that E(K(t) 2 |K(u), 0 ≤ u ≤ s) = K(s) 2 + λ(t − s).