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

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).