By James C. Spall
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That is, for any s ≥ 0, P X(t + s) − X(s) = n = exp(−λt)(λt)n /n!. Indeed, the process is a Markov process and ∆m P X(t + ∆t) ≤ m + ∆m|X(t) = m = exp(−λ∆t)(λ∆t)l /l! l=0 and the probability distribution at time t + ∆t only depends on the state of the system at time t and not on the history of the system. 4, dP0 (t) = −λP0 (t), dt dPn (t) = −λPn (t) + λPn−1 (t) for n ≥ 1, dt E(X(t)) = λt, and Var(X(t)) = λt. 3 are finite-difference approximations to the above differential equations and approach these differential equations as ∆t → 0.
Prove that Xn → X + X 2 in the mean square sense. That is, prove that Xn − (X + X 2 ) RV → 0 as n → ∞. 10. Consider the random number generator Xn+1 = (8Xn + 9)mod(7) for n = 0, 1, 2, . . , with Un = Xn /7. Let X0 = 2 and calculate X1 , X2 , . . , X10 . Determine the period of the generator. 11. Assume that the function f is integrable and maps [0, 1] into [0, 1]. Con1 sider estimating 0 f (x) dx using two different Monte Carlo approaches. 9 Monte Carlo 29 for i = 1, 2, . . , n. In this approach, σ12 = E(f 2 ) − (E(f ))2 .
Thus, 52 2 Stochastic Processes W (tk ) = X(tk ) for k = 0, 1, 2, . . , N . 11. In particular, W (t) ≈ X(tk ) t − tk tk+1 − t + X(tk+1 ) h h for tk ≤ t ≤ tk+1 . 14. Simulation of a Wiener process by a discrete process Let tk = kh for k = 0, 1, 2, . . , N where h = T /N . Define the discrete stochastic process {Xn }N n=0 on the partition 0 = t0 < t1 < · · · < tN = T in the following way. Let X0 = 0 and let the transition probabilities pi,k = P {Xn+1 = kδ|Xn = iδ} be ⎧ for k = i − 1 ⎨ λ∆t/2δ 2 , pi,k (t) = 1 − λ∆t/δ 2 , for k = i ⎩ λ∆t/2δ 2 , for k = i + 1 assuming that λ∆t/δ 2 < 1.