- Stochastic Modeling 1 (70.72 KiB) Viewed 79 times
STOCHASTIC MODELING
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STOCHASTIC MODELING
STOCHASTIC MODELING
Question 6 In this question assume that {W(1): 2 0} is the standard Wiener process (or Brownian motion). (a) Let 0<u<t. Write down the distribution of W(6) -W(u). (b) Write down E[W(t) and E[W(t)?) (c) The Brownian bridge on the interval (0,1) is the stochastic process (B(t): t€ (0,11} defined by B(t) = W(t) – W(1) where (W(t): t 0} is the standard Brownian motion or Wiener process. Find the mean and autocovariance functions of the process (B(t): t € (0,11), that is, find po(t) =E[B(t)) and KB(1,8) = E((B(t) - MB(t)}(B(3) – HB(s))] for 0 <8t<1. (d) Write down the probability density function (pdf) of the random variable B(1/2). А
Question 6 In this question assume that {W(1): 2 0} is the standard Wiener process (or Brownian motion). (a) Let 0<u<t. Write down the distribution of W(6) -W(u). (b) Write down E[W(t) and E[W(t)?) (c) The Brownian bridge on the interval (0,1) is the stochastic process (B(t): t€ (0,11} defined by B(t) = W(t) – W(1) where (W(t): t 0} is the standard Brownian motion or Wiener process. Find the mean and autocovariance functions of the process (B(t): t € (0,11), that is, find po(t) =E[B(t)) and KB(1,8) = E((B(t) - MB(t)}(B(3) – HB(s))] for 0 <8t<1. (d) Write down the probability density function (pdf) of the random variable B(1/2). А