# Question

Suppose X (t) is a Weiner process with diffusion parameter λ = 1 as described in Section 8.5.

(a) Write the joint PDF of X1 = X (t1) and X2 = X (t2) for t2 < t1 by evaluating the covariance matrix of X = [X1, X2] T and using the general form of the joint Gaussian PDF in Equation 6.22.

(b) Evaluate the joint PDF of X1 = X (t1) and X2 = X (t2) for t2 < t1 indirectly by defining the related random variables and. Y1 = X1 and Y2 = X2 – X1.Noting that Y1 and Y2 are independent and Gaussian, write down the joint PDͯF of Y1 and Y2 and then form the joint PDF of Y1 and Y2 be performing the appropriate 2 * 2 transformation.

(c) Using the technique outlined in part (b), find the joint PDF of three samples of a Wiener process, X1 = X (t1), X2 = X (t2), and X3 = X (t3) for t1 < t2 < t3.

(a) Write the joint PDF of X1 = X (t1) and X2 = X (t2) for t2 < t1 by evaluating the covariance matrix of X = [X1, X2] T and using the general form of the joint Gaussian PDF in Equation 6.22.

(b) Evaluate the joint PDF of X1 = X (t1) and X2 = X (t2) for t2 < t1 indirectly by defining the related random variables and. Y1 = X1 and Y2 = X2 – X1.Noting that Y1 and Y2 are independent and Gaussian, write down the joint PDͯF of Y1 and Y2 and then form the joint PDF of Y1 and Y2 be performing the appropriate 2 * 2 transformation.

(c) Using the technique outlined in part (b), find the joint PDF of three samples of a Wiener process, X1 = X (t1), X2 = X (t2), and X3 = X (t3) for t1 < t2 < t3.

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