Problem 32 (Correlated Brownian Motions) Let W(t) and U(t)...

Problem 32 (Correlated Brownian Motions) Let W(t) and U(t) be two independent standard Brownian motions. Let -1 64 c1 64 1. Define the random process X(t) as X(t) = c1 W(t) + sqrt{1 - c1^2}, U(t), for all t 64 [0, infty). a. Show that X(t) is a standard Brownian motion. b. Find the covariance and correlation coefficient of X(t) and W(t). That is, find Cov(X(t), W(t)) and c1(X(t), W(t)).

Transcript

00:01 Hello students, part a, w of t and u of t are independent standard brownian motion.

00:29 Thus, w of 0 is equal to u of 0 is equal to 0, that is initial state is 0.

00:42 For 0 less than or equal to t1 less than t2, that is w of t2 minus w of t1 follows normal with mean 0, variance t2 minus t1 and u of t2 minus u of t1 also follows normal with 0, t2 minus t1.

01:15 W of t and u of t have independent increments.

01:36 Now, x of 0 is equal to rho into w of 0 plus square root of 1 minus rho square into u of 0.

01:53 So, that is equal to rho into 0 plus square root of 1 minus rho square into 0, which is 0 only.

02:02 Now, x of t2 minus x of t1 that is equal to rho into w of t2 minus w of t1 plus square root of 1 minus rho square into u minus t2 minus u minus t1.

02:37 So, expectation of x of t2 minus x of t1 is equal to rho into expectation of w of t2 minus w of t1 plus square root of 1 minus rho square into expectation of u of t2 minus u of t1.

03:00 So, that is equal to rho into 0 plus square root of 1 minus rho square into 0, which is 0.

03:07 Now, variance of x of t2 minus x of t1 is equal to rho square into variance of w of t2 minus w of t1 plus 1 minus rho square into variance of u of t2 minus u of t1 that is equal to rho square into t2 minus t1 plus 1 minus rho square into t2 minus t1.

03:32 That is equal to t2 minus t1.

03:37 We know that the linear combination of normal random variable is normal random variable.

03:45 Thus x of t2 minus x of t1 follows normal with 0, t2 minus t1.

03:51 X of t is a linear combination of independent variables where w of t and u of t with independent increments.

03:58 So, x of t have independent increments...

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