Question: 4. (a) Let the process (B(t)) t>0 be a standard Brownian motion and suppose c > 0 is a constant. Let be a stopping time

4. (a) Let the process (B(t)) t>0 be a standard Brownian motion and suppose c > 0 is a constant. Let be a stopping time such that B()=c and B(u)

(i) Explain what is meant by saying is a stopping time.

(ii) Define Z(u) = (B(u), for 0 < u < , Z(u)= 2cB(u), for u>

Describe the Brownian motion Z(u) in relation to B(t). Show that Z(u) is also a Brownian motion and calculate its mean and variance.

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