1. Let (B)120 be a standard Brownian motion with its right continuous filtration (Fi)120. As usual,...

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1. Let (B)120 be a standard Brownian motion with its right continuous filtration (Fi)120. As usual, let E, denote expectation under the measure of (B)e20 with Bo = x. For a e R, let Ta := inf{t > 0: Bị = a}, and for a < b set Tab := inf{t >0: Bị ¢ (a, b)} = min{Ta. T,}. (b) Show that for any integrable random variable Z, E,[ZI[T, < Ta]|Fr.] = I[T, < Ta]E, [Z\Fr,]. 1. Let (B)120 be a standard Brownian motion with its right continuous filtration (Fi)120. As usual, let E, denote expectation under the measure of (B)e20 with Bo = x. For a e R, let Ta := inf{t > 0: Bị = a}, and for a < b set Tab := inf{t >0: Bị ¢ (a, b)} = min{Ta. T,}. (b) Show that for any integrable random variable Z, E,[ZI[T, < Ta]|Fr.] = I[T, < Ta]E, [Z\Fr,].