 Solution

ID: past-exam-of-the-random-matrix-theory-course-of-the-university-of-oxford/2025/3/c/i/solution

B (t) \sim N (0, t)

gives

E e^{2 i B (t)} = e^{- 2 t}

. Taking real parts and using

sin^{2} b = (1 - cos 2 b) /2

E Δ (t) = \frac{1 - e ^{- 2 t}}{2}, t \to \infty lim E Δ (t) = \frac{1}{2} .

There is also a direct check from the stochastic differential equation. Its Itô integral has zero expectation because its integrand is bounded. Thus

m (t) = E Δ (t)

satisfies

m^{'} (t) = 1 - 2 m (t)

and

m (0) = 0

, whose solution is the same expression.

 New to topics? Read the docs here!