15.5 Let Bt a standard Brownian motion on R.

a) Show that for all f such that the integral exists we have E[f(Bt)] = 1

√2πt  ∞

−∞

f(x)e− x2 2t dx

.

b) Apply the previous part to f(x) = x2k and find a recurrence relation between E[B2k t ] and E[B2(k−1)

t ].

c) Use the previous part and induction to show that E[B2k t ] = (2k)!

2kk!

t k, k ∈ N.