 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2008/p/2/solution

Let

M_{j} = s^{''} (j)

M_{0} = M_{2} = 0

. The interior spline equation on equal unit intervals is

M_{0} + 4 M_{1} + M_{2} = 6 [(4 - 0) - (0 - 0)] = 24,

M_{1} = 6

. The standard second-derivative interpolation formula, or direct integration of the linear second derivative, gives

s (x) = {x^{3} - x, - x^{3} + 6 x^{2} - 7 x + 2, 0 \leq x \leq 1, 1 \leq x \leq 2.

The two pieces have equal value, first derivative, and second derivative at one, and

s^{''}

vanishes at zero and two.

x

changes the nodal data by

(0, 1, 2)

without affecting second derivatives. By uniqueness, the second requested natural cubic spline is $s (x) + x$ , with pieces

x^{3}

and

- x^{3} + 6 x^{2} - 6 x + 2

respectively.

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