 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/a/2/b/ii/solution

Conditional on

F_{n}

, the increment is

- X_{n} /2

1 - X_{n}

, and

(1 - X_{n}) /2

X_{n}

. Therefore

E ((X_{n + 1} - X_{n})^{2} ∣ F_{n}) = \frac{( 1 - X _{n} ) X _{n}^{2} + X _{n} ( 1 - X _{n} ) ^{2}}{4} = \frac{X _{n} ( 1 - X _{n} )}{4} .

Taking expectations using the tower property yields

E (X_{n + 1} - X_{n})^{2} = \frac{1}{4} E [X_{n} (1 - X_{n})] .

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