Past exam of the probability, measure and martingales course of the University of Oxford / 2023 / 3 / b / ii

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- Solution ii

Solution

ϕ_{a} (t) = t \land a

is a concave function, and

X \land a

is integrable since

∣ X \land a ∣ \leq ∣ X ∣ + a

. The concave form of the conditional Jensen inequality gives

E [X \land a ∣ G] \leq (E [X ∣ G]) \land a = Y \land a almost surely .

The random variables

X

and

Y

have the same probability distribution, so

E [X \land a] = E [Y \land a]

. Using the law of total expectation, the nonnegative difference between the right and left sides of the inequality has zero expectation. It must vanish almost surely. Therefore

E [X \land a ∣ G] = Y \land a almost surely .

The equality comes from the distributional assumption as well as the conditional Jensen inequality.

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