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IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices

X(t) satisfies dX = 2X dt + 3X dW. Using Ito's lemma, which SDE does Y = X² satisfy?

Y = X² follows dY = 13Y dt + 6Y dW. The drift is 2·2 plus the Ito correction 3² = 9, giving 13, and the volatility doubles to 6 because the derivative factor is 2X.

  1. AdY = 4Y dt + 6Y dW
  2. BdY = 13Y dt + 6Y dWCorrect
  3. CdY = 4Y dt + 9Y dW
  4. DdY = 11Y dt + 6Y dW
  5. dY = 13Y dt + 9Y dW

Explanation

dY = 2X dX + (dX)² = 2X(2X dt + 3X dW) + 9X² dt = (4+9)Y dt + 6Y dW = 13Y dt + 6Y dW. Omitting the (dX)² term gives 4Y dt, and subtracting it would give 11.

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