Chapter 10 Question Page 6-7

Discussion in 'CM2' started by Chirag Wadhwa, May 19, 2022.

  1. Chirag Wadhwa

    Chirag Wadhwa Keen member

    In this question, can anyone explain how is the distribution of the integral expression obtained in the given image?
    Image Link:-https://ibb.co/xGQWrpV
     
  2. CapitalActuary

    CapitalActuary Very Active Member

    The integral is shown to be equal to the sum of 2 independent random variables with distributions N(0, 0.5) and N(0, 2). You know they are normally distributed because that's in the definition of the Wiener process, and you know they are independent because that's the independent increments property of the Wiener process, also in the definition of the Wiener process (i.e. how it behaves between time 1.5 and 2 is independent of how it behaves between time 2 and 2.5).

    The sum of independent normally distributed random variables is also a random variable with a normal distribution, see here: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

    i.e. X~N(0, 0.5) and Y~N(0, 2), so X+Y~N(0, 2.5)
     

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