Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

1. 14. Generalised Itō Integral. Let be a probability space and let be a standard Wiener process. Given that is a simple process, show
   

   and

   

   Solution

   For the first result, using Taylor's theorem on and subsequently applying Itō's formula we have

   

   Taking integrals from to ,

   

   and rearranging the terms, finally

   

   since .

   As for the second result, from Taylor's theorem and Itō's formula

   Taking integrals ...