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

3.2 Problems and Solutions

3.2.1 Itō Calculus

1. 1. Itō Integral. Let be a probability space and let be a standard Wiener process. Let the Itō integral of be defined as the following limit
   

   where , for .

   Show that the quadratic variation of is

   

   and hence

   

   Finally, show that the Itō integral is a martingale.

   Solution

   For the first part of the solution, see Problem 2.2.6.1 (page 89).

   Given and ...