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

2.2 Problems and Solutions

2.2.1 Basic Properties

1. 1. Let be a probability space. We consider a symmetric random walk such that the -th step is defined as
   

   where , . By setting , we let

   

   where .

   Show that the symmetric random walk has independent increments such that the random variables

   

   are independent.

   Finally, show that and .

   Solution

   By definition

   and for , ,

   Because and since is independent ...