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

1. 24. Let be a probability space and let be a Poisson process with intensity and be a standard Wiener process relative to the same filtration , . By considering the process
   

   where , use Itō's formula to find an SDE for .

   By setting show that solution to the SDE can be expressed as

   

   Solve the differential equation to find and deduce that .

   Solution

   By expanding using Taylor's theorem and taking ...