David Applebaums' "Lévy processes and stochastic calculus"
It looks like integrating with respect to an $\alpha$-stable process is addressed directly on page 212 with the Ito formula to follow starting on page 218. Sadly the Ito formula section is not included in the Google preview.
I have not read this particular book but I have read his older "Levy Processes" in graduate school and I found it quite readable.