APPENDIX: ITÔ'S LEMMA ON THE RATIO OF TWO PROCESSES

In this appendix we state without proof ⁵ a couple of useful results, in the development of this paper, for mixed diffiision–jump processes. Given the homogeneous linear stochastic differential equations:

where dQ_X are dQ_Y uncorrelated Poisson processes and dW_X, dW_Y are Wiener processes satisfying Cov(dW_X, dW_Y) = ρ_XY dt, and the Poisson processes are independent of trie Wiener processes, then the stochastic differentials of the ratio X_t/Y_t and of the product X_t Y_t , are given, respectively, by:

Note

⁵ For the proofs, we refer the reader to Gihman and Skorohod (1972).