Concerning this fact, the paper by [Carmona et al.][1] refers (on p. 28) to "Example 2 of Section 4.9.13 of Liptser and Shyriaev [19]". 

A proof of this fact can also be found e.g. in Section 9.3 of [this project, p. 66][2]. 


  [1]: http://www.numdam.org/item/AIHPB_2003__39_1_27_0/
  [2]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2087921