The 25-year-old Irishman, who could be banned for six months, denied taking the drug and is awaiting the result of a B sample following the test at Chantilly in France on July 19.
"I have never taken cocaine in my life and I will do everything that I can to prove that I have not taken cocaine," said the jockey, top rider for Sheikh Fahad Al Thani's Qatar Racing.
"I want to thank those who are supporting me and in the meantime I want to keep riding winners and focus on my career."
The PJA said Murphy, who rode 168 winners in 2019 and has been on course to retain his title this year, had been informed by France Galop on Aug. 19 of the positive result of his urine sample.
"On the same day, Oisin organised for an independent laboratory to undertake hair sampling to prove his innocence," it added.
"...On 26th August the results of the hair test returned completely negative for metabolites of cocaine, which was entirely expected."
Kieren Fallon and Frankie Dettori both served six-month bans after testing positive for cocaine in 2006 and 2012 respectively. (Reporting by Alan Baldwin, editing by...)