Sometimes romance languages use only a definite article because the possessive is supposed to be understood, like when speaking about body parts or other things that could only belong to the speaker. It's hard to know when they want you to follow this though, so I got it wrong.
Both are correct according to this reference, which usually seems to include outdated or regional but still correct variations, but you would need a native speaker to comment on which is most used and where.
The following reference only has perso, so I assume that is the safe version of the participle to use.
conjugations verb "perdere"
"perdo" is the conjgation of first person singular of simple present (io perdo)
"perso" (or perduto) is the past participle used in compound tenses