In Russian we do not use possessive pronouns when they are implied by other words or the context. There is no need saying that the ticket I lost was mine, unless I have lost your ticket.
So, we can simply say
Я потеряла билет. - I have lost my ticket.
Я приду с мужем/другом/коллегой. - I will come with my
Она разговаривала с сестрой. - She was speaking with her sister.
Я помою руки. - I will wash my hands. (Whose hands shall I wash? There is no need stating that unless I decide to wash somebody else's hands)
If it is still unclear, whose ticket, colleauge or sister it is, we can use the universal reflexive possessive pronoun свой. It stands for мой, твой, их, ваш, наш, etc., but is reflexive.
Я потеряла свой билет.
Я приду со своим коллегой.
Она разговаривала со своей сестрой
In English you just don't have such a short and convenient word which substitutes for all the possessive pronouns at once. That's why it is translated as a phrase.
We do not use свой in certain situations. E.g., when introducing sth or sb:
Это мой сын.
Это моя работа.
Вот наш дом.
I would say, that we can hardly use свой in the Nominative case, as it usually refers to the objects of a sentence, and objects require objective cases (all the rest but Nominative).
Sometimes свой stands for own. And in this meaning it can be used in the Nominative case.
У них свой дом. - They have their own house (they do not live in a block or share a house with anybody).
I think that in English the possessive pronouns everywhere just spell the articles, that's why they are used in such sentences as "He came with his wife". Otherwise you should say "He came with the wife" which would sound less polite or clear, maybe? In Russian there are no articles, which simplifies the situation.