This sentence doesn't sound correct but it must be. Can anyone explain why it uses "your" instead of "you" considering...

The sentence reads: "I have completed a comprehensive training program and would appreciate your considering me for this position."

Thank you.
The possessive (your) is correct before a gerund ( -ing word).

People do use the objective form (you) in those cases, but the possessive form is usually considered better English. your is more formal with the gerund, so it's especially good in this sentence because it's an application for a job, and you don't want to sound too casual in that situation.

Ah! Thank you kindly!