From: McGraw Hill’s 2010 GRE :
English possessive determiners (my, our, your, his/her/its, their – sometimes called possessive adjectives) must match the person and number of the possessor and not the noun phrase to which they are linked.

Richard likes his hot dogs with lots of relish. The word his is third person singular to match with Richard, NOT third person their to match with hotdogs.