"their expert" is correct and usable in written English.
You can use it to refer to someone else's expert, such as in the sentence "The company asked their expert to review the proposal."
They deserve applause for their expert bidding.
And I must try to be their expert friend.
He was chosen by the UK government as their expert.
They should write articles giving their expert analysis.
But no amount of their expert heavy lifting can get this turkey to fly.
The fair also featured speakers who took the stage to offer some of their expert advice.
Some promise to evaluate pictures quickly and to call back with their expert opinion.
Each judge scores each dance independently, based on its merits and in their expert opinion".
Stevenson's novellas are strongly plotted, particularly in their expert thematic dovetailing.
HOW did investment banks, paid for their expert advice, pick such lemons?
Otherwise, this was the antithesis of their expert dissection of Liverpool on Monday.
