During the dispute each party is responsible for their own costs. If the parties reach a settlement between themselves, they can agree between them who should pay those costs.
If the matter goes to a trial a Judge will decide who pays the costs. The general rule is that the losing party will pay the successful party’s costs. A Judge has a wide discretion on costs and can reduce the successful party’s costs if he or she believes they may be disproportionate. There is no automatic provision that the deceased’s Estate pays the costs. It is important that you consider the impact of costs before taking on a claim.