Ohio Rev. Code Ann. § 2323.36

After final judgment in an action in which security for costs has been given, on motion of the defendant or a person having a right to such costs, or a part thereof, after ten days' notice of such motion, the court may render judgment in the name of the defendant, or his legal representatives, against the surety, his executors, or administrators, for the amount of costs adjudged against the plaintiff, or so much thereof as is unpaid. Execution may be issued on such judgment, as in other cases, for the use and benefit of the persons entitled to such costs.