The final decision to pursue or not to pursue charges is with the prosecuting office handling the case.  Sometimes they will decide to pusue charges, even when the victim wants them dropped.  The reasoning behind this process is to put the decision making on the prosecutor--- so that (at least in theory) victims won't be pressured by defendants to drop charges.  If a victim really wants to drop a charge, the state can still go forward if they can prove the case.  If a case is a particularly bad case or if the defendant has been a problem to the community, then they will often go forward with the charges.  If the case wasn't that big of deal and the victim really doesn't want to go forward, many prosecutors will honor the request of the victim.  So.... yes, the state can pick up the charge for any case, including a personal property claim.