You don't need a cryptocurrency specifically to do this. Rather you can piggy back on a different consensus system: https://petertodd.org/2017/scalable-single-use-seal-asset-tr...

For example, while it'd be a bit of a hack, you could fairly easily re-use even the various Certificate Transparency blockchains to trade stocks.

Also, I wouldn't say that the incentive mechanisms keep the ledger running. Rather they keep the PoW consensus mechanism running; in all existing trustless decentralized currencies it's your node that verifies the ledger is working correctly, and other people are voluntarily distributing the data without a reward.