I have thought a bit about that question. At a first glance, it makes no sense for a bank to use neither a centralized blockchain - because it would offer no advantage to traditional database accounting systems - nor an open blockchain with PoW or other consensus mechanism - because they would give up control about the transactions and expose them to certain risks.
An interesting post has been written
in the Ethereum blog. There are two main categories where "restricted" blockchains could make sense.
First as an "inter-banking" transfer mechanism ("consortium blockchain") where the trust between institutions can be lowered a little bit (blocks must be signed by the majority of connected institutions, but not by all, and one malbehaviouring node - e.g. after a hack - would not be able to bring down the system).
And then as a private "in-bank" blockchain - but such private blockchains would not be actually very different from the traditional databases, only that they would add a kind of "cryptographic proof" for the state of the system. It would be a kind of additional security layer. The author of the blog post, however, thinks that with less complex measures like
libsnark that could also be achieved.
What would have to be analyzed if these two ways really represent a more efficient approach to security than already known traditional security mechanisms. I think the real benefit of the blockchain is that it enables decentralized systems to work - so "bank applications" would only be a side-effect and not be at all important for Bitcoin.
I've read the linked ethereum post and found it pretty interesting, thanks.
In connection with the above mentioned in-bank blockchain, I don't think that a kind of solution like that will be implemented at all. My opinion is that banks don't need a blockchain like solution just for intenal purposes. They already have their own system that's accurate, fast and has security functions implemented. In-bank transfers usually complete in a second, or even less, it's just modifying 2 values in a database.
Making the system redundant (2 server farms in the country) can provide enough security for disaster recovery purposes with a proper backup procedure implemented and used. If they create a new blockchain solution to eliminate the 2 server farms and they provide dedicated computers in each branch to keep the database always up-to-date and updated, they can save the price of the server farms but current blockchain solutions will lead to slower service, which is unacceptable now.
Blockchain solution is needed in case of inter-bank transfers because the current system is relatively slow. In my country, transfers (same currency, same country) completes in an hour usually, but in peak time it's a bit more. This could be done faster using a restricted blockchain, used by the banks in the same country. Blockchain for international transfers can be a real challenge.