I posed this question thinking of the "cashless society". As many of you have commented on the cash -v- card discussion, our financial transactions now consist mainly of the transfer of digital information (representing cash) from your accumulation to the accumulation of the recipient, represented by digital information. There is no transfer of any physical objects.
When we think of "finding" something, we tend to associate that with the finding of a physical object. If any of you found a wallet with either some cash or credit cards and other things necessary to exist in society, I'm sure that your ethics would lead you to having the owner found and the wallet returned. But how do you deal with the situation of finding an unexpected increase in your financial accumulation through the receipt of digital financial information? My thought is that, in our modern society, digital financial information has replaced physical banknotes. Therefore, the entry in your account represents a "thing".
Let's have a look at the way larceny is defined at Law. In New South Wales, The High Court defined larceny in the leading case of Ilich v R in 1987:
“A person steals who, without the consent of the owner, fraudulently and without a claim of right made in good faith, takes and carries away anything capable of being stolen with intent, at the time of such taking, permanently to deprive the owner thereof.”
The offence is committed if a person intends, "to deprive the owner thereof.” For an allegation of "stealing by finding" to be proved, it must be shown that the accused made no, or insufficient, attempt to find the owner. Merely to leave it there and say nothing is proof of making no or insufficient attempt to find the owner.
What would I do?
Contact the police and have them make a "Found Property" report and provide you with a reference number for that report. In NSW that is called an Event Number. Trying to explain how what you want to report as finding something could be a bit hard, but you should be able to convince even the most lethargic constable to create the Event. A benefit of making that report is that, if you say when making the report that you would like to claim the money after all enquiries have been made to find an owner have failed, then a Court could give it to you. Always a good idea if you happen to find a backpack full of money and some drugs.
As soon as practicable, contact your financial institution's Fraud Unit and let them know what has happened. Because the deposit had to have been made digitally, the Fraud Unit should have the ways and means to locate the source of the deposit.
While talking to the Fraud Unit, ask them how to have the deposit transferred from your account to a secure account of the financial institution pending resolution of the matter. That removes your possible liability under the Law for "permanently depriving" the owner of the possible interest the amount could be earning.