Is it a solved issue for equities? How do you know the shares you purchased are valid?

It's solved to an extraordinarily high degree. 99.999% or greater. There are fringe cases that pop up from time to time and make headlines; the reason they make headlines, is because trillions of dollars in equities trade hands every year in the US, and such cases are very rare.

It's because there's an established system of trust, liability, responsibility, regulation and oversight in the network, assuming you're using any number of several dozen popular brokers (and that you're a common investor). Microsoft has a vested interest in knowing how many shares it actually has outstanding and who owns them; there's a chain that follows, that includes the CEO, CFO, board, other large powerful investors, regulators, common investors, and so on and so forth.

The scale and solved nature of it is roughly 100x that of Bitcoin in contrast.

IIRC there was the issue with fake shares of Overstock being traded:

http://investors.overstock.com/phoenix.zhtml?c=131091&p=irol...

> It's because there's an established system of trust, liability, responsibility, regulation and oversight in the network

You can just admit you don’t know.

Essentially, my understanding is that all the shares are held by a common trusted corporation, the Depository Trust & Clearing Corporation (DTCC), or something like it. In this way, transfers in ownership are as fast as updates to their ledger about who owns what.

I suspect it's only a matter of time before something like the DTCC is established for Bitcoin. Yes, from one perspective it would defeat the "trust-less" part of the system, but from another perspective it would enable instantaneous trades both within exchanges and between them. All that's necessary for this scheme to work is for the exchanges and other market participants to trust that the depository company will deliver on any assets it claims to owe them.