This is why exchanges with public asset lists are important (for example, BitFunder). Even if BitFunder suddenly shuts down, asset issuers will be able to know who owned what.
Ah yes, thank you for this link.
But I guess you also have to rely on the asset issuers themselves honouring the contracts made on any individual stock exchanges - ultimately they could just neglect to pay a dividend for 1 month. How would you enforce that? And without any formal contracts or if a stock exchange went down, how would you trade out of the stock at a future date?