Finance Option Spread Thesis
The trick is to adjust the terminal payoff function of the (knock-in) barrier security in such a way that the simple contingent claim with this payoff function has the same price as the barrier security.
The same authors also have a paper on complex barrier options and lookbacks.
Nowadays very liquid markets exist for "plain vanilla" options (put- and call-options) with a variety of characteristics (expirations and strike-prices).
In many cases such options are much more natural hedging instruments for "exotic" options.
There is a lot a papers by Paul Glasserman, Brodie & Glasserman, for instance. No such problems with the ingenious least-squares idea of Longstaff & Schwartz . It is possible to keep the thesis completely theoretical (i.e.
A "fundamental analysis" approach to finding abnormal returns ("good odds/bets") is given by Dixon and Coles.
The "official" reason is that betting markets are quite close to the idealized markets with uncertainty we usually analyze (eg.
there is a clear terminal at which the outcome is decided), so we may get valuable information about decisions under uncertainty. One very common feature (try a Google-search) in the analyses of betting markets is the so-called "favorite/longshot-bias". Or in other words: You get a higher rate of return from betting on favorites than on "longshots".
Well-definedness of the problem: Bensoussan (1984), Karatzas (1989), Myneni (1992).
Even in complete(ly) discrete models (as for instance I&F-teori) American securities usually aren't rigorously defined & analyzed (there's no need to because "it's obvious").