I know two applications of local time in option pricing theory.
First, it allows a derivation of Dupire's formula on local volatility in a neat way (i.e. without resorting to differential operator theory which I'm not really acquainted with).
Second, it can be used to show that you can't perfectly replicate (for example) a call option final payoff by the naive strategy of letting conditional orders on the market consistent with the strike in a Black-Scholes context.
Does anyone is aware of other applications of local times in finance, answers with references would be most appreciated ?