Is discrete time stochastic processes enough to solve finance problems in practice? There is a lot more work with continuous-time stochastic analysis than discrete time. A lot of work is done in the Itô-theory. 
Is this theory needed in practice? Or could we just work with discrete times, and do numerical calculations? Are there some problems/examples which we have to use continuous time, and it would be more difficult to do in discrete time? It seems that when we solve problems in practice, we almost always discretize the time when we problem, so why not just work with discrete models from the start? Then we do not have to worry about path-properties and the proofs would be simpler?
Do you think it is worth doing the continuous time other than it beeing interesting mathematically? If so, which problems is it used for?
 A: Most production implementations of stochastic financial models are discretisations of continuous time models as you point out. Having worked on desks trading credit, rates and inflation derivatives, am pretty sure every single model that we ever used was of this form. The discretisations can obviously be monte carlo, or finite difference schemes. 
Re starting with continuous models, mathematically, continuous treatments are convenient frankly more so than just interesting - consider the simplicity of simple old black scholes versus eg writing down the price a lattice is going to come up with. Now if this does make, in certain cases proofs more difficult, then clearly some of that convenience is undermined.
Continuous definitely trumps closed form btw (the latter is a nice to have but have only seen analytic formulae used on desk for quick calculations, not actually to price and risk trades - practically everything uses numerical methods).
So I would say continuous modelling for ease of conceptual understanding and clarity of exposition, and subsequent discretisation as an implementation paradigm of this is here to stay. The fact is that the calibration instruments for any model are discrete, so discretisation is an inherent property of any calibration to pricing scheme. That said, knowing an analytic formula for a delta is useful even if the implementation will just reprice on perturbed market data. 
Just to caveat and not to generalise, I guess the range of models I am talking about are deterministic reduced form or stochastic intensity processes for credit, everything from short rate to LIBOR market models for rates, american monte carlo techniques (like longstaff-schwartz regressions) for nonlinear continuation value payoffs like CVA etc etc, and hybrid treatments of these (for example correlated diffusive credit/rates for credit-rate hybrids). So in some sense re your last question, continuous time modelling is used almost surely (don't think I've ever seen anything else for exotic derivatives), even if implementations are almost surely discrete.
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