Continuous time can sometimes give you closed form solutions for stuff, which is nice. Moving to the computer is a little harder, though. Besides that, they're pretty similar. Maybe get comfortable with Hamilton Jacobi Bellman equations and viscosity solutions?

That's the problem with continuous-time stuff, it's a big investment. Getting comfortable with HJB and viscosity solutions will take time if you don't have a good background in stochastic calculus...

What? You can solve HJBs and find viscosity solutions without any stochastic calculus. Conversely, you can do stochastic calculus without touching HJBs or viscosity solutions. (Of course, if you’re doing stochastic calculus in economics, you will probably need an HJB, or even a viscosity solution, but my point stands.)

This post conflates two different things that are separate.

Yes of course but who cares, this is an economics forum... I mentioned HJB because above poster did so, but uf you insist, I should have said "learning optimal control in continuous-time take time". And if OP comes from econ undergrad, I don't think it's worth it now, unless there is a special idea in mind.

My research agenda (OP here) is heterogeneous agents models (I am interested in the incorporation of a richer description of firms' portfolio choice and financial sector (especially maturity transformation) to these models). Because I know, that these models are computationally very challenging, If you want to build something bigger & more realistic, and hence I get interested in continuous time, because (according to people like JFV, Judd, Moll,...) it allows to solve these models much more efficiently than what you can achieve in discrete-time settings.