A simple roadmap for solving HJBE and optimal stopping problems

[jlperla]

@ChrisRackauckas @FernandoKuwer
More later, but this should give a sense of where we will eventually want to go, adding on pieces to test each time. First, solve everything in the stationary setup (i.e. only a singleton in the time dimension, which leads to an ODE).

• stationary, univariate diffusion with reflecting barriers, a uniform grid, and a constant discount rate. Posted as Simple example of discretizing a univariate diffusion with reflecting boundaries #44
• add in absorbing barriers instead of reflecting barriers (especially an absorbing at the bottom and a reflecting at the top). Roughly shown in (10) and (11) of Finite Differences
• add in a jump-diffusion process with a jump size as a function of the current state. The reflecting barriers there are tricky.
• add in support for non-uniform grids. Discussed in detail in Finite Differences

In all cases, the easiest test is to look at the generated discretized matrix and to solve a simple HJBE, as discussed in (54) to (56) of Finite Differences

While this is the most important set of features for our immediate project, there are plenty of other useful features. First, look at the time varying versions, which generate a PDE:

• Add in time-variation in all parameters/functions of the univariate jump-diffusion process (including boundaries). The assumption is that the final step the parameters have converged (and hence it is at the stationary level). See Section 2 of Finite Differences
• Add in time-varying discount rates

Next, expand the set of stochastic processes

• Add in an additional discretely-valued state variable which evolves according to a continuous-time markov chain.
• Add in 2 dimensional diffusion processes. Need to be careful with the upwind process there, since monotonicity is subtle.