I am trying to wrap my head around some features of singular stochastic control, and one of the things that has been bothering me is that authors sometimes take the singular control to be left-continuous with right-limits (see, for example, [1]). Is there a particular reason for that? For me it seems more natural to consider RCLL processes, specially because the theory of stochastic integration (as far as my knowledge goes) is developed for integrators which are RCLL.
Of course, in most papers on singular stochastic control the singular control is also restricted to be non-decreasing, so stochastic integrals where it is the integrator become regular path-wise Lebesgue-Stieltjes integrals. But the question still remains.
I have a hunch that maybe if one defines the singular control to be RCLL, then its left limit at the starting time should become part of the "data" of the problem, thus adding an argument to the value function, but this is only based on some intuition and might be wrong.
I appreciate any insights/references on this matter.
On a similar note, but also of independent interest, is there a general theory of stochastic integration with LCRL processes as integrators?
References
[1] Haussmann, U. G., & Suo, W. (1995). Singular Optimal Stochastic Controls II: Dynamic programming. SIAM Journal on Control and Optimization, 33(3), 937–959. https://doi.org/10.1137/S0363012993250529
