It is “not too difficult” in the sense that we can recycle components from the literature. I would remove the H flow by considering a time zero investment in the riskless asset of H/r, that will generate an interest stream of H dt. Then, upon hitting the lower boundary we get the principal back, so a payment of H/r. Likewise, at the upper boundary we get H/r plus the payment from exercising the option [$] S^* - X [$]. Each of the two known and constant payments can then be valued as “rebates” in a perpetual double barrier model. Jamshidian may have been the first to give an explicit solution for this, involving a ratio of hyperbolic sines (but no infinite sum, which one would expect). Optimizing the barrier locations looks like a numerical problem, though. And, when we’re almost done, we must remember to subtract H/r and check that the answer is still positive.

I don't like this approach. Unfortunately, it feels like a fudge. (btw H is like property tax, so seeing it as a kind of rebate is hard to get my head around),

It has commonality with first exit times but the boundaries are unknowns.

Anyways, I thought about it logically and I solved it as a ODE free boundary value problem. We solve 2nd order ODE

[$]V(S) = AU_{+} + BU_{-}[$] where A and B are to be found and so on. In general we have four things to find

. A,B

. [$]S_*,[$][$]S^*[$]

These are found from the instantaneous stopping condition for V and the smooth fit for [$]V'[$]. We take the derivative but not for optimising stuff but for continuity reasons.

Then afterwards I found the same results from the work of my comrade Shiryaev.