I don't understand: the situation here is one where your only option is to be all in cash or all in the stock. The Kelly criterion only makes sense when you can choose an arbitrary fraction to be in each.
(And the Kelly criterion amounts to maximizing E(delta log wealth), which is exactly what I'm proposing. If you have to wager your entire bankroll, any gamble with a nonzero chance of bankrupting you has E(delta log wealth) = -infinity and just sitting on your cash is better.)
Ah, I missed that part of the OP. So then I think your argument is correct.
Let's suppose you start with $1000 to invest, and the only thing you can invest it in is stock ABC. You are only permitted to occupy two states:
* All assets in cash
* All assets in stock ABC
You incur a $2 transaction fee every time you buy or sell.
Kind of annoying limitations to operate under. But you have a powerful advantage as well. You have a perfect crystal ball that each day gives you the [probability density function](http://en.wikipedia.org/wiki/Probability_density_function) of ABC's closing price for the following day (but no further ahead in time).
What would be an optimal decision rule for when to buy and sell?