You need an exponentially increasing reward for your argument to go through. In particular, this doesn’t prove enough:
Since at each moment in time, you face the exact same problem (linearly increasing reward, α-exponentially decaying survival rate)
The problem isn’t exactly the same, because the ratio of (linear) growth rate to current value is decreasing over time. At some point, the value equals β/α (is the right expression, I think?), and your marginal value of waiting is 0 (and decreasing), and you sell.
If the ratio of growth rate to current value is constant over time, then you’re in the same position at each step, but then it’s either the St. Petersburg paradox or worthless.
You need an exponentially increasing reward for your argument to go through. In particular, this doesn’t prove enough:
The problem isn’t exactly the same, because the ratio of (linear) growth rate to current value is decreasing over time. At some point, the value equals β/α (is the right expression, I think?), and your marginal value of waiting is 0 (and decreasing), and you sell.
If the ratio of growth rate to current value is constant over time, then you’re in the same position at each step, but then it’s either the St. Petersburg paradox or worthless.