Well, yes. If you buy a stock with positive drift and hold it, the model predicts "infinite growth" (in the sense that for any number N you give me I can give y

Well, yes. If you buy a stock with positive drift and hold it, the model predicts "infinite growth" (in the sense that for any number N you give me I can give you a time t at which the E[S(t)] > N). But it might take quite some time, and it's still random, it might be much smaller or much bigger. You could be tempted to employ leverage. However, that introduces the chance of being wiped out. ETA: Real rates are normally positive. So you can achieve the same result by investing in long term bonds with less risk. Just have to wait even longer.