Kelly Criterion - if I bet 20% of my bankroll, outcome is undecided and I need to make another 10% bet - is 2nd bet 10% of total or of remaining 80%?

I am trying to understand what the "correct"/optimal way of betting is using the Kelly Criterion but for bets that don't immediately give an outcome.

For example, I have 100 dollars. I get a buy signal, I run Kelly Criterion and it says optimal is 20% bet, so I put 20 dollars on that buy signal. Now, let's say in the next timestep, I get another buy signal on something else, but the first position hasn't been closed yet - so I have 80 dollars cash and 20 locked up in a position. The Kelly Criterion says 10% is optimal - would I bet 10 dollars on this new position (10% of my total portfolio value) or 8 (10% of remaining free cash)?

Any references or justification would be appreciated. Thank you!

• The conceptual generalization you may be looking for is a "growth optimal portfolio" which can be obtained by maximizing the expected logarithm of terminal wealth: $\max \operatorname{E}[\log W]$. (My answer here might get you on the right track. In theory, you could solve a sequential problem with backwards induction: i.e. solve all possible cases of the last period $t$ problem then use that to solve all possible cases of $t-1$ problem etc.... Commented Feb 23, 2023 at 1:34
• In most any practical investment setting though, you can't solve for that kind of optimal solution because you don't have anything resembling enough information. So a practical way forward may look really quite different. Commented Feb 23, 2023 at 1:36
• Yes. It sounds to me like quite a challenging problem to formulate and solve. Much more complicated than what Mr. Kelly considered in his 1956 paper. Commented Feb 23, 2023 at 19:38