Given a continuous probability distribution of betting payoff returns, what is the strategy to maximize expected geometric growth of wealth?

The Kelly criterion is a mathematical formula used to determine the optimal fraction of one's capital to wager on bets with known risk and known payoff, in order to maximize expected geometric growth. Typically, it is formulated for bets that have a probability p to return a fixed fraction r of the amount wagered, and a probability q = 1-p to lose all of the wager.

However, the Kelly criterion can be generalized for bets with continuous payoff structures, such as a bet which returns a fraction of the wager ranging from -1/2 to 2 with uniform probability.

To determine the fraction f to bet in order to maximize the expected growth, we determine the expected logarithmic growth of wealth E as a function of f and maximize it by using analysis. Let P(r) be the probability density function of the bet returning a payoff of r. Then in which a and b are the lower and upper bounds of the payoff, respectively.

The integral is only defined if (1+fr) > 0 for all r in [a,b], implying that one should never bet a fraction more than -1/a for a negative lower bound a of the payoff. Practically, this shows mathematically that one should never directly short a stock, which is well known to have the potential of infinite loss. (However, the legal option of declaring bankruptcy may make this moot.)

Maximizing the expected growth, Unfortunately, at this point, numerical methods must be used to solve for f in general. (At least I wasn't able to find a solution. I tried various manipulations, including Laplace transforms, but had no luck. If you have a solution or insight, I'd love to hear about it!)

An alternative form involving the cumulative distribution function of the payoff F(r) can be found using integration by parts: The CDF F(r) = 0 at the lower bound r = a, and F(r) = 1 at the upper bound r = b, implying another relationship that can be used to find f numerically