The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate.
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0} The figure plots the amount gained with a win on the x-axis against the fraction of portfolio to bet on the y-axis.
1 a=0.
according to the weak law of large numbers.
∑ i p i < 1 , {\displaystyle \sum _{i}{p_{i}}<1,} ∑ i β i < 1 \sum _{i}{\beta _{i}}<1
E log ( S t ) = log ( S 0 ) + ( μ − σ 2 2 ) t .
S t ′ = S 0 ′ exp ( ( f μ − ( f σ ) 2 2 ) t + f σ W t ) {\displaystyle S'_{t}=S'_{0}\exp \left(\left(f\mu -{\frac {(f\sigma )^{2}}{2}}\right)t+f\sigma W_{t}\right)}
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