Gaps change everything

Modify the swing system of 3.1 so that 3% of trades gap through the stop and lose $4R$: R-multiples $+2.5$ w.p. 0.40, $-1$ w.p. 0.57, $-4$ w.p. 0.03. (a) Compute the expectancy and the Kelly fraction (numerically). Compare with the no-gap system ($f^* = 0.16$). (b) For risk fractions 1%, 2% and 5%, compute the Lundberg exponent and the bounds on $P$(ever lose 20%) and $P$(ever halve). Compare with the no-gap table in 3.7. (c) Verify that $\theta = 1$ at the new Kelly fraction (Proposition 3.14). (d) What does this imply for a trader who estimates the R distribution from a few dozen trades, none of which gapped?