= Finite-horizon drawdown of a two-sign-memory process
{title2=$D_{\max}(N)=2\lfloor N/2\rfloor+2\quad(N\ge1)$}
For increments $X_t=a_{t-2}a_{t-1}+a_t$ with freely admissible starting sign history, the worst absolute <financial drawdown> over $N$ steps is the displayed value. A four-state recursion $d_{n+1}(b,c)=\min_{a=\pm1}[d_n(a,b)+ab+c]$ computes the minimum cumulative increment, with zero initial costs. The minimum is attained by nonpositive increments: $(-2,0)^k,-2$ for odd $N=2k+1$, and $(-2,0)^{k-1},-2,-2$ for even $N=2k$. The horizon-free bound is infinite. At most two strictly negative increments can be consecutive, but confusing that losing-run bound with drawdown would be incorrect.
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