Modern Stochastics: Theory and Applications

We determine the quadratic variational cost that yields the large-deviation upper bound for the mean exit time from $(-1,1)$ in the small-noise regime for an autoregressive process driven by a continuous two-slope map f lying strictly below the diagonal on $(0,1)$. The Klebaner–Liptser reduction expresses this constant as an infimum, over all finite horizons, of quadratic path actions. Högnäs and Jung studied several nonlinear and piecewise-linear autoregressive maps and evaluated the corresponding variational problem for the two-slope family in the boundary cases in which one of the slopes vanishes; the case in which both slopes p and q are nonzero was left open. We solve this remaining case over the full admissible parameter range, including the finite-horizon attainment question, and give an exact parameter classification of the cost. We also settle the remaining boundary regimes $q=1$ and $p=0,\hspace{2.5pt}q\gt 1$, as well as the limiting case $f(1)=1$. Together with the earlier boundary formulas, this completes the exact variational analysis of continuous two-slope maps satisfying $f(x)\lt x$ for $0\lt x\lt 1$.