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For a class of additive processes driven by the affine recursion $X_{n+1} =A_n X_n + B_n$, we develop a sample-path large deviations principle in the$M_1'$ topology on $D [0,1]$. We allow $B_n$ to have both signs and focus onthe case where Kesten's condition holds on $A_1$, leading to heavy-taileddistributions. The most likely paths in our large deviations results are stepfunctions with both positive and negative jumps.

Sample-path large deviations for a class of heavy-tailed Markov additive processes