arXiv2026
In this paper, we study a class of Markov-modulated multiplicative Lindley-type recursions of the form $W_{n+1}=[V_nW_n+Y_n(V_n)]^+$, where the additive term may depend on the multiplicative coefficient and on a common finite-state background Markov chain. The main feature of the model is that the multiplicative coefficient is allowed to take negative values, which destroys the usual monotonicity structure of Lindley-type recursions. We consider two cases. In the first one, $V_n$ may take the values $1$, a constant $a\in(0,1)$, or a negative value, according to specific probabilities, while $Y_n(V_n)=S_n-A_{n+1}$ and the relevant Laplace--Stieltjes transforms are assumed to be rational. In the second one, $V_n\in\{1,-1\}$, and the additive term has a more general form depending on $V_n$. For both models, we derive vector-valued functional equations for the stationary transform. For the first model, we obtain a recursive transform representation, while for the second model we derive a finite-dimensional characterization based on root-cancellation conditions. We also provide a recursive procedure for computing stationary moments and discuss the light-tailed asymptotic behaviour. An extension based on the Farlie--Gumbel--Morgenstern copula is also presented. Numerical experiments show how the transform representation can be used to compute stationary moments, tail probabilities and quantiles, and validate the results against Monte Carlo simulation.