Markov Additive Processes II. Large Deviations

Let \(Xₙ, Sₙ); n = 0,1,…\ be a Markov additive process, ₙ\ taking values in a general state space E, while ₙ\ ⊂ Rᵈ. The large deviation principle is shown to hold for Pₓ\(Xₙ, Sₙ) ∈ A × nΓ\, A ⊂ E, Γ ⊂ Rᵈ, the upper bound holding for closed sets Γ, the lower bound for open sets. The only hypothesis for the lower bound is irreducibility of ₙ\, and nonsingularity of ₙ\. The rate function is characterized in terms of the transform kernel of Pₓ.