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Motivated by a risk process with positive and negative premium rates, weconsider a real-valued Markov additive process with finitely many backgroundstates. This additive process linearly increases or decreases while thebackground state is unchanged, and may have upward jumps at the transitioninstants of the background state. It is known that the hitting probabilities ofthis additive process at lower levels have a matrix exponential form. We herestudy the hitting probabilities at upper levels, which do not have a matrixexponential form in general. These probabilities give the ruin probabilities inthe terminology of the risk process. Our major interests are in their analyticexpressions and their asymptotic behavior when the hitting level goes toinfinity under light tail conditions on the jump sizes. To derive thoseresults, we use a certain duality on the hitting probabilities, which may havean independent interest because it does not need any Markovian assumption

Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes