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In this paper, we study the asymptotic of exit problem for controlled Markovdiffusion processes with random jumps and vanishing diffusion terms, where therandom jumps are introduced in order to modify the evolution of the controlleddiffusions by switching from one mode of dynamics to another. That is,depending on the state-position and state-transition information, the dynamicsof the controlled diffusions randomly switches between the different drift anddiffusion terms. Here, we specifically investigate the asymptotic exit problemconcerning such controlled Markov diffusion processes in two steps: (i) First,for each controlled diffusion model, we look for an admissible Markov controlprocess that minimizes the principal eigenvalue for the correspondinginfinitesimal generator with zero Dirichlet boundary conditions -- where suchan admissible control process also forces the controlled diffusion process toremain in a given bounded open domain for a longer duration. (ii) Then, usinglarge deviations theory, we determine the exit place and the type ofdistribution at the exit time for the controlled Markov diffusion processescoupled with random jumps and vanishing diffusion terms. Moreover, theasymptotic results at the exit time also allow us to determine the limitingbehavior of the Dirichlet problem for the corresponding system of ellipticpartial differential equations containing a small vanishing parameter.

On the asymptotic of exit problems for controlled Markov diffusion processes with random jumps and vanishing diffusion terms