We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterise the value function via HJB equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.

This paper studies the eigenvalue problem on R d for a class of second order, elliptic operators of the form L f = a ij ∂ xi ∂ xj + b i ∂ xi + f , associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function f fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.

Zero-sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton-Jacobi-Isaacs equation.

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A system of N weakly interacting particles whose dynamics is given in terms of jumpdiffusions with a common factor is considered. The common factor is described through another jump-diffusion and the coefficients of the evolution equation for each particle depend, in addition to its own state value, on the empirical measure of the states of the N particles and the common factor. A Central Limit Theorem, as N → ∞, is established. The limit law is described in terms of a certain Gaussian mixture. An application to models in Mathematical Finance of self-excited correlated defaults is described.