We study an optimal investment control problem for an insurance company having two business branches, life annuity insurance and non-life insurance. The company can invest its surplus into a risk free asset and a risky asset with the price dynamics given by a geometric Brownian motion. The optimization objective is to maximize the survival probability of the total portfolio on the infinite time interval. In the absence of investments, the portfolio surplus is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes and upward jumps are interpreted as random gains that arise at the final moments of the life annuity contracts realizations (i.e. at the moments of death of policyholders) as a result of the release of unspent funds. The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. The solving to the optimization problem that yields the maximal survival probability, as well as the optimal strategy, is related to the classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation, if this solution exists. In the considered risk model, HJB includes integral operators of two types: classical Volterra operator and singular Volterra one, but their sum is not a Volterra-type operator. It makes the asymptotic analysis of the solution quite complicated. However, for the case of small jumps (when the jumps have exponential distributions), we obtained asymptotic representations of solutions for both small and large values of the initial surplus. Also, in the case of zero interest rate, we obtain a Lundberg-type bound for the survival probability corresponding to the optimal choice among strategies with constant amount of money invested to a risky asset. It turns out that this bound coincides with the asymptotic representation at infinity for the value function in the case of zero interest.

Discrete time control systems whose dynamics and observations are described by stochastic equations are common in engineering, operations research, health care, and economics. For example, stochastic filtering problems are usually defined via stochastic equations. This talk discusses sufficient conditions for solving such problems by value iterations, when the goal is to minimize expected total costs. The model is usually defined by a pair of stochastic equations and by a one-step cost function. Stochastic equations describe the state and observation processes, and these equations are defined by transition and observation functions. We formulate sufficient conditions on observation, transition, and one-step cost functions for convergence of value-iteration algorithms for problems with finite and infinite horizons. It is well-known that nonlinear and linear filtering problems can be presented as Partially Observable Markov Decision Processes (POMDPs). The paper applies contemporary results on convergence of value iterations for Markov Decision Processes (MDPs) and for POMDPs to filtering problems. It formulates conditions on observation and transition functions which imply weak continuity of the filter. Weak continuity of the filter means weak continuity of transition probabilities between belief states. The sufficient condition on one-step functions is their K-inf-compactness. Weak continuity of the filter and K-inf-compactness of costs imply the existence of optimal policies, validity of optimality equations, and convergence of value iterations. The sufficient conditions for weak continuity of filters described in the talk hold for broad classes of nonlinear filters and for Kalman filters. In a broader sense, this talk describes new methods for studying and optimizing discrete-time nonlinear filtering problems.