||1.Since the fundamental work of V.E. Benes et.al. Sochastics (1980) 4, 39-82, where the problem of tracking a standard Brownian motion by a finite variation process was explicitly solved, the problems of singular control have found multiple applications in a number of fields, like queues, networks, insurance and mathematical finance. These applications are multiplied when the previous problem is accompanied by another player, which can collaborate or compete, using a discretionary stopping time. In this talk we shall present some recent results for controlled Levy processes, putting special emphasis in the applications.
2. In this talk, I will present recent results on phase transitions arising in certain stochastic optimal control problems. More precisely, we are concerned with the maximization problem of long-run-average for controlled Markov chains, or Markov decision processes, on infinite graphs. Some phase transition phenomena are observed in terms of the associated optimality equation. The results provide a stochastic control interpretation for the discrete model of homopolymers, as well as the principal eigenvalue of discrete Schrodinger operators on graphs.