The stochastic goodwill problem

Stochastic control problems related to optimal advertising under uncertainty are considered. In particular, we determine the optimal strategies for the problem of maximizing the utility of goodwill at launch time and minimizing the disutility of a stream of advertising costs that extends until the launch time for some classes of stochastic perturbations of the classical Nerlove-Arrow dynamics. We also consider some generalizations such as problems with constrained budget and with discretionary launching

Martingale approach to stochastic differential games of control and stopping

We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.Comment: Published in at http://dx.doi.org/10.1214/07-AOP367 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default

In this paper we study the optimal m-states switching problem in finite horizon as well as infinite horizon with risk of default. We allow the switching cost functionals and cost of default to be of polynomial growth and arbitrary. We show uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem with risk of default. This problem is connected with the valuation of a power plant in the energy market.Comment: 25 pages; Real options, Backward stochastic differential equations, Snell envelope, Stopping times, Switching, Viscosity solution of PDEs, Variational inequalities. arXiv admin note: text overlap with arXiv:0805.1306 and arXiv:0904.070