Bernt Øksendal

University of Oslo

Optimal pricing strategies and Stackelberg equilibria in time- delayed stochastic differential games.

Abstract: In the classical newsvendor problem there are two agents:

(i) The manufacturer, who today (i.e. at time t-\delta) decides the unit price to sell the manufactured goods for to the retailer, with delivery tomorrow (at time t)

(ii) The retailer, who then today (at time t-\delta) decides the quantity to order from the manufacturer and the price to sell each item for to the public the next day.

What is the optimal price set by the manufacturer and the optimal quantity to order and the optimal retailer price? The problem is that neither of these agents know what the demand will be the next day, only its probabilistic distribution. This is a problem that occurs in many situations, for example in the pricing of electricity in a liberated electricity market. We generalize this classical newsvendor problem to continuous time and a jump diffusion setting, and formulate it as a problem to find the Stackelberg equilibrium of a stochastic differential game with delayed information flow. We find a maximum principle for this type of control problem, and use it to solve the optimal pricing problem in some specific cases.

The presentation is based on recent joint work with Leif Sandal and Jan Ubøe, both at NHH, Bergen, Norway.