  When randomness in demand affects the sales of a product, retailers use
dynamic pricing strategies to maximize their profits. In this article, we
formulate the pricing problem as a continuous-time stochastic optimal control
problem and find the optimal policy by solving the associated
Hamilton-Jacobi-Bellman (HJB) equation. We propose a new approach to modelling
the randomness in the dynamics of sales based on diffusion processes. The model
assumes a continuum approximation to the stock levels of the retailer which
should scale much better to large-inventory problems than the existing Poisson
process models in the revenue management literature. The diffusion process
approach also enables modelling of the demand volatility, whereas Poisson
process models do not.
  We present closed-form solutions to the HJB equation when there is no
randomness in the system. It turns out that the deterministic pricing policy is
near-optimal for systems with demand uncertainty. Numerical errors in
calculating the optimal pricing policy may, in fact, result in a lower profit
on average than with the heuristic pricing policy.
