When the Bellman Equation Cannot Be Solved Analytically

Many questions in managerial decision making imply —if uncertainty is involved— stochastic optimization problems of the form V (S) = max x E {∫ ∞ 0 e−ρtf(S, x) dt } where the state transition dS = g(S, x) dt + σ(S) dz describes the underlying stochastic process S as an Itō process and dz is a Brownian motion. For very simple problems, results can be obtained analytically by solving the corresponding Bellman equation. In all other cases, V (S) has to be computed numerically. Unfortunately, conventional numerical methods are either slow, or completely fail as the solution is a saddlepoint path so that all neighboring solution paths diverge. This thesis applies methods developed in “Applied Computational Economics and Finance” by M.J. Miranda and P.L. Fackler to solve selected problems that arise in modeling managerial decision making, optimal pricing of nondurables, modeling the ups and downs in oil prices, and modeling the impact of CO2 emission into the atmosphere.