Ambiguous Jump-Diffusions and Optimal Stopping

Abstract

An ambiguity averse decision-maker contemplates investment of a fixed size capital into a project with a stochastic profit stream under the Knightian uncertainty. Multiple priors are modeled as a "cloud" of diffusion processes with embedded compound Poisson jumps. The "cloud" contains the Brownian motion (BM) as a process with zero density of jumps. The decision-maker has recursive multiple priors utility as in Epstein and Schneider (2003) and chooses the optimal investment timing. We demonstrate that if the expected present value (EPV) of the project is the same for each jump-diffusion prior at the moment of investment, then the BM is the worst prior in the waiting region. The same conclusion holds for some parameter values even when the BM gives the highest EPV of the project. For other parameter values, it is possible that the local dynamics of the worst case prior is given by a jump-diffusion in a vicinity of the investment threshold and by the BM in a vicinity of negative infinity. Explicit formulas for the value functions and investment thresholds are derived.