The Financial Bubble Model with L\'evy Jump Processes
In this work we consider an extension of the Berestycki--Monneau--Scheinkman (BMS) model for speculative financial bubbles, in which the investor disagreement process is allowed to have jumps. While the original BMS framework assumes that disagreement evolves along continuous paths, our model accounts for sudden shifts in market sentiment through an independent L\'evy jump process. Using optimal stopping theory and the It\^o--L\'evy formula, we show that the speculative bubble premium satisfies a non-local partial integro-differential equation (PIDE) with a moving obstacle. We develop a viscosity solution theory for this non-local obstacle problem. We prove a comparison principle by a doubling-of-variables argument adapted to the non-local jump integral, and we prove existence and uniqueness of the bubble price by Perron's method, constructing explicit continuous sub- and supersolutions. We then introduce a monotone Implicit--Explicit finite difference scheme for the bubble premium. Following the Barles--Souganidis framework, we show that the discrete operator preserves the M-matrix property and that the scheme converges locally uniformly to the unique viscosity solution under a state-dependent Courant--Friedrichs--Lewy (CFL) condition. At the end of the paper we implement the scheme via a PSOR--Picard algorithm and present numerical tests for four finite- and infinite-activity L\'evy models.