Department of Mathematics University d'Evry, France
About the pricing equation in finance
We consider the pricing equation in finance in a rather general Jump–Diffusion Setting with Regimes (see [2, 3]). This model may be viewed as a suitable generalization of the interacting Itˆo and Point Process model considered by Becherer and Schweizer in [1] (we consider in effect an interacting Itˆo–L´evy and Point Process model). But as opposed to the set-up of [1] where linear reaction-diffusion systems of parabolic equations (pricing equations of European Contingent Claims, from the point of view of the financial interpretation) are considered from the point of view of classic solutions, here the application we have in mind consists of more general optimal stopping or optimal game problems (pricing equations of American or Game Contingent Claims, see [2]) for which the related reaction-diffusion systems typically do not have classic solutions (and even less so, that there are also jumps in the Itˆo–L´evy component of our model). This leads us to study solutions in the viscosity sense, using related reflected BSDE techniques as the main tool. We establish in particular the convergence of stable, monotone and consistent approximation schemes to the unique viscosity solution of the pricing equation.