13. Analytic Approach

In this chapter we derive the companion variational inequality approach to the reflected BSDEs of Chap. 12. First we introduce systems of partial integro-differential variational inequalities associated with these BSDEs and we state suitable definitions of viscosity solutions for related problems. Remember that BSDEs are used to model nonlinear phenomena, meaning that the equivalent PDEs (or systems of them, or PIDEs) are nonlinear too. They therefore don’t have classical solutions, but only solutions in weaker senses, viscosity solutions being the notion of choice for the kind of nonlinearities we face in pricing (or more general control) problems, which at least have some kind of comparison property (recall the sub- versus super-martingale story sketched in the discussion of Chap. 2).

We then deal with the corresponding existence, uniqueness and stability issues. The value processes (first components) in the solutions of the BSDEs is characterized in terms of the value functions for related optimal stopping or Dynkin game problems. We then establish a discontinuous viscosity solutions comparison (again) principle, which is the deterministic counterpart of the BSDEs comparison theorem alluded to above. In particular, this comparison principle implies uniqueness of viscosity solutions for the related obstacle problems. The comparison principle is also used for proving the convergence of stable, monotone and consistent deterministic approximation schemes. The notion of viscosity solutions is nice because everything happens as if it wasn’t there: all the classical results which apply to linear problems can be extended to nonlinear problems endowed with a comparison property, provided one switches to the notion of viscosity solutions for these problems. But the underlying mathematics are nontrivial, which is why we need Chap. 13!