A stochastic approach to a new type of parabolic variational inequalities

Abstract

We study the following quasilinear partial differential equation with two subdifferential operators:

where for and

The operator (resp. ) is the subdifferential of the convex lower semicontinuous function (resp. ). We define the viscosity solution for such kind of partial differential equation and prove the uniqueness of the viscosity solution when does not depend on . To prove the existence of a viscosity solution, a stochastic representation formula of Feymann–Kac type will be developed. For this end, we investigate a fully coupled forward–backward stochastic variational inequality.

Keywords::

• forward–backward stochastic differential equations
• variational inequalities
• subdifferential operators
• viscosity solutions

AMS Subject Classification::

• 60H10
• 60H30
• 49J40

Acknowledgements

The author wishes to express his thanks to Rainer Buckdahn and Aurel Ră¸canu for their useful suggestions and discussions and to Lucian Maticiuc for his remarks.

Notes

^1. Let and . Denote by the set of triples ( denotes the set of all symmetric non-negative matrices), such that we call the parabolic super-jet of at . Similarly, we define the parabolic sub-jet of at denoted by as the set of triples such that

Additional information

Funding

The work of this author is supported by the Marie Curie ITN Project, ‘Deterministic and Stochastic Controlled Systems and Application’ FP7-PEOPLE-2007-1-1-ITN [grant number 213841-2].