Path dependent partial differential equations and related topics

2014-07-03The aim of this thesis is to extend the viscosity solutions theory of partial differential equations to the space of continuous paths. It is well‐known that, when Markovian, through the Feynman‐Kac formula, the BSDE theories of various kinds provide a stochastic representation for viscosity solutions of parabolic PDEs. The wellposedness of BSDEs also holds in a non‐Markovian framework. It is then reasonable to wonder if one can write a non‐Markovian Feynman‐Kac formula and provide a relation between non‐Markovian BSDEs and the so‐called path‐dependent partial differential equations(PPDEs). In this thesis we give a definition of viscosity solutions for PPDEs for several kind of equations. When the equations are fully nonlinear, this definition requires results on optimal stopping theory under nonlinear expectation that we establish. These results allows us, under assumptions, to prove the wellposedness for viscosity solutions of PPDEs and show that several non‐Markovian control problems can be studied using PPDEs