Parabolic Classical and Path-Dependent Partial Differential Equations: Stochastic Methods

Start: 04/11/2012 - 1:15pm
End  : 04/11/2012 - 2:15pm

Applied Math Seminar

Henry Schellhorn, CGU


In the first part of the talk I will review some well-known correspondences between parabolic partial differential equations (PDEs) and stochastic processes, in increasing order of generality. Whenever the solution exists and is unique: (i)  the fundamental solution of the heat equation is a Gaussian density, (ii) the solution of semilinear parabolic PDEs is given by the Feynman-Kac representation, (iii) the solution of quasilinear parabolic PDEs can be represented by the solution of forward-backward stochastic differential equations (FBSDE), while (iv) the solution of the general nonlinear parabolic PDE can be represented as the solution of a second-order FBSDE (2-BSDEs). 2BSDEs were introduced around 2006. The main tool for establishing these correspondences is the chain rule of stochastic calculus, known as Ito’s lemma.

In the second part of the talk, I describe path-dependent PDEs (PPDEs), which generalize classical PDEs. PPDEs correspond to FBSDEs where the terminal condition depends on the path. This is a very common problem in many applications ranging from stochastic control to mathematical finance. One way to view this problem as a PDE is to augment the state space, and describe it as an infinite system of PDEs. Another one, namely the PPDE formulation, was proposed by Dupire, and developed in Cont and Fournie (Journal of Functional Analysis 2011). It consists of introducing a functional derivative, which acts only on the “last coordinate” of the path. As a result a “functional” Ito’s formula can be obtained.

I conclude the talk by showing a new or independently rediscovered result for PPDEs, which generalizes the classical semi-group theory of parabolic PDEs, albeit only for “smooth” problems: the solution of a PPDE can be represented as an exponential of the initial condition. The generator of this exponential is equal to one half times the second-order Malliavin derivative, and thus the exponential can be easily calculated by Dyson series. Dyson series are an important tool in quantum field theory, but they are not so well-known in the area of stochastic processes and PDEs.


RN 103