For a d-dimensional diffusion of the form dX t = µ(X t )dt + σ(X t )dW t , and continuous functions f and g, we study the existence and the uniqueness of adapted processes Y , Z, Γ and A solving the second order backward stochastic differential equation (2BSDE) , admits a C 3 -solution, it follows directly from Itô's lemma that the processes solve the 2BSDE, where L is the Dynkin operator of X without the drift term. The main result of the paper shows that if the PDE has comparison as in the theory of viscosity solutions and if f is Lipschitz in Y and decreasing in Γ, the existence of a solution (Y, Z, Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v, and Y t = v(t, X t ), t ∈ [0, T ]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully non-linear parabolic PDEs. As a consequence, the numerical treatment of such PDE's can now be approached by Monte Carlo methods.
It then follows directly from Ito's formula REF that the processes
Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs
arXiv: Probability
Mathematics