Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

Abstract

Abstract 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 uniqueness of adapted processes Y , Z , Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula 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 f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then 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 the process Y is of the form 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 nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.