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

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) dY_t = f(t, X_t, Y_t, Z _t, Γ_t)dt + Z′_t o dX_t, t ∈ [0, T), dZ_t = A_tdt + Γ_tdX _t, t ∈[0, T), Y_T = g(X_T). If the associated PDE -v_t(t, x) + f(t, x, v(t, x), Dv(t, x), D ²v(t, x)) = 0, (t, x) ∈ [0, 7) × ℝ^d, v(T, x) = g(x), has a sufficiently regular solution, then it follows directly from Itô's formula that the processes v(t, X_t), Dv(t, X _t), D²v(t, X_t), ℒDv(t, X_t), t ∈ [0, T], 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.