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

For a d-dimensional diffusion of the form dX(t) = mu(X-t)dt + sigma(X-t)dW(t) and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Gamma, and A solving the second-order backward stochastic differential equation (2BSDE) dY(t) = f (t, X-t, Y-t, Z(t), Gamma(t))dt + Z(t)' circle dX(t), t epsilon [0, T), dZ(t) = A(t) dt + Gamma(t) dX(t), t epsilon [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(2)v (t, x)) = 0, (t, x) epsilon [0, T) x R-d, v (T, x) = g (x), has a sufficiently regular solution, then it follows directly from Ito's formula that the processes v(t, X-t), Dv(t, X-t), D(2)v(t, X-t), LDv(t, X-t), t epsilon [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 Gamma and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z, Gamma, 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 epsilon [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. (c) 2006 Wiley Periodicals, Inc.