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)$$dY_{t} = f(t,X_{t}, Y_{t}, Z_{t}, \Gamma_{t}) dt + Z_t'\circ dX_{t}, \quad t \in [0,T),$$$$dZ_{t} = A_{t} dt + \Gamma_{t}dX_{t}, \quad t \in [0,T),$$$$Y_{T} = g(X_{T}).$$ If the associated PDE$$- v_{t}(t,x) + f(t,x,v(t,x), Dv(t,x), D^{2}v(t,x)) = 0,$$$$(t,x) \in [0,T) \times {\cal R}^{d}, \quad v(T,x) = g(x),$$ has a sufficiently regular solution, then it follows directly from Itô's formula that the processes$$v(t,X_{t}), Dv(t,X_{t}), D^{2}v(t,X_t), {\cal L} Dv(t,X_{t}), \quad t \in [0,T],$$ solve the 2BSDE, where 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.