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

Abstract

For a d-dimensional diffusion of the form dXt = μ(X t)dt + σ(Xt)dWt 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) dYt = f(t, Xt, Yt, Z t, Γt)dt + Z′t o dXt, t ∈ [0, T), dZt = Atdt + ΓtdX t, t ∈[0, T), YT = g(XT). If the associated PDE -vt(t, x) + f(t, x, v(t, x), Dv(t, x), D 2v(t, x)) = 0, (t, x) ∈ [0, 7) × ℝd, v(T, x) = g(x), has a sufficiently regular solution, then it follows directly from Itô's formula that the processes v(t, Xt), Dv(t, X t), D2v(t, Xt), ℒDv(t, Xt), 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 Yt = v(t, Xt), 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.