## 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) dY_{t} = f(t, X_{t}, Y_{t}, Z _{t}, Γ_{t})dt + Z′_{t} o dX_{t}, t ∈ [0, T), dZ_{t} = A_{t}dt + Γ_{t}dX _{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 ^{2}v(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}), D^{2}v(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.

Original language | English (US) |
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Pages (from-to) | 1081-1110 |

Number of pages | 30 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 60 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2007 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics