Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs |
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Authors: | Patrick Cheridito H Mete Soner Nizar Touzi Nicolas Victoir |
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Institution: | 1. Princeton University, E‐Quad, E‐416, Princeton, NJ 08544;2. Ko? University, College of Administrative Sciences and Economics, Fener Yolu Caddesi, Sariyer 80910, Istanbul, Turkey;3. Centre de Mathématiques Appliquées, école Polytechnique, 91128 Palaiseau Cedex, France;4. University of Oxford Wellington Square, Oxford OX1 2JD, United Kingdom |
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Abstract: | For a d‐dimensional diffusion of the form dXt = μ(Xt)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) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes 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 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. |
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