[Paper Review] Second order backward stochastic differential equations and fully non-linear parabolic PDEs

Patrick Cheridito, H. Meté Soner+2 more|ArXiv.org|Sep 14, 2005

Stochastic processes and financial applications24 references4 citations

TL;DR

This paper establishes a stochastic representation for solutions to fully non-linear parabolic PDEs via second-order backward stochastic differential equations (2BSDEs). It proves that under viscosity solution comparison and monotonicity conditions, the unique solution to the 2BSDE corresponds to the viscosity solution of the associated PDE, enabling Monte Carlo numerical methods for such PDEs.

ABSTRACT

We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte Carlo methods for their numerical treatment.

Motivation & Objective

• To establish a stochastic representation for solutions of fully non-linear parabolic PDEs using second-order backward stochastic differential equations (2BSDEs).
• To prove that the solution of a 2BSDE corresponds to the viscosity solution of an associated fully non-linear PDE under comparison and monotonicity conditions.
• To extend the connection between stochastic equations and PDEs beyond semi-linear and quasi-linear cases, which are limited by linear second-order terms.
• To enable numerical solution of fully non-linear PDEs via Monte Carlo methods through the 2BSDE formulation.

Proposed method

• Formulates a second-order backward stochastic differential equation (2BSDE) with adapted processes (Y, Z, Γ, A) driven by a diffusion process X_t.
• Uses Itô’s lemma to show that if a C^3 solution v exists for the associated PDE, then (v(t,X_t), Dv(t,X_t), D^2v(t,X_t), L Dv(t,X_t)) solves the 2BSDE.
• Applies viscosity solution theory to the PDE, assuming the comparison principle holds for the PDE's value function.
• Imposes Lipschitz continuity in Y and decreasing behavior in Γ for the generator f to ensure uniqueness and stability.
• Derives a stochastic representation where the solution process Y_t equals v(t,X_t), linking the 2BSDE solution to the PDE solution.
• Adapts the framework to boundary value problems by introducing a stopping time at the domain boundary, modifying the terminal condition accordingly.

Experimental results

Research questions

• RQ1 Can second-order backward stochastic differential equations (2BSDEs) provide a stochastic representation for solutions of fully non-linear parabolic PDEs?
• RQ2 Under what conditions does the solution of a 2BSDE uniquely correspond to a viscosity solution of the associated PDE?
• RQ3 How can Monte Carlo methods be applied to numerically solve fully non-linear parabolic PDEs via 2BSDEs?
• RQ4 What are the necessary regularity and structural conditions on the generator f and the terminal condition g for the existence and uniqueness of 2BSDE solutions?
• RQ5 Can the 2BSDE framework be extended to boundary value problems with lateral boundary conditions?

Key findings

• If the associated PDE admits a unique continuous viscosity solution under the comparison principle, then the 2BSDE has at most one solution, ensuring uniqueness.
• The solution process Y_t of the 2BSDE satisfies Y_t = v(t, X_t), where v is the viscosity solution of the PDE.
• The existence of a C^3 solution to the PDE implies the existence of a solution to the 2BSDE, providing a sufficient condition.
• The generator f must be Lipschitz in Y and decreasing in Γ to ensure the solution's uniqueness and stability.
• The 2BSDE framework allows for the numerical approximation of fully non-linear PDEs using Monte Carlo simulations.
• The method extends to boundary value problems by incorporating a stopping time at the domain boundary, preserving the correspondence between 2BSDE and PDE solutions.