Coupled FBSDEs with Measurable Coefficients and its Application to Parabolic PDEs

TL;DR

This paper establishes the existence and uniqueness of solutions for coupled FBSDEs with measurable coefficients using probabilistic methods, and applies these results to solve certain classes of parabolic PDEs and real-world problems.

Contribution

It introduces a novel probabilistic approach to coupled FBSDEs with discontinuous coefficients and links these to the well-posedness of semilinear parabolic PDEs.

Findings

Proved existence and uniqueness of solutions for coupled FBSDEs with measurable coefficients.

Established well-posedness of certain semilinear parabolic PDEs with discontinuous data.

Applied results to pandemic policy-making and carbon derivative pricing.

Abstract

Using purely probabilistic methods, we prove the existence and the uniqueness of solutions fora system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous coefficients. As a corollary, we obtain the well-posedness of semilinear parabolic partial differential equations (PDEs) $$ \begin{aligned} &\mathcal{L} u(t,x)+F(t,x,u,\partial_x u)=0;\qquad u(T,x)=h(x)\\ &\mathcal{L}:=\partial_t+\frac{1}{2}\sum_{i,j=1}^m(\sigma\sigma^\intercal)_{ij}(t,x)\partial^2_{x_ix_j} \end{aligned} $$ in the natural domain of the second-order linear parabolic operator $\mathcal{L}$. We allow $F$ and $h$ to be discontinuous with respect to $x$. Finally, we apply the result to optimal policy-making for pandemics and pricing of carbon emission financial derivatives.