Forward-backward stochastic differential equations associated with systems of quasilinear parabolic equations and comparison theorems

Ya. I. Belopolskaya

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 19, Tome 412 (2013), pp. 15-46

Résumé

We develop a probabilistic approach to construction of a viscosity solution of the Cauchy problem for a system of quasilinear parabolic equations with respect to a vector function $u(t,x)\in R^{d_1}$, $x\in R^d$. Our approach is based on a possibility to reduce the original quasilinear parabolic system to a quasilinear parabolic equation in an alternative phase space and derive forward-backward stochastic differential equations associated with it. This reduction shows the way to prove some comparison theorems for BSDEs and as a result to construct a probabilistic representation of a viscosity solution of the original Cauchy problem.

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