Geodesic
Forward-backward stochastic differential equations associated with systems of quasilinear parabolic equations and comparison theorems
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 19, Tome 412 (2013), pp. 15-46 
Ya. I. Belopolskaya. Forward-backward stochastic differential equations associated with systems of quasilinear parabolic equations and comparison theorems. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 19, Tome 412 (2013), pp. 15-46. http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a1/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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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