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

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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. 
@article{ZNSL_2013_412_a1,
     author = {Ya. I. Belopolskaya},
     title = {Forward-backward stochastic differential equations associated with systems of quasilinear parabolic equations and comparison theorems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {15--46},
     publisher = {mathdoc},
     volume = {412},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a1/}
}
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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/