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) ∈ R d1 , x ∈ R d. Our approach is based on the 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 allows us 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. Bibliography: 16 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 412, 2013, pp. 15–46.
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Belopolskaya, Y.I. Forward-Backward Stochastic Equations Associated with Systems of Quasilinear Parabolic Equations and Comparison Theorems. J Math Sci 204, 7–27 (2015). https://doi.org/10.1007/s10958-014-2183-6
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DOI: https://doi.org/10.1007/s10958-014-2183-6