FORWARD-BACKWARD STOCHASTIC EQUATIONS ASSOCIATED WITH SYSTEMS OF QUASILINEAR PARABOLIC EQUATIONS AND COMPARISON THEOREMS

摘要

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~(d_1), 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.