Existence of a weak solution to a Markovian BSDE with discontinuous coefficients

Abstract We establish the existence of weak solutions for a decoupled system of a forward stochastic differential equation (SDE) and a backward stochastic differential equation (BSDE). The generator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H(x,y,z)} is assumed continuous in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(y,z)} but possibly discontinuous in x . The drift of the forward component is merely measurable drift and the diffusion coefficient can be discontinuous. Our approach is based on partial differential equations.