Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

Abstract

We establish the existence and uniqueness as well as the stability of p -integrable solutions to multidimensional backward stochastic differential equations (BSDEs) with super-linear growth coefficient and a p -integrable terminal condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . The generator could neither be locally monotone in the variable y nor locally Lipschitz in the variable z . As application, we establish the existence and uniqueness of weak (Sobolev) solutions to the associated systems of semilinear parabolic PDEs. The uniform ellipticity of the diffusion matrix is not required. Our result covers, for instance, certain systems of PDEs with logarithmic nonlinearities which arise in physics.