Multivalued backward stochastic differential equations with oblique subgradients

Abstract

We study the existence and uniqueness of the solution for the following backward stochastic variational inequality with oblique reflection (for short, $BSVI\left(H(t,y),\varphi,F\right)$), written under differential form \[ \left\{\begin{array} [c]{l}% -dY_{t}+H\left(t,Y_{t}\right) \partial\varphi\left(Y_{t}\right) \left(dt\right) \ni F\left(t,Y_{t},Z_{t}\right) dt-Z_{t}dB_{t},\quad t\in\left[ 0,T\right] ,\smallskip\\ Y_{T}=\eta, \end{array} \right. \] where $H$ is a bounded symmetric smooth matrix and $\varphi$ is a proper convex lower semicontinuous function, with $\partial\varphi$ being its subdifferential operator. The presence of the product $H\partial\varphi$ does not permit the use of standard techniques because it does conserve neither the Lipschitz property of the matrix nor the monotonicity property of the subdifferential operator. We prove that, if we consider the dependence of $H$ only on the time, the equation admits a unique strong solution and, allowing the dependence also on the state of the system, the above $BSVI\left(H(t,y),\varphi,F\right)$ admits a weak solution in the sense of the Meyer-Zheng topology. However, for that purpose we must renounce at the dependence on $Z$ for the generator function and we situate our problem in a Markovian framework.