Reflected backward stochastic differential equations in an orthant (Q698364)

The aim of this article is to study a backward stochastic differential equation with oblique reflection in an orthant. More precisely one considers the following system \[ Z(t)=\xi+ \int^T_t\bigl(s,Z(s) \bigr)ds+ \int^T_t R\bigl(s, Z(s)\bigr) dY(s)- \int^T_t\bigl \langle U(s),dB(s)\bigr \rangle, \] with \(Z_i(\cdot)\geq 0\), \(Y_i(\cdot)\) nondecreasing, increasing only when \(Z_i(\cdot) =0\), \(1\leq i\leq d\), where \(B\) is a \(d\)-dimensional Brownian motion, \(\xi\) bounded, measurable, with \(\xi_i\geq 0\), and the parameters \(b: \Omega\times [0,T]\times \mathbb{R}^d\to \mathbb{R}^d\) and \(R:\Omega\times [0,T]\times \mathbb{R}^d\to M_d(\mathbb{R})\) are bounded and satisfy the usual Lipschitz assumptions. The existence and uniqueness of a solution \((Y,Z)\) is established under the following uniform spectral radius condition: For \(i\neq j\), there exists a constant \(v_{ij}\) such that \(|r_{ij} (\omega,t,z) |\leq v_{ij}\), where the spectral radius \(\sigma(V)\) of the matrix \(V=(v_{ij})\) (with \(v_{ii}=0)\) satisfies \(\sigma(V)<1\).