Backward stochastic difference equations for a single jump process (Q1930453)

A discrete time single jump process \(X=X_t,\; t\in\{0, 1, \dotsc, L\}\) with values in a measurable space \((\mathbb E, \mathcal E)\) is used as the basis for the model. Here, \(L\) is a finite terminal time, and the process remains at its initial point \(z_0\in\mathbb E\) until a random time \(T\), when it jumps to a new random position \(z\in\mathbb E\). A backward stochastic difference equation based on the martingale random measure \(q\) is defined; it is of the form \[ Y_t+\sum_{u=t}^{L-1}H(\omega, u, Y_u, Z(u, \cdot)) +\sum_{u=t}^{L-1}\int_{\mathbb E}Z(u, y)q(u+1, d y)=Q \] for \(t\in\{0,\dotsc,L-1\}\), where \(Q\) is a real valued terminal condition, i.e., \(Y_L(\omega)=Q(\omega)\), \(\omega\in\{0, \dotsc, L\}\times\mathbb E\). The existence and uniqueness of solutions is proved under some assumptions on the adapted map \(H\). Applications to nonlinear expectations are presented, as well.