The authors present a very useful theory of backward stochastic difference equations on a space related to discrete time, finite state processes. Their results follow an analogy with the famous theory of backward stochastic differential equations, but, these stochastic (discrete) processes are not some approximations to the continuous case [see for example (*) J. Ma, Ph. Protter, J. San Martín and S. Torres, Ann. Appl. Probab. 12, No. 1, 302–316 (2002; Zbl 1017.60074)], they are studied as constructions in their own right. Therefore, they became more adequate models for real applications of backward stochastic evolution equations.
The dynamics are given by the following backward stochastic equation
\[ Y_t(\omega)-\sum_{t\geq u<T}F(\omega, u, Y_u(\omega),Z_u(\omega))+ \sum_{t\geq u <T} Z_u(\omega) M_{u+1}(\omega)=Q(\omega), \]
where \(T\) is a finite deterministic terminal time, \(F\) an adapted map \(F:\Omega\times \{0,\dots,T\}\times \mathbb R^K\times \mathbb R^{K\times N}\to \mathbb R^{K}\), \(Q\) an \(\mathbb R^k\) valued \({\mathcal F}_T\)-measurable terminal condition and \(\{M\}\) is a (discrete) martingale with respect to the filtration \(\{{\mathcal F}_t\}\), \(t\in\{0,\dots, T\}\).
The main result of the third section is given in Theorem 2 which proves the existence and uniqueness of a solution of the above equation. The proof is based on a (discrete) martingale representation theorem (given by R. J. Elliot and H. Yang [Appl. Math. Optimization 30, No. 1, 51–78 (1994; Zbl 0810.93062)] and presented in the second section). The obtained result extends a similar result given in [(*)] .
In the fourth section, a comparison theorem is given (Theorem 3). This result extends some result given in [S. Peng, Dynamically consistent nonlinear evaluations and expectations. Preprint No. 2004-1, Institute of Mathematics, Shandong University (2005), arXiv:math/0501415]. Two interesting corollaries follow from this theorem. Also, three useful examples are presented.
In the fifth section, the authors study the inverse problem, i.e., given a solution of the backward stochastic equation, can we determine the values of the driven function \(F\)? This problem of interest in applications [see S. Peng, Stochastics Stochastics Rep. 38, No. 2, 119–134 (1992; Zbl 0756.49015) or P. Briand, F. Coquet, Y. Hu, J. Mémin and S. Peng, Electron. Commun. Probab. 5, 101–117 (2000; Zbl 0966.60054)]. The main results are given in Theorem 4, Theorem 5 and Theorem 6, which assure the existence and uniqueness of the driven function \(F\).
In the next section, the authors present a consistent theory for the application of backward stochastic difference equations to risk theory. The authors draw a parallel with the classical (continuous) risk theory (see S. N. Cohen and R. J. Elliott [Ann. Appl. Probab. 20, No. 1, 267–311 (2010; Zbl 1195.60077)], H. Föllmer and A. Schied [Stochastic finance. An introduction in discrete time. Berlin: de Gruyter (2011; Zbl 1213.91006)], P. Barrieu and N. El Karoui [Finance Stoch. 9, No. 2, 269–298 (2005; Zbl 1088.60037)] or E. Rosazza Gianin [Insur. Math. Econ. 39, No. 1, 19–34 (2006; Zbl 1147.91346)]). The main result is given in Theorem 7.
In the last section, an answer is given to the following question: given a particular map \(q\mapsto Y_0\), is it possible to find an \({\mathcal F}_t\)-consistent nonlinear expectation which agrees with it? The result presented in Theorem 8 represents the main result of this section. A very good example ends the paper.