Discrete-time stochastic processes on Riesz spaces.
Measure and integration theory has been as inspiring source for the development of the theory of Riesz spaces and positive operators. For example, the space of all measurable functions on some measurable space is a Riesz space and the integral is a positive operator on the space of all integrable functions. Moreover, even the construction of the Lebesgue integral is based on the notions of positivity and order convergence. The authors translate several notions of discrete martingale theory into the framework of Riesz spaces having a weak order unit; weak order units are needed as a substitute for the constants in spaces of measurable functions. The basic notion is that of a conditional expectation which is defined as an order continuous positive projection having a Dedekind complete range and preserving weak order units. Conditional expectations are then used to define filtrations and stopping times: A filtration is defined as a sequence of conditional expectations \( (T_i) \) satisfying \( T_jT_i = T_i = T_iT_j \) for all \( i \leq j \), and a stopping time with respect to the filtration \( (T_i) \) is defined as an increasing sequence of conditional expectations \( (P_i) \) satisfying \( P_i \leq I \) for all \( i \) and \( T_jP_i = P_iT_j \) for all \( i \leq j \). The definition of a (sub-/super-)martingale is then straightforward. In this abstract setting, the authors prove, among other results, the Doob-Meyer decomposition and the optional stopping theorem. The paper is pleasant to read since the authors take care to prepare the abstract definitions in the Riesz space setting by a thorough discussion of the corresponding traditional notions from martingale theory. It is likely that the general theory developed here can also be applied to asymptotic martingales.
conditional expectation, martingales, stopping times
scientific article; zbMATH DE number 2124781