On One Extension of a Martingale

In this paper we introduce an ε‐martingale and a strongε‐martingale. The first is defined by the inequality $|{\bf E}(X_t\,|\,{\cal F}_s)- X_s|\le \varepsilon$, and the second one can be obtained from the ε‐martingale by replacing in the definition fixed time moments with stopping times. The paper proves that a right‐continuous ε‐martingale is a strong 2ε‐martingale. At the same time we construct an example of a right‐continuous ε‐martingale which is not a strong ε‐martingale for any $a<2$. We show that the dependence between ε‐martingales and strong ε‐martingales has no analogues for ε‐submartingales. We also give the criterion for testing if a right‐continuous with left limits process is a strong ε‐martingale or not. The criterion is based on the possibility of uniform approximation of the process by a martingale with precision $\varepsilon/2$.