On one extension of a martingale

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