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We give a survey of the results related to extensions of the Kolmogorov inequality for the distribution of the absolute value of the maximum of the sum of centered independent random variables to the case of martingales considered at random stopping times. [ABSTRACT FROM AUTHOR]

The family Q = (Q(n),Fn)n=0∞ of probability measures Q(n) defined on Fn is considered. It is called a deformation of the first kind if for all n ∊ {0, 1, 2,... }, Q(n+1)1)∣Fn ⩽ Q(n), and a deformation of the second kind if Q(n+1)∣Fn ⩽ Q(n). For finite stopping times τ measures Q(τ) are introduced by the formula Q(t)(A) = Σ i∞ =0 Q(i)(A{t = i}), where A ϵ Ft . With the help of these measures a generalization of Doob's optional sampling theorem is formulated and proved for deformed submartingales of the first kind in the case of neighboring stopping times, and of the second kind in the case of boundedly spaced stopping times. [ABSTRACT FROM AUTHOR]

The optimal stopping problem with incomplete data for the Kalman-Bucy scheme is reduced to the optimal stopping problem with complete data. In the case of a continuous gain function it is proven that price convergence has an order (ϵ1 +ϵ2) when small perturbation parameters ϵ1, ϵ2 of an observable process tend to zero. [ABSTRACT FROM AUTHOR]

A two-person game Γ is considered which is specified by the following random walks. Let x[SUBn] and y[SUBn] be independent symmetric random walks on the set E = {0, 1, . . . , K}. Assume they start from the states a and b respectively (1 ≦ a < b ≦ K - 1), are absorbed with probability 0.5 at points 0 and K, and are reflected to the points 1 and K - 1, respectively, with the same probability 0.5. Players I and II observe the random walks x[SUBn] and y[SUBn], respectively, and stop them at Markov times τ and σ being strategies of the game. Each player knows the values of K, a, and b but has no information about the behavior of the other player. The rules of the game are as follows. If x[SUB&tau] > y[SUBσ] then player II pays player I, say, $1; if x[SUBτ < y[SUBσ] then I pays II $1; and if x[SUB&tau] = y[SUBσ] then the outcome of the game is said to be a draw. The aim of each player is to maximize the expected value of his income. We find the equilibrium situation and the value of the game. [ABSTRACT FROM AUTHOR]