Random walks maximizing the probability to visit an interval

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We consider random walks, say Wₙ=(M₀, M₁,, Mₙ), of length n starting at 0 and based on the martingale sequence Mₖ with differences Xₘ=Mₘ-Mₘ₋₁. Assuming that the differences are bounded, |Xₘ|≤ 1, we solve the problem equation Dₙ(x) P {Wₙ visits an interval[x,∞)}, x∈ R, piirma equation where is taken over all possible Wₙ. In particular, we describe random walks which maximize the probability in piirma. We also extend the result to super-martingales.