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We formulate and solve a finite horizon full balance sheet of a two-mode optimal switching problem related to trade-off strategies between expected profit and cost yields. Given the current mode, this model allows for either a switch to the other mode or termination of the project, and this happens for both sides of the balance sheet. A novelty in this model is that the related obstacles are nonlinear in the underlying yields, whereas, they are linear in the standard optimal switching problem. The optimal switching problem is formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We prove the existence of a continuous minimal solution of this system using an approximation scheme and fully characterize the optimal switching strategy. [ABSTRACT FROM PUBLISHER]

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order. [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]