A Markovian decision model with general state space, compact action space, and the average cost as criterion is considered. The existence of an optimal policy is shown via an optimality inequality in terms of the minimal average cost g and a relative value function w. The existence of some w is usually shown via relative compactness in a space of real-valued functions on the state space. Here it shall be shown that one can instead do with pointwise relative compactness in the set of real numbers if one makes use of a (generalized) lower limit of functions. An application to an inventory model is given.

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Consider an option with maturity time T corresponding to a contingent claim H (in an incomplete market). A fair hedging price for H should take into account an optimal dynamical hedging plan against H. Let Ct be the cumulative cost and ℑt be the set of events of the history up to time t. You can choose the plan at time t such that you minimize (i) E[{Ct+1 − Ct}2 ∣ ℑt], (ii) E[{CT − Ct}2 ∣ ℑt], or (iii) E[{CT − C0}2]. Sufficient conditions on the underlying stochastic process (in discrete time) are provided such that the fair hedging price does not depend on the choice of (i), (ii), or (iii), which fact should increase its acceptability.

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