Abstract: The authors present in this paper a new concept of risk measure defined at an arbitrary stopping time τ, the F_τ-coherent risk measure. For an F_τ-coherent risk adjusted value (the negative of a F_τ-coherent risk measure) ф_т: L^∞ (F) →L^∞ (F_τ), which satisfies the Fatou property, its representation theorem is obtained by using a special F_τ-convex set of probability measures. It is also proven that an F_τ-coherent risk adjusted value can be represented by its acceptance set.

Key words:
Risk measure, F_τ-coherent, Representation theorem, F_τ-covex, Acceptance set.