Convex risk measure

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function ${\displaystyle \varrho }$ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

Consider a random outcome ${\displaystyle X}$ viewed as an element of a linear space ${\displaystyle {\mathcal {L}}}$ of measurable functions, defined on an appropriate probability space. A functional ${\displaystyle \varrho :{\mathcal {L}}}$ → ${\displaystyle \mathbb {R} \cup \{+\infty \}}$ is said to be coherent risk measure for ${\displaystyle {\mathcal {L}}}$ if it satisfies the following properties:

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