Dynamic risk measure

In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra. In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra. A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures. A different approach to dynamic risk measurement has been suggested by Novak. Consider a portfolio's returns at some terminal time T {displaystyle T} as a random variable that is uniformly bounded, i.e., X ∈ L ∞ ( F T ) {displaystyle Xin L^{infty }left({mathcal {F}}_{T} ight)} denotes the payoff of a portfolio. A mapping ρ t : L ∞ ( F T ) → L t ∞ = L ∞ ( F t ) {displaystyle ho _{t}:L^{infty }left({mathcal {F}}_{T} ight) ightarrow L_{t}^{infty }=L^{infty }left({mathcal {F}}_{t} ight)} is a conditional risk measure if it has the following properties for random portfolio returns X , Y ∈ L ∞ ( F T ) {displaystyle X,Yin L^{infty }left({mathcal {F}}_{T} ight)} : If it is a conditional convex risk measure then it will also have the property: A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies: The acceptance set at time t {displaystyle t} associated with a conditional risk measure is If you are given an acceptance set at time t {displaystyle t} then the corresponding conditional risk measure is where ess inf {displaystyle { ext{ess}}inf } is the essential infimum.