Representation theorems for dynamic convex risk measures

2025/10/23 by Zheng, Shiqiu
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.20660

Abstract

In this paper, we prove that under the domination condition: \calE^-μ,-ν[-ξ|\calF_t]≤ρ_t(ξ)≤\calE^μ,ν[-ξ|\calF_t], ∀ξ∈ L^exp_T (resp. L²(F_T)), ∀ t∈[0,T], where \calE^μ,ν is the g-expectation with generator μ|z|+ν|z|², μ≥0, ν≥0, the dynamic convex (resp. coherent) risk measure ρ admits a representation as a g-expectation, whose generator g is convex (resp. sublinear) in the variable z and has a quadratic (resp. linear) growth. As an application, we show that such dynamic convex (resp. coherent) risk measure ρ admits a dual representation, where the penalty term (resp. the set of probability measures) is characterized by the corresponding generator g.

Citations

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