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Representation theorems for dynamic convex risk measures

Shiqiu Zheng

TL;DR

The paper studies dynamic convex (and coherent) risk measures under a domination condition by quadratic $g$-expectations and proves that such DRMs admit a representation as a $g$-expectation with a generator $g$ convex in $z$ and of quadratic growth (linear growth in the coherent case). It further derives a dual representation in terms of a family of measures and a Legendre-Fenchel transform of $g$, with the dual domain characterized by the subdifferential $\partial g(t,0)$ for coherent DRMs. The results apply to unbounded terminal variables and do not require independent increments, extending prior Lipschitz results and providing a robust link between DRMs and BSDE/g-expectations. A nonlinear Doob-Meyer decomposition for $\rho$-supermartingales is developed to support the convergence and representation arguments, and a dynamic entropy example illustrates the framework's concreteness and applicability to risk assessment over time.

Abstract

In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-μ,-ν}[-ξ|{\cal{F}}_t]\leqρ_t(ξ)\leq{\cal{E}}^{μ,ν}[-ξ|{\cal{F}}_t],\quad \forallξ\in \mathcal{L}^{\exp}_T\ (\text{resp.}\ L^2(\mathcal{F}_T)),\ \forall t\in[0,T], \end{equation*} where ${\cal{E}}^{μ,ν}$ is the $g$-expectation with generator $μ|z|+ν|z|^2, μ\geq0, ν\geq0$, 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 $g$.

Representation theorems for dynamic convex risk measures

TL;DR

The paper studies dynamic convex (and coherent) risk measures under a domination condition by quadratic -expectations and proves that such DRMs admit a representation as a -expectation with a generator convex in and of quadratic growth (linear growth in the coherent case). It further derives a dual representation in terms of a family of measures and a Legendre-Fenchel transform of , with the dual domain characterized by the subdifferential for coherent DRMs. The results apply to unbounded terminal variables and do not require independent increments, extending prior Lipschitz results and providing a robust link between DRMs and BSDE/g-expectations. A nonlinear Doob-Meyer decomposition for -supermartingales is developed to support the convergence and representation arguments, and a dynamic entropy example illustrates the framework's concreteness and applicability to risk assessment over time.

Abstract

In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-μ,-ν}[-ξ|{\cal{F}}_t]\leqρ_t(ξ)\leq{\cal{E}}^{μ,ν}[-ξ|{\cal{F}}_t],\quad \forallξ\in \mathcal{L}^{\exp}_T\ (\text{resp.}\ L^2(\mathcal{F}_T)),\ \forall t\in[0,T], \end{equation*} where is the -expectation with generator , the dynamic convex (resp. coherent) risk measure admits a representation as a -expectation, whose generator is convex (resp. sublinear) in the variable and has a quadratic (resp. linear) growth. As an application, we show that such dynamic convex (resp. coherent) risk measure admits a dual representation, where the penalty term (resp. the set of probability measures) is characterized by the corresponding generator .
Paper Structure (6 sections, 12 theorems, 162 equations)

This paper contains 6 sections, 12 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Quadratic $g$-expectations
  3. Representations of dynamic convex risk measures
  4. Some properties of convex DRMs
  5. A Doob-Meyer decomposition for $\rho$-supermartingales
  6. The proofs of Theorem \ref{['rmt']} and Corollary \ref{['rmt2']}

Key Result

Lemma 2.3

Let $\xi\in\mathcal{L}^{\exp}_T$ and $g,f\in{\cal{G}}^{\mu,\nu}\cap{\cal{G}}^{\mu,\nu}_{\theta}$. Then the following hold: (i) For all $t\in[0,T]$ and $s\in[0,t]$, ${\cal{E}}^g[{\cal{E}}^g[\xi|{\cal{F}}_t]|{\cal{F}}_s]={\cal{E}}^g[\xi|{\cal{F}}_s]$; (ii) For all $t\in[0,T]$ and $\zeta\in\mathcal{L}^ (v) For all $\beta>1$, there exists a constant $C>0$ depending only on $\beta$, $\mu$, $\nu$ and $T

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 25 more