Source-linked AI summary
Representation of the penalty term of dynamic concave utilities
Freddy Delbaen, Shige Peng, Emanuela Rosazza Gianin
TL;DR
The paper addresses representation of penalty terms for general dynamic concave utilities and dynamic convex risk measures. It applies g-expectation theory in a Brownian-filtration setting to establish the representation, with the main result stated in Theorem 5. The paper also notes boundaries involving incomplete markets and equivalent-measure assumptions.
Problem
The paper addresses how to represent the penalty term of general dynamic concave utilities, hence of dynamic convex risk measures.
Method
The paper applies g-expectation theory to dynamic concave utilities in a Brownian filtration with fixed finite horizon T and an equivalent probability measure with zero penalty.
Results
The paper proves that the penalty term has the stated g-expectation-based form, with the exact statement given in Theorem 5.
Takeaways & Limitations
The representation connects suitable dynamic concave utilities with conditional g-expectations and thereby supports their analysis through this theory.
Takeaways & Limitations
In an incomplete market, the lower-price utility can satisfy the paper's properties without being given by a g-expectation.
Abstract
from arXiv · showhide
In this paper we will provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.
1 Introduction
The paper connects dynamic risk measures and g-expectations, then targets a representation of penalty terms for general dynamic concave utilities under a Brownian-filtration setting.
- g-expectations arise as solutions of nonlinear backward stochastic differential equations and have financial applications.
- Conditional g-expectations generate dynamic risk measures that quantify risk at intermediate times relative to a maturity date.
- Dynamic risk measures induced by conditional g-expectations satisfy a time-consistency property.
- The paper represents penalty terms of general dynamic concave utilities, and therefore of dynamic convex risk measures, using g-expectations.
- The representation is developed in a Brownian filtration with a fixed finite horizon and an equivalent probability measure having zero penalty.
- Section 3 contains the main representation result, obtained by applying the theory of g-expectations.
2 Notation and preliminaries
The preliminaries define the Brownian and BSDE framework, state assumptions on the generator, and summarize conditions under which g-expectations induce convex or coherent risk measures.
- The framework uses a standard multidimensional Brownian motion and its augmented filtration on a fixed probability space.
- The generator assumptions include Lipschitz continuity, predictability, square-integrability, independence from y, and convexity in z.
- A BSDE with terminal condition ξ and generator g has a unique predictable solution under the stated setting.
- The analysis subsequently restricts terminal random variables ξ to the essentially bounded space L∞(Ω, FT, P).
- An F-consistent expectation that is translation invariant and dominated by some E^μ is induced by a conditional g-expectation.
- Under the usual assumptions, g-expectations induce convex risk measures, and positive homogeneity in z further yields coherence.
- The paper works with concave utilities by associating each monetary utility functional with the negative of its risk measure.
3 Representation of the penalty term of dynamic concave utilities
The section represents penalties of dynamic concave utilities under Brownian filtration, finite horizon, and equivalent-measure assumptions by applying g-expectation theory. The representation connects time-consistent utilities with convex penalty functions obtained through duality and approximation.
- Main representation: Theorem 5 represents the penalty term of a dynamic concave utility through a nonnegative function f that is proper, convex, and lower semi-continuous in its final argument.The theorem applies to utilities satisfying the section’s stated assumptions and covers all stopping-time intervals.
- Main representation: Time-consistency is equivalent to the representation in Theorem 5(i) under the listed assumptions, linking recursive utility evaluation with the penalty representation.The associated penalty also satisfies the cocycle property, which is equivalent to decomposition of acceptance sets in this setting.
- Approximation by g-expectations: Truncated utilities u_n are induced by conditional g_n-expectations and satisfy BSDEs, with convex generators whose dual functions f_n are proper, convex, and lower semi-continuous.The functions g_n increase with n, while f_n decrease and eventually stabilize pointwise to f.
- Approximation by g-expectations: For equivalent measures Q with bounded density process q, the truncated penalty admits a conditional expectation representation involving f_n evaluated at q.The dual function f_n is induced by the convex generator g_n.
- Passage to the limit: The limiting function f is obtained as the decreasing limit of f_n, with f_n eventually equal to f pointwise where finite.The construction uses bounded-density approximations and convergence of the corresponding penalties.
- Passage to the limit: The approximation theorem provides probability measures Q_n with bounded q_n whose penalties converge to the penalty of any equivalent probability measure Q.This convergence is used to establish the final representation for the untruncated penalty.
4 Appendix
The appendix establishes regularity properties of the dynamic penalty process, including supermartingale behavior, continuity along stopping times, and a càdlàg modification. These properties support the appendix’s representation arguments.
- Setup: For an equivalent measure Q with finite initial penalty, the appendix studies the process (c_t,T(Q)) over the finite horizon.The setup assumes c_0,T(Q) < +∞ and considers stopping times up to T.
- Supermartingale structure: The penalty family satisfies the conditional supermartingale inequality c_σ,T(Q) ≥ E^Q[c_τ,T(Q)|F_σ] whenever σ ≤ τ.This property is obtained from the cocycle property of the penalty.
- Continuity: For decreasing stopping times σ_n ↓ σ, the expected incremental penalty satisfies E^Q[c_σ,σ_n(Q)] → 0.The result is established through a contradiction argument using near-optimal elements of the acceptance set.
- Continuity: For decreasing stopping times σ_n ↓ σ, the total expected penalty converges: E^Q[c_σ_n,T(Q)] → E^Q[c_σ,T(Q)].This follows from the cocycle property and the preceding continuity result.
- Regularization: The process (c_t,T(Q)) admits a càdlàg modification, and its values at stopping times agree with the corresponding penalty variables.The proof uses supermartingale regularization and approximation of general stopping times by rational-valued stopping times.