Multivariate systemic optimal risk transfer equilibrium

Abstract

A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in: “Systemic optimal risk transfer equilibrium”, Mathematics and Financial Economics (2021), for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical Bühlmann’s notion of a risk exchange equilibrium with a capital allocation principle based on systemic expected utility optimization. In this paper we extend such a notion to the case when the value function to be optimized is multivariate in a general sense, and it is not simply given by the sum of univariate utility functions. This takes into account the fact that preferences of single agents might depend on the actions of other participants in the game. Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.

Short Communication: Are Shortfall Systemic Risk Measures One Dimensional?

“…+ , so that Z2 − Z1 = π(W) for some W ∈ (L+) N by (6). Hence, W2 := W1 + W ≥ W1 satisfies π(W2) = Z2 and U(W1) ≤ U(W2) ≤ π U(Z2).…”

“…Indeed, Z = Z + − Z − ∈ L M , for Z ± the componentwise positive and negative parts, so that Z ± = π(X±) for X± ∈ L N + (by (6)) and by linearity Z = π(X+ − X−). The same works with deterministic vectors in particular, yielding the second equality.…”

“…Take z = 0 ∈ R M and take a maximizing sequence (xn)n for π U (0), w.l.o.g. assuming that U (xn) ≥ π U (0) − 1 for every n. By [6]…”

“…A function Φ : [0, +∞) N → R is called multivariate Orlicz function if it is null in 0, convex, continuous, increasing in the usual componentwise order and satisfies: there exist A > 0, B constants such that Φ(x) ≥ A N j=1 x j − B for every x ∈ [0, +∞) N . We refer to [1] and [6] for further details. Inspired by [6] Definition 3.4, we say that a function U : R N → R is well controlled if there exist a multivariate Orlicz function Φ : R N → R and a function h : [0, +∞) → R such that U (x) ≤ − Φ((x) − ) + ε N j=1 x j + h(ε) for every ε > 0.…”

“…We refer to [1] and [6] for further details. Inspired by [6] Definition 3.4, we say that a function U : R N → R is well controlled if there exist a multivariate Orlicz function Φ : R N → R and a function h : [0, +∞) → R such that U (x) ≤ − Φ((x) − ) + ε N j=1 x j + h(ε) for every ε > 0.…”