Optimization of convex risk functions

What are convex measures of risk?
A convex risk measure is defined as follows: Definition 1.1 (Convex risk measure) A convex risk measure is a function ρ : X → R which satisfies the following for each X, Y ∈ X: (i) (Convexity) ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for 0 ≤ λ ≤ 1. (ii) (Monotonicity) If X ≤ Y , then ρ(X) ≥ ρ(Y )..
A convex risk measure is defined as follows: Definition 1.1 (Convex risk measure) A convex risk measure is a function ρ : X → R which satisfies the following for each X, Y ∈ X: (i) (Convexity) ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for 0 ≤ λ ≤ 1. (ii) (Monotonicity) If X ≤ Y , then ρ(X) ≥ ρ(Y ).
Definition 1.1 (Convex risk measure) A convex risk measure is a function ρ : X → R which satisfies the following for each X, Y ∈ X: (i) (Convexity) ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for 0 ≤ λ ≤ 1. (ii) (Monotonicity) If X ≤ Y , then ρ(X) ≥ ρ(Y ).

We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and opti mization theory in vector spaces of

How can es be induced by a convex risk measure?

Assume that the insurer uses ES at level α∈(0,1)

Define a convex function β:Φ1→R∪{+∞}via β(h)=0if h(t)=hα(t)≜1αtI[0,α](t)+I(α,1](t)otherwise β(h)=+∞

Then, ESαcan be induced by substituting this particular βinto the expression (11)for the convex risk measure in Theorem 2

What are optimization problems involving convex risk functions?

We consider optimization problems involving convex risk functions

By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions, we develop new representation theorems for risk models, and optimality and duality theory for problems with convex risk functions

What are some examples of convex risk functionals?

Below we list some classic examples of convex risk functionals

The first interesting examples are the standard deviation and the variance, both well known to be convex (Deprez and Gerber, 1985) and they have a representation in Theorem 2

2 Example 2 1Standard Deviation

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have.
A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position.
Many risk measures have hitherto been proposed, each having certain characteristics.
The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality.
The EVaR can also be represented by using the concept of relative entropy.
Because of its connection with the VaR and the relative entropy, this risk measure is called entropic value at risk.
The EVaR was developed to tackle some computational inefficiencies of the CVaR.
Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures.
Both the CVaR and the EVaR are members of this class.

Type of mathematical function

Optimization of convex risk functions

What are convex measures of risk?

How can es be induced by a convex risk measure?

What are optimization problems involving convex risk functions?

What are some examples of convex risk functionals?