Minimizing measures of risk by saddle point conditions

Abstract

MSC:Keywords: Risk minimization Saddle point condition Actuarial and financial applications a b s t r a c tThe minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.

V@R representation theorems in ambiguous frameworks

“…These theorems may play a critical role in real applications of risk analysis because they significantly simplify many risk optimization problems. 15,16,22 Therefore, let us give a representation theorem for the robust value at risk.…”

“…Actually, there was a very important precedent since the representation of coherent and convex risk measures played a remarkable role to optimize them. 15,16 Ambiguous settings are characterized by the absence of a perfect knowledge about the probability space that the decision-maker must consider, and they are becoming more and more usual in applications of risk analysis since the lack or data or committed measurement errors may generate significant differences between the real probabilities of the states of the world and the estimated ones. Although there are different approaches to introduce ambiguity, [17][18][19][20][21] we have chosen the one by Balbás et al 22 since, as justified by these authors, their analysis is quite general, contains the nonambiguous framework as a particular case, and deals with a set of priors intuitive and easy to interpret in real applications.…”

Vector Risk Functions

“…Similarly, L ∞ (μ, E) will be the Banach space of E-valued essentially bounded and integrable functions, endowed with the norm 2 and the natural inclusion is continuous. If p < ∞ and E * satisfies the Radon-Nikodym property then, L q (μ, E * ) is the dual space of L p (μ, E).…”

“…In both cases, the Representation Theorems of Section 4 permit us to extend the findings of Balbás et al (2010a), in such a way that the minimization of vector risks become a differentiable problem, and then, standard Lagrangian linked and saddle point linked necessary and sufficient optimality conditions may be given. We will not present a detailed analysis because this is a straightforward extension of Balbás et al (2010a), if one bears in mind Theorems 4.5. and 4.7.…”

Minimax strategies and duality with applications in financial mathematics

“…Therefore, it is worthwhile to study whether the Balbás et al This paper focuses on a general minimax convex problem and provides both a dual approach and necessary and sufficient optimality conditions, which easily apply in practical applications. In Section 2 we will introduce the general framework and will prove a Mean Value Theorem significantly extending that in Balbás et al (2010a). It will characterize some specially important linear and continuous real valued functions on a general Banach space, and the Hahn Banach and the Banach Steinhause Theorems will play a crucial role in the proof.…”