Complex risk statistics with scenario analysis
aa r X i v : . [ q -f i n . R M ] N ov Complex risk statistics with scenario analysis
Fei Sun
School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China
Yichuan Dong ∗ National Supercomputing Center in Shenzhen, Shenzhen 518055, China
Abstract
Complex risk is a critical factor for both intelligent systems and risk management. In this paper, weconsider a special class of risk statistics, named complex risk statistics. Our result provides a new approachfor addressing complex risk, especially in deep neural networks. By further developing the properties relatedto complex risk statistics, we are able to derive dual representation for such risk.
Keywords: complex risk, intelligent systems, risk statistic, deep neural networks, dual representation
1. Introduction
Research on complex risk is a popular topic in both intelligent systems and theoretical research, andcomplex risk models have attracted considerable attention, especially in deep neural networks. The quanti-tative calculation of risk involves two problems: choosing an appropriate complex risk model and allocatingcomplex risk to individual components. This has led to further research on complex risk.In a seminal paper, Artzner et al. [4, 5] first introduced the class of coherent risk measures. Later Sunet al. [15] and Sun and Hu [16] focused on set-valued risk measures. However, traditional risk measuresmay fail to describe the characteristics of complex risk. This concept has promoted the study of complexrisk measures. Systemic risk measures were axiomatically introduced by Chen et al. [8]. Other studies ofcomplex risk measures include those of Acharya et al. [1], Armenti et al. [2], Biagini et al. [6], Brunnermeierand Cheridito [7], Feinstein et al. [11], Gauthier et al. [12], Tarashev et al. [18], and the references therein.From the statistical point of view, the behaviour of a random variable can be characterized by its obser-vations, the samples of the random variable. [13] and [14] first introduced the class of natural risk statistics,the corresponding dual representations are also derived. An alternative proof of the dual representationof the natural risk statistics was also derived by [3]. Later, [20] obtained dual representations for convexrisk statistics, and the corresponding results for quasiconvex risk statistics were obtained by [19]. Deng andSun [9] focused on the regulator-based risk statistics for portfolios. However, all of these risk statistics aredesigned to quantify risk of simple component (i.e. a random variable) by its samples. A natural questionis determining how to quantify complex risk by its samples.The main focus of this paper is a new class of risk statistics, named complex risk statistics. In thiscontext, we divide the measurement of complex risk into two steps. Our results illustrate that each complexrisk statistic can be decomposed into a clustering function and a simple risk statistic, which provides a newapproach for addressing complex risk. By further developing the axioms related to complex risk statistics,we are able to derive their dual representations.The remainder of this paper is organized as follows. In Sect. 2, we derive the definitions related tocomplex risk statistics. Sect. 3 discusses a new measurement of complex risk statistics. Finally, in Sect. 4,we consider the dual representations of complex risk statistics. ∗ Corresponding author
Email addresses: [email protected] (Fei Sun), [email protected] (Yichuan Dong)
Preprint submitted to Complexity December 1, 2020 . The definition of complex risk statistics
In this section, we state the definitions related to complex risk statistics. Let R d be the d -dimensionalEuclidean space, d ≥
1. For any x = ( x , . . . , x d ), y = ( y , . . . , y d ) ∈ R d , x ≤ y means x i ≤ y i , 1 ≤ i ≤ d .For any positive integer k i , the element X in product Euclidean space R k × R k × . . . × R k d is denoteby X := ( X , . . . , X k , X , . . . , X k , . . . . . . X d , . . . , X dk d ). For any X, Y ∈ R k × R k × . . . × R k d and 1 ≤ i ≤ d , X (cid:23) Y means P k i j =1 X ij ≤ P k i j =1 Y ij . From now on, the addition and multiplication are all definedpointwise. h X, Y i = P di =1 ( P k i j =1 X ij P k i j =1 Y ij ). h x, y i = P di =1 x i y i For any X ∈ R k × R k × . . . × R k d , X [ k i ] := (0 , . . . . . . , , X i , . . . , X ik i , , . . . . . . , ∈ k × . . . × k i − × R k i × k i +1 × . . . × k d . Definition 2.1.
A simple risk statistic is a function ̺ : R d → R ∪ { + ∞} that satisfies the followingproperties, A1 Monotonicity: for any x, y ∈ R d , x ≥ y implies ̺ ( x ) ≥ ̺ ( y ) ; A2 Convexity: for any x, y ∈ R d and λ ∈ [0 , , ̺ (cid:0) λx + (1 − λ ) y (cid:1) ≤ λ̺ ( x ) + (1 − λ ) ̺ ( y ) . Remark 2.1.
The properties A1 − A2 are very well known and have been studied in detail in the study ofrisk statistics. Definition 2.2.
A clustering function is a function φ : R k × R k × . . . × R k d → R d that satisfies the followingproperties, B1 Monotonicity: for any
X, Y ∈ R k × R k × . . . × R k d , X (cid:23) Y implies φ ( X ) ≥ φ ( Y ) ; B2 Convexity: for any
X, Y ∈ R k × R k × . . . × R k d and λ ∈ [0 , , φ ( λX +(1 − λ ) Y ) ≤ λφ ( X )+(1 − λ ) φ ( Y ) ; B3 Correlation: for any X ∈ R k × R k × . . . × R k d , there exists a simple risk statistic ̺ such that (cid:0) ( ̺ ◦ φ )( X [ k ] ) , ( ̺ ◦ φ )( X [ k ] ) , . . . , ( ̺ ◦ φ )( X [ k d ] ) (cid:1) = φ ( X ) . Definition 2.3.
A complex risk statistic is a function ρ : R k × R k × . . . × R k d → R that satisfies thefollowing properties, C1 Monotonicity: for any
X, Y ∈ R k × R k × . . . × R k d , X (cid:23) Y implies ρ ( X ) ≥ ρ ( Y ) ; C2 Convexity: for any
X, Y ∈ R k × R k × . . . × R k d and λ ∈ [0 , , ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X )+ (1 − λ ) ρ ( Y ) ; C3 Statistical convexity: for any
X, Y, Z ∈ R k × R k × . . . × R k d , λ ∈ [0 , and ≤ i ≤ d , if ρ ( Z [ k i ] ) = λρ ( X [ k i ] ) + (1 − λ ) ρ ( Y [ k i ] ) , then ρ ( Z ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) .
3. How to measure complex risk
In this section, we derive a new approach to measure complex risk in intelligent systems. To this end, weshow that each complex risk statistic can be decomposed into a simple risk statistic varrho and a clusteringfunction φ . In other words, the measurement of complex risk statistics can be simplified into two steps. Theorem 3.1.
A function ρ : R k × R k × . . . × R k d → R is a complex risk statistic in the case of thereexists a clustering function φ : R k × R k × . . . × R k d → R d and a simple risk statistic ̺ : R d → R such that ρ is the composition of ̺ and φ , i.e. ρ ( X ) = ( ̺ ◦ φ )( X ) for all X ∈ R k × R k × . . . × R k d . (3.1) Proof.
We first derive the ‘ only if ’ part. We suppose that ρ is a complex risk statistic and define a function φ by φ ( X ) := (cid:0) ρ ( X [ k ] ) , ρ ( X [ k ] ) , . . . , ρ ( X [ k d ] ) (cid:1) (3.2)for any X ∈ R k × R k × . . . × R k d . Since ρ satisfies the convexity C2 , it follows φ (cid:0) λX + (1 − λ ) Y (cid:1) = (cid:0) ρ ( λX [ k ] + (1 − λ ) Y [ k ] ) , . . . , ρ ( λX [ k d ] + (1 − λ ) Y [ k d ] ) (cid:1) ≤ λ (cid:0) ρ ( X [ k ] ) , . . . , ρ ( X [ k d ] ) (cid:1) + (1 − λ ) (cid:0) ρ ( Y [ k ] ) , . . . , ρ ( Y [ k d ] ) (cid:1) = λφ ( X ) + (1 − λ ) φ ( Y )2or any X, Y ∈ R k × R k × . . . × R k d and λ ∈ [0 , φ satisfies the convexity B2 . Similarly, themonotonicity B1 of φ can also be implied by the monotonicity C1 of ρ . Next, we consider a function ̺ : φ ( R k × R k × . . . × R k d ) → R that is defined by ̺ ( x ) := ρ ( X ) where X ∈ R k × R k × . . . × R k d with φ ( X ) = x. (3.3)Thus, we immediately know that φ satisfies the correlation B3 , which means φ defined above is a clusteringfunction. Next, we illustrate that the ̺ defined above is a simple risk statistic. Suppose x, y ∈ φ ( R k × R k × . . . × R k d ) with x ≥ y , there exists X, Y ∈ R k × R k × . . . × R k d such that φ ( X ) = x , φ ( Y ) = y . Then, wehave φ ( X ) ≥ φ ( Y ) , which means X ≥ Y by the monotonicity of φ . Thus, it follows from the property C1 of ρ that ̺ ( x ) = ρ ( X ) ≥ ρ ( Y ) = ̺ ( y )which implies ̺ satisfies the monotonicity A1 . Let x, y ∈ φ ( R k × R k × . . . × R k d ) with φ ( X ) = x , φ ( Y ) = y for any X, Y ∈ R k × R k × . . . × R k d , which implies ̺ ( x ) = ρ ( X ) and ̺ ( y ) = ρ ( Y ). We also consider z := λx + (1 − λ ) y for any λ ∈ [0 , ̺ , there exists a Z ∈ R k × R k × . . . × R k d such that ̺ ( λx + (1 − λ ) y ) = ρ ( Z )with φ ( Z ) = λx + (1 − λ ) y . Hence, from the statistic convexity C3 of ρ , we know that ̺ ( λx + (1 − λ ) y ) = ρ ( Z ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y )= λ̺ ( x ) + (1 − λ ) ̺ ( y ) , which implies the convexity A2 of ̺ . Thus, ̺ is a simple risk statistic and from (3.2) and (3.3), we have ρ = ̺ ◦ φ . Next, we derive the ‘ if ’ part. We suppose that φ is a clustering function and ̺ is a simple riskstatistic. Furthermore, define ρ = ̺ ◦ φ . Since ̺ and φ are monotone and convex, it is relatively easy to checkthat ρ satisfies monotonicity C1 and convexity C2 . We now suppose that X, Y, Z ∈ R k × R k × . . . × R k d which satisfies ρ ( Z [ k i ] ) = λρ ( X [ k i ] ) + (1 − λ ) ρ ( Y [ k i ] )for any λ ∈ [0 , B3 of φ implies φ ( Z ) = λφ ( X ) + (1 − λ ) φ ( Y ) . Thus, we have ρ ( Z ) = ̺ ( λφ ( X ) + (1 − λ ) φ ( Y )) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) , which indicates ρ satisfies the property C3 . Thus, the ρ defined above is a complex risk statistic. Remark 3.1.
Theorem 3.1 not only provide a decomposition result for complex risk statistics, but alsopropose a approach to deal with complex risk especially in large scale integration. Notably, we first use theclustering function φ to convert the complex system risk into simple, then we quantify the simplified risk bythe simple risk statistic. Therefore, an engineer who deal with the measurement of complex risk in large scaleintegration can construct a reasonable complex risk statistic by choosing an appropriate clustering functionand an appropriate simple risk statistic. The clustering function should reflect his preferences regarding theuncertainty of large scale integration. In the following section, we derive the dual representation of complex risk statistics with the acceptancesets of φ and ̺ . 3 . Dual representation Before we study the dual representation of complex risk statistics on R k × R k × . . . × R k d , the acceptancesets should be defined. Since each complex risk statistic ρ can be decomposed into a clustering function φ and a simple risk statistic ̺ , we need only to define the acceptance sets of φ and ̺ , i.e. A ̺ := (cid:8) ( c, x ) ∈ R × R d : ̺ ( x ) ≤ c (cid:9) (4.1)and A φ := (cid:8) ( y, X ) ∈ R d × ( R k × R k × . . . × R k d ) : φ ( X ) ≤ y (cid:9) . (4.2)We will see later on that these acceptance sets can be used to provide complex risk statistics on R k × R k × . . . × R k d . The following properties are needed in the subsequent study. Definition 4.1.
Let M and N be two ordered linear spaces. A set A ⊂ M × N satisfies f-monotonicity if ( m, n ) ∈ A , q ∈ N and n ≥ q imply ( m, q ) ∈ A . A set A ⊂ M × N satisfies b-monotonicity if ( m, n ) ∈ A , p ∈ M and p ≥ m imply ( p, n ) ∈ A . Proposition 4.1.
We suppose that ρ = ̺ ◦ φ is a complex risk statistic with a clustering function φ : R k × R k × . . . × R k d → R d and a simple risk statistic ̺ : R d → R . The corresponding acceptance sets A ̺ and A φ are defined by (4.1) and (4.2). Then, A φ and A ̺ are convex sets and they satisfy the f-monotonicityand b-monotonicity. Proof.
It is easy to check the above properties from definitions of φ and ̺ .The next proposition provides the primal representation of complex risk statistics on R k × R k × . . . × R k d considering the acceptance sets. Proposition 4.2.
We suppose that ρ = ̺ ◦ φ is a complex risk statistic with a clustering function φ : R k × R k × . . . × R k d → R d and a simple risk statistic ̺ : R d → R . The corresponding acceptance sets A ̺ and A φ are defined by (4.1) and (4.2). Then, for any X ∈ R k × R k × . . . × R k d , ρ ( X ) = inf (cid:8) c ∈ R : ( c, x ) ∈ A ̺ , ( x, X ) ∈ A φ (cid:9) (4.3) where we set inf ∅ = + ∞ . Proof.
Since ρ = ̺ ◦ φ , we have ρ ( X ) = inf (cid:8) c ∈ R : ( ̺ ◦ φ )( X ) ≤ c (cid:9) . (4.4)Using the definition of A ̺ , we know that ̺ ( x ) = inf (cid:8) c ∈ R : ( c, x ) ∈ A ̺ (cid:9) (4.5)for any x ∈ R d . Then, from (4.4) and (4.5), ρ ( X ) = inf (cid:8) c ∈ R : ( c, φ ( X )) ∈ A ̺ (cid:9) . It is easy to check that (cid:8) c ∈ R : ( c, φ ( X )) ∈ A ̺ (cid:9) = (cid:8) c ∈ R : ( c, x ) ∈ A ̺ , ( x, X ) ∈ A φ (cid:9) . Thus, ρ ( X ) = inf (cid:8) c ∈ R : ( c, x ) ∈ A ̺ , ( x, X ) ∈ A φ (cid:9) . With Proposition 4.2, we now introduce the main result of this section: the dual representation of complexrisk statistics on R k × R k × . . . × R k d . 4 heorem 4.1. We suppose that ρ = ̺ ◦ φ is a complex risk statistic characterized by a continue clusteringfunction φ and a continue simple risk statistic ̺ . Then, for any X ∈ R k × R k × . . . × R k d , ρ ( X ) has thefollowing form ρ ( X ) = sup ( b y, b X ) ∈P n h b X, X i − α ( b y, b X ) o (4.6) where α : R d × ( R k × R k × . . . × R k d ) → R is defined by α ( b y, b X ) := sup ( c,x ) ∈A ̺ ( y,Y ) ∈A φ n − c − h b y, ( y − x ) i + h b X, Y i o and P := (cid:8) ( b y, b X ) ∈ R d × ( R k × R k × . . . × R k d ) , α ( b y, b X ) < ∞ (cid:9) . Proof.
Using Proposition 4.2, we have ρ ( X ) = inf (cid:8) c ∈ R : ( c, x ) ∈ A ̺ , ( x, X ) ∈ A φ (cid:9) for any X ∈ R k × R k × . . . × R k d . Furthermore, we can rewrite this formula as ρ ( X ) = inf ( c,x ) ∈ R × R d (cid:8) c + I A ̺ ( c, x ) + I A φ ( x, X ) (cid:9) (4.7)where the indicator function of a set A ∈ X × Y is defined by I A ( a, b ) := (cid:26) , ( a, b ) ∈ X × Y∞ , otherwise.From Proposition 4.1, we know that A ̺ and A φ are convex sets. Thus, I ′A ̺ ( b c, b x ) = sup ( c,x ) ∈A ̺ (cid:8)b cc + h b x, x i (cid:9) , b c ∈ R , b x ∈ R d and I ′A φ ( b y, b X ) = sup ( y,X ) ∈A φ (cid:8)b yy + h b X, X i (cid:9) , b y ∈ R d , b X ∈ R k × R k × . . . × R k d . Next, since ̺ is continue, it follows that A ̺ is closed. Thus, by the duality theorem for conjugate functions,we have I A ̺ ( c, x ) = I ′′A ̺ ( c, x )= sup ( b c, b x ) ∈ R × R d (cid:8)b cc + h b x, x i − I ′A ̺ ( b c, b x ) (cid:9) = sup ( b c, b x ) ∈ R × R d nb cc + h b x, x i − sup ( c,x ) ∈A ̺ (cid:8)b cc + h b x, x i (cid:9)o . Similarly, we have I A φ ( x, X ) = I ′′A φ ( x, X )= sup ( b y, b X ) ∈ R d × ( R k × R k × ... × R kd ) (cid:8) h b y, x i + h b X, X i − I ′A φ ( b y, b X ) (cid:9) = sup ( b y, b X ) ∈ R d × ( R k × R k × ... × R kd ) n h b y, x i + h b X, X i − sup ( y,X ) ∈A φ (cid:8) h b y, y i + h b X, X i (cid:9)o . Thus, we know that ρ ( X ) = inf ( c,x ) ∈ R × R d (cid:8) c + I A ̺ ( c, x ) + I A φ ( x, X ) (cid:9) = inf ( c,x ) ∈ R × R d sup ( b c, b x ) ∈ R × R d ( b y, b X ) ∈ R d × ( R k × R k × ... × R kd ) n c (1 + b c ) + h b x + b y, x i + h b X, X i − I ′A ̺ ( b c, b x ) − I ′A φ ( b y, b X ) o . ̺ and the continuity of φ , we can interchange the supremum and the infimum above,i.e. ρ ( X ) = sup ( b c, b x ) ∈ R × R d ( b y, b X ) ∈ R d × ( R k × R k × ... × R kd ) inf ( c,x ) ∈ R × R d n c (1 + b c ) + h b x + b y, x i + h b X, X i − I ′A ̺ ( b c, b x ) − I ′A φ ( b y, b X ) o = sup ( b y, b X ) ∈ R d × ( R k × R k × ... × R kd ) n h b X, X i − sup ( c,x ) ∈A ̺ ( y,X ) ∈A φ (cid:8) − c − h b y, y − x i + h b X, X i (cid:9)o . With α ( b y, b X ) is defined by α ( b y, b X ) : = sup ( c,x ) ∈A ̺ ( y,X ) ∈A φ (cid:8) − c − h b y, y − x i + h b X, X i (cid:9) = sup ( c,x ) ∈A ̺ ( y,Y ) ∈A φ n − c − h b y, ( y − x ) i + h b X, Y i o and P := (cid:8) ( b y, b X ) ∈ R d × ( R k × R k × . . . × R k d ) , α ( b y, b X ) < ∞ (cid:9) , it immediately follows that ρ ( X ) = sup ( b y, b X ) ∈P n h b X, X i − α ( b y, b X ) o . Remark 4.1.
Note that, the proof of Theorem 4.1 above utilized the primal representation of complex riskstatisticss in Proposition 4.2, which indicates that the acceptance sets A ̺ and A φ play a vital role. Thus, thedual representation of complex risk statistics ρ still dependent on the clustering function φ and the simplerisk statistic ̺ .
5. Conclusions
In this paper, we derive a new class of risk statistics in intelligent systems, especially in deep neuralnetworks, named complex risk statistics. Our results illustrate that an engineer who deal with the measure-ment of complex risk in intelligent systems can construct a reasonable complex risk statistic by choosing anappropriate clustering function and an appropriate simple risk statistic.