Dynamic risk measures on variable exponent Bochner--Lebesgue spaces
aa r X i v : . [ q -f i n . R M ] J u l Noname manuscript No. (will be inserted by the editor)
Dynamic risk measures on variable exponent Bochner–Lebesgue spaces
Fei Sun · Yijun HuAbstract
In this paper, we consider dynamic risk measures on L p ( · ) . Keywords risk measure · dynamic Mathematics Subject Classification (2010)
The remainder of this paper is organized as follows. In Section 2, we briefly review the definition and mainproperties of variable exponent Bochner–Lebesgue spaces. In Section 3, we consider the dual representationof convex risk measures on variable exponent Bochner–Lebesgue spaces. Section 4 discusses a class of specificrisk measures known as optimized certainty equivalents. Finally, in Section 5, we study dual representationof dynamic risk measures on the variable exponent Bochner–Lebesgue spaces. The related time consistencyis also studied.
In this section, we briefly introduce properties of variable exponent Bochner–Lebesgue spaces. p( · ) The main aim of this paper is to study the dual representation of dynamic risk measures on variableexponent Bochner–Lebesgue spaces. To this end, this section first considers convex risk measures, which willbe used later for the dynamic risk measures. The convex risk measure was first introduced by F¨ollmer andSchied (2002) and Frittelli and Rosazza Gianin (2002).In the absence of ambiguity, we denote the variable exponent Bochner–Lebesgue space by L p ( · ) := L p ( · ) ( Ω, E ). Let T be a discrete time horizon which can reach infinity and consider a filtered probabil-ity space ( Ω, F , ( F t ) Tt =0 , P ) with {∅ , Ω } = F ⊂ F ⊂ . . . ⊂ F T = F . Let L p ( · ) ( F t ) be the space of allstrongly F t -measurable functions f t which satisfy Definition ?? . Note that L p ( · ) = L p ( · ) ( F T ). We denote L p ( · ) ( K ) := { f ∈ L p ( · ) : f : Ω → K } and L p ′ ( · ) ( K ) := { g ∈ L p ′ ( · ) : g : Ω → K } . We denote the space of allessentially bounded F t -measurable random variables by L ∞ t := L ∞ ( Ω, F t , µ ). F. SunSchool of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, ChinaE-mail: [email protected]. HuSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaE-mail: [email protected] Fei Sun, Yijun Hu
Remark 31
As the Banach space E is partially ordered by K , the definition of L p ( · ) implies that each f ∈ L p ( · ) is an E -valued measurable function. Thus, in the absence of ambiguity, we also consider L p ( · ) tobe partially ordered by K . Next, the definition of a convex risk measure on L p ( · ) is introduced using anaxiomatic approach. Definition 31
Let E be a Banach space ordered by the partial ordering relation induced by a cone K withinterior point z , and L p ( · ) be a variable exponent Bochner–Lebesgue space. A function ̺ : L p ( · ) → R is saidto be a p ( · )-convex risk measure if it satisfies the following:A1 Monotonicity: for any f , f ∈ L p ( · ) , f ≤ K f implies ̺ ( f ) ≥ ̺ ( f );A2 Translation invariance: for any m ∈ R and f ∈ L p ( · ) , ̺ ( f + mz ) = ̺ ( f ) − m ;A3 Convexity: for any f , f ∈ L p ( · ) and λ ∈ (0 , ̺ ( λf + (1 − λ ) f ) ≤ λ̺ ( f ) + (1 − λ ) ̺ ( f ). Remark 32
The order in A1 is the partial order under a cone K which is defined by Remark ?? . Theinterior point z of K in A2 is considered to be the numeraire asset, which means that mz ∈ E for any m ∈ R . Before we study the dual representation of the p ( · )-convex risk measures, the acceptance sets shouldbe defined. Definition 32
The acceptance set of the p ( · )-risk measure ̺ is defined as A ̺ := (cid:8) f ∈ L p ( · ) : ̺ ( f ) ≤ (cid:9) and we denote A ̺ by A ̺ := n g ∈ (cid:0) L p ( · ) (cid:1) ∗ : h g, f i ≥ f ∈ A ̺ o . Remark 33
It is relatively easy to check that A ̺ is a convex set if ̺ satisfies the convexity property. A ̺ can be considered as the positive polar cone of A ̺ .Now, we provide the dual representation of p ( · )-convex risk measures which will be used in the proof of p ( · )-dynamic risk measures in Section 5. Theorem 31 If ̺ : L p ( · ) → R is a p ( · )-convex risk measure, then for any f ∈ L p ( · ) , ̺ ( f ) = sup g ∈ Q p ( · ) {h g, − f i − α ( g ) } where Q p ( · ) := n g ∈ (cid:0) L p ( · ) (cid:1) ∗ : h dgdµ , z i = 1 , dgdµ ∈ L p ′ ( · ) ( K ) o and the minimal penalty function α min is denoted by α min ( g ) := sup f ∈ L p ( · ) (cid:8) h g, − f i − ̺ ( f ) (cid:9) = sup f ∈A ̺ (cid:8) h g, − f i (cid:9) . Proof.
For any g ∈ Q p ( · ) , we denote α ( g ) = sup f ∈ L p ( · ) (cid:8) h g, − f i − ̺ ( f ) (cid:9) and α min ( g ) = sup f ∈A ̺ (cid:8) h g, − f i (cid:9) . We now show that α ( g ) = α min ( g ) for any g ∈ Q p ( · ) . Note that α ( g ) ≥ α min ( g ). Indeed, for any f ∈ A ̺ , h g, − f i − ̺ ( f ) ≥ h g, − f i . Hence,sup f ∈ L p ( · ) (cid:8) h g, − f i − ̺ ( f ) (cid:9) ≥ sup f ∈A ̺ (cid:8) h g, − f i − ̺ ( f ) (cid:9) ≥ sup f ∈A ̺ (cid:8) h g, − f i (cid:9) . We now prove that α ( g ) ≤ α min ( g ). For any f ∈ L p ( · ) , consider f = f + ̺ ( f ) z ∈ A ̺ . Thus, α min ( g ) ≥ h g, − f i = h g, − f i − ̺ ( f ) h g, z i = h g, − f i − ̺ ( f ) Z Ω h dgdµ , z i dµ = h g, − f i − ̺ ( f ) . ynamic risk measures on variable exponent Bochner–Lebesgue spaces 3 Hence, we have α ( g ) = α min ( g ), and it is easy to check that ̺ ( f ) ≥ sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) . Next, we show that the above inequality only holds in the case of equality. Suppose there is some f ∈ L p ( · ) such that ̺ ( f ) > sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) . Hence, there exists some m ∈ R such that ̺ ( f ) > m > sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) . Thus, we have ̺ ( f + mz ) = ̺ ( f ) − m > , which means that f + mz / ∈ A ̺ . As { f + mz } is a singleton set, it is also a convex set. On the contrary, A ̺ is also a closed convex set because ̺ is a p ( · )-convex risk measure. Then, by the Strong Separation Theoremfor convex sets, there exists some π ∈ ( L p ( · ) ) ∗ such that h π, f + mz i > sup f ∈A ̺ h π, f i . (3.1)By Remark ?? , h π, f i = R Ω h h, f i dµ , where h ∈ L p ′ ( · ) ( Ω, E ∗ ). It is easy to check that h takes negative valueson L p ( · ) ( K ). Then, we have that − h ∈ L p ′ ( · ) ( K ). For any − π ∈ Q p ( · ) , h h, z i = −
1. Thus, by (3.1), we canconclude that h π, f + mz i > sup f ∈A ̺ h π, f i ⇒ Z Ω h h, f + mz i dµ > sup f ∈A ̺ Z Ω h h, f i dµ ⇒ Z Ω (cid:0) h h, f i − m (cid:1) dµ > sup f ∈A ̺ Z Ω h h, f i dµ ⇒ Z Ω h h, f i dµ − m > sup f ∈A ̺ Z Ω h h, f i dµ ⇒ h π, f i − sup f ∈A ̺ h π, f i > m ⇒ h π, f i − α ( − π ) > m. Replacing − π by g , we have h g , − f i − α ( g ) > m. This is a contradiction, because in this case m > sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) ≥ h g , − f i − α ( g ) > m. The contradiction arises from the assumption that some f ∈ L p ( · ) exists such that ̺ ( f ) > sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) . Hence, we have ̺ ( f ) = sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) . For the opposite direction, it is relatively simple to check that ̺ satisfies the properties of a p ( · )-convex riskmeasure. This completes the proof of Theorem 31.The p ( · )-convex risk measures can be considered as extensions of the convex risk measures studied byFrittelli and Rosazza Gianin (2002). A special example of p ( · )-convex risk measures, the so-called OCE, isdiscussed in the next section. Finally, in Sections 5, the p ( · )-convex risk measures are used to study the dualrepresentation of the p ( · )-dynamic risk measures. Fei Sun, Yijun Hu p( · ) In this section, we study a special class of p ( · )-convex risk measures: the Optimized Certainty Equivalent(OCE), which will be used as an example of dynamic risk measures in Section 5. The OCE was first introducedby Ben-Tal and Teboulle (1986) and later developed by the same researchers (Ben-Tal and Teboulle 2007). Inthis section, we define the OCE on variable exponent Bochner–Lebesgue spaces L p ( · ) . Further, we establishits main properties, and show how it can be used to generate p ( · )-convex risk measures. Note that the OCEcan be used as an application of convex risk measures. Definition 41
Let u : E → [ −∞ , + ∞ ] be a closed, concave, and non-decreasing (partially ordered by K )function. Suppose that u ( θ ) = 0, where θ is the zero element of E . We denote the set of such u by U . Remark 41
For any u ∈ U and f ∈ L p ( · ) , we denote by u ( f ) : Ω → R and E u ( f ) the expectation of u ( f )with respect to a probability measure µ . Definition 42
For any u ∈ U and f ∈ L p ( · ) , the OCE of some uncertain outcome f is defined by the map S u : L p ( · ) → R , S u ( f ) = sup η ∈ R { η + E u ( f − ηz ) } where the domain of S u is defined as dom S u = { f ∈ L p ( · ) | S u ( f ) > −∞} 6 = ∅ and S u is finite on dom S u . Theorem 41
For any u ∈ U , the following properties hold for S u :(a) For any f ∈ L p ( · ) and m ∈ R , S u ( f + mz ) = S u ( f ) + m ;(b) For any f , f ∈ L p ( · ) , f ≤ K f implies that S u ( f ) ≤ S u ( f );(c) For any f , f ∈ L p ( · ) and λ ∈ (0 , S u ( λf + (1 − λ ) f ) ≥ λS u ( f ) + (1 − λ ) S u ( f ). Proof. (a) For any f ∈ L p ( · ) , m ∈ R , S u ( f + mz ) = sup η ∈ R { η + E u ( f + mz − ηz ) } = m + sup η ∈ R { η − m + E u ( f − ( η − m ) z ) } = m + S u ( f ) . (b) For any f , f ∈ L p ( · ) with f ≤ K f , we have f − ηz ≤ K f − ηz . As u is non-decreasing, we have S u ( f ) = sup η ∈ R { η + E u ( f − ηz ) } ≤ sup η ∈ R { η + E u ( f − ηz ) } = S u ( f ) . (c) For any f , f ∈ L p ( · ) and λ ∈ (0 , S u ( λf + (1 − λ ) f ) = sup η ∈ R n η + E u (cid:0) λf + (1 − λ ) f − ηz (cid:1)o . We take η = λη + (1 − λ ) η . Then, S u ( λf + (1 − λ ) f )= sup η ,η ∈ R n λη + (1 − λ ) η + E u (cid:0) λ ( f − η z ) + (1 − λ )( f − η z ) (cid:1)o ≥ sup η ,η ∈ R n λη + (1 − λ ) η + λ E u ( f − η z ) + (1 − λ ) E u ( f − η z ) o = sup η ,η ∈ R n λ (cid:0) η + E u ( f − η z ) (cid:1) + (1 − λ ) (cid:0) η + E u ( f − η z ) (cid:1)o = λS u ( f ) + (1 − λ ) S u ( f ) . This completes the proof of Theorem 41.
Theorem 42
The function ̺ , defined as ̺ ( f ) := − S u ( f ) for any f ∈ L p ( · ) , is a p ( · )-convex risk measure. ynamic risk measures on variable exponent Bochner–Lebesgue spaces 5 Proof:
The proof of Theorem 42 is straightforward from Theorem 41.
Proposition 41
For any u ∈ U , α ∈ R + , and f ∈ L p ( · ) , the OCE S u ( f ) is sub-homogeneous, i.e.(a) S u ( αf ) ≤ αS u ( f ) , ∀ α > S u ( αf ) ≥ αS u ( f ) , ∀ ≤ α ≤ Proof.
Denote S ( α ) := α S u ( αf ). Then, S ( α ) = 1 α S u ( αf ) = sup η ∈ R n η + E α u (cid:0) α ( f − ηz ) (cid:1)o . (4.1)Next, we show that S ( α ) is non-increasing in α > f ∈ L p ( · ) . For α ≥ α ≥
0, we have u ( α t ) − u ( α t ) α − α ≤ u ( α t ) − u ( θ ) α − , for any t ∈ E by the concavity of u . As u ( θ ) = 0, we have1 α u ( α t ) ≤ α u ( α t ) . Then, from (4.1), we have S ( α ) ≥ S ( α ), which clearly implies ( a ) and ( b ). Proposition 42 (Second-order stochastic dominance) We denote C u ( f ) := u − E u ( f ) for any u ∈ U and f ∈ L p ( · ) . We also assume that the supremum in the definition of S u is attained. Then, for any f , f ∈ L p ( · ) , S u ( f ) ≥ S u ( f ) if and only if C u ( f ) ≥ C u ( f ) . Proof.
We first show the “if” part. If C u ( f ) ≥ C u ( f ), we have E u ( f ) ≥ E u ( f ) by the fact that u isnon-decreasing. Then, from the definition of S u , it follows that S u ( f ) ≥ S u ( f ). We now show the “only if”part. Let ℓ f , ℓ f be the points where the suprema of S u ( f ) and S u ( f ) are attained, respectively. Then, forany u ∈ U , S u ( f ) = ℓ f + E u ( f − ℓ f z ) ≥ ℓ f + E u ( f − ℓ f z ) ≥ ℓ f + E u ( f − ℓ f z ) , where the first inequality comes from S u ( f ) ≥ S u ( f ). Therefore, for any u ∈ U , E u ( f − ℓ f z ) ≥ E u ( f − ℓ f z ), which implies E u ( f ) ≥ E u ( f ). Then, C u ( f ) ≥ C u ( f ). p( · ) A person’s risk assessments may change over time. This observation motivated us to study the dynamic p ( · )-convex risk measures on the variable exponent Bochner–Lebesgue spaces.In fact, in dynamic cases, the risk measures can be regarded as both the minimum capital requirementof some real number and the hedging of some financial positions denoted by bounded random variables.Conditional p ( · )-convex risk measures are now introduced using an axiomatic approach. Definition 51
A map ̺ t : L p ( · ) → L ∞ t is called a conditional p ( · )-convex risk measure if it satisfies thefollowing properties for all f, f , f ∈ L p ( · ) :i. Monotonicity: f ≤ K f implies ̺ t ( f ) ≥ ̺ t ( f );ii. Conditional cash invariance: for any m t ∈ L ∞ t , ̺ t ( f + m t z ) = ̺ t ( f ) − m t ;iii. Conditional convexity: for any λ ∈ L ∞ t with λ ∈ [0 , ̺ t ( λf + (1 − λ ) f ) ≤ λ̺ t ( f ) + (1 − λ ) ̺ t ( f );iv. Normalization: ̺ t ( θ ) = 0, ̺ t ( f ) < ∞ .Additionally, a conditional p ( · )-convex risk measure is coherent if it satisfies the following property:v. Conditional positive homogeneity: for any λ ∈ L ∞ t with λ > ̺ t ( λf ) = λ̺ t ( f ). Fei Sun, Yijun Hu
Remark 51
Note that any element in L ∞ t := L ∞ ( Ω, F t , µ ) is a random variable, where F t is a sub- σ -algebra of F . As stated by Detlefsen and Scandolo (2005), if the additional information is described bya sub- σ -algebra F t of the total information F T , then a conditional risk measure is a map assigning an F t -measurable random variable ̺ t ( f ), representing the conditional riskiness of f , to every F T -measurablefunction f , representing a final payoff.The acceptance set of a conditional p ( · )-convex risk measure ̺ t is defined as A t := (cid:8) f ∈ L p ( · ) : ̺ t ( f ) ≤ (cid:9) for any 0 ≤ t ≤ T. (5.1)The corresponding stepped acceptance set is defined as A t,t + s := (cid:8) f ∈ L p ( · ) ( F t + s ) : ̺ t ( f ) ≤ (cid:9) for any 0 ≤ t < t + s ≤ T. (5.2) Proposition 51
The acceptance set A t of a conditional p ( · )-convex risk measure ̺ t has the followingproperties:1. Conditional convexity: for any f , f ∈ A t , and an F t -measurable function α with 0 ≤ α ≤
1, we have αf + (1 − α ) f ∈ A t ;2. Solidity: for any f ∈ A t with f ≤ K f , we have f ∈ A t ;3. Normalization: 0 ∈ A t . Proof.
It is easy to check properties 1–3 using Definition 51.
Definition 52
A sequence ( ̺ t ) Tt =0 is called a dynamic p ( · )-convex risk measure if each ̺ t is a conditional p ( · )-convex risk measure for any 0 ≤ t ≤ T .We now study the dual representation of a conditional p ( · )-convex risk measure. First, the notion of the F t -conditional inner product related to L p ( · ) should be defined. Definition 53
For any f ∈ L p ( · ) and g ∈ ( L p ( · ) ) ∗ , we define the F t -conditional inner product h g, − f i t by Z A h g, − f i t dµ = h g, − f i for any A ⊆ F t . (5.3)We also define the minimal penalty function α min t as α min t ( g ) := ess sup f ∈A t h g, − f i t . (5.4) Lemma 51
For any g ∈ Q p ( · ) , 0 ≤ t ≤ T , and A ⊆ F t , Z A α min t ( g ) dµ = sup f ∈A t h g, − f i . (5.5) Proof.
We first show that there exists a sequence ( f n ) n ∈ N in A t such thatess sup f ∈A t h g, − f i t = lim n →∞ h g, − f n i t . (5.6)Indeed, for any f , f ∈ A t , we define b f := f I B + f I B c where B := {h g, − f i t ≥ h g, − f i t } . By property 1of Proposition 51, we know that b f ∈ A t . Hence, by the definition of b f , h g, − b f i t = max {h g, − f i t , h g, − f i t } . Thus, (5.6) holds. We now have Z A α min t ( g ) dµ = Z A ess sup f ∈A t h g, − f i t dµ = Z A lim n →∞ h g, − f n i t dµ = lim n →∞ Z A h g, − f n i t dµ = lim n →∞ h g, − f n i≤ sup f ∈A t h g, − f i . ynamic risk measures on variable exponent Bochner–Lebesgue spaces 7 The converse inequality is easy to check.The following theorem gives the dual representation of conditional p ( · )-convex risk measures. Theorem 51
Suppose ̺ t is a conditional p ( · )-convex risk measure. Then, the following statements areequivalent.(1) ̺ t has the robust representation ̺ t ( f ) = ess sup g ∈ Q p ( · ) (cid:8) h g, − f i t − α t ( g ) (cid:9) for any f ∈ L p ( · ) , (5.7)where Q p ( · ) := n g ∈ (cid:0) L p ( · ) (cid:1) ∗ : h dgdµ , z i = 1 , dgdµ ∈ L p ′ ( · ) ( K ) o , and α t is a map from Q p ( · ) to the set of F t -measurable random variables such that ess sup g ∈ Q p ( · ) {− α t ( g ) } =0;(2) ̺ t has a robust representation in terms of the minimal function, i.e. ̺ t ( f ) = ess sup g ∈ Q p ( · ) (cid:8) h g, − f i t − α min t ( g ) (cid:9) for any f ∈ L p ( · ) ; (5.8)(3) ̺ t is continuous from above under K , i.e. f n ց f ⇒ ̺ t ( f n ) ր ̺ t ( f ) . (5.9) Proof. (2) ⇒ (1) is obvious. We first prove (1) ⇒ (3). Using Lemma 5 of Cheng and Xu (2013), supposethat f n ց f . Then, by the monotonicity of ̺ t , we have ̺ t ( f n ) ր ̺ t ( f ).Next, we show (3) ⇒ (2). The inequality ̺ t ( f ) ≥ ess sup g ∈ Q p ( · ) (cid:8) h g, − f i t − α mint ( g ) (cid:9) is a direct consequence of the definition of α min t . Now, we need only show the inverse inequality. To this end,we define a map e ̺ : L p ( · ) → R as e ̺ ( f ) = R A ̺ t ( f ) dµ . It is easy to check that e ̺ is a p ( · )-convex risk measureas defined in Section 3 which is continuous from above. Hence, by Theorem 31, we know that e ̺ has the dualrepresentation e ̺ ( f ) = sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) , f ∈ L p ( · ) , where the minimum penalty function α min is given by α min ( g ) := sup f ∈A e ̺ (cid:8) h g, − f i (cid:9) . By Lemma 51, we have Z A α min t ( g ) dµ = sup f ∈A t h g, − f i for any g ∈ Q p ( · ) . As e ̺ ( f ) ≤ f ∈ A t , Z A α min t ( g ) dµ = sup f ∈A t h g, − f i ≤ α ( g )for any g ∈ Q p ( · ) . Thus, we have Z A ̺ t ( f ) dµ = e ̺ ( f )= sup g ∈ Q p ( · ) (cid:8) h g, − f i − α ( g ) (cid:9) ≤ sup g ∈ Q p ( · ) n Z A h g, − f i t dµ − Z A α min t ( g ) dµ o = sup g ∈ Q p ( · ) n Z A (cid:0) h g, − f i t − α min t ( g ) (cid:1) dµ o ≤ Z A ess sup g ∈ Q p ( · ) (cid:8) h g, − f i t − α min t ( g ) (cid:9) dµ. Fei Sun, Yijun Hu
Thus, (5.8) holds.Now, with the definition and dual representation, we consider the time consistency of dynamic p ( · )-convexrisk measures. Definition 54
A dynamic p ( · )-convex risk measure ( ̺ t ) Tt =0 is said to be time consistent if, for all f , f ∈ L p ( · ) and 0 ≤ t < t + s ≤ T , ̺ t + s ( f ) ≤ ̺ t + s ( f ) ⇒ ̺ t ( f ) ≤ ̺ t ( f ) . (5.10) Remark 52
Time consistency means that if two payoffs will have the same riskiness tomorrow in everystate of nature, then the same conclusion should be drawn today.
Theorem 52
Let ( ̺ t ) Tt =0 be a dynamic p ( · )-convex risk measure such that each ̺ t is continuous from above.Then, the following conditions are equivalent for any 0 ≤ t < t + s ≤ T :1). ( ̺ t ) Tt =0 is time consistent;2). A t = A t,t + s + A t + s ;3). ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) = ̺ t ( f ) for any f ∈ L p ( · ) . Proof.
We first show the equivalence between 1) and 3). Suppose that 3) holds and ̺ t + s ( f ) ≤ ̺ t + s ( f ) forany f , f ∈ L p ( · ) . Then, by the monotonicity of ̺ t , ̺ t ( f ) = ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≤ ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) = ̺ t ( f ) . Next, suppose that ( ̺ t ) Tt =0 is time consistent, and set f := − ̺ t + s ( f ) z = − ̺ t + s ( f ) z for any f ∈ L p ( · ) .Thus, ̺ t ( f ) = ̺ t ( f ) = ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) = ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) . We now show the equivalence between 2) and 3). To this end, suppose that 3) holds and let f ∈ A t,t + s , f ∈ A t + s . Then, setting f := f + f , we have ̺ t + s ( f ) = ̺ t + s ( f + f ) = ̺ t + s ( f ) − f z ≤ − f z . Thus, by the monotonicity of ̺ t , we know that ̺ t ( f ) = ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≤ ̺ t ( f ) ≤ , which implies A t ⊇ A t,t + s + A t + s . For the inverse relation, let f ∈ A t and define f := f + ̺ t + s ( f ) z , f := f − f = − ̺ t + s ( f ) z . Then, by theconditional cash invariance of ̺ t , it is easy to check that f ∈ A t,t + s , f ∈ A t + s , which implies A t ⊆ A t,t + s + A t + s . Let us now suppose that 2) holds and f ∈ A t . It is easy to check that f + ̺ t + s ( f ) z ∈ A t + s . Then, with A t ⊆ A t,t + s + A t + s , we have − ̺ t + s ( f ) z ∈ A t,t + s . Hence, we know that ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≤
0, which implies ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≤ ̺ t ( f ) . Now, we need only show the inverse inequality. Indeed, for any f ∈ L p ( · ) such that − ̺ t + s ( f ) z ∈ A t,t + s , wehave ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≤
0. It is easy to check that f + ̺ t + s ( f ) z ∈ A t + s . Thus, by A t ⊇ A t,t + s + A t + s , wehave f ∈ A t , which implies ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) ≥ ̺ t ( f ) . The case for the recursive property strongly relies on the validity of conditional cash invariance for ̺ t ,and hence on the interpretation as conditional capital requirements. In fact, if ̺ t + s ( f ) is the conditionalcapital requirement that has to be set aside at date t + s in view of the final payoff f , then the risky positionis equivalently described, at date t , by the payoff ̺ t (cid:0) − ̺ t + s ( f ) z (cid:1) occurring in t + s .We end this section with a special example of conditional p ( · )-convex risk measures. ynamic risk measures on variable exponent Bochner–Lebesgue spaces 9 Example 51 (Conditional OCE) Let u : E → R be a closed, concave, and non-decreasing (partially orderedby K ) function and suppose that u ( θ ) = 0, where θ is the zero element of E . Then, for any f ∈ L p ( · ) , theconditional OCE of some uncertain outcome f is defined by the map S u : L p ( · ) → L ∞ t : S u ( f ) = ess sup η ∈ L ∞ t n η + E (cid:2)(cid:0) u ( f − ηz ) (cid:1) |F t (cid:3)o . Thus, by Definition 51, it is easy to check that the function ̺ t defined as ̺ t ( f ) := − S u ( f ) for any f ∈ L p ( · ) is a conditional p ( · )-convex risk measure. References
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