Time consistency for scalar multivariate risk measures

Proof of Lemma 5.7..

We will prove this result by demonstrating that (iii)  ⇒  (ii)  ⇒  (i)  ⇒  (iii). Following this we will provide notes on any differences in the proof that are utilized when considering the opposite orderings that are stated in parentheses in the three assertions of Lemma 5.7.

(iii) ⇒ (ii) First, consider the representation of ρ t w for any w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) given by (A.1). For any X ∈ L p ⁢ ( ℝ d ) ,

We are able to move the essential supremum inside the conditional expectation since

is ℱ s -decomposable.

(ii) ⇒ (i) Let X ∈ A t and assume X ∉ A t , s + A s (which is closed and convex by Assumption 4.4). Then there exists a Y ∈ L q ⁢ ( ℝ d ) such that 𝔼 ⁢ [ Y 𝖳 ⁢ X ] < inf Z ∈ A t , s + A s ⁡ 𝔼 ⁢ [ Y 𝖳 ⁢ Z ] . By the monotonicity property of acceptance sets, it follows that Y ∈ L q ⁢ ( ℝ + d ) and, in fact, there exists ( ℚ , w + m ⊥ ) ∈ 𝒲 t ⊆ 𝒲 t , s such that Y = w t T ⁢ ( ℚ , w + m ⊥ ) . By the appropriate decomposability of the acceptance sets, we find that

In particular, this implies that either w ∈ R t ⁢ ( 0 ) + or 𝔼 ⁢ [ - α t , s ⁢ ( ℚ , w + m ⊥ ) ] ≤ inf u ∈ R t ⁢ ( 0 ) ⁡ 𝔼 ⁢ [ w 𝖳 ⁢ u ] = - ∞ which is a contradiction to the strict inequality with 𝔼 ⁢ [ w t T ⁢ ( ℚ , w + m ⊥ ) 𝖳 ⁢ X ] constructed initially. Similarly, we can conclude that w t s ⁢ ( ℚ , w + m ⊥ ) ∈ R s ⁢ ( 0 ) + . Then it follows that

This produces a contradiction, and thus X ∈ A t , s + A s . Note that the last step uses X ∈ A t if and only if ρ t w ⁢ ( X ) ≤ 0 for every w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) .

(i) ⇒ (iii) Let w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) , ( ℚ , m ⊥ ) ∈ 𝒲 t , s ⁢ ( w ) , and ( ℝ , n ⊥ ) ∈ 𝒲 s ⁢ ( w t s ⁢ ( ℚ , w + m ⊥ ) )

Note that M s ⊥ = L s q ⁢ ( M ⊥ ) for all times s, thus n ⊥ 𝖳 ⁢ Z t , s = 0 almost surely for every Z t , s ∈ M s . Additionally, in the last line we can interchange the essential supremum and the conditional expectation by the ℱ s -decomposability of A s .

For the opposite orderings:

1. In the implication from (iii) to (ii) we take advantage of the inequality on the penalty functions

   𝔼 ⁢ [ β t , s w ⁢ ( ℚ , m ⊥ , ℝ , n ⊥ ) ] ≥ 𝔼 ⁢ [ α t , s ⁢ ( ℚ , w + m ⊥ ) + α s ⁢ ( ℝ , w t s ⁢ ( ℚ , w + m ⊥ ) + n ⊥ ) ]

   and that, by definition,

   𝔼 ⁢ [ α s ⁢ ( ℝ , w t s ⁢ ( ℚ , w + m ⊥ ) + n ⊥ ) ] ≥ sup u ∈ R s ⁢ ( 0 ) ⁡ 𝔼 ⁢ [ w t s ⁢ ( ℚ , w + m ⊥ ) 𝖳 ⁢ ( - u ) ] = + ∞

   if w t s ⁢ ( ℚ , w + m ⊥ ) ∉ R s ⁢ ( 0 ) + . Thus by considering both (A.1) and (A.2), we find

   ρ t w ( X ) = ess ⁢ sup ( ℚ , m ⊥ ) ∈ 𝒲 t , s ⁢ ( w ) w t s ⁢ ( ℚ , w + m ⊥ ) ∈ R s ⁢ ( 0 ) + ess ⁢ sup ( ℝ , n ⊥ ) ∈ 𝒲 s ⁢ ( w t s ⁢ ( ℚ , w + m ⊥ ) ) ( - β t , s w ( ℚ , m ⊥ , ℝ , n ⊥ )

   + 𝔼 [ w s T ( ℝ , w t s ( ℚ , w + m ⊥ ) + n ⊥ ) 𝖳 ( - X ) ∣ ℱ t ] )

   for every X ∈ L p ⁢ ( ℝ d ) and w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) .

2. In the implication from (ii) to (i) we take advantage of Lemma B.1. This implies that

   0 ≥ - α t , s ( ℚ , w + m ⊥ ) + 𝔼 [ ρ s w t s ⁢ ( ℚ , w + m ⊥ ) ( X ) ∣ ℱ t ]

   for every X ∈ A t , s + A s , ( ℚ , m ⊥ ) ∈ 𝒲 t , s ⁢ ( w ) , w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) , and w t s ⁢ ( ℚ , w + m ⊥ ) ∈ R s ⁢ ( 0 ) + . Thus, it follows that

   0 ≥ ess ⁢ sup ( ℚ , m ⊥ ) ∈ 𝒲 t , s ⁢ ( w ) w t s ⁢ ( ℚ , w + m ⊥ ) ∈ R s ⁢ ( 0 ) + ( - α t , s ( ℚ , w + m ⊥ ) + 𝔼 [ ρ s w t s ⁢ ( ℚ , w + m ⊥ ) ( X ) ∣ ℱ t ] ) ≥ ρ t w ( X )

   for every w ∈ L t q ⁢ ( R ~ t ⁢ ( 0 ) + \ M ⊥ ) . And this is true if and only if X ∈ A t .

∎