11Dynamic risk measures

In this chapter we return to the quantification of financial risk in terms of monetary risk measures. As we saw in Chapter 4, a monetary risk measure specifies the capital which is needed in order to make a given financial position acceptable. In our present dynamic setting, the capital requirement at a given time will depend on the available information and will thus become a random variable. In Section 11.1 we introduce the notion of a conditional monetary risk measure. In the convex case we prove a conditional version of the robust representation theorem which involves the conditional expected losses under various models Q and a conditional penalization of these models.

As time varies, both the capital requirement and the penalization of a given probabilistic model become stochastic processes. The structure of these processes depends on how the risk assessments at different times are connected with each other. In Section 11.2 we focus on a strong notion of time consistency which amounts to a recursive property of the successive capital requirements. For a typical model Q, time consistency implies that the corresponding penalization process is a super-martingale under Q, and that its Doob decomposition takes a special form. Time consistency is also characterized by a combined supermartingale property of the capital requirements and the penalization process.

11.1Conditional risk measures and their robust representation

As in the preceding chapters we fix a filtration (Ft)t=0,...,T on our probability space (Ω,F, P) such that F0 = {∅, Ω} and FT = F. As the set X of all financial positions we take the space L∞ := L∞(Ω,F, P). The subspace ∞consists of those positions whose outcome only depends on the history up to time t. All inequalities and equalities applied to random variables are meant to hold P-a.s. if not stated otherwise.

Definition 11.1. A map will be called a monetary conditional risk measure if it satisfies the following properties for all X, Y ∈ L∞:

–Conditional cash invariance∞: ρt(X + Xt) = ρt(X) − Xt for any

–Monotonicity: X ≤ Y ⇒ ρt(X) ≥ ρt(Y).

–Normalization: ρt(0) = 0.

A monetary conditional risk measure will be called convex if it satisfies

–Conditional ∞convexity: with 0 ≤ λ ≤ 1.

A convex conditional risk measure will be called coherent if it satisfies in addition

–∞Conditional positive homogeneity: ρt(λX) = λρt(X) for such that λ ≥ 0.

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For t = 0 we recover our previous definition of (unconditional and normalized) monetary, convex, and coherent risk measures given in Section 4.1, since the space reduces to the real line.

To any conditional monetary risk measure ρt we associate its acceptance set

One easily checks that At has the following properties:

–X ∈ At , Y ≥ X ⇒ Y ∈ At ,

–ess infand 0 ∈ At.

Note that ρt is uniquely determined by its acceptance set since

A conditional monetary risk measure can thus be viewed as the conditional capital requirement needed at time t to make a financial position X acceptable at that time. Moreover, ρt is a convex conditional risk measure if and only if At satisfies the additional property of

–Conditional convexity: X, Y ∈ At and with 0 ≤ λ ≤ 1 implies λX + (1 − λ) ∞Y ∈ At.

Conversely, one can use acceptance sets to define conditional monetary risk measures: If a given acceptance set At ⊆ L∞ satisfies the above conditions then the functional defined via (11.2) is a conditional monetary risk ∞measure.

Exercise 11.1.1. Let be a conditional monetary risk ∞measure. For X ∈ L∞ we define

Show that for X, Y ∈ L∞

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Exercise 11.1.2. Show that every conditional ∞monetary risk measure has the following property:

Then show that (11.3) is equivalent to the the following local property: if X, Y ∈ L∞ and A ∈ Ft, then

Hint: It can be helpful to use Exercise 11.1.1.

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By M1(P) we denote the set of all probability measures on (Ω,F) that are absolutely continuous with respect to P. As we have shown in Theorem 4.33, an unconditional convex risk measure that has the Fatou property admits a robust representation of the form

with some penalty function α : M1(P) → ℝ ∪ {+∞}. In fact we can take the minimal penalty function

In this section we are going to prove a robust representation for conditional convex risk measures which is analogous to (11.4). Here the penalty function will depend on the history up to time t, and so it will be given by a map αt from (a subset of) M1(P) to the set of Ft-measurable random variables with values in ℝ ∪{+∞}. In analogy to (11.5), we are going to take the minimal penalty function, defined as the worst conditional loss over all positions which are acceptable at time t:

The conditional expectations on the right are a priori defined only Q-a.s., but they also need to be well-defined under the reference measure P. This is the case when Q belongs to the set

Theorem 11.2. For a convex conditional risk measure ρt the following properties are equivalent:

(a) ρt has the representation

where for Q ∈ Qt the penalty function is given by (11.6). In fact the representation (11.7) also holds if we replace Qt in (11.7) by the smaller set

(b) ρt has the following Fatou property:

for any bounded sequence (Xn) ⊆ L∞ which converges P-a.s. to X ∈ L∞.

(c) ρt is continuous from above, i.e.,

for any sequence (Xn) ⊆ L∞ and X ∈ L∞.

Proof. (a)⇒(b): Lebesgue’s dominated convergence theorem for conditional expectations yields

and hence P-a.s., for each Q ∈ Qt. Thus, P-a.s.,

and so the representation (11.7) implies ρt(X) ≤ lim infn ρt(Xn).

(b)⇒(c): The Fatou property yields lim infn ρt(Xn)≥ ρt(X). On the other hand we have lim supn ρt(Xn)≤ ρt(X) by monotonicity, and this implies ρt(Xn) → ρt(X).

(c)⇒(a): Since X + ρt(X) ∈ At for any X ∈ L∞ by conditional cash invariance, the definition (11.6) of implies

hence

for any Q ∈ Qt. This yields the inequality

In order to prove the equality in (11.7), both for Qt and for the smaller set Pt, it is therefore enough to show that

where

To this end, note first that

Next, we consider the map ρ : L∞ → ℝ defined by

It is easy to check that ρ is a convex risk measure. Property (c) implies that ρ is continuous from above; equivalently, it has the Fatou property. Thus, Theorem 4.33 states that ρ has the robust representation

where the penalty function α(Q) is given by

Next we prove that Q = P on Ft , and hence Q ∈ Pt, if α(Q) < ∞. Indeed, take A ∈ Ft and λ > 0. Then

hence

for any λ > 0. Thus α(Q) < ∞ implies P[ A ]≤ Q[ A ] for any A ∈ Ft, and hence P = Q on Ft.

Furthermore,

holds for every Q ∈ Pt. Indeed,

holds by equation (11.17) in Lemma 11.3 below, and so (11.12) and (11.11) imply (11.13) since ρ(X) ≤ 0 for all X ∈ At.

Thus we have shown

We can now conclude the proof of inequality (11.9) as follows, using first (11.14) and then (11.13):

The following lemma was used in the preceding proof.

Lemma 11.3. For Q ∈ Qt and 0 ≤ s ≤ t,

and in particular

Proof. First we show that the family

is directed upward for any Q ∈ (see Appendix A.5). Indeed, for X, Y ∈ we can Qt At take where A := {EQ[ −X |Ft ]≥ EQ[ −Y |Ft ]} ∈ Ft. Conditional convexity of ρt implies Z ∈ At, and clearly we have

Thus the family is directed upward. Hence, Theorem A.37 implies that its essential supremum can be computed as the supremum of an increasing sequence within this family, i.e., there exists a sequence in At such that

By monotone convergence we get

The converse inequality follows directly from the definition of This shows (11.16), and for s = 0 we get (11.17).

Remark 11.4. The inequalities (11.8) and (11.9) in the proof of the implication “(c)⇒(a)” in Theorem 11.2 show that the representation (11.7) also holds if we replace Qt in (11.7) not only by Pt but by the even smaller set

In view of (11.8) and (11.9), we can conclude that the representation (11.7) holds for any set Q such that

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Example 11.5. Suppose that preferences are characterized by an exponential utility function u(x) = 1 − exp(−βx) with β > 0. At time t the conditional expected utility of a financial position X ∈ L∞ is then given by the Ft-measurable random variable

The set

satisfies the conditions required from an acceptance set. In analogy to Example 4.34, the induced convex conditional risk measure ρt is given by

i.e.,

Clearly, ρt is a monetary conditional risk measure. It will be shown in Exercise 11.1.3 that it is also convex and admits a robust representation with minimal penalty function

where

denotes the conditional relative entropy of Q ∈ Pt with respect to P, given Ft. For this reason, ρt is called the conditional entropic risk measure.

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Exercise 11.1.3. Argue as in Lemma 3.29 to show that for any Q ∈ Pt the conditional relative entropy (11.20) satisfies the following variational identity:

where the second supremum is attained by Then proceed as in Example 4.34 to conclude that the conditional entropic risk measure is convex and that (11.19) describes indeed its minimal penalty function for Q ∈ Pt.

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In the coherent case we obtain the following representation result:

Corollary 11.6. Let ρt be a coherent conditional risk measure that has the Fatou property. Then ρt has the representation

where the setis given by

and coincides with the set in (11.18).

Proof. In the coherent case the penalty function can only take the values 0 or ∞ for Q ∈ Qt. Indeed, for and any λ > 0 we have λA X ∈ At due to conditional positive homogeneity of ρt, hence

and the lower bound converges to infinity on A as λ ↑ ∞. Thus on A.Next, letbe as in (11.18). Then and this implies P[ A ] = 0. Thus coincides with the set of all Q ∈ Qt for which P-a.s., and so the representation (11.7), where we are free to take instead of Qt by Remark 11.4, reduces to (11.21).

Example 11.7. For any λ ∈ (0, 1), the acceptance set

defines a monetary conditional risk measure, which we call conditional Value at Risk at level λ:

The conditional risk measure V@Rλ( · |Ft) is conditionally positively homogenous, but it is not conditionally convex. While the terminology ‘conditional Value at Risk’ seems quite natural, one should be aware of a possible confusion because the unconditional risk measure AV@R, which was introduced in Definition 4.48, is sometimes also called Conditional Value at Risk and denoted by CV@R.

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Definition 11.8. For λ ∈ (0, 1] let denote the set of all measures Q ∈ Pt whose density dQ/dP is P-a.s. bounded by 1/λ. The resulting coherent conditional risk measure

is called conditional Average Value at Risk at level λ.

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We have the following representation result for the coherent conditional risk measure AV@Rλ( · |Ft).

Proposition 11.9. For X ∈ L∞, the essential supremum in (11.23) is attained for any whose density satisfies

and the set of such maximizers is nonempty. In particular,

Proof. Fix X ∈ L∞ and note that

where we have used (11.22) in the last step. On the other hand, (11.22) also implies that

for each n, and hence that

The two inequalities (11.26) and (11.27) imply that the set of measures satisfying (11.24) is nonempty. Now let ZX = dQX /dP be the density of such a measure, and let Z be the density of any other We furthermore define := X +V@Rλ(X|Ft). Then V@Rλ(|Ft) = 0 and, arguing as in the proof of the general Neyman-Pearson lemma in the form of Theorem A.35,

with P-a.s. equality if and only if Z is also of the form (11.24). The conditional cash invariance of AV@Rλ(·|Ft) thus yields the first part of the assertion. The identity (11.25) now follows from the following chain of identities:

11.2Time consistency

In this section we consider a sequence of convex conditional risk measures

In such a dynamic setting the key question is how the risk assessments of a financial position at different times are connected to each other.

Definition 11.10. A sequence of conditional risk measures ( ρt)t=0,1,...,T is called (strongly) time-consistent if for any X, Y ∈ L∞ and for all t ≥ 0 the following condition holds:

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Lemma 11.11. Time consistency is equivalent to each of the following two properties:

(a) ρt+1(X) = ρt+1(Y) ρt(X) = ρt(Y) for t = 0,1, . . . T − 1.

(b) Recursiveness: ρt = ρt(−ρt+1) for t = 0,1, . . . , T − 1.

Proof. Time consistency clearly implies (a).

(a) ⇒ (b): Note that ρt+1(−ρt+1(X)) = ρt+1(X), due to conditional cash-invariance and normalization. Applying (a) with Y := −ρt+1(X) we thus obtain ρt(X) = ρt(−ρt+1(X)).

(b) implies time consistency: If ρt+1(X) ≤ ρt+1(Y) then ρt(−ρt+1(X)) ≤ ρt(−ρt+1(Y)) by monotonicity, and so ρt(X) ≤ ρt(Y) follows from (b).

Example 11.12. For fixed β > 0, the sequence of conditional entropic risk measures

is time-consistent. Let us check recursiveness:

Note, however, that time consistency will be lost if the constant risk aversion parameter β is replaced by an adapted process (βt). Moreover, under a suitable condition of dynamic law-invariance, the conditional entropic risk measure is in fact the only time-consistent dynamic convex risk measure; see [196], where this result is reduced to an application of Proposition 2.46. Here we include the ordinary conditional expectation with respect to P as the limiting case β = 0.

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Example 11.13. The sequence of coherent conditional risk measures

given by conditional Average Value at Risk at some level λ ∈ (0, 1) is not time-consistent, and the same is true for conditional Value at Risk and for conditional mean-standard deviation risk measures defined in terms of Sharpe ratios. To see this, note first that all these risk measures are well defined on L2(P), and that they are of the form

if X has a conditional Gaussian distribution with respect to Ft (compare Examples 4.11, 4.12 and Exercise 4.4.3 (b)). Now consider a position of the form X = X1+ X2 for two independent Gaussian random variables X i with distribution N (0, σ) and i2assume that F1 is the σ-field generated by X1. Then

hence

On the other hand,

and so we have ρ0(−ρ1(X)) > ρ0(X) unless X1 or X2 are constant and therefore σ1 = 0 or σ2 = 0. Note that the preceding argument works also if it is possible to represent ρt in the form

for some set M of Borel probability measures on (0, 1]. Such a representation would extend the representation of law-invariant coherent risk measures given in Corollary 4.63 to the conditional setting.

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Exercise 11.2.1. Show that recursiveness as defined in part (b) of Lemma 11.11 is equivalent to the following condition: for 0 ≤ s < t ≤ T and X ∈ L∞,

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The next exercise relies on and generalizes the preceding one.

Exercise 11.2.2. For a stopping time τ : Ω → {0, . . . , T} we define

Show that recursiveness is equivalent to the following property: if σ is another stopping time such that σ ≤ τ, then

Hint: Use Exercise 11.2.1 and (11.3).

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Exercise 11.2.3. Let ρt, t = 0, . . . , T, be any sequence of convex conditional risk measures. Show that the recursive definition

yields a time-consistent sequence of convex conditional risk measures.

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We are now going to characterize time consistency of the sequence ( ρt)t=0,1,...,T both in terms of the corresponding acceptance sets (At)and in terms of the penalty t=0,1,...,T processes Our main goal is to prove a supermartingale criterion which characterizes time consistency in terms of the joint behavior of the stochastic processes ( ρt(X))t=0,1,...,T and (αt(Q))t=0,1,...,T.

To this end we introduce some notation. Suppose that we look just one step ahead and assess the risk only for those positions whose outcome will be known by the end of the next period. This means that we restrict the conditional convex risk measure ρt to the space The ∞corresponding one-step acceptance set is given by

and

is the resulting one-step penalty function.

The following lemma holds for any sequence of monetary conditional risk measures. The equivalences between set inclusions for the acceptance sets and inequalities for the risk measures can be used as starting points for various departures from the strong notion of time consistency which we are considering here; see, e.g., [234], [271], and [274]. For two sets A , B ⊂ L∞ we will use the following notation:

Lemma 11.14. Let ( ρt)t=0,1,...,T be a sequence of monetary conditional risk measures. Then the following equivalences hold for all t = 0, . . . , T − 1 and all X ∈ L∞:

(a) X ∈ At,t+1 + At+1 ⇐⇒ −ρt+1(X) ∈ At,t+1.

(b) At ⊆ At,t+1 + At+1 ⇐⇒ ρt(−ρt+1) ≤ ρt.

(c) At ⊇ At,t+1 + At+1 ⇐⇒ ρt(−ρt+1) ≥ ρt.

Proof. (a) To prove “⇒” take X = Xt,t+1 + Xt+1 with Xt,t+1 ∈ At,t+1 and Xt+1 ∈ At+1. Then

by cash invariance, and monotonicity implies

hence −ρt+1(X) ∈ At+1. The converse implication follows immediately from the decomposition X = X + ρt+1(X) − ρt+1(X), since X + ρt+1(X) ∈ At+1 for all X ∈ L∞ and −ρt+1(X) ∈ At,t+1 by assumption.

(b) In order to show “⇒”, take X ∈ L∞. Since X + ρt(X) ∈ At ⊆ At,t+1 + At+1, we obtain

by (a) and by cash invariance. This implies

To prove “⇐” take X ∈ At. Then −ρt+1(X) ∈ At,t+1 by the right-hand side of (b), and hence X ∈ At,t+1 + At+1 by (a).

(c) Take X ∈ L∞ and assume At ⊇ At,t+1 + At+1. Then

belongs to At. This implies

by cash invariance, and so we have shown “⇒”. For the converse implication take X ∈ At,t+1 + At+1. Since −ρt+1(X) ∈ At,t+1 by (a), we obtain

hence X ∈ At.

The preceding lemma implies immediately the following result.

Proposition 11.15. Let ( ρt)t=0,1,...,T be a sequence of convex conditional risk measures such that each ρt has the Fatou property. Then the following conditions are equivalent:

(a) ( ρt)t=0,1,...,T is time-consistent.

(b) At = At,t+1 + At+1 for t = 0, . . . , T − 1.

In the sequel, we will investigate the time consistency of a sequence ( ρt)t=0,1,...,T of convex conditional risk measures in terms of the dynamics of their penalty functions. In doing this, we need to assume that every element of the sequence can be represented in terms of the same set Q of probability measures:

These representations are only well-defined if Q ⊂ Qt for all t. Since T < ∞, every Q ∈ Q must be equivalent to P on FT = F, and so we may as well assume

Definition 11.16. A sequence ( ρt)t=0,1,...,T of convex conditional risk measures is called sensitive if the representation (11.28) holds in terms of the set Q in (11.29).

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Sensitivity is also called relevance. It generalizes the concept of sensitivity introduced in Section 4.3 to the case of conditional risk measures, and it can be characterized along the lines of Theorem 4.43. In fact, it can be shown that a time-consistent sequence ( ρt)t=0,1,...,T of conditional convex risk measures is sensitive as soon as ρ0 is sensitive in the sense of Definition 4.42; see [128].

The following theorem, and in particular the equivalence of its conditions (a) and (c), is the main result of this section.

Theorem 11.17. Let ( ρt)t=0,1,...,T be a sensitive sequence of convex conditional risk measures. Then the following conditions are equivalent:

(a) ( ρt)t=0,1,...,T is time-consistent.

(b) For any Q ∈ Q,

(c) For any Q ∈ Q and any X ∈ L∞, the process

satisfies

and is a Q-supermartingale for any Q with

We prepare the proof with the following two lemmas. The first lemma deals with the pasting of two probability measures Q1 and Q2 in the deterministic time t. It is a special case of the results in Section 6.4, but we state and prove it here to keep the discussion self-contained.

Lemma 11.18. For Q1, Q2 ∈ Q and t ∈ {0, . . . , T} there exists Q ∈ Q such that Q = Q1 on Ft and

In particular we have

Proof. When setting

then EQ1 [ Z | Ft ] = 1, and so dQ = Z dQ1 defines a measure Q ∈ Q that coincides with Q1 on Ft. Moreover, Proposition A.16 yields (11.30). The identity (11.31) now follows from the definition of

Lemma 11.19. The family

is directed upward.

Proof. By applying Lemma 11.18 with Q1 := P we see that it is enough to show that the family

is directed upward. To this end, fix Q1, Q2 ∈ Q ∩ Pt and let Yi := EQi[ −X | Ft ] − and A := {Y2 ≥ Y1}. We then define

This random variable is P-a.s. strictly positive and satisfies

It follows that Z is the density of a probability measure Q ∈ Q ∩ Pt. Under Q, we have

Similarly,

Therefore,

which concludes the proof.

Proof of Theorem 11.17: (a) ⇒ (b): By Proposition 11.15, any X ∈ At can be written as the sum of some Xt,t+1 ∈ At,t+1 and some Xt+1 ∈ At+1. Hence, we obtain

for any Q ∈ Q, using Lemma 11.3 in the last step.

(b) ⇒ (c): Fix X ∈ L∞. By Lemma 11.19 and Theorem A.37 we can find a sequence Qn ∈ Q such that ρt+1(X) can be identified as the limit of an increasing sequence:

Now take Q ∈ Q and write We have to show that Ut ≥ EQ[ Ut+1 | Ft ]. Using property (b) we see that

Now we use the sequence of measures Qn appearing in (11.32). By Lemma 11.18, we are free to assume that Qn = Q on Ft+1, and this implies

Using the approximation (11.32) and monotone convergence for conditional expectations, and applying property (b) together with equation (11.34) to each Qn, we obtain

Together with (11.33) this yields

If in addition α0(Q) < ∞ then (b) implies Moreover, ρt(X) is bounded by X∞, and so Ut is integrable and hence a supermartingale.

(c) ⇒ (a): Take X, Y ∈ L∞ such that ρt+1(X) ≤ ρt+1(Y); we have to show ρt(X) ≤ ρt(Y). For each Q ∈ Q,

The first inequality follows from the supermartingale property in (c), and in the third step we have used the inequality

which is valid for any Q ∈ Q; see (11.8). Thus

for all Q ∈ Q, and this implies the desired inequality ρt(Y) ≥ ρt(X) due to our assumption that ρt admits the representation (11.28).

Remark 11.20. It follows from property (b) of Theorem 11.17 that

This in turn implies for all Thus, the process is a Q-supermartingale for any such Q. This can be seen as a built-in learning effect: If the dynamics are in fact driven by the probability measure Q then the penalization of these measures decreases “on average” in the sense that it is a supermartingale under Q. Note that property (b) provides more information than just the supermartingale property: It actually yields an explicit description of the predictable increasing process in the Doob decomposition of the supermartingale in terms of the “one-step” penalty functions More precisely,

where M Q is a martingale under Q.

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Exercise 11.2.4. Show that the cost of superhedging defines a time-consistent sequence of conditional convex risk measures. More precisely, in the situation and with the notation of Chapter 9, let

for Y S∈ L∞. Show that is a time-consistent sequence of conditional convex risk measures if the conditions of Corollary 9.32 are satisfied.

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Remark 11.21. The weaker version (11.35) of property (b) is in fact equivalent to a weaker notion of time consistency, called weak time consistency, that is defined by the implication

and which is equivalent to the following weaker version

of property (b) in Proposition 11.15; see Exercise 11.2.5

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Exercise 11.2.5. Let ( ρt)t=0,1,...,T be a sensitive sequence of convex conditional risk measures. Show that the following conditions are equivalent.

(a) ( ρt)t=0,1,...,T is weakly time consistent in the sense that ρt+1(X) ≤ 0 implies ρt(X) ≤ 0.

(b) The acceptance sets satisfy the inclusion At+1 ⊆ At for t = 0, . . . , T − 1.

(c) The penalty functions satisfy

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We will now consider the coherent case in Theorem 11.17. An initial discussion of this case was already given at the end of Section 6.5 in connection with our analysis of stability under pasting. Recall from Definition 6.39 that the pasting of two equivalent probability measures Q1 and Q2 in a stopping time σ ≤ T is the probability measure

According to Definition 6.42, a set Qρ of equivalent probability measures is called stable if, for any Q1, Q2 ∈ Q and σ ∈ T , also their pasting in σ is belongs to Q. We can now state the following characterization of time consistency for coherent dynamic risk measures. It extends, and provides a converse of, Theorem 6.52.

Theorem 11.22. Let ( ρt)t=0,1,...,T be a sensitive sequence of convex conditional risk measures, and assume that ρ0 is coherent. Then the following conditions are equivalent.

(a) ( ρt)t=0,1,...,T is time-consistent.

(b) There exists a stable set Qρ ⊂ Q such that

In particular, each ρt is coherent if these equivalent conditions hold.

Proof. It was shown in Theorem 6.52 that (b) implies recursiveness and in turn (a).

To show the implication “(a)⇒(b)”, let

Then is a nonnegative Q-supermartingale for ∈ Qρ any Q by Theorem 11.17. Since ρ0 is coherent, we have and hence Q-a.s. (and hence P-a.s.) for each t, due to the supermartingale property (compare Exercise 6.1.2). Therefore the representation 11.36 follows with Remark 11.4. In particular, each ρt is coherent.

To show the stability of Qρ, we recall from Proposition 6.44 that the stability of Qρ is equivalent to the following fact: For Q1, Q2 ∈ Qρ, t ∈ {0, . . . , T}, and B ∈ Ft, also the probability measure

belongs to Qρ. Clearly, this measure Q is equivalent to P and hence belongs to Q. To prove that and in turn Q ∈ Qρ, we show first that Indeed, for X ∈ At,

because for i = 1,2

This shows our claim Next, part (b) of Theorem 11.17 yields that

where we have used the fact that Q = Q1 on Ft in the fourth identity. An iteration of this argument shows

Remark 11.23. As in the other chapters of Part II, we have limited the discussion to a finite horizon T < ∞. If we pass to an infinite horizon, the characterization of time consistency in terms of supermartingale properties in Theorem 11.17 yields convergence results for risk measures which can be seen as nonlinear extensions of martingale convergence; cf. [128].

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