Time-consistency of risk measures: how strong is such a property?

Abstract

Quite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen in Memoriam Paul-André Meyer, Lecture notes in mathematics, vol 1874, pp 215–258, 2006; Föllmer and Penner in Stat Decis 14(1):1–15, 2006; Bion-Nadal in Stoch Process Appl 119:633–654, 2009; Delbaen et al. in Finance Stoch 14(3):449–472, 2010; Laeven and Stadje in Math Oper Res 39:1109–1141, 2014, among many others). However, almost all the papers address to coherent or convex risk measures satisfying cash-additivity. In the present work, we study time-consistency for more general dynamic risk measures where either only cash-invariance or both cash-invariance and convexity are dropped. This analysis is motivated by the recent papers of El Karoui and Ravanelli (Math Finance 19:561–590, 2009) and Cerreia-Vioglio et al. (Math Finance 21(4):743–774, 2011) who discussed and weakened the axioms above by introducing cash-subadditivity and quasi-convexity. In particular, we investigate and discuss whether the notion of time-consistency is too restrictive, when considered in the general framework of quasi-convex and cash-subadditive risk measures. Finally, we provide some conditions guaranteeing time-consistency in this more general framework.

Appendix: Proof of Theorem 2

In the following, we will restrict to a discrete-time setting. To prove Theorem 2, the idea is to proceed by steps by using an approach similar to El Karoui and Ravanelli (2009) and Acciaio et al. (2012), that is to define a suitable setting where the original risk measure $\pi$ can be transformed into a cash-additive risk measure $\hat{\pi }$. We will then apply to $\hat{\pi }$ the results on time-consistency already known for cash-additive risk measures and then, after having shown the relation between the penalty terms of $\pi$ and $\hat{\pi }$, we will formulate the necessary condition in terms of c. Differently from El Karoui and Ravanelli (2009) and Acciaio et al. (2012), respectively, we will consider a dynamic setting and a finite time horizon.

The main difficulties and reasons why we consider a discrete-time setting are that in continuous time, there are some additional technicalities concerning the Itô-Watanabe decomposition [with consequences on (23)] and, moreover, it is more difficult to relate the expected values between the initial and the extended probability spaces [see (22)].

Before proving Theorem 2, we introduce the setting and recall some results that will be used in the proof.

1.1 A.1 Setting

We consider a discrete-time setting with finite time horizon T. We denote by $\mathbb {T}$ the set of times, i.e. $\mathbb {T}=\left\{ 0,1,\ldots ,T\right\}$, and by $\hat{\Omega }\triangleq \Omega \times \mathbb {T}$ the enlarged sample space. Any discrete-time stochastic process $\hat{X}=(X_t)_{t\in \mathbb {T}}$ with $X_t\in L^{\infty }(\mathcal {F}_t)$ can be therefore understood as defined on the enlarged space $\hat{\Omega }= \Omega \times \mathbb {T}$, that is

$$\begin{aligned} \hat{X}(\omega ,t)=X_t(\omega ). \end{aligned}$$

Let

$$\begin{aligned} \hat{\mathcal {F}}= \sigma (\left\{ A_t \times \left\{ t\right\} : A_t \in \mathcal {F}_t, t \in \mathbb {T}\right\} ) \end{aligned}$$

be the $\sigma$-algebra $\hat{\mathcal {F}}$ considered on $\hat{\Omega }$.

Furthermore, let $(\hat{\Omega },\hat{\mathcal {F}})$ be endowed with the filtration $(\hat{\mathcal {F}}_t)_{t\in \mathbb {T}}$ given by

$$\begin{aligned} \hat{\mathcal {F}}_t=\sigma (\left\{ A_j \times \left\{ j\right\} , A_t \times \left\{ t,\ldots ,T\right\} : A_j \in \mathcal {F}_j,j<t,A_t \in \mathcal {F}_t\right\} ), \quad t \in \mathbb {T}, \end{aligned}$$

and with the probability measure $\hat{P}\triangleq P\otimes \mu$, where $\mu =(\mu _t)_{t\in \mathbb {T}}$ is some adapted process satisfying $\sum _{t\in \mathbb {T}} \mu _t=1$, $\mu _t >0$ for all $t\in \mathbb {T}$ and

$$\begin{aligned} E_{\hat{P}}[\hat{X}]\triangleq E_P \left[ \sum \limits _{t\in \mathbb {T}} X_t \mu _t\right] \end{aligned}$$

for any $\hat{X} \in L^{\infty } (\hat{\mathcal {F}})$. A random variable $\hat{X}=(X_t)_{t\in \mathbb {T}}$ on $(\hat{\Omega },\hat{\mathcal {F}},\hat{P})$ is then $\hat{\mathcal {F}}_t$-measurable if and only if $X_s$ is $\mathcal {F}_s$-measurable for all $s=0,1\ldots ,t$ and $X_s=X_t$ for all $s>t$.

Notice that the present setting is exactly the same as in Acciaio et al. (2012) with the only difference that we use a finite time horizon.

Denote by $\hat{\mathcal {P}}$ (respectively, $\mathcal {P}$) the set of all probability measures that are absolutely continuous with respect to $\hat{P}$ (resp. P). For any $Q\in \mathcal {P}$, denote by $\Gamma (Q)$ the set of positive random measures $\gamma$ on $\mathbb {T}$ such that

$$\begin{aligned} \sum _{t\in \mathbb {T}} \gamma _t =1, \quad Q{\hbox {-}}a.s. \end{aligned}$$

and by $\Delta (Q)$ the set of predictable non-increasing processes $\delta =(\delta _t)_{t\in \mathbb {T}}$ with $\delta _0=1$ and $\delta _T= \lim _{t \rightarrow T} \delta _t$, Q-a.s.. We recall from Acciaio et al. (2012) the following result guaranteeing that any $\hat{Q}$ in $\hat{\mathcal {P}}$ admits a decomposition $\hat{Q}(\mathrm{d}\omega , t)= Q(\mathrm{d}\omega ) \otimes \gamma (\omega , t)$ for some probability measure $Q \in \mathcal {P}$ and some random measure $\gamma \in \Gamma (Q)$.

Theorem 15

(see Lemma 3.2, Theorem 3.4 and Corollary B.3 in Acciaio et al. 2012)

• (i) Fix any probability measure $Q\in \mathcal {P}$. To any $\gamma \in \Gamma (Q)$ it corresponds a process $\delta \in \Delta (Q)$ and vice versa, by means of

  $$\begin{aligned} \delta _t&=1-\sum _{s=0}^{t-1}\gamma _s= \sum _{s=t}^T \gamma _s, \quad Q{\hbox {-}}a.s., \, \forall t\in \mathbb {T}\\ \gamma _t&=\delta _t-\delta _{t+1}, \quad t\in \mathbb {T}. \end{aligned}$$

• (ii) Any probability measure $\hat{Q}\in \hat{\mathcal {P}}$ admits a decomposition $\hat{Q}= Q \otimes \gamma = Q \otimes \delta$ by a probability measure $Q\in \mathcal {P}$ and a measure $\gamma \in \Gamma (Q)$ (resp. $\delta \in \Delta (Q)$), meaning that

  $$\begin{aligned} E_{\hat{Q}}[\hat{X}] = E_Q\left[ \sum _{t\in \mathbb {T}} X_t \gamma _t\right] = E_Q\left[ \sum _{t\in \mathbb {T}} (X_t-X_{t+1})\delta _t\right] \end{aligned}$$

  (22)

  holds for any $\hat{X} \in L^{\infty } (\hat{\mathcal {F}})$. Conversely, given any $Q\in \mathcal {P}$ and any $\gamma \in \Gamma (Q)$ (resp. any $\delta \in \Delta (Q)$), then

  $$\begin{aligned} \hat{Q}= Q\otimes \gamma = Q \otimes \delta \end{aligned}$$

  (23)

  is a probability measure in $\hat{\mathcal {P}}$.

• (iii) For any $\hat{Q}\in \hat{\mathcal {P}}$ with decomposition $\hat{Q}= Q \otimes \gamma = Q \otimes \delta$, the conditional expectation of $\hat{X}\in L^\infty (\hat{\mathcal {F}})$ given $\hat{\mathcal {F}}_t$ is

  $$\begin{aligned} E_{\hat{Q}}[ \hat{X}|\hat{\mathcal {F}}_t] = X_0 \mathbb {1}_{\left\{ 0 \right\} } + \cdots +X_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } +E_{Q}\left[ \sum _{s=t}^T \frac{\gamma _s}{\delta _t}X_s \bigg |\mathcal {F}_t\right] \mathbb {1}_{\left\{ t,\ldots ,T\right\} }, \end{aligned}$$

  (24)

  where the conditional expectation under Q is well defined Q-a.s. on $\left\{ \delta _t>0\right\}$.

1.2 A.2 Proof of Theorem 2

In order to prove Theorem 2, we proceed by steps as follows. First, we transform our dynamic convex cash-subadditive risk measure $\pi$ into a convex and cash-additive one $\hat{\pi }$. Then, we apply to $\pi$ the results already known for cash-additive risk measures. Finally, we translate such results by going back to $\pi$.

Given a dynamic risk measure $\pi _{t,T} : L^\infty (\mathcal {F}) \rightarrow L^\infty (\mathcal {F}_t)$, we define the dynamic risk measure $\hat{\pi }_{t,T}: L^\infty (\hat{\mathcal {F}}) \rightarrow L^\infty (\hat{\mathcal {F}}_t)$ as

$$\begin{aligned} \hat{\pi }_{t,T}(\hat{X}) \triangleq X_0 \mathbb {1}_{\left\{ 0 \right\} } + X_1 \mathbb {1}_{\left\{ 1 \right\} } +\cdots + X_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } + \big (\pi _{t,T}(X_T-X_t)+X_t \big ) \mathbb {1}_{\left\{ t,t+1,\ldots ,T\right\} }\nonumber \\ \end{aligned}$$

(25)

for any $\hat{X}=(X_t)_{t\in \mathbb {T}} \in L^{\infty } (\hat{\mathcal {F}})$. Similarly, $\hat{\pi }_{t,v}: L^\infty (\hat{\mathcal {F}}_v) \rightarrow L^\infty (\hat{\mathcal {F}}_t)$ can be defined starting from $\pi _{t,v}$ as

$$\begin{aligned} \hat{\pi }_{t,v}(\hat{X}) \triangleq X_0 \mathbb {1}_{\left\{ 0 \right\} } + X_1 \mathbb {1}_{\left\{ 1 \right\} } +\cdots + X_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } + \big (\pi _{t,v}(X_v-X_t)+X_t \big ) \mathbb {1}_{\left\{ t,t+1,\ldots ,T\right\} }\nonumber \\ \end{aligned}$$

(26)

for any $\hat{X}=(X_t)_{t\in \mathbb {T}} \in L^{\infty } (\hat{\mathcal {F}})$.

The following proposition shows how the properties on $\pi _{t,v}$ will be translated in those of $\hat{\pi }_{t,v}$.

Proposition 16

Let $\hat{\pi }_{t,T}: L^\infty (\hat{\mathcal {F}}) \rightarrow L^\infty (\hat{\mathcal {F}}_t)$ be defined starting from $\pi _{t,T} : L^\infty (\mathcal {F}) \rightarrow L^\infty (\mathcal {F}_t)$ as in (25).

• (i) If $(\pi _{t,T})_{t}$ is a normalized, cash-subadditive and convex dynamic risk measure, then $(\hat{\pi }_{t,T})_{t}$ is a normalized, cash-additive and convex dynamic risk measure.

• (ii) If $\pi _{t,T}$ is regular, then $\hat{\pi }_{t,T}$ is regular.

• (iii) If $\pi _{t,T}$ is continuous from below, then $\hat{\pi }_{t,T}$ is continuous from below.

• (iv) Items (i)–(iii) hold true also when $\hat{\pi }_{t,T}$ (resp. $\pi _{t,T}$) is replaced by $\hat{\pi }_{t,v}$ (resp. $\pi _{t,v}$).

• (v) $(\hat{\pi }_{s,t})_{s,t \in \mathbb {T},s \le t}$ is time-consistent if and only if

  $$\begin{aligned} (\pi _{s,t}(\cdot - X_s) +X_s)_{0\le s\le t\le T} \end{aligned}$$

  is time-consistent for all $X_s \in L^\infty (\mathcal {F}_s)$, that is: if and only if

  $$\begin{aligned} \pi _{s,t}(\pi _{t,v}(X_v -X_t) +X_t -X_s)+X_s = \pi _{s,t}(X_v-X_s)+X_s, \end{aligned}$$

  (27)

  for all $X_v \in L^{\infty }(\mathcal {F}_v),X_t \in L^{\infty }(\mathcal {F}_t), X_s \in L^{\infty }(\mathcal {F}_s)$.

In the following, $(\pi _{s,t})_{s,t \in \mathbb {T},s \le t}$ satisfying (27) will be called X-time-consistent. Notice that such a property implies the classical time-consistency and is implied by time-consistency and cash-additivity together. The main reason why we assume such a stronger condition on $\pi$ is that, by definition of $\hat{\pi }$, we need to manage the term $\pi _{t,v}(X_v-X_t)+X_t$ making $\hat{\pi }$ cash-additive.

Proof

• (i) First of all, we notice that $\hat{\pi }_{t,T}$ is $\hat{\mathcal {F}}_t$-measurable. Take, indeed, an arbitrary $\hat{X}\in L^\infty (\hat{\mathcal {F}})$ and set $\hat{Y}=\hat{\pi }_{t,T}(\hat{X})$. By definition of $\hat{\pi }$, it then follows that, for any $u<t$, $Y_u= X_u$ (hence $\mathcal {F}_u$-measurable), while for any $u\ge t$ it holds that $Y_u=Y_t=\pi _{t,T}(X_T-X_t)+X_t$ is $\mathcal {F}_t$-measurable. Thus $\hat{\pi }_{t,T}$ is well defined as a mapping from $L^\infty (\hat{\mathcal {F}})$ to $L^\infty (\hat{\mathcal {F}}_t)$. It remains to check the other properties. Normalization of $\hat{\pi }_{t,T}$ follows immediately by that of $\pi _{t,T}$. Cash-additivity. Let $\hat{X}\in L^\infty (\hat{\mathcal {F}})$ and $\hat{m}_t\in \hat{\mathcal {F}}_t$. Then

  $$\begin{aligned} \hat{m}_t=m_0 \mathbb {1}_{\left\{ 0 \right\} } + \cdots + m_{t-1} \mathbb {1}_{\left\{ t-1\right\} } + m_t \mathbb {1}_{\left\{ t,t+1,\ldots ,T\right\} }, \end{aligned}$$

  with $m_u \in \mathcal {F}_u$ for all $u\le t$ and $m_u=m_t \in \mathcal {F}_t$ for all $u> t$. By definition of $\hat{\pi }$, it follows that

  $$\begin{aligned} \hat{\pi }_{t,T}(\hat{X}+ \hat{m}_t) = \,&(X_0+m_0) \mathbb {1}_{\left\{ 0 \right\} } + \cdots + (X_{t-1}+m_{t-1} )\mathbb {1}_{\left\{ t-1\right\} } \\&+ (\pi _{t,T}(X_T+m_t -X_t-m_t )+X_t+m_t)\mathbb {1}_{\left\{ t,t+1,\ldots ,T\right\} }\\ = \,&\hat{\pi }_{t,T}(\hat{X})+\hat{m}_t. \end{aligned}$$

  Convexity of $\hat{\pi }_{t,T}$ is a direct consequence of convexity of $\pi _{t,T}$. Monotonicity Consider any $\hat{X}, \hat{Y}\in L^\infty (\hat{\mathcal {F}})$ such that $\hat{X}\le \hat{Y}$, that is $X_u\le Y_u$P-a.s. for any $u\in \left\{ 0,1,\ldots ,T \right\}$. It follows that

  $$\begin{aligned} \hat{\pi }_{t,T}(\hat{X})&=X_0 \mathbb {1}_{\left\{ 0 \right\} } +\cdots + X_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } + (\pi _{t,T}(X_T-X_t)+X_t) \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&\le Y_0 \mathbb {1}_{\left\{ 0 \right\} } +\cdots + Y_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } + (\pi _{t,T}(X_T-X_t)+X_t) \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&\le Y_0 \mathbb {1}_{\left\{ 0 \right\} } +\cdots + Y_{t-1} \mathbb {1}_{\left\{ t-1 \right\} } + (\pi _{t,T}(X_T-Y_t)+Y_t-X_t+X_t) \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&\le \hat{\pi }_{t,T}(\hat{Y}), \end{aligned}$$

  where the last two inequalities are due, respectively, to cash-subadditivity (together with $Y_t \ge X_t$) and to monotonicity of $\pi _{t,T}$.

• (ii) Regularity Because of normalization of $\hat{\pi }_{t,T}$, it is sufficient to verify that $\hat{\pi }_{t,T}(\hat{X}\mathbb {1}_{{\hat{A}}}) = \hat{\pi }_{t,T}(\hat{X}) \mathbb {1}_{{\hat{A}}}$ for any $\hat{X}\in L^\infty (\hat{\mathcal {F}})$ and ${\hat{A}}\in \hat{\mathcal {F}}_t$. By definition of $\hat{\mathcal {F}}_t$, it is sufficient to prove regularity of $\hat{\pi }_{t,T}$ on events of the form

  $$\begin{aligned} A_s\times \left\{ s\right\} \quad \text{ with } A_s\in \mathcal {F}_s \, (s<t), \quad \quad \text{ or } \quad \quad A_t\times \left\{ t,\ldots ,T \right\} \quad \text{ with } A_t\in \mathcal {F}_t. \end{aligned}$$

  Let us start by considering a set ${\hat{A}}= A_s\times \left\{ s\right\} \in \hat{\mathcal {F}}_t$ with $s<t$ and $A_s\in \mathcal {F}_s$. Then $\hat{X}\mathbb {1}_{{\hat{A}}} = X_s\mathbb {1}_{A_s \times \left\{ s\right\} }$ and, consequently,

  $$\begin{aligned} \hat{\pi }_{t,T}(\hat{X}\mathbb {1}_{{\hat{A}}}) = X_s\mathbb {1}_{A_s \times \left\{ s\right\} } + (\pi _{t,T}(0 ) +0) \mathbb {1}_{\left\{ t, \ldots , T\right\} } = \hat{\pi }_{t,T}(\hat{X}) \mathbb {1}_{{\hat{A}}}. \end{aligned}$$

  Consider now a set ${\hat{A}}=A_t\times \left\{ t,\ldots ,T \right\}$ with $A_t \in \mathcal {F}_t$. Hence

  $$\begin{aligned} \hat{X}\mathbb {1}_{{\hat{A}}} = 0 \mathbb {1}_{\left\{ 0, \ldots , t-1 \right\} }+X_u\mathbb {1}_{A_t \times \left\{ t,\ldots ,T\right\} } \end{aligned}$$

  and, because of the regularity of $\pi _{t,T}$,

  $$\begin{aligned} \hat{\pi }_{t,T}(\hat{X}\mathbb {1}_{{\hat{A}}})&= 0 \mathbb {1}_{\left\{ 0, \ldots , t-1 \right\} }+ \big (\pi _{t,T}(X_T \mathbb {1}_{A_t }-X_t \mathbb {1}_{A_t } ) +X_t \mathbb {1}_{A_t }\big ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \\&= 0 \mathbb {1}_{\left\{ 0, \ldots , t-1\right\} } +(\pi _{t,T}(X_T-X_t ) +X_t)\mathbb {1}_{A_t } \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \\&= \hat{\pi }_{t,T}(\hat{X}) \mathbb {1}_{{\hat{A}}}. \end{aligned}$$

• (iii) Continuity from below Let $(\hat{X}^n)_{n \ge 0}$ be an arbitrary sequence with $\hat{X}^n \nearrow _n \hat{X}$, i.e. $X_u^n (\omega ) \nearrow _n X_u(\omega )$ for $\hat{P}$-a.e. $(\omega ,u) \in \hat{\Omega }$. Then

  $$\begin{aligned} 0 \le \,&\hat{\pi }_{t,T}(\hat{X}) - \hat{\pi }_{t,T}(\hat{X}^n) \\ = \,&(X_0-X_0^n) \mathbb {1}_{\left\{ 0\right\} } + \cdots + (X_{t-1}-X_{t-1}^n) \mathbb {1}_{\left\{ t-1\right\} } \\&+ \big ( \pi _{t,T}(X_T-X_t ) + X_t - \pi _{t,T}(X_T^n- X_t^n ) -X_t^n\big ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \\ \le \,&(X_0-X_0^n) \mathbb {1}_{\left\{ 0\right\} } + \cdots + (X_{t-1}-X_{t-1}^n) \mathbb {1}_{\left\{ t-1\right\} } \\&+ \big ( \pi _{t,T}(X_T-X_t ) + X_t - (\pi _{t,T}(X_T^n- X_t ) +X_t^n)\big ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} }, \end{aligned}$$

  where the last inequality is due to monotonicity of $\pi _{t,T}$ and to $X^n_t \le X_t$ for any n. By continuity from below of $\pi _{t,T}$ and by the assumption on $X_u^n$, we can conclude that

  $$\begin{aligned} 0 \le \hat{\pi }_{t,T}(\hat{X}) - \hat{\pi }_{t,T}(\hat{X}^n) \rightarrow _n 0, \end{aligned}$$

  hence continuity from below of $\hat{\pi }_{t,T}$.

• (iv) can be checked similarly as in (i)–(iii).

• (v) Time-consistency First, assume that $(\pi _{s,t}(\cdot - Z_s)+Z_s)_{0\le s\le t\le T}$ is time-consistent for any random variable $Z_s\in L^{\infty }(\mathcal {F}_s)$. Consider an arbitrary $\hat{X}\in L^\infty (\hat{\mathcal {F}}_v)$ and set $\hat{Y}= \hat{\pi }_{t,v}(\hat{X})$. Hence, $Y_u=X_u$ for $u<t$, and $Y_u=\pi _{t,v}(X_v-X_t)+X_t$ for $u\ge t$. Then

  $$\begin{aligned} \hat{\pi }_{s,t}(\hat{\pi }_{t,v}(\hat{X}))&= \hat{\pi }_{s,t}(\hat{Y})\\&= Y_u \mathbb {1}_{\{0,1,\ldots ,s-1\}} + \big (\pi _{s,t}( Y_t-Y_s)+Y_s\big )\mathbb {1}_{\left\{ s,\ldots ,T\right\} } \\&=X_u \mathbb {1}_{\{0,1,\ldots ,s-1\}} + \big (\pi _{s,t}( \pi _{t,v}(X_v-X_t)+X_t-Y_s)+Y_s\big )\mathbb {1}_{\left\{ s,\ldots ,T\right\} } \\&= \hat{\pi }_{s,v}(\hat{X}), \end{aligned}$$

  where the last equality is due to X-time consistency of $\pi$. Conversely, it is easy to check that time-consistency of $(\hat{\pi }_{s,t})_{0\le s\le t\le T}$ implies

  $$\begin{aligned} \pi _{s,t}\big (\pi _{t,v}(X_v-X_t)+X_t-X_s \big )=\pi _{t,v}(X_v-X_s)+X_s, \end{aligned}$$

  for all $(X_s)_{s\in \mathbb {T}}$ with $X_s$ being $\mathcal {F}_s$-measurable, that is X-time-consistency of $(\pi _{s,t}(\cdot -X_s)+X_s)_{0\le s\le t\le T}$.

$\square$

In the next proposition, we show that the minimal penalty function of $(\pi _{s,t})_{s,t \in \mathbb {T},s \le t}$ and the minimal penalty function of $(\hat{\pi }_{s,t})_{s,t \in \mathbb {T},s \le t}$ coincide. We recall that the minimal penalty term $(c_{s,t})_{s,t \in \mathbb {T},s \le t}$ of a dynamic convex and cash-additive (respectively, cash-subadditive) risk measure $(\pi _{s,t})_{s,t \in \mathbb {T},s \le t}$ which is regular and continuous from below is given by

$$\begin{aligned} c_{s,t}(Q)= {\text {ess.sup}}_{X \in L^{\infty }(\mathcal {F}_t)} \{E_Q [X \big | \mathcal {F}_t] - \pi _{s,t}(X) \}, \end{aligned}$$

(28)

respectively

$$\begin{aligned} c_{s,t}(DQ)= {\text {ess.sup}}_{X \in L^{\infty }(\mathcal {F}_t)} \{D_{s,t} E_Q [X \big | \mathcal {F}_t] - \pi _{s,t}(X) \}. \end{aligned}$$

(29)

See Detlefsen and Scandolo (2005) and Mastrogiacomo and Rosazza Gianin (2015), respectively.

Proposition 17

Let $c_{t,s}$ (resp. $\hat{c}_{t,s}$) be the minimal penalty function of $\pi _{t,s}$ (resp. $\hat{\pi }_{t,s}$). For any probability measure $\hat{Q}$ with decomposition $\hat{Q}=Q\otimes \delta =Q \otimes \gamma$ where $Q\in \mathcal {P}$, $\delta \in \Delta (Q)$ and $\gamma \in \Gamma (Q)$, it holds that

$$\begin{aligned} \hat{c}_{t,t+1}(\hat{Q}) =c_{t,t+1}(\tilde{\delta }_t Q)\mathbb {1}_{\{t,\ldots ,T\}} \end{aligned}$$

(30)

and, under the assumption that $\gamma _{t+1}= \cdots = \gamma _{T-1}=0$,

$$\begin{aligned} \hat{c}_{t,T}(\hat{Q}) =c_{t,T}(\tilde{\delta }_t Q)\mathbb {1}_{\{t,\ldots ,T\}} \end{aligned}$$

(31)

where $\tilde{\delta }_t= 1- \tilde{\gamma }_t =1- \frac{\gamma _t}{\delta _t}$.

Proof

Let $Q\in \mathcal {P}$, $\delta \in \Delta (Q)$ and $\hat{Q}=Q\otimes \delta$. Since $\hat{\pi }_{t,t+1}$ is a convex and cash-additive risk measure, its minimal penalty function is given by

$$\begin{aligned}&\hat{c}_{t,t+1}(\hat{Q}) \\&\quad = {\text {ess.sup}}_{\hat{X}\in L^\infty (\hat{\mathcal {F}}_{t+1})} \left\{ E_{\hat{Q}}[\hat{X}|\hat{\mathcal {F}}_t]- \hat{\pi }_{t,t+1}(\hat{X}) \right\} \\&\quad = {\text {ess.sup}}_{\hat{X}\in L^\infty (\hat{\mathcal {F}}_{t+1})} \nonumber \\&\qquad \times \left\{ E_{\hat{Q}}[\hat{X}|\hat{\mathcal {F}}_t]- X_u \mathbb {1}_{\left\{ 0, \ldots , t-1 \right\} } -\bigg (\pi _{t,t+1}(X_{t+1}-X_t)+X_t \bigg ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \right\} \\&\quad = {\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), X_{t+1} \in L^{\infty }(\mathcal {F}_{t+1})} \nonumber \\&\qquad \times \left\{ E_Q\left[ \frac{\gamma _t}{\delta _t}X_t + \sum _{s=t+1}^T\frac{\gamma _{s}}{\delta _t}X_{t+1} \bigg |\mathcal {F}_t\right] - \bigg (\pi _{t,t+1}(X_{t+1}-X_t) +X_t\bigg ) \right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&\quad ={\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), X_{t+1} \in L^{\infty }(\mathcal {F}_{t+1})} \nonumber \\&\qquad \times \left\{ (1- \tilde{\gamma }_t) E_Q\left[ X_{t+1} - X_t\big |\mathcal {F}_t\right] - \pi _{t,t+1}(X_{t+1}-X_t) \right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \\&\quad ={\text {ess.sup}}_{Y \in L^{\infty }(\mathcal {F}_{t+1})} \left\{ (1- \tilde{\gamma }_t) E_Q\left[ Y \big | \mathcal {F}_t\right] - \pi _{t,t+1}(Y) \right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&\quad =c_{t,t+1}(\tilde{\delta }_t Q) \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \end{aligned}$$

where the third equality follows from (24), the forth follows by the definition of $\hat{\mathcal {F}}_{t+1}$ and where $\tilde{\gamma _s}\triangleq \frac{\gamma _s}{\delta _t}\ge 0$ for any $s \ge t$ and $\sum _{s=t}^T \tilde{\gamma }_s = 1$.

Similarly, the minimal penalty function of $\hat{\pi }_{t,T}$ is given by

$$\begin{aligned} \hat{c}_{t,T}(\hat{Q})&= {\text {ess.sup}}_{\hat{X}\in L^\infty (\hat{\mathcal {F}})} \left\{ E_{\hat{Q}}[\hat{X}|\hat{\mathcal {F}}_t]- \hat{\pi }_{t,T}(\hat{X}) \right\} \\&= {\text {ess.sup}}_{\hat{X}\in L^\infty (\hat{\mathcal {F}})}\nonumber \\&\qquad \times \left\{ E_{\hat{Q}}[\hat{X}|\hat{\mathcal {F}}_t]- X_u \mathbb {1}_{\left\{ 0, \ldots , t-1 \right\} } -\bigg (\pi _{t,T}(X_T-X_t)+X_t \bigg ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \right\} \\&= {\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), \ldots , X_T \in L^{\infty }(\mathcal {F})} \nonumber \\&\qquad \times \left\{ E_Q\left[ \sum _{s=t}^T \frac{\gamma _s}{\delta _t}X_s\bigg |\mathcal {F}_t\right] \mathbb {1}_{\left\{ t,\ldots ,T\right\} }- \bigg (\pi _{t,T}(X_T-X_t) +X_t\bigg )\mathbb {1}_{\left\{ t,\ldots ,T\right\} } \right\} \\&={\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), \ldots , X_T \in L^{\infty }(\mathcal {F})} \nonumber \\&\qquad \times \left\{ E_Q\left[ \sum _{s=t}^T \tilde{\gamma }_sX_s\bigg |\mathcal {F}_t\right] - \bigg (\pi _{t,T}(X_T-X_t) +X_t\bigg ) \right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \end{aligned}$$

By the assumption on $\gamma$, it follows that ${\tilde{\gamma }}_{t+1}=\cdots ={\tilde{\gamma }}_{T-1}=0$ and, consequently,

$$\begin{aligned} \hat{c}_{t,T}(\hat{Q})&= {\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), X_T \in L^{\infty }(\mathcal {F})} \\&\qquad \times \left\{ \tilde{\gamma }_t X_t + E_Q\left[ {\tilde{\gamma }}_T X_{T}\big |\mathcal {F}_t\right] - \bigg (\pi _{t,T}(X_{T}-X_t) +X_t\bigg )\right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} } \\&={\text {ess.sup}}_{X_t \in L^{\infty }(\mathcal {F}_t), X_T \in L^{\infty }(\mathcal {F})} \\&\qquad \left\{ (1- \tilde{\gamma }_t )E_Q\left[ X_{T}-X_t\big |\mathcal {F}_t\right] - \pi _{t,T}(X_{T}-X_t) \right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&={\text {ess.sup}}_{Y \in L^\infty (\mathcal {F})} \left\{ \tilde{\delta }_t E_Q\left[ Y\big |\mathcal {F}_t\right] - \pi _{t,T}(Y)\right\} \mathbb {1}_{\left\{ t,\ldots ,T\right\} }\\&= c_{t,T}(\tilde{\delta }_tQ) \mathbb {1}_{\left\{ t,\ldots ,T\right\} }. \end{aligned}$$

$\square$

We are now ready to prove that the generalized cocycle is a necessary condition for a convex and cash-subadditive dynamic risk measure (in a discrete-time setting) to be X-time-consistent.

Proof (Proof of Theorem 2)

Assume first that $(\pi _{t,v})_{t, v\in \mathbb {T}, t\le v}$ is normalized. From Proposition 16 it follows that $(\hat{\pi }_{t,v})_{t, v\in \mathbb {T}, t\le v}$ associated with $(\pi _{t,v})_{t, v\in \mathbb {T}, t\le v}$ as in (26) is a normalized, cash-additive and convex dynamic risk measure which is regular, continuous from below and time-consistent. Consequently, by Bion-Nadal (2008, Theorem 3.3) $(\hat{c}_{t,T})_{t\in \mathbb {T}}$ associated with $(\hat{\pi }_{t,T})_{t \in \mathbb {T}}$ satisfies the cocycle condition

$$\begin{aligned} \hat{c}_{t,T}(\hat{Q})= \hat{c}_{t,t+1}(\hat{Q})+ E_{\hat{Q}}\left[ \hat{c}_{t+1,T}(\hat{Q})\big |\hat{\mathcal {F}}_t\right] , \end{aligned}$$

(32)

for any $\hat{Q}\in \hat{\mathcal {P}}$. Take then any $\hat{Q}\in \hat{\mathcal {P}}$ such that $\hat{Q}= Q \otimes \gamma$ with $\gamma$ as in Proposition 17 and any $Q \in \mathcal {P}$. Then by

$$\begin{aligned} \hat{c}_{t,t+1}(\hat{Q})&= c_{t,t+1}(\tilde{\delta }_t Q)\mathbb {1}_{\{t,\ldots ,T\}}\nonumber \\ \hat{c}_{t,T}(\hat{Q})&= c_{t,T}(\tilde{\delta }_t Q)\mathbb {1}_{\{t,\ldots ,T\}} \end{aligned}$$

(33)

and

$$\begin{aligned}&E_{\hat{Q}}\left[ \hat{c}_{t+1,T}(\hat{Q})\big |\hat{\mathcal {F}}_t\right] \\&\quad = 0 \mathbb {1}_{\left\{ 0\right\} } +\cdots + 0 \mathbb {1}_{\left\{ t-1\right\} } + E_{Q}\left[ \sum _{s=t}^T {\tilde{\gamma }}_s \hat{c}_{t+1,T}(\hat{Q})\big |_s\bigg |\mathcal {F}_t\right] \mathbb {1}_{\{t,\ldots ,T\}}\\&\quad = E_{Q}\left[ \underbrace{{\tilde{\gamma }}_t \cdot 0 + {\tilde{\gamma }}_{t+1} c_{t+1,T}(\tilde{\delta }_t Q) + \cdots + }_{=0}{\tilde{\gamma }}_{T} c_{t+1,T}(\tilde{\delta }_t Q)\bigg |\hat{\mathcal {F}}_t\right] \mathbb {1}_{\{t,\ldots ,T\}} \\&\quad = E_{Q}\left[ (1-{\tilde{\gamma }}_{t}) c_{t+1,T}(\tilde{\delta }_t Q)\big |\mathcal {F}_t\right] \mathbb {1}_{\{t,\ldots ,T\}}\\&\quad =\tilde{\delta }_t E_{Q}\left[ c_{t+1,T}(\tilde{\delta }_t Q)\big |\mathcal {F}_t\right] \mathbb {1}_{\{t,\ldots ,T\}}, \end{aligned}$$

it follows that

$$\begin{aligned} E_{\hat{Q}}\left[ \hat{c}_{t+1,T}(\hat{Q})\big |\hat{\mathcal {F}}_t\right] =\tilde{\delta }_t E_{Q}\left[ c_{t+1,T}(\tilde{\delta }_t Q)\big |\mathcal {F}_t\right] \mathbb {1}_{\{t,\ldots ,T\}} \end{aligned}$$

(34)

Combining (32)–(34) we obtain

$$\begin{aligned} c_{t,T}(\tilde{\delta }_t Q)= c_{t,t+1}(\tilde{\delta }_t Q) + \tilde{\delta }_t E_{Q}\left[ c_{t+1,T}(\tilde{\delta }_t Q)\big |\mathcal {F}_t\right] . \end{aligned}$$

(35)

Since Q and $\gamma$ were arbitrarily chosen, condition (35) holds for any $Q \in \mathcal {P}$ and $\tilde{\delta }_t \in \mathcal {D}$. Suppose now that $(\pi _{t,u})_{t, u\in \mathbb {T}, t\le u}$ is not necessarily normalized. Since $\bar{\pi }_{s,u}(X) \triangleq \pi _{s,u}(X) - \pi _{s,u}(0)$ is a normalized, convex and cash-subadditive risk measure with minimal penalty term $\bar{c}_{s,u}(DQ) = c_{s,u}(DQ) + \pi _{s,u}(0) = c_{s,u}(DQ) - {\text {ess.inf}}_{D ,Q} c_{s,u}(DQ)$, the last statement of Theorem 2 then follows by applying the first part of the result.$\square$