A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective

The aim of this section is to give a chronological survey of the developments of the theory of dynamic risk and performance measures. Although it is not an obvious task to establish the exact lineup, we tried our best to account for the most relevant works according to adequate chronological order.

We trace back the origins of the research regarding time consistency to Koopmans (1960) who put on the precise mathematical footing, in terms of the utility function, the notion of persistency over time of the structure of preferences.

Subsequently, in the seminal paper, Kreps and Porteus (1978) treat the time consistency at a general level by axiomatising the “choice behavior” of an agent by taking into account how choices at different times are related to each other; in the same work, the authors discuss the motivations for studying the dynamic aspect of choice theory.

Before we move on to reviewing the works on dynamic risk and performance measures, it is worth mentioning that the robust expected utility theory proposed by Gilboa and Schmeidler (1989) can be viewed as a more comprehensive theory than the one discussed in (Artzner et al. 1999); we refer to (Roorda et al. 2005) for the relevant discussion.

The set \(\mathcal {Q}\) can be viewed as a set of generalized scenarios, and a coherent risk measure is equal to the worst expected loss under various scenarios. By relaxing the set of axioms, the static coherent risk measures were generalized to static convex risk measures and to an even more general class called monetary risk measures. See, for instance, (Szegö 2002) for a survey of static risk measures, as well as (Cheridito and Li 2009; 2008). On the other hand, axiomatic theory of performance measures was originated in (Cherny and Madan 2009). A general theory of risk preferences and their robust representations, based on only two generic axioms, was studied in (Drapeau 2010; Drapeau and Kupper 2013).

Moving to the dynamic setup, we first introduce an underlying filtered probability space \((\Omega,\mathcal {F},\{\mathcal {F}_{t}\}_{t\geq 0}, P)\), where the increasing collection of σ-algebras \(\mathcal {F}_{t}, \ t\geq 0,\) models the flow of information that is accumulated through time.

The property (4) represents what has become known in the literature as the strong time consistency property. For example, if \(\mathcal {Q}=\{P\}\), then the strong time consistency reduces to the tower property for conditional expectations. From a practical point of view, this property essentially means that assessment of risks propagates in a consistent way over time: assessing at time t future risk, represented by random variable X, is the same as assessing at time t a risky assessment of X done at time t+1 and represented by −ρ _t+1(X). Additionally, the property (4) is closely related to the Bellman principle of optimality or to the dynamic programming principle (see, for instance, (Bellman and Dreyfus 1962; Carpentier et al. 2012).

Delbaen (2006) studies the recursivity property in terms of m-stable sets of probability measures, and also describes the time consistency of dynamic coherent risk measures in the context of martingale theory. The recursivity property is equivalent to properties known as time consistency and the rectangularity in the multi-prior Bayesian decision theory. Epstein and Schneider (2003) study time consistency and rectangularity property in the framework of “decision under ambiguity.”

It needs to be said that several authors refer to (Wang 2002) for an alternative axiomatic approach to time consistency of dynamic risk measures.

This means that if tomorrow we assess the riskiness of X and Y at the same level, then today X and Y must have the same level of riskiness. It can be shown that for dynamic coherent risk measures, or more generally for dynamic monetary risk measures, property (5) is equivalent to (4).

Motivated by results regarding the pricing procedure in incomplete markets, based on use of risk measures, Roorda et al. (2005) study dynamic coherent risk measures (for the case of random variables on finite probability space and discrete time) and introduce the notion of (strong) time consistency; note that their work was similar and contemporaneous to (Riedel 2004). They show that strong time consistency entails recursive computation of the corresponding optimal hedging strategies. Moreover, time consistency is also described in terms of the collection of probability measures that satisfy the “product property,” similar to the rectangularity property mentioned above.

Similarly, as in the static case, the dynamic coherent risk measures were extended to dynamic convex risk measures by replacing sub-additivity and positive homogeneity properties with convexity. In the continuous time setup, Rosazza Gianin (2002) links dynamic convex risk measures to nonlinear expectations or g-expectations, and to Backward Stochastic Differential Equations (BSDEs). Strong time consistency plays a crucial role and, in view of (4), it is equivalent to the tower property for conditional g-expectations. These results are further studied in a sequel of papers (Frittelli and Rosazza Gianin 2004; Peng 2004; Rosazza Gianin 2006), as well as in Coquet et al. (2002).

where \(\mathcal {M}(P)\) is the set of all probability measures absolutely continuous with respect to P, and α ^min is the minimal penalty function.⁴ The natural question of describing (strong) time consistency in terms of properties of the minimal penalty functions was studied by Scandolo (2003). Also in (Scandolo 2003), the author discusses the importance in the dynamic setup of the special property called locality. It should be mentioned that locality property was part of the earlier developments in the theory of dynamic risk measures. For example, it was called dynamic relevance axiom in (Riedel 2004), and zero-one law in (Peng 2004). Similarly to previous studies, (Scandolo 2003) finds a relationship between time consistency, the recursive construction of dynamic risk measures, and the supermartingale property. These results are further investigated in Detlefsen and Scandolo (2005). Also in these works, it was shown that the dynamic entropic risk measure is a strongly time consistent convex risk measure.

Weber (2006) continues the study of dynamic convex risk measures for random variables in a discrete time setup and introduces weaker notions of time consistency acceptance and rejection time consistency. Mainly, the author studies the law invariant risk measures, and characterizes time consistency in terms of the acceptance indicator and in terms of the acceptance sets of the form \(\mathcal {N}_{t}=\{X \mid \rho _{t}(X)\leq 0\}\). Along the same lines, Föllmer and Penner (2006) investigate the dynamic convex risk measures, representation of strong time consistency as a recursivity property, and they relate it to the Bellman principle of optimality. They also prove that the supermartingale property of the penalty function corresponds to the weak or acceptance/rejection time consistency. Moreover, the authors study the co-cycle property of the penalty function for the dynamic convex risk measures that admit robust representation (see Definition 50).

Artzner et al. (2007) continue to study the strong time consistency for dynamic risk measures, its equivalence with the stability property of test probabilities and with the optimality principle.

It is worth mentioning that Bion-Nadal (2004) studies dynamic monetary risk measures in a continuous time setting and their time consistency property in the context of model uncertainty when the class of probability measures is not specified.

Motivated by optimization subject to risk criterion, Ruszczynski and Shapiro (2006a) elevate the concepts from (Ruszczyński and Shapiro 2006b) to the dynamic setting, with the main goal to establish conditions under which the dynamic programming principle holds.

Cheridito and Kupper (2011) introduce the notion of aggregators and generators for dynamic convex risk measures and give a thorough discussion about the composition of time-consistent convex risk measures in the discrete time setup, for both random variables and stochastic processes. They link time consistency to one step dynamic penalty functions. In this regard, we also refer to (Cheridito et al. 2006; Cheridito and Kupper 2009).

Jobert and Rogers (2008) take the valuation concept as the starting point, rather than the dynamics of acceptance sets, with the valuation functional being the negative of a risk measure. To quote the authors (strong) “time consistency is the heart of the matter.” Kloeppel and Schweizer (2007) use dynamic convex risk measures for valuation in incomplete markets, where the time consistency plays a key role. Cherny (2007) uses dynamic coherent risk measure for pricing and hedging European options; see also (Cherny and Madan 2006).

Roorda and Schumacher (2007) study the weak form of time consistency for dynamic convex risk measure.

Bion-Nadal (2006) continues to study various properties of dynamic risk measures, both in discrete and in continuous time, mainly focusing on the composition property mentioned above, and thus on the strong time consistency. The composition property is characterized in terms of stability of probability sets. The author defines the co-cycle condition for the penalty function and shows its equivalence to strong time consistency. In the followup paper, (Bion-Nadal 2008), the author continues to study the characterization of time consistency in terms of the co-cycle condition for minimal penalty function. For further related developments in the continuous time framework see (Bion-Nadal 2009b).

Observing that Value at Risk (V@R) is not strongly time consistent, Boda and Filar (2006), and Cheridito and Stadje (2009) construct a strongly time consistent alternative to V@R by using a recursive composition procedure.

Tutsch (2008) gives a different perspective on time consistency of convex risk measures by introducing the update rules⁵ and generalizes the strong and weak form of time consistency via test sets.

The theory of dynamic risk measures finds its application in areas beyond the regulatory capital requirements. For example, Cherny (2010) applies dynamic coherent risk measures to risk-reward optimization problems and in (Cherny 2009) to capital allocation; Bion-Nadal (2009a) uses dynamic risk measures for time consistent pricing; Barrieu and El Karoui (2004; 2005; 2007) study optimal derivatives design under dynamic risk measures; Geman and Ohana (2008) explore the time consistency in managing a commodity portfolio via dynamic risk measures; Zariphopoulou and Zitkovic (2010) investigate the maturity independent dynamic convex risk measures.

In Delbaen et al. (2010), the authors establish a representation of the penalty function of dynamic convex risk measure using g-expectation and its relation to the strong time consistency.

There exists a significant literature on a special class of risk measures that satisfy the law invariance property. Kupper and Schachermayer (2009) prove that the only relevant, law invariant, strongly time consistent risk measure is the entropic risk measure.

For a fairly general study of dynamic convex risk measures and their time consistency we refer to (Bion-Nadal and Kervarec 2012) and (Bion-Nadal and Kervarec 2010). Acciaio et al. (2012) give a comprehensive study of various forms of time consistency for dynamic convex risk measures in a discrete time setup. This includes strong and weak time consistency, representations of time consistency in terms of acceptance sets, and the supermartingale property of the penalty function. We would like to point out the survey by Acciaio and Penner (2011) of discrete time dynamic convex risk measures. This work deals with (essentially bounded) random variables and examines most of the papers mentioned above from the perspective of the robust representation framework.

Although the connection between BSDEs and the dynamic convex risk measures in a continuous time setting had been established for some time, it appears that Stadje (2010) was the first author to create a theoretical framework for studying dynamic risk measures in discrete time via the Backward Stochastic Difference Equations (BS ΔEs). Due to the backward nature of BS ΔEs, the strong time consistency of risk measures played a critical role in characterizing the dynamic convex risk measures as solutions of BS ΔEs. In a series of papers, Cohen and Elliott further studied the connection between dynamic risk measures and BS ΔEs (Cohen and Elliott 2010, 2011; Elliott et al. 2015).

Föllmer and Penner (2011) developed the theory of dynamic monetary risk measures under Knightian uncertainty, where the corresponding probability measures are not necessarily absolutely continuous with respect to the reference measure. See also Nutz and Soner (2012) for a study of dynamic risk measures under volatility uncertainty and their connection to G-expectations.

From a slightly different point of view, Ruszczynski (2010) studies Markov risk measures, that enjoy strong time consistency, in the framework of risk-averse preferences; see also (Shapiro 2009, 2011, 2012; Fan and Ruszczyński 2014). Some concepts from the theory of dynamic risk measures are adopted to the study of the dynamic programming for Markov decision processes.

In the recent paper, Mastrogiacomo and Rosazza Gianin (2015) provide several forms of time consistency for sub-additive dynamic risk measures and their dual representations.

Finally, we want to mention that during the last decade significant advances were made towards developing a general theory of set-valued risk measures (Hamel and Rudloff 2008; Hamel et al. 2011; Feinstein and Rudloff 2013; Hamel et al. 2013; Feinstein and Rudloff 2015), including the dynamic version of them, where mostly the corresponding form of strong time consistency is considered.

We recall that the main objective of use of risk measures for financial applications is mapping the level of risk of a financial position to a regulatory monetary amount expressed in units of the relevant currency. Accordingly, the key property of any risk measure is cash-additivity ρ(X−m)=ρ(X)+m. Clearly, one can think of the risk measures as generalizations of V@R.

A concept that is, in a sense, complementary to the concept of risk measures, is that of performance measures, which can be thought as generalizations of the well known Sharpe ratio. In similarity with the theory of risk measures, the development of the theory of performance measures followed an axiomatic approach. This was initiated by Cherny and Madan (2009), where the authors introduced the (static) notion of the coherent acceptability index–a function on L ^∞ with values in \(\mathbb {R}_{+}\) that is monotone, quasi-concave, and scale invariant. As a matter of fact, scale invariance is the key property of acceptability indices that distinguishes them from risk measures, and, typically, acceptability indices are not cash-additive. The dynamic version of coherent acceptability indices was introduced by Bielecki et al. (2014b), for the case of stochastic processes, finite probability space, and discrete time. From now on, we will use as synonyms the terms measures of performance, performance measures, and acceptability indices.

where X∈L ^∞ , and m _t ,n _t are \(\mathcal {F}_{t}\)-measurable random variables. Biagini and Bion-Nadal (2014) study dynamic performance measures in a fairly general setup that generalize the results of (Bielecki et al. 2014b). Later, using the theory of dynamic coherent acceptability indices developed in (Bielecki et al. 2014b), Bielecki et al. (2013) propose a pricing framework, called dynamic conic finance, for dividend paying securities in discrete time. The time consistency property was at the core of establishing the connection between dynamic conic finance and classical arbitrage pricing theory. The static conic finance, that served as motivation for (Bielecki et al. 2013), was introduced in (Cherny and Madan 2010). Finally, in recent papers (Bielecki et al. 2015b; Rosazza Gianin and Sgarra 2013), the authors elevate the notion of dynamic coherent acceptability indices to the case of sub-scale invariant performance measures. For that, BSDEs are used in (Rosazza Gianin and Sgarra 2013) and BS ΔEs are used in (Bielecki et al. 2015b).

For a general theory of robust representations of quasi-concave maps that covers both dynamic risk measures and dynamic acceptability indices, see (Frittelli and Maggis 2011; Bielecki et al. 2016; Frittelli and Maggis 2014; Bion-Nadal 2016). Also in (Bielecki et al. 2016), the authors study the strong time consistency of quasi-concave maps via the concept of certainty equivalence; see also (Frittelli and Maggis 2010).

To our best knowledge, (Bielecki et al. 2014a) is the only paper that combines into a unified framework the time consistency for dynamic risk measures and dynamic performance measure. It uses the concept of update rules that serve as a vehicle for connecting preferences at different times. We take the update rules perspective as the main tool for surveying the existing forms of time consistency.

We conclude this literature review by listing works, which in our opinion, are most relevant to this survey (not all of which are mentioned above though).

Dynamic monetary risk measures, strong time consistency:

(discrete time) (Cheridito and Kupper (2011); Cheridito et al. (2006)); (continuous time) (Bion-Nadal 2004; Föllmer and Penner 2011).

Dynamic acceptability indices: (Bielecki et al. (2014b); Biagini and Bion-Nadal (2014); Bielecki et al. (2013); Rosazza Gianin and Sgarra (2013); Bielecki et al. (2016); Frittelli and Maggis (2014); Bielecki et al. (2014a); Bielecki et al. (2015b)).

Let \((\Omega,\mathcal {F},\mathbb {F}=\{\mathcal {F}_{t}\}_{t\in \mathbb {T}},P)\) be a filtered probability space, with \(\mathcal {F}_{0}=\{\Omega,\emptyset \}\), and \(\mathbb {T}=\{0,1,\ldots, T\}\), where \(T\in \mathbb {N}\) is a fixed and finite time horizon. We will also use the notation \(\mathbb {T}'=\{0,1,\ldots, T-1\}\).

For \(\mathcal {G}\subseteq \mathcal {F}\) we denote by \(L^{0}(\Omega,\mathcal {G},P)\) and \(\bar {L}^{0}(\Omega,\mathcal {G},P)\) the sets of all \(\mathcal {G}\)-measurable random variables with values in (−∞,∞) and [−∞,∞], respectively. In addition, we use the notation \(L^{p}(\mathcal {G}):=L^{p}(\Omega,\mathcal {G},P)\), \(L^{p}_{t}:=L^{p}(\mathcal {F}_{t})\), and \(L^{p}:=L_{T}^{p}\), for p∈{0,1,∞}; analogously we define \(\bar {L}^{0}_{t}\). We also use the notation \(\mathbb {V}^{p}:=\{(V_{t})_{t\in \mathbb {T}}: V_{t}\in L^{p}_{t}\}\), for p∈{0,1,∞}.⁶ Moreover, we use \(\mathcal {M}(P)\) to denote the set of all probability measures on \((\Omega,\mathcal {F})\) that are absolutely continuous with respect to P, and we set \(\mathcal {M}_{t}(P):=\{Q\in \mathcal {M}(P)\,:\, Q|_{\mathcal {F}_{t}}=P|_{\mathcal {F}_{t}}\}\).

For any \(m\in \bar L^{0}_{t}\), the value m1_{t} corresponds to a cash flow of size m received at time t. We use this notation for the case of random variables to present more unified definitions (see Appendix “Dynamic LM-measures”).

for any \(A\in \mathcal {F}_{t}\). We call this random variable the \(\mathcal {F}_{t}\)-conditional essential infimum of X. Similarly, we define ess sup _t (X):=−ess inf _t (−X), and we call it the \(\mathcal {F}_{t}\)-conditional essential supremum of X. Again, see Appendix “Conditional expectation and conditional essential supremum/infimum” for more details and some elementary properties of conditional essential infimum and supremum.

The next definition introduces the main object of this work.

It is well recognized that locality and monotonicity are two properties that must be satisfied by any reasonable dynamic measure of performance and/or measure of risk, and in fact are shared by most, if not all, of such measures studied in the literature. The monotonicity property is natural for any numerical representation of an order between the elements of \(\mathcal {X}\). The locality property (also referred to as regularity, or zero-one law, or relevance) essentially means that the values of the LM-measure restricted to a set \(A\in \mathcal {F}\) remain invariant with respect to the values of the arguments outside of the same set \(A\in \mathcal {F}\); in particular, the events that will not happen in the future do not affect the value of the measure today.

for any \(t\in \mathbb {T}\). We impose this (technical) assumption to ensure that the maps φ _t that we consider are not degenerate in the sense that they are not taking infinite values for all \(X\in \mathcal {X}\) on some set \(A_{t}\in \mathcal {F}_{t}\) of positive probability, for any \(t\in \mathbb {T}\); in the literature, sometimes such maps are referred to as proper (Kaina and Rüschendorf 2009). If this is the case, then there exists a family \(\{Y_{t}\}_{t\in \mathbb {T}}\), where \(Y_{t}\in \mathcal {X}\), such that \(\varphi _{t}(Y_{t})\in L^{0}_{t}\) for any \(t\in \mathbb {T}\), and so we can consider maps \(\tilde {\varphi }\) given by \(\tilde {\varphi }_{t}(\cdot):=\varphi _{t}(\cdot)-\varphi _{t}(Y_{t})\), that satisfy assumption (10) and preserve the same order as the maps φ _t do. Typically, in the risk measure framework, one assumes that φ _t (0)=0, which implies (10). However, here we cannot assume that φ _t (0)=0, as we will also deal with dynamic performance measures for which φ _t (0)=∞.

Finally, let us note that in the literature, traditionally the dynamic risk measures are monotone decreasing. On the other hand, the measures of performance are monotone increasing. In view of condition 2) in Definition 1, whenever our LM-measure corresponds to a dynamic risk measure, it needs to be understood as the negative of that risk measure. In such cases, in order to avoid confusion, we refer to the respective LM-measure as to dynamic (monetary) utility measure rather than as dynamic (monetary) risk measure. See Appendix “Dynamic LM-measures” for details.

In this section, we present a brief survey of approaches to time consistent assessment of preferences, or to time consistency—for short, that were studied in the literature. As discussed in the Introduction, time consistency is studied via numerical representations of preferences. Various numerical representations will be surveyed below, and discussed in the context of dynamic LM-measures.

To streamline the presentation, we focus our attention on the case of random variables, that is \(\mathcal {X}=L^{p}\), for p∈{0,1,∞}.⁷ Usually, the risk measures and the performance measures are studied on spaces smaller than L ⁰, such as L ^p , p∈[1,∞]. This is motivated by the aim to obtain so called robust representation of such measures (see Appendix “Dynamic LM-measures”), since a certain topological structure is required for that (cf. Remark 13). On the other hand, time consistency refers only to consistency of measurements in time, where no particular topological structure is needed, and thus most of the results obtained here hold true for p=0.

In Section “Generic approaches”, we outline two generic approaches to time consistent assessment of preferences: an approach based on update rules and an approach based on benchmark families. These two approaches are generic in the sense that nearly all types of time consistency can be represented within these two approaches. On the contrary, the approaches outlined in Section “Idiosyncratic approaches” are specific. That is to say, those approaches are suited only for specific types of time consistency, specific classes of dynamic LM-measures, specific spaces, etc.

Generic approaches

In this section, we outline two concepts that underlie the generic approaches to time consistent assessment of preferences: the update rules and the benchmark families. It will be seen that different types of time consistency can be characterized in terms of these concepts.

Update rules

The approach to time consistency using update rules was developed in Bielecki et al. (2014a). An update rule is a tool that is applied to preference levels, and used for relating assessments of preferences done using a dynamic LM-measure at different times.

Definition 2

A family \(\mu =\{\mu _{t,s}:\, t,s\in \mathbb {T},\, t<s\}\) of maps \(\mu _{t,s}:\bar {L}^{0}_{s}\to \bar {L}^{0}_{t}\) is called an update rule if μ satisfies the following conditions:

1. 1)

   (Locality) ;

    
2. 2)

   (Monotonicity) if m≥m ^′, then μ _t,s (m)≥μ _t,s (m ^′);

    

for any s>t, \(A\in \mathcal {F}_{t}\), and \(m,m'\in \bar {L}^{0}_{s}\).

Next, we give a definition of time consistency in terms of update rules.

Definition 3

Let μ be an update rule. We say that the dynamic LM-measure φ is μ-acceptance (resp. μ-rejection) time consistent if

$$ \varphi_{s}(X)\geq m_{s} \quad(\mathrm{resp.} \leq)\quad \Longrightarrow\quad \varphi_{t}(X)\geq \mu_{t,s}(m_{s})\quad(\mathrm{resp.} \leq), $$

(11)

for all s>t, \(s,t\in \mathbb {T}\), \(X\in \mathcal {X}\), and \(m_{s}\in \bar {L}^{0}_{s}\). If property (11) is satisfied for s=t+1, \(t\in \mathbb {T}'\), then we say that φis one-step μ-acceptance (resp. one-step μ-rejection) time consistent.

We see that m _s and μ _t,s (m _s ) serve as benchmarks to which the measurements of φ _s (X) and φ _t (X) are compared, respectively. Thus, the interpretation of acceptance time consistency is straightforward: if \(X\in \mathcal {X}\) is accepted at some future time \(s\in \mathbb {T}\), at least at level m _s , then today, at time \(t\in \mathbb {T}\), it is accepted at least at level μ _t,s (m _s ). Similar reasoning holds for the rejection time consistency. Essentially, the update rule μ converts the preference levels at time s to the preference levels at time t.

We started our survey of time consistency with Definition 3 since, as we will demonstrate below, this concept of time consistency covers various cases of time consistency for risk and performance measures that can be found in the existing literature. In particular, it allows to establish important connections between different types of time consistency. The time consistency property of an LM-measure, in general, depends on the choice of the updated rule; we refer to Section “Time consistency for random variables” for an in-depth discussion.

It is useful to observe that the time consistency property given in terms of update rules can be equivalently formulated as a version of the dynamic programming principle (see (Bielecki et al. 2014a), Proposition 3.6): φ is μ-acceptance (resp. μ-rejection) time consistent if and only if

$$ \varphi_{t}(X)\geq \mu_{t,s}(\varphi_{s}(X))\quad (\mathrm{resp.} \leq), $$

(12)

for any \(X\in \mathcal {X}\) and \(s,t\in \mathbb {T}\), such that s>t. The interpretation of (12) is as follows: if the numerical assessment of preferences about X is given in terms of a dynamic LM-measure φ, then this measure is μ-acceptance time consistent if and only if the numerical assessment of preferences about X done at time t is greater than the value of the measurement done at any future time s>t and updated at time t via μ _t,s . The analogous interpretation applies to the ejection time consistency.

Next, we define two interesting and important classes of update rules.

Definition 4

Let μ be an update rule. We say that μ is

1. 1)

   s-invariant, if there exists a family \(\{\mu _{t}\}_{t\in \mathbb {T}}\) of maps \(\mu _{t}:\bar {L}^{0}\to \bar {L}^{0}_{t}\), such that μ _t,s (m _s )=μ _t (m _s ) for any \(s,t\in \mathbb {T}\), s>t, and \(m_{s}\in \bar {L}^{0}_{s}\);

    
2. 2)

   projective, if it is s-invariant and μ _t (m _t )=m _t , for any \(t\in \mathbb {T}\), and \(m_{t}\in \bar {L}^{0}_{t}\).

    

Remark 3

If an update rule μ is s-invariant, then it is enough to consider only the corresponding family \(\{\mu _{t}\}_{t\in \mathbb {T}}\). Hence, with slight abuse of notation, we write \(\mu =\{\mu _{t}\}_{t\in \mathbb {T}}\) and call it an update rule as well.

Example 1

The families \(\mu ^{1}=\{\mu ^{1}_{t}\}_{t\in \mathbb {T}}\) and \(\mu ^{2}=\{\mu ^{2}_{t}\}_{t\in \mathbb {T}}\) given by

$$\mu^{1}_{t}(m)=E[m|\mathcal{F}_{t}],\quad \text{and}\quad \mu^{2}_{t}(m)=\mathrm{ess\,inf}_{t} m,\quad m\in \bar{L}^{0}, $$

are projective update rules. It will be shown in Example 5 that there is a dynamic LM-measure that is μ ²–time consistent but not μ ¹–time consistent.

Benchmark families

The approach to time consistency based on families of benchmark sets was initiated by (Tutsch 2008), where the author applied this approach in the context of dynamic risk measures. Essentially, a benchmark family is a collection of subsets of \(\mathcal {X}\) that contain reference or test objects. The idea of time consistency in this context, is that the preferences about objects of interest must compare in a consistent way to the preferences about the reference objects.

Definition 5

1. (i)
   A family \(\mathcal {Y}=\{\mathcal {Y}_{t}\}_{t\in \mathbb {T}}\) of sets \(\mathcal {Y}_{t}\subseteq \mathcal {X}\) is a benchmark family if

   $$0\in\mathcal{Y}_{t}\quad \mathrm{and }\quad \mathcal{Y}_{t}+\mathbb{R}=\mathcal{Y}_{t}, $$

   for any \(t\in \mathbb {T}\).
    
2. (ii)
   A dynamic LM-measure φ is acceptance (resp. rejection) time consistent with respect to the benchmark family \(\mathcal {Y}\), if

   $$ \varphi_{s}(X)\geq \varphi_{s}(Y)\quad (resp. \leq)\quad \Longrightarrow\quad \varphi_{t}(X)\geq \varphi_{t}(Y)\quad (resp. \leq), $$

   (13)

   for all s≥t, \(X\in \mathcal {X}\), and \(Y\in \mathcal {Y}_{s}\).

    

Informally, the “degree” of time consistency with respect to \(\mathcal {Y}\) is measured by the size of \(\mathcal {Y}\). Thus, the larger the sets \(\mathcal {Y}_{s}\) are, for each \(s\in \mathbb {T}\), the stronger the degree of time consistency of φ.

Example 2

The families of sets \(\mathcal {Y}^{1}=\{\mathcal {Y}^{1}_{t}\}_{t\in \mathbb {T}}\) and \(\mathcal {Y}^{2}=\{\mathcal {Y}^{2}_{t}\}_{t\in \mathbb {T}}\) given by

$$\mathcal{Y}^{1}_{t}=\mathbb{R}\quad \text{and}\quad \mathcal{Y}^{2}_{t}=\mathcal{X}, $$

are benchmark families. They relate to weak and strong types of time consistency, as will be discussed later on.

For future reference, we recall from (Bielecki et al. 2014a, Proof of Proposition 3.9) that φ is acceptance (resp. rejection) time consistent with respect to \(\mathcal {Y}\), if and only if φ is acceptance (resp. rejection) time consistent with respect to the benchmark family \(\mathcal {\widehat {Y}}\) given by

Relation between update rule approach and the benchmark approach

The difference between the update rule approach and the benchmark family approach is that the preference levels are chosen differently. Specifically, in the former approach, the preference level at time s is chosen as any \(m_{s}\in \bar {L}^{0}_{s}\), and then updated to the preference level at time t, using an update rule. In the latter approach, the preference levels at both times s and t are taken as φ _s (Y) and φ _t (Y), respectively, for any reference object \(Y\in \mathcal {Y}_{s}\), where \(\mathcal {Y}_{s}\) is an element of the benchmark family \(\mathcal {Y}\).

These two approaches are strongly related to each other. Indeed, for any LM-measure φ and for any benchmark family \(\mathcal {Y}\), one can construct an update rule μ such that φ is time consistent with respect to \(\mathcal {Y}\) if and only if it is μ-time consistent.

For example, in case of acceptance time consistency of φ with respect to \(\mathcal {Y}\), using the locality of φ, it is easy to note that (13) is equivalent to

where and \(\widehat {\mathcal {Y}}=\{\widehat {\mathcal {Y}}_{s} \}_{s\in \mathbb {T}}\) is defined in (14). Consequently, setting

and using (12), we deduce that φ satisfies (13) if and only if φ is time consistent with respect to the update rule \(\widetilde \mu _{t,s}\) (see (Bielecki et al. 2014a, Proposition 3.9) for details). The analogous argument works for rejection time consistency.

Generally speaking, the converse implication does not hold true; the notion of time consistency given in terms of update rules is more general. For example, time consistency of a dynamic coherent acceptability index cannot be expressed in terms of a single benchmark family.

In this section, we survey the time consistency of LM-measures applied to random variables. Accordingly, we assume that \(\mathcal {X}=L^{p}\), for a fixed p∈{0,1,∞}. We proceed with the discussion of various related types of time consistency, without much reference to the existing literature. Such references are provided in Section “Literature review”.

Taxonomy of results

For the convenience of the reader, in Fig. 1 below, we summarize the results surveyed in Section “Time consistency for random variables”. For transparency, we label (by circled numbers) each arrow (implication or equivalence) in the flowchart, and we relate the labels to the relevant results, also providing comments on converse implications whenever appropriate.

1. ■■■

   Proposition 2, 2)

    
2. ■■■

   Proposition 1, 4)

    
3. ■■■

   Remark 8 and Proposition 3. The converse implication is not true in general, see Example 8.

    
4. ■■■

   Proposition 3. Generally speaking, the converse implication is not true. See Example 8: the negative of Dynamic Entropic Risk Measure with γ<0 is weakly acceptance time consistent, but it is not supermaringale time consistent, i.e., it is not acceptance time consistent with respect to the projective update rule \(\mu _{t}=E_{t}[m|\mathcal {F}_{t}]\).

    
5. ■■■

   Proposition 4, 4). The converse implication is not true in general. For the counterexample, see (Acciaio and Penner 2011, Proposition 37).

    
6. ■■■

   Proposition 3, and see also . In general, strong time consistency does not imply weak acceptance time consistency, see Remark 6.

    
7. ■■■

   Proposition 7, 3)

    
8. ■■■

   This is heuristic statement. See Remark 6.(ii).

    
9. ■■■

   Proposition 4, 5)

    

We preserve the same names for various types of time consistency for both the random variables and the stochastic processes. However, we stress that the nature of time consistency for stochastic processes is usually much more intricate. If φ is an LM-measure, and \(V\in \mathbb {V}^{p}\), then in order to compare φ _t (V) and φ _s (V), for s>t, one also needs to take into account the cash flows between times t and s.

In order to account for the intermediate cash flows, we modify appropriately the concept of the update rule.

Note that the update rule introduced in Definition 3 may be considered as the generalized update rule, which is constant with respect to X, i.e., μ(·,X)=μ(·,Y) for any \(X,Y\in \mathcal {X}\). In what follows, if there is no ambiguity, we drop the term generalized.

As before, we say that the update rule μ is s-invariant, if there exists a family \(\{\mu _{t}\}_{t\in \mathbb {T}}\) of maps \(\mu _{t}:\bar {L}^{0}\times \mathcal {X} \to \bar {L}^{0}_{t}\), such that μ _t,s (m _s ,X)=μ _t (m _s ,X) for any \(s,t\in \mathbb {T}\), s>t, \(X\in \mathcal {X}\), and \(m_{s}\in \bar {L}^{0}_{s}\).

We now arrive at the corresponding definition of time-consistency.

where \(\tilde \mu \) is the one-step update rule for random variables, and \(f:\bar {\mathbb {R}}\to \bar {\mathbb {R}}\) is a Borel measurable function such that f(0)=0. Property (24) is postulated primarily to allow establishing a direct connection between our results and the existing literature. Moreover, when using one-step update rules of form (24), the one-step time consistency for random variables is a particular case of one-step time consistency for stochastic processes by considering cash flows with only terminal payoff, namely stochastic processes such that V=(0,…,0,V _T ).

we have that μ-acceptance (resp. μ-rejection) time consistency is equivalent to one step μ-acceptance (resp. μ-rejection) time consistency. This is another reason why we consider only one step update rules for stochastic processes.

Semi-weak time consistency

In this section, we introduce the concept of semi-weak time consistency for stochastic processes. We have not discussed semi-weak time consistency in the case of random variables, since, in that case, semi-weak time consistency coincides with the weak time consistency.

As it was shown, (Bielecki et al. 2014b), none of the forms of time consistency existing in the literature at the time when that paper was written were suitable for scale-invariant maps such as acceptability indices. In fact, even the weak acceptance and the weak rejection time consistency for stochastic processes (as defined in the present paper) are too strong in the case of scale invariant maps. This is a reason why we introduce yet a weaker notion of time consistency, which we will refer to as semi-weak acceptance and semi-weak rejection time consistency. The notion of semi-weak time consistency for stochastic processes, introduced next, is well suited for scale-invariant maps; we refer the reader to (Bielecki et al. 2014b) for a detailed discussion on time consistency for such maps and their dual representations.¹³

Definition 13

Let φ be a dynamic LM-measure on \(\mathbb {V}^{p}\). Then, φ is semi-weakly acceptance time consistent if

and it is semi-weakly rejection time consistent if

Clearly, weak acceptance/rejection time consistency for stochastic processes implies semi-weak acceptance/rejection time consistency.

Next, we will show that the definition of semi-weak time consistency is indeed equivalent to the time consistency introduced in (Bielecki et al. 2014b).

Proposition 11

Let φ be a dynamic LM-measure on \(\mathbb {V}^{p}\). The following properties are equivalent

1. 1)

   φ is semi-weakly acceptance time consistent.

    
2. 2)
   φ is one step μ-acceptance time consistent, where the (generalized) update rule is given by

   $$\mu_{t,t+1}(m,V) =1_{\{V_{t}\geq 0\}}\mathrm{ess\,inf}_{t} m+1_{\{V_{t}< 0\}}(-\infty). $$
    
3. 3)
   For all \(V\in \mathbb {V}^{p}\), \(t\in \mathbb {T}\), t<T, and \(m_{t}\in \bar {L}^{0}_{t}\), such that V _t ≥0

   $$\varphi_{t+1}(V)\geq m_{t}\quad \Longrightarrow\quad\varphi_{t}(V)\geq m_{t}. $$
    

A similar result is true for semi-weak rejection time consistency.

For the proof, see (Bielecki et al. 2014a, Proposition 4.8).

Property 3) in Proposition 11, which is the definition of the (acceptance) time consistency given in (Bielecki et al. 2014b), best illustrates the financial meaning of semi-weak acceptance time consistency: if tomorrow we accept the dividend stream \(V\in \mathbb {V}^{p}\) at level m _t , and if we get a positive dividend V _t paid today at time t, then today we accept the cash flow V at least at level m _t as well. A similar interpretation is valid for semi-weak rejection time consistency.

The next two results are important. In particular, they generalize the work done in (Bielecki et al. 2014b) regarding duality between cash-additive risk measures and acceptability indices.

Proposition 12

Let \(\{\varphi ^{x}\}_{x\in \mathbb {R}_{+}}\) be a decreasing family¹⁴ of dynamic LM-measures on \(\mathbb {V}^{p}\). Assume that for each \(x\in \mathbb {R}_{+}\), φ ^x is weakly acceptance (resp. weakly rejection) time consistent. Then, the family \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) of maps \(\alpha _{t}:\mathbb {V}^{p}\to \bar {L}^{0}_{t}\) defined by

is a semi-weakly acceptance (resp. semi-weakly rejection) time consistent dynamic LM-measure.

For the proof, see (Bielecki et al. 2014a, Proposition 4.9). It will be useful to note that α _t (V) defined in (28) can also be written as

$$ \alpha_{t}(V)=\sup\{x \in \mathbb{R}^{+} \mid \varphi_{t}^{x}(V)\geq 0\}. $$

(29)

Proposition 13

Let \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) be a dynamic LM-measure, which is independent of the past and translation invariant.¹⁵ Assume that \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) is semi-weakly acceptance (resp. semi-weakly rejection) time consistent. Then, for any \(x\in \mathbb {R}_{+}\), the family \(\varphi ^{x}=\{\varphi _{t}^{x}\}_{t\in \mathbb {T}}\) of maps \(\varphi ^{x}_{t}:\mathbb {V}^{p}\to \bar {L}^{0}_{t}\) defined by

is a weakly acceptance (resp. weakly rejection) time consistent dynamic LM-measure.

For the proof, see (Bielecki et al. 2014a, Proposition 4.10). In what follows, we will use the fact that \(\varphi ^{x}_{t}(V)\) defined in (30) can also be written as

This type of dual representation, i.e., (28) and (30), or, equivalently, (29) and (31), first appeared in (Cherny and Madan 2009) where the authors studied the static (one period of time) case. Subsequently, in (Bielecki et al. 2014b), the authors extended these results to the case of stochastic processes with special emphasis on the time consistency property. In contrast to the results of (Bielecki et al. 2014b), Propositions 12 and 13 consider an arbitrary probability space, not just a finite one.

Other types of time consistency

Other types of time consistency for stochastic processes may be defined in analogy to what is done in Section “Other types of time consistency” for the case of random variables. For brevity, we limit our discussion here to the update rules derived from dynamic LM-measures.

First, given a dynamic LM-measure φ on \(\mathbb {V}^{p}\), we denote by \(\widetilde {\varphi }\) the family of maps \(\widetilde {\varphi }_{t}: L_{t+1}^{p}\to \bar {L}^{0}_{t}\) given by

$$ \widetilde{\varphi}_{t}(X):=\varphi_{t}(1_{\{t+1\}}X),\ \text{for}\ t\in\mathbb{T}'. $$

(32)

Since φ is monotone and local on \(\mathbb {V}^{p}\), then, clearly, \(\widetilde {\varphi }_{t}\) is local and monotone on \(L_{t+1}^{p}\).

Next, for any \(t\in \mathbb {T}'\), we extend \(\widetilde {\varphi }_{t}\) to \(\bar {L}_{t+1}^{0}\), preserving locality and monotonicity (see Remark 12), and this extension produces a one-step update rule.

For example, the middle acceptance time consistency is obtained by taking the update rule μ given as

$$\mu_{t,t+1}(m,V)=\widetilde{\varphi}^{-}_{t}(m+V_{t}),\quad t\in\mathbb{T}', $$

where \(\widetilde {\varphi }_{t}^{-}:\bar {L}^{0}_{t+1}\to \bar {L}^{0}_{t}\) is defined as in (55), with the sets \(\mathcal {Y}^{-}_{A}(X)\) replaced by

Taxonomy of results

In Fig. 2, we summarize the results surveyed in Section “Time consistency for stochastic processes”. We label each arrow (implication or equivalence) in the flowchart with numbers in squares and we relate the labels to the relevant results. Additionally, we provide comments on converse implications whenever appropriate.

1. ■■■

   Proposition 9, 5)

    
2. ■■■

   Proposition 9, 4)

    
3. ■■■

   Proposition 11, 3)

    
4. ■■■

   Proposition 10

    
5. ■■■

   Proposition 10, and see also .

    
6. ■■■

   Proposition 14.

    

Remark 10

The converse of implications and in Flowchart 2 do not hold true in general; one can use the same counterexamples as in the case of random variables. For a counterexample showing that the converse of does not hold true in general, see Example 5.

In this section, we present examples that illustrate the different types of time consistency for dynamic risk measures and dynamic performance measures, as well as the relationships between them.

We recall that according to the convention adopted in this paper, the dynamic LM-measures representing risk measures are the negatives of their classical counterparts. With this understanding, in the titles of the examples representing risk measures below we will skip the term “negative”.

Example 3

(Value at Risk (V@R)) Let \(\mathcal {X}=L^{0}\) and α∈(0,1). We denote by \(\varphi ^{\alpha }_{t}(X)\) an \(\mathcal {F}_{t}\)-conditional α-quantile of X

$$ \varphi^{\alpha}_{t}(X):= \mathrm{ess\,sup}\{Y\in L_{t}^{0} \mid P[X\leq Y \mid \mathcal{F}_{t}]\leq\alpha \}. $$

(33)

According to our convention, the conditional V@R is defined by \(\mathrm {V}@\mathrm {R}^{\alpha }_{t}(X):= -\varphi ^{\alpha }_{t}(X)\).

The family of maps \(\{\varphi ^{\alpha }_{t}\}_{t\in \mathbb {T}}\) is a dynamic monetary utility measure. It is well known that \(\{\varphi ^{\alpha }_{t}\}_{t\in \mathbb {T}}\) is not strongly time consistent; see (Cheridito and Stadje 2009) for details. However, it is both weakly acceptance and weakly rejection time consistent. Indeed, if \(\varphi ^{\alpha }_{s}(X)\geq 0\), for some s>t, and X∈L ⁰, then for any ε<0 we get and

Since ε<0 was chosen arbitrarily, we get \(\varphi ^{\alpha }_{t}(X)\geq 0\), and thus, in view of Proposition 2, \(\{\varphi ^{\alpha }_{t}\}_{t\in \mathbb {T}}\) is weakly acceptance time consistent.

Now, let us assume that \(\varphi ^{\alpha }_{s}(X)\leq 0\). Then, due to the locality of the conditional expectation, we have that for any ε>0. In fact, if , then there exists an \(\mathcal {F}_{s}\)-measurable set A with positive measure on which

Taking any \(Y'\in L_{s}^{0}\) such that we know that for \(\mathcal {F}_{s}\)-measurable random variable ’ we get

Thus,

$$0\geq \mathrm{ess\,sup}\{Y\in L_{s}^{0} \mid P[X\leq Y \mid \mathcal{F}_{s}]\leq\alpha\} \geq Z, $$

which leads to the contradiction.

Consequently, for any \(Y\in L^{0}_{t}\) and ε>0, we get

and, consequently, on \(\mathcal {F}_{t}\)-measurable set {Y>ε}. Hence,

Since ε>0 was chosen arbitrary, we conclude that \(\varphi ^{\alpha }_{t}(X)\leq 0\), thus \(\{\varphi ^{\alpha }_{t}\}_{t\in \mathbb {T}}\) is weakly rejection time consistent.

Example 5

(Dynamic TV@RAcceptability Index for Processes) Tail Value at Risk Acceptability Index was introduced in (Cherny and Madan 2009), as an example of static scale invariant performance measure for the case of random variables. Here, along the lines of (Bielecki et al. (Bielecki et al. 2014b)), we extend this notion to the dynamic setup and apply it to the case of stochastic processes. Let \(\mathcal {X}=\mathbb {V}^{0}\), and for a fixed α∈(0,1], we consider the sets \(\{D^{\alpha }_{t}\}_{t\in \mathbb {T}}\) defined in (34). We consider the distortion function \(g(x)=\frac {1}{1+x}\), \(x\in \mathbb {R}^{+}\), and we define \(\rho ^{x}=\{\rho _{t}^{x}\}_{t\in \mathbb {T}}, \ x\in \mathbb {R}_{+}\), as follows

$$ \rho^{x}_{t}(V)=\underset{Z\in D^{g(x)}_{t}}{ess\,inf}E\left[Z\sum_{i=t}^{T}V_{i}\Big|\mathcal{F}_{t}\right], \quad V\in\mathbb{X}, \ t\in\mathcal{T}. $$

(38)

Then, ρ ^x is an increasing (with respect to x) family of dynamic coherent utility measures for processes, and the map \(\alpha =\{\alpha _{t}\}_{t\in \mathbb {T}}\) given by

$$ \alpha_{t}(V)=\sup\{x\in\mathbb{R}_{+} \mid \rho_{t}^{x}(V)\leq0\}, $$

(39)

is a dynamic acceptability index for processes (see (Cherny and Madan 2009) and (Bielecki et al. (Bielecki et al. 2014b))). Moreover,

Clearly, (39) and (40) are the counterparts of (29) and (31), respectively.

Considering the above, then, similarly to Example 4, one can show that ρ ^x is weakly rejection time consistent, but it is not weakly acceptance time consistent, for any fixed \(x\in \mathbb {R}_{+}\), and hence, by Proposition 12 and Proposition 13, α is semi-weakly rejection time consistent but not semi-weakly acceptance time consistent.

Example 7

(Dynamic Gain Loss Ratio) Dynamic Gain Loss Ratio (dGLR) is another popular measure of performance, which essentially improves on some drawbacks of Sharpe Ratio (such as penalizing for positive returns), and it is given by the ratio of expected return over expected losses. Formally, for \(\mathcal {X}=\mathbb {V}^{1}\), dGLR is defined as

$$ \varphi_{t}(V):=\left\{ \begin{array}{ll} \frac{E\left[\sum_{i=t}^{T}V_{i}|\mathcal{F}_{t}\right]}{E\left[(\sum_{i=t}^{T}V_{i})^{-}|\mathcal{F}_{t}\right]}, &\quad \text{if}~E\left[\sum_{i=t}^{T}V_{i}|\mathcal{F}_{t}\right]>0,\\ 0, &\quad \text{otherwise}. \end{array}\right. $$

(42)

For various properties and dual representations of dGLR see (Bielecki et al. 2014b, 2016). In (Bielecki et al. (Bielecki et al. 2014b)), assuming that Ω is finite, the authors showed that dGLR is both semi-weakly acceptance and semi-weakly rejection time consistent. For the sake of completeness, we will show here that dGLR is semi-weakly acceptance time consistent.

Assume that \(t\in \mathbb {T}'\), and \(V\in \mathcal {X}\). In view of Definition 13, it is enough to show that

On the set {V _t <0}, the inequality (43) is trivial. Since φ _t is non-negative and local, without loss of generality, we may assume that ess inf _t (φ _t+1(V))>0. Since, φ _t+1(V)≥ess inf _t (φ _t+1(V)), we have that

$$ E\left[\sum_{i=t+1}^{T}V_{i}\Big|\mathcal{F}_{t+1}\right]\geq \mathrm{ess\,inf}_{t}(\varphi_{t+1}(V))\cdot E\left[\left(\sum_{i=t+1}^{T}V_{i}\right)^{-}\Big|\mathcal{F}_{t+1}\right]. $$

(44)

Using (44) we obtain

Note that ess inf _t (φ _t+1(V))>0 implies that φ _t+1(V)>0, thus \(E\left [\sum _{i=t+1}^{T} V_{i}|\mathcal {F}_{t+1}\right ]>0\). Hence, on the set {V _t ≥0}, we have

$$E\left[ \sum_{i=t}^{T} V_{i} \Big|\mathcal{F}_{t} \right] \geq E\left[E\left[\sum_{i=t+1}^{T}V_{i}\Big|\mathcal{F}_{t+1}\right]\Big|\mathcal{F}_{t}\right]>0. $$

We conclude the proof by combining the last inequality with (45).

¹ An LM-measure is a function that is local and monotone; see Definition 1. These two properties have to be satisfied by any reasonable dynamic measure of performance and/or measure of risk, and are shared by most such measures in the existing literature.

² In the original paper (Artzner et al. 1999), the authors considered finite probability spaces, but later the theory was elevated to a general probability space (Delbaen 2000; 2002).

³ In (Riedel 2004), the author considered discounted dividend processes, but for simplicity here we write the time consistency for random variables.

⁴ See Appendix “Robust representations for dynamic monetary utility measures” for the definition of minimal penalty functions, up to a sign, and for the corresponding robust representations.

⁵ In the present manuscript, we also use the name ‘update rules’, although the concept used here is different from that introduced in (Tutsch 2008).

⁶ Unless otherwise specified, it will be understood in the rest of the paper that p∈{0,1,∞}.

⁷ Most of the concepts discussed in this Section can be modified to deal with the case of stochastic processes, as we will do in Section “Time consistency for stochastic processes”.

⁸ See Section “Dynamic LM-measures” for details.

⁹ By \(\mathcal {F}_{t}\)-convex we mean that for any \(Z_{1},Z_{2}\in \mathcal {D}_{t}\), and \(\lambda \in L^{0}_{t}\) such that 0≤λ≤1 we get \(\lambda Z_{1}+(1-\lambda)Z_{2}\in \mathcal {D}_{t}\).

¹⁰ The term robust is inspired by robust representations of risk measures.

¹¹ See Appendix “LM-extensions” for the definition of φ ⁻

¹² We recall that the elements of \(\mathbb {V}^{p}\) are interpreted as discounted dividend processes.

¹³ In (Bielecki et al. 2014b), the authors combine both semi-weak acceptance and rejection time consistency into one single definition and call it time consistency.

¹⁴ A family, indexed by \(x\in \mathbb {R}_{+}\), of maps \(\{\varphi _{t}^{x}\}_{t\in \mathbb {T}}\), is called decreasing, if \(\varphi _{t}^{x}(X)\leq \varphi _{t}^{y}(X)\) for all \(X\in \mathcal {X}\), \(t\in \mathbb {T}\) and \(x,y\in \mathbb {R}_{+}\), such that x≥y.

¹⁵ See Appendix “Dynamic LM-measures” for details.

¹⁶ That is closed with respect to topology η; if η will be clear from the context, we will simply write that f is lower semi-continuous. If \(\mathcal {X}= L^{p}\), then we use the topology induced by ∥·∥ _p norm (see (Föllmer and Schied 2004, Appendix A.7) for details).

¹⁷ This means that there exist \(Y\in \mathcal {X}\) such that for all \(n\in \mathbb {N}\) we have |X _n |≤|Y|.

¹⁸ That is, it satisfies monotonicity and locality on \(\bar {L}_{0}\), as in 5) and 6) in Proposition 16.

¹⁹ We will use the convention ess sup ∅=−∞ and ess inf ∅=∞.

²⁰ That is, \(\mathcal {F}_{t}\)-local and monotone on \(\mathcal {B}\).

Here we provide a brief exposition of the three fundamental concepts used in the paper: the dynamic LM-measures, the conditional essential suprema/infima, and the LM-extensions.

LM-extensions

In this part of the appendix, we introduce the concept of an LM-extension of a dynamic LM-measure for random variables.

Definition 16

Let φ be a dynamic LM-measure on L ^p . We call a family \(\widehat {\varphi }=\{\widehat {\varphi }_{t}\}_{t\in \mathbb {T}}\) of maps \(\widehat {\varphi }_{t}:\bar {L}^{0}\to \bar {L}^{0}_{t}\) an LM-extension of φ, if for any \(t\in \mathbb {T}\), \(\widehat {\varphi }_{t}|_{\mathcal {X}}\equiv \varphi _{t}\), and \(\widehat {\varphi }_{t}\) is local and monotone on \(\bar {L}_{0}\).¹⁸

We will show below that such extension exists, for which we will make use of the following auxiliary sets:

defined for any \(X\in \bar {L}^{0}\) and \(A\in \mathcal {F}\).

Definition 17

Let φ be a dynamic LM-measure. The collection of functions \(\varphi ^{+}=\{\varphi _{t}^{+}\}_{t\in \mathbb {T}}\), where \(\varphi ^{+}_{t}:\bar {L}^{0}\to \bar {L}^{0}_{t}\) is defined as¹⁹

is called the upper LM-extension of φ. Respectively, the collection of functions \(\varphi ^{-}=\{\varphi _{t}^{-}\}_{t\in \mathbb {T}}\), where \(\varphi ^{-}_{t}:\bar {L}^{0}\to \bar {L}^{0}_{t}\), and

is called the lower LM-extension of φ.

The next result shows that φ ^± are two “extreme” extensions, and any other extension is sandwiched between them.

Proposition 17

Let φbe a dynamic LM-measure. Then, φ ⁻ and φ ⁺ are LM-extensions of φ. Moreover, let \(\widehat {\varphi }\) be an LM-extension of φ. Then, for any \(X\in \bar {L}^{0}\) and \(t\in \mathbb {T}\),

$$ \varphi_{t}^{-}(X)\leq\widehat{\varphi}_{t}(X)\leq \varphi_{t}^{+}(X). $$

(56)

Clearly, in general, the maps (54) and (55) are not equal, and thus the extensions of an LM-measure are not unique.

Remark 12

Let \(t\in \mathbb {T}\) and \(\mathcal {B}\subseteq \bar {L}^{0}\) be such that, for any \(A\in \mathcal {F}_{t}\), and As a generalization of Proposition 17, one can show that for any \(\mathcal {F}_{t}\)-local and monotone mapping²⁰ \(f:\mathcal {B}\to \bar {L}_{t}^{0}\), the maps f ^± defined analogously as in (54) and (55) are extensions of f to \(\bar {L}^{0}\), preserving locality and monotonicity.

Remark 13

For a large class of LM-measures, as mentioned earlier, there exists a “robust representation” type theorem—essentially a representation, via convex duality, as a function of conditional expectation. We refer the reader to (Bielecki et al. 2016) and references therein, where the authors present a general robust representation for dynamic quasi-concave upper semi-continuous LM-measures. Hence, an alternative construction of extensions can be obtained through the robust representations of LM-measures, by considering conditional expectations defined on the extended real number line, etc.