A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LMmeasure perspective
 Tomasz R. Bielecki^{1}Email author,
 Igor Cialenco^{1} and
 Marcin Pitera^{2}
DOI: 10.1186/s4154601700129
© The Author(s) 2017
Received: 30 March 2016
Accepted: 18 December 2016
Published: 1 March 2017
Abstract
In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LMmeasure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a unified framework.
Keywords
Time consistency Update rule Dynamic LMmeasure Dynamic risk measure Dynamic acceptability index Measure of performanceMSC2010
91B30 62P05 97M30 91B06“The dynamic consistency axiom turns out to be the heart of the matter.” A. Jobert and L. C. G. Rogers
Valuations and dynamic convex risk measures, Math Fin 18(1), 2008, 1–22.
Introduction
The goal of this work is to give a comprehensive overview of the time consistency property of dynamic risk and performance measures. We focus on discrete time setup, since most of the existing literature on this topic is dedicated to this case.
The time consistency surveyed in this paper is related to dynamic decision making subject to various uncertainties that evolve in time. Typically, decisions are made subject to the decision maker’s preferences, which may change in time and thus they need to be progressively assessed as an integral part of the decision making process. Naturally, the assessment of preferences should be done in such a way that the future preferences are assessed consistently with the present ones. This survey is focusing on this aspect of time consistency of a dynamic decision making process.
Traditionally, in finance and economics, the preferences are aimed at ordering cash and/or consumption streams. A convenient way to study preferences is to study them via numerical representations, such as (dynamic) risk measures, (dynamic) performance measures, or, more generally, dynamic LMmeasures^{1} (Bielecki et al. 2014a). Consequently, the study of the time consistency of preferences is also conveniently done in terms of their numerical representations. This work is meant to survey various approaches to modelling and analysis of the time consistency of numerical representations of preferences.
As stated above, the objects of our survey—the dynamic LMmeasures—are meant to “put a preference order” on the sets of underlying entities. There exists a vast literature on the subject of preference ordering, with various approaches towards establishing an order of choices, such as the decision theory or the expected utility theory, that trace their origins to the mid 20th century. We focus our attention, essentially, on the axiomatic approach to defining risk or performance measures.
The axiomatic approach to measuring risk of a financial position was initiated in the seminal paper by Artzner et al. (1999), and has been going through a flourishing development since then. The measures of risk introduced in (Artzner et al. 1999), called coherent risk measures, were meant to determine the regulatory capital requirement by providing a numerical representation of the riskiness of a portfolio of financial assets. In this framework, from mathematical point of view, the financial positions are understood as either discounted terminal values (payoffs) of portfolios, that are modeled in terms of random variables, or they are understood as discounted dividend processes, cumulative or bullet, that are modeled as stochastic processes. Although stochastic processes can be viewed as random variables (on appropriate spaces), and vice versa  random variables can be treated as particular cases of processes—it is convenient, and in some instances necessary, to treat these two cases separately—the road we are taking in this paper.
In the paper (Artzner et al. 1999), the authors considered the case of random variables, and the risk measurement was done at time zero only. This amounts to considering a one period time model in the sense that the measurement is done today of the cash flow that is paid at some fixed future time (tomorrow). Accordingly, the related risk measures are referred to as static measures. Since then, two natural paths were followed: generalizing the notion of risk measure by relaxing or changing the set of axioms, or/and considering a dynamic setup. By dynamic setup we mean that the measurements are done throughout time and are adapted to the flow of available information. In the dynamic setup, both discrete and continuous time evolutions have been studied, for both random variables and stochastic processes as the inputs. In the present work, we focus our attention on the discrete time setup, although we briefly review the literature devoted to continuous time.
This survey is organized as follows. We start with the literature review relevant to the dynamic risk and performance measures focusing on the time consistency property in the discrete time setup. In Section “Mathematical preliminaries”, we set the mathematical scene; in particular, we introduce the main notations used in this paper and the notion of LMmeasures. Section “Approaches to time consistent assessment of preferences” is devoted to the time consistency property. There we discuss two generic approaches to time consistent assessment of preferences and point out several idiosyncratic approaches. We put forth in this section the notion of an update rule that, we believe, is the key tool for studying time consistency in a unified framework. Sections “Time consistency for random variables” and “Time consistency for stochastic processes” survey some concepts and results regarding time consistency in the case of random variables and in the case of stochastic processes, respectively. Our survey is illustrated by numerous examples that are presented in Section “Examples”. We end the survey with two appendices. In “Appendix” we provide a brief exposition of the three fundamental concepts used in the paper: the dynamic LMmeasures, the conditional essential suprema/infima, and LMextensions. Finally, in Appendix “Proofs” we collect proofs of several results stated throughout our survey.
Literature review
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 σalgebras \(\mathcal {F}_{t}, \ t\geq 0,\) 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 mstable 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 multiprior 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 subadditivity and positive homogeneity properties with convexity. In the continuous time setup, Rosazza Gianin (2002) links dynamic convex risk measures to nonlinear expectations or gexpectations, 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 gexpectations. 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 zeroone 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 cocycle 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 BionNadal (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 timeconsistent 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.
BionNadal (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 cocycle condition for the penalty function and shows its equivalence to strong time consistency. In the followup paper, (BionNadal 2008), the author continues to study the characterization of time consistency in terms of the cocycle condition for minimal penalty function. For further related developments in the continuous time framework see (BionNadal 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 rules^{5} 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 riskreward optimization problems and in (Cherny 2009) to capital allocation; BionNadal (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 gexpectation 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 (BionNadal and Kervarec 2012) and (BionNadal 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 Gexpectations.
From a slightly different point of view, Ruszczynski (2010) studies Markov risk measures, that enjoy strong time consistency, in the framework of riskaverse 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 subadditive 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 setvalued 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 cashadditivity 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, quasiconcave, 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 cashadditive. 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 BionNadal (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 subscale 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 quasiconcave maps that covers both dynamic risk measures and dynamic acceptability indices, see (Frittelli and Maggis 2011; Bielecki et al. 2016; Frittelli and Maggis 2014; BionNadal 2016). Also in (Bielecki et al. 2016), the authors study the strong time consistency of quasiconcave 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).

random variables, strong time consistency: (discrete time) (BionNadal 2006; Boda and Filar 2006; Cheridito and Stadje 2009; Detlefsen and Scandolo 2005; Föllmer and Penner 2006; Frittelli and Scandolo 2006; Geman and Ohana 2008; Ruszczyński and Shapiro 2006a; Scandolo 2003), (Acciaio and Penner 2011; Acciaio et al. 2012; Bielecki et al. 2014a; Bielecki et al. 2016; BionNadal 2008; Cheridito and Kupper 2009; Cohen and Elliott 2010, 2011; Elliott et al. 2015; Fasen and Svejda 2012; Iancu et al. 2015; Kupper and Schachermayer 2009; Mastrogiacomo and Rosazza Gianin 2015; Roorda and Schumacher 2015; Stadje 2010);
(continuous time) (Barrieu and El Karoui 2004, 2007; BionNadal 2006, 2008, 2009b; BionNadal and Kervarec 2012; Delbaen 2012; Delbaen et al. 2010; Frittelli and Rosazza Gianin 2004; Jiang 2008; Klöppel and Schweizer 2007; Nutz and Soner 2012; Penner and Réveillac 2014; Rosazza Gianin 2002, 2006; Sircar and Sturm 2015).

random variables, supermartingale time consistency: (Scandolo 2003; Detlefsen and Scandolo 2005).

random variables, acceptance/rejection time consistency: (Acciaio et al. 2012; Bielecki et al. 2014a; Föllmer and Penner 2006; Roorda and Schumacher 2007, 2015; Tutsch 2008; Weber 2006).

stochastic processes, strong and supermartingale time consistency: (discrete time) (Bielecki et al. 2014a; Scandolo 2003), (continuous time) (Jobert and Rogers 2008)
Dynamic monetary risk measures, strong time consistency:
(discrete time) (Cheridito and Kupper (2011); Cheridito et al. (2006)); (continuous time) (BionNadal 2004; Föllmer and Penner 2011).
Dynamic acceptability indices: (Bielecki et al. (2014b); Biagini and BionNadal (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)).
Mathematical preliminaries
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 \(\mathbb {T}'=\{0,1,\ldots, T1\}\).
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 LMmeasures”).
Remark 1
We note that the space \(\mathbb {V}^{p}\), endowed with the multiplication ·_{ t }, does not define a proper L ^{0}–module (Filipovic et al. 2009; Vogelpoth 2009) (e.g., in general, 0·_{ t } V≠0). However, in what follows, we will adopt some concepts from L ^{0}module theory, which naturally fit into our study. We refer the reader to (Bielecki et al. 2015a, 2016) for a thorough discussion on this matter.
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.
Definition 1
 1)
 2)
(Monotonicity) X≤Y⇒φ _{ t }(X)≤φ _{ t }(Y);
for any \(t\in \mathbb {T}\), \(X,Y\in \mathcal {X}\) and \(A\in \mathcal {F}_{t}\).
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 zeroone law, or relevance) essentially means that the values of the LMmeasure 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.
Remark 2
While in most of the literature the axiom of locality is not stated directly, it is very often implied by other assumptions. For example, if \(\mathcal {X}=L^{\infty }\), then monotonicity and cashadditivity imply locality (cf. (Pitera 2014), Proposition 2.2.4). Similarly, any convex (or concave) map is also local (cf. (Detlefsen and Scandolo 2005)). It is also worth mentioning that locality is strongly related to time consistency. In fact, in some papers locality is considered as a part of the time consistency property discussed below (see e.g. (Detlefsen and Scandolo 2009)).
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 LMmeasure 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 LMmeasure as to dynamic (monetary) utility measure rather than as dynamic (monetary) risk measure. See Appendix “Dynamic LMmeasures” for details.
Approaches to time consistent assessment of preferences
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 LMmeasures.
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 LMmeasures”), 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 LMmeasures, 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 LMmeasure at different times.
Definition 2
 1)
 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
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 onestep μacceptance (resp. onestep μrejection) 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 LMmeasure, in general, depends on the choice of the updated rule; we refer to Section “Time consistency for random variables” for an indepth discussion.
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 LMmeasure φ, then this measure is μacceptance 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
 1)
sinvariant, 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)
projective, if it is sinvariant 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 sinvariant, 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
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
 (i)A family \(\mathcal {Y}=\{\mathcal {Y}_{t}\}_{t\in \mathbb {T}}\) of sets \(\mathcal {Y}_{t}\subseteq \mathcal {X}\) is a benchmark family iffor any \(t\in \mathbb {T}\).$$0\in\mathcal{Y}_{t}\quad \mathrm{and }\quad \mathcal{Y}_{t}+\mathbb{R}=\mathcal{Y}_{t}, $$
 (ii)A dynamic LMmeasure 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
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 LMmeasure 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 μtime 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.
Idiosyncratic approaches
Each such approach to time consistency of a given LMmeasure exploits the idiosyncratic properties of this LMmeasure, which are not necessarily shared by other LM measures, and typically is suited only for a specific subclass of dynamic LMmeasures. For example, in case of dynamic convex or monetary risk measures the time consistency can be characterized in terms of the relevant properties of associated acceptance sets and/or the dynamics of the penalty functions and/or the rectangular property of the families of probability measures. These idiosyncratic approaches, and the relevant references, were mentioned and briefly discussed in Section “Literature review”. Detailed analysis of each of these approaches is beyond the scope of this survey.
Time consistency for random variables
In this section, we survey the time consistency of LMmeasures 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”.
Weak time consistency
The main idea behind this type of time consistency is that if “tomorrow”, say at time s, we accept X∈L ^{ p } at level φ _{ s }(X), then “today”, say at time t, we would accept X at any level less than or equal to φ _{ s }(X), adjusted by the information \(\mathcal {F}_{t}\) available at time t. Similarly, if tomorrow we reject X at level φ _{ s }(X), then today, we should also reject X at any level greater than or equal to φ _{ s }(X), adapted to the information \(\mathcal {F}_{t}\).
Definition 6
Propositions 1 and 2 provide some characterizations of weak acceptance time consistency.
Proposition 1
 1)
φ is weakly acceptance time consistent.
 2)φ is μacceptance time consistent, where μ is a projective update rule, given by$$\mu_{t}(m)=\mathrm{ess\,inf}_{t}m. $$
 3)The following inequality is satisfied$$ \varphi_{t}(X)\geq \underset{Q\in\mathcal{M}_{t}(P)}{\mathrm{ess\,inf}}E_{Q}[\varphi_{s}(X)\mathcal{F}_{t}], $$(15)
for any X∈L ^{ p }, \(s,t\in \mathbb {T}\), s>t.
 4)For any X∈L ^{ p }, \(s,t\in \mathbb {T}\), s>t, and \(m_{t}\in \bar {L}^{0}_{t}\), it holds that$$\varphi_{s}(X)\geq m_{t} \Rightarrow \varphi_{t}(X)\geq m_{t}. $$
Similar results hold true for weak rejection time consistency.
It is worth mentioning that Property 4) in Proposition 1 was suggested as the notion of (weak) acceptance and (weak) rejection time consistency in the context of scale invariant measures, called acceptability indices (cf. (Biagini and BionNadal 2014; Bielecki et al. 2014b)).
Usually, the weak time consistency is considered for dynamic monetary risk measures on L ^{ ∞ } (cf. (Acciaio and Penner 2011) and references therein). This case lends itself to even more characterizations of this property.
Proposition 2
 1)
φ is weakly acceptance time consistent.
 2)
φ is acceptance time consistent with respect to \(\{\mathcal {Y}_{t}\}_{t\in \mathbb {T}}\), where \(\mathcal {Y}_{t}=\mathbb {R}\).
 3)For any X∈L ^{ p } and \(s,t\in \mathbb {T}\), s>t,$$ \varphi_{s}(X)\geq 0 \Rightarrow \varphi_{t}(X)\geq 0. $$(16)
 4)
\(\mathcal {A}_{t+1}\subseteq \mathcal {A}_{t}\), for any \(t\in \mathbb {T}\), such that t<T.
 5)For any \(Q\in \mathcal {M}(P)\) and \(t\in \mathbb {T}\), such that t<T,where α ^{min} is the minimal penalty function in the robust representation of φ.$$\alpha^{\min}_{t}(Q)\geq E_{Q}[\alpha^{\min}_{t+1}(Q)\,\, \mathcal{F}_{t}], $$
Analogous results are obtained for weak rejection time consistency.
We note that equivalence of properties 1), 2), and 3) also holds true in the case of \(\mathcal {X}=L^{0}\), and not only for representable, but for any dynamic monetary utility measure; for the proof, see (Bielecki et al. 2014a, Proposition 4.3). Property 4) is a characterisation of weak time consistency in terms of acceptance sets, and property 5) gives a characterisation in terms of the supermartingale property of the penalty function. For the proof of the equivalence of 3), 4), and 5), see (Acciaio and Penner 2011, Proposition 33).
The next result shows that weak time consistency is indeed one of the weakest forms of time consistency, in the sense that the weak time consistency is implied by any time consistency generated by a projective update rule; we refer to (Bielecki et al. 2014a, Proposition 4.5) for the proof.
Proposition 3
Let φ be a dynamic LMmeasure on L ^{ p }, and let μ be a projective update rule. If φ is μacceptance (resp. μrejection) time consistent, then φ is weakly acceptance (resp. weakly rejection) time consistent.
Remark 4
An important feature of the weak time consistency is its invariance with respect to monotone transformations. Specifically, let \(g:\bar {\mathbb {R}}\to \bar {\mathbb {R}}\) be a strictly increasing function and let φ be a weakly acceptance/rejection time consistent dynamic LMmeasure. Then, \(\{g\circ \varphi _{t}\}_{t\in \mathbb {T}}\) is also a weakly acceptance/rejection time consistent dynamic LMmeasure.
Remark 5
In the case of general LMmeasures, the weak time consistency may not be characterized as in 2) of Proposition 2. For example, if φ is a (normalized) acceptability index, then \(\varphi _{t}(\mathbb {R})=\{0,\infty \}\), for \(t\in \mathbb {T}\), which does not agree with 4) in Proposition 1.
Strong time consistency
As already stated in the Introduction, the origins of the strong form of time consistency can be traced to (Koopmans 1960). Historically, this is the first and the most extensively studied form of time consistency in dynamic risk measures literature. It is fair to mention, that this form of time consistency also appears in the insurance literature, as the iterative property, and it is related to the mean value principle (Gerber 1974; Goovaerts and De Vylder 1979).
We start with the definition of strong time consistency.
Definition 7
for any X,Y∈L ^{ p } and \(s,t\in \mathbb {T}\), such that s>t.
Strong time consistency gains its popularity and importance due to its equivalence to the dynamic programming principle. This equivalence, as well as other characterisations of strong time consistency, are the subject of the following two propositions.
Proposition 4
 1)
φ is strongly time consistent.
 2)
There exists an update rule μ such that φ is both μacceptance and μrejection time consistent.
 3)
φ is acceptance time consistent with respect to \(\{\mathcal {Y}_{t}\}_{t\in \mathbb {T}}\), where \(\mathcal {Y}_{t}=L^{p}\).
 4)There exists an update rule μ such that for any X∈L ^{ p }, \(s,t\in \mathbb {T}\), s>t,$$ \mu_{t,s}(\varphi_{s}(X))=\varphi_{t}(X). $$(18)
 5)There exists a onestep update rule μ such that for any X∈L ^{ p }, \(t\in \mathbb {T}\), t<T,$$\mu_{t,t+1}(\varphi_{t+1}(X))=\varphi_{t}(X). $$
See Appendix “Proofs” for the proof of Proposition 4. Property 4) in this proposition is referred to as Bellman’s principle or the dynamic programming principle. Also, note that 5) implies that any strongly time consistent dynamic LMmeasure can be constructed using a backward recursion starting from φ _{ T }:=ϱ, where ϱ is an LMmeasure. See (Cheridito and Kupper 2011) where the recursive construction for dynamic risk measures is discussed in details.
An important, and frequently studied, type of strong time consistency is the strong time consistency for dynamic monetary risk measures on L ^{ ∞ } (cf. (Acciaio and Penner 2011) and references therein). As the next result shows, there are more equivalences that are valid in this case.
Proposition 5
 1)
φ is strongly time consistent.
 2)φ is recursive, i.e., for any X∈L ^{ p }, \(s,t\in \mathbb {T}\), s>t,$$\varphi_{t}(X)= \varphi_{t}(\varphi_{s}(X)). $$
 3)
\(\mathcal {A}_{t}=\mathcal {A}_{t,s}+\mathcal {A}_{s}\), for all \(t,s\in \mathbb {T}\), s>t.
 4)For any \(Q\in \mathcal {M}(P)\), \(t,s\in \mathbb {T}\), s>t,$$\alpha_{t}^{\min}(Q)=\alpha_{t,s}^{\min}(Q)+E_{Q}[\alpha^{\min}_{s}(Q)\,\, \mathcal{F}_{t}]. $$
 5)For any X∈L ^{ p }, \(Q\in \mathcal {M}(P)\), \(s,t\in \mathbb {T}\), s>t,$$\varphi_{t}(X)\alpha^{\min}_{t}(Q)\leq E_{Q}[\varphi_{s}(X)\alpha^{\min}_{s}(Q)\,\, \mathcal{F}_{t}]. $$
For the proof see, for instance, (Acciaio and Penner 2011, Proposition 14).
Remark 6

(i) In general, for dynamic LMmeasures, the strong time consistency does not imply either the weak acceptance or weak rejection time consistency. Indeed, let us consider \(\varphi =\{\varphi _{t}\}_{t\in \mathbb {T}}\), such that φ _{ t }(X)=t (resp. φ _{ t }(X)=−t) for all X∈L ^{0}. Since φ _{ t }(0)=t≧̸ess inf_{ t } φ _{ s }(0)=s (resp. −t≦̸−s), for s>t, we conclude that φ is not weakly acceptance (resp. weakly rejection) time consistent. However, since φ _{ t }(X)=φ _{ t }(φ _{ s }(X)) for any X∈L ^{0}, then φ is strongly time consistent. We note, that if the update rule in Definition 7 is projective, as it is usually the case for dynamic monetary risk measures, then, due to Proposition 3, the strong time consistency implies the weak time consistency.

(ii) It is worth mentioning that, in principle, strong time consistency is not suited for acceptability indices (Bielecki et al. 2014b, 2015b; Cherny and Madan 2009). Let φ be a scale invariant dynamic LMmeasure, and let \(A\in \mathcal {F}_{s}\) be such that P[A]=1/2, for some s>0, \(s\in \mathbb {T}\). Additionally, assume that \(\mathcal {F}_{0}\) is trivial. We consider the sequence of random variables By locality and scale invariance of φ, we have that φ _{ s }(X _{ n })=φ _{ s }(X _{1}), for \(n\in \mathbb {N}\). If φ is strongly time consistent, then we also have that φ _{0}(X _{ n })=φ _{0}(X _{1}), \(n\in \mathbb {N}\). On the other hand, any reasonable measure of performance should assess X _{ n } at the higher level as n increases, which contradicts the fact that φ _{0}(X _{ n }) is a constant sequence.
Robust expectations, submartingales, and supermartingales
The concept of a projective update rule is connected with the concept of the (conditional) nonlinear expectation (see, for instance, (Peng 1997) for the definition and properties of nonlinear expectation). In (Peng 2004; Rosazza Gianin 2006), the authors established a link between nonlinear expectations and dynamic risk measures. One particularly important example of an projective update rule is the standard conditional expectation operator. Time consistency in L ^{ ∞ } framework, defined in terms of conditional expectation, was studied in (Detlefsen and Scandolo 2005, Section 5) and associated with the super(sub)martingale property.
The next result introduces a general class of updates rules that are generated by conditional expectations and determining families of sets. First, we recall the concept of the determining family of sets (see, for instance, (Cherny 2006) for more details).
Proposition 6
 1)
the family ϕ is a projective update rule;
 2)
if φ is ϕacceptance time consistent, then \(\{g\circ \varphi _{t}\}_{t\in \mathbb {T}}\) is also ϕacceptance time consistent, for any increasing and concave function \(g:\bar {\mathbb {R}}\to \mathbb {R}\).
Remark 7
Classical (static) coherent risk measures defined on L ^{ ∞ } admit robust representation of the form (1) for some set of probability measures \(\mathcal {Q}\). It is known that the set \(\mathcal {Q}\) might not be unique. Consequently, there may exist multiple extensions of ρ to a map defined on \(\bar {L}^{0}\) (see Appendix “LMextensions” for the concept of the extension). Nevertheless, as in (Cherny 2006), one can consider the maximal set \(\mathcal {D}\) called determining set of a risk measure, which guarantees the uniqueness of such extension. The family of maps defined in (19) is an example of a family of such extensions. Consequently, we see that the coherent risk measures constitute a good starting point for generation of update rules.
For the proof of Proposition 6, see Appendix “Proofs”. The counterpart of Proposition 6 for rejection time consistency is obtained by taking ess sup instead of ess inf in (19), and assuming that g is convex.
In the particular case of determining family with \(\mathcal {D}_{t}=\{1\}\), for any \(t\in \mathbb {T}\), the projective update rule takes the form \(\mu _{t}(m)=E[m\mathcal {F}_{t}]\), \(m\in \bar {L}^{0}\). This is an important case, as it produces the concept of supermartingale and submartingale time consistency.
Definition 8
Remark 8

(i) Note that any dynamic LMmeasure that is ϕacceptance time consistent, where ϕ is given in (19), is also weakly acceptance time consistent, as ϕ is projective. In particular, any supermartingale time consistent LMmeasure is also weakly acceptance time consistent. A similar statement holds true for rejection time consistency.

(ii) As mentioned in (Bielecki et al. 2014a), the idea of update rules might be used to weight the preferences. Intuitively speaking, the risk of loss in the far future might be more preferred than the imminent risk of loss. This idea was used in (Cherny 2010). For example, the update rule μ of the form$$ \mu_{t,s}(m,X)=\left\{ \begin{array}{ll} \alpha^{st} E[m\mathcal{F}_{t}] & \text{on} ~\{E[m\mathcal{F}_{t}] \geq 0\},\\ \alpha^{ts} E[m\mathcal{F}_{t}] & \text{on}~ \{E[m\mathcal{F}_{t}] < 0\}. \end{array}\right. $$(20)
for a fixed α∈(0,1) would achieve this goal.
Other types of time consistency
The weak, strong, and super/submartingale forms of time consistency have attracted the most attention in the existing literature. In this section, we present other forms of time consistency that have been studied.
Middle time consistency
The notion of middle time consistency was originally formulated for dynamic monetary risk measures on L ^{ ∞ } (cf. (Acciaio and Penner 2011)). The main idea is to replace the equality in (17) by an inequality. The term middle acceptance or middle rejection is used depending on the direction of the inequality.
Definition 9
The middle acceptance (resp. middle rejection) time consistency is equivalent to the acceptance (resp. rejection) time consistency with respect to the benchmark family \(\mathcal {Y}=\{\mathcal {Y}_{t}\}_{t\in \mathbb {T}}\), given by \(\mathcal {Y}_{t}=L^{p}\cap {L}^{0}_{t}\). In the case of dynamic convex risk measures, other characterizations of middle acceptance time consistency are available, as the following proposition shows.
Proposition 7
 1)
φ is middle acceptance time consistent.
 2)
φ is φ ^{−}acceptance time consistent.^{11}
 3)For any X∈L ^{ p }, \(s,t\in \mathbb {T}\), s>t,$$ \varphi_{t}(X)\geq \varphi_{t}(\varphi_{s}(X)). $$(21)
 4)For any X∈L ^{ p } and \(t\in \mathbb {T}\), such that t<T,$$\varphi_{t+1}(X)\varphi_{t}(X)\in\mathcal{R}_{t,t+1}. $$
 5)For any \(X\in \mathcal {R}_{t}\) and \(t\in \mathbb {T}\), such that t<T,$$\varphi_{t+1}(X)\in\mathcal{R}_{t}. $$
 6)
For any \(t\in \mathbb {T}\), such that t<T, \( \mathcal {A}_{t}\supseteq \mathcal {A}_{t,t+1}+\mathcal {A}_{t+1}\).
 7)For any \(Q\in \mathcal {M}(P)\) and \(t\in \mathbb {T}\), such that t<T,$$\alpha_{t}^{\min}(Q) \geq \alpha_{t,t+1}^{\min}(Q)+E_{Q}[\alpha_{t+1}^{\min}(Q)  \mathcal{F}_{t}]. $$
 7)For any \(Q\in \mathcal {M}(P)\) and \(t\in \mathbb {T}\), such that t<T,$$\varphi_{t}(X)\geq E_{Q}[\varphi_{t+1}(X)\mid\mathcal{F}_{t}]+\alpha_{t,t+1}^{\min}(Q). $$
Since φ ^{−} is an LMextenstion of φ, and φ _{ s }(Y)=Y, for any \(Y\in L^{p} \cap \bar {L}^{0}_{s}\), the equivalence between 1) and 2) is immediate. For all other equivalences see (Acciaio and Penner 2011, Section 4.2) and references therein. Property 1) in Proposition 7 is sometimes called prudence (see (Penner 2007)).
Time consistency induced by LMmeasure
It turns out that any dynamic LMmeasure generates an update rule. Indeed, as the next result shows, any LMextension of an LMmeasure (see Appendix “LMextensions” for the definition of LMextension) is an sinvariant update rule.
Proposition 8
Any LMextension \(\widehat {\varphi }\) of a dynamic LM–measure φ is an sinvariant update rule. Moreover, \(\widehat {\varphi }\) is projective if and only if φ _{ t }(X)=X, for \(t\in \mathbb {T}\) and \(X\in L^{p}\cap \bar {L}^{0}_{t}\).
The proof is deferred to Appendix “Proofs”.
LMextensions may be used to give stronger forms of strong and middle time consistency, that are especially well suited in the case of dynamic monetary risk measures.
where \(\hat {\varphi }\) is an extension of φ from \(\mathcal {X}\) to \(\bar {L}^{0}\). Accordingly, we say that φ is strongly ^{∗} time consistent, if there exists an LMextension \(\hat {\varphi }\), of φ, such that φ is both \(\hat {\varphi }\)acceptance and \(\hat {\varphi }\)rejection time consistent.
Note that since \(\hat {\varphi }\) is an update rule, the strong ^{∗} time consistency implies strong time consistency in the sense of Definition 7. In general, the converse implication is not true; to see this, it is enough to consider strong time consistency for an update rule that is not sinvariant.
In the same fashion, we say that φ is middle ^{∗} acceptance time consistent, if there exists an LMextension of φ, say \(\hat {\varphi }\), such that φ is \(\hat {\varphi }\)acceptance time consistent. In view of Proposition 17, this is equivalent to saying that φ is middle ^{∗} acceptance time consistent if it is φ ^{−}acceptance time consistent. Likewise, to define middle ^{∗} rejection time consistency we use the mapping φ ^{+}.
Taxonomy of results
 ■■■
Proposition 2, 2)
 ■■■
 ■■■
 ■■■
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 \(\mu _{t}=E_{t}[m\mathcal {F}_{t}]\).
 ■■■
Proposition 4, 4). The converse implication is not true in general. For the counterexample, see (Acciaio and Penner 2011, Proposition 37).
 ■■■
Proposition 3, and see also . In general, strong time consistency does not imply weak acceptance time consistency, see Remark 6.
 ■■■
 ■■■
 ■■■
Time consistency for stochastic processes
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 LMmeasure, 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.
Definition 10
The family \(\mu =\{\mu _{t,s}:\, t,s\in \mathbb {T},\, t<s\}\) of maps \(\mu _{t,s}:\bar {L}^{0}_{s}\times \mathcal {X}\to \bar {L}^{0}_{t}\) is called a generalized update rule if for any \(X\in \mathcal {X}\) the family \(\mu (\cdot,X)=\{\mu _{t,s}(\cdot,X):\, t,s\in \mathbb {T},\, t<s\}\) is an 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 sinvariant, 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 timeconsistency.
Definition 11
for all \(s,t\in \mathbb {T}\), s>t, \(X\in \mathcal {X}\), and \(m_{s}\in \bar {L}^{0}_{s}\). In particular, if property (23) is satisfied for s=t+1, t=0,…,T, then we say that φis onestep μacceptance (resp. onestep μrejection) time consistent.
where \(\tilde \mu \) is the onestep 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 onestep update rules of form (24), the onestep time consistency for random variables is a particular case of onestep 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 μacceptance (resp. μrejection) time consistency is equivalent to one step μacceptance (resp. μrejection) time consistency. This is another reason why we consider only one step update rules for stochastic processes.
Weak time consistency
We start with the following definition.
Definition 12
The next result is the counterpart of Proposition 1 and Proposition 2.
Proposition 9
 1)
φ is weakly acceptance time consistent.
 2)φ is μacceptance time consistent, where μ is an sinvariant update rule, given by$$\mu_{t}(m,V)=ess\,{inf}_{t}m+V_{t}. $$
 3)For any \(V\in \mathbb {V}^{p}\) and t<T$$ \varphi_{t}(V)\geq \underset{Q\in\mathcal{M}_{t}(P)}{\mathrm{ess\,inf}}~E_{Q}[\varphi_{t+1}(V)\mathcal{F}_{t}]+V_{t}. $$(26)
 4)For any \(V\in \mathbb {V}^{p}\), t<T, and \(m_{t}\in \bar {L}^{0}_{t}\),$$\varphi_{t+1}(V)\geq m_{t} \Rightarrow \varphi_{t}(V)\geq m_{t}+V_{t}. $$
 5)For any \(V\in \mathbb {V}^{p}\) and t<T,$$\varphi_{t+1}(V)\geq 0 \Rightarrow \varphi_{t}(V)\geq V_{t}. $$
Analogous equivalences are true for weak rejection time consistency.
The proof of Proposition 9 is analogous to the proofs of Proposition 1 and Proposition 2, and we omit it.
As mentioned earlier, the update rule, and consequently time consistency for stochastic processes, depends also on the value of the process (the dividend paid) at time t. In the case of weak time consistency this feature is interpreted as follows: if tomorrow, at time t+1, we accept \(V\in \mathbb {V}^{p}\) at the level greater than \(m_{t+1}\in \mathcal {F}_{t+1}\), then today at time t, we will accept V at least at the level ess inf_{ t } m _{ t+1} (i.e., the worst level of m _{ t+1} adapted to the information \(\mathcal {F}_{t}\)) plus the dividend V _{ t } received today.
Finally, we present the counterpart of Proposition 3 for the case of stochastic processes.
Proposition 10
If φis a dynamic onestep LMmeasure on \(\mathbb {V}^{p}\), which is μacceptance (resp. μrejection) time consistent, then φ is weakly acceptance (resp. weakly rejection) time consistent.
Proposition 10 can be proved in a way analogous to the proof of Proposition 3.
Remark 9
Semiweak time consistency
In this section, we introduce the concept of semiweak time consistency for stochastic processes. We have not discussed semiweak time consistency in the case of random variables, since, in that case, semiweak 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 scaleinvariant 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 semiweak acceptance and semiweak rejection time consistency. The notion of semiweak time consistency for stochastic processes, introduced next, is well suited for scaleinvariant 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 LMmeasure on \(\mathbb {V}^{p}\). Then, φ is semiweakly acceptance time consistent if
and it is semiweakly rejection time consistent if
Clearly, weak acceptance/rejection time consistency for stochastic processes implies semiweak acceptance/rejection time consistency.
Next, we will show that the definition of semiweak time consistency is indeed equivalent to the time consistency introduced in (Bielecki et al. 2014b).
Proposition 11
 1)
φ is semiweakly acceptance time consistent.
 2)φ is one step μacceptance 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)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 semiweak 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 semiweak 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 semiweak rejection time consistency.
The next two results are important. In particular, they generalize the work done in (Bielecki et al. 2014b) regarding duality between cashadditive risk measures and acceptability indices.
Proposition 12
Let \(\{\varphi ^{x}\}_{x\in \mathbb {R}_{+}}\) be a decreasing family^{14} of dynamic LMmeasures 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 semiweakly acceptance (resp. semiweakly rejection) time consistent dynamic LMmeasure.
Proposition 13
Let \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) be a dynamic LMmeasure, which is independent of the past and translation invariant.^{15} Assume that \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) is semiweakly acceptance (resp. semiweakly 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 LMmeasure.
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.
Strong time consistency
Let us start with the definition of strong time consistency.
Definition 14
Now, let us present the counterpart of Proposition 4.
Proposition 14
 1)
φ is strongly time consistent.
 2)
There exists an update rule μ such that: for any \(t\in \mathbb {T}'\), \(m\in \bar {L}^{0}_{t}\), and \(V,V'\in \mathbb {V}^{p}\), satisfying \({V}_{t}={V}^{\prime }_{t}\), we have μ _{ t,t+1}(m,V)=μ _{ t,t+1}(m,V ^{′}); the family φ is both onestep μacceptance and onestep μrejection time consistent.
 3)There exists an update rule μ such that for any t<T and \(V\in \mathbb {V}^{p}\)$$\varphi_{t}(V)=\mu_{t,t+1}(\varphi_{t+1}(V),1_{\{t\}}V_{t}). $$
As in the case of random variables, strong time consistency is usually considered for dynamic monetary risk measures on \(\mathbb {V}^{\infty }\). In this case, additional equivalent properties can be established. For brevity, we skip the details, and only show the general idea for deriving a litany of equivalent properties. This idea is rooted in a specific construction of strongly time consistent dynamic LMmeasures.
Corollary 1
For a more detailed explanation of this idea and other equivalent properties see, e.g., (Cheridito and Kupper 2011) or (Ruszczyński and Shapiro 2006b).
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 LMmeasures.
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 onestep update rule.
Taxonomy of results
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Examples
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 LMmeasures 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
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
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 4
Combining (36) and (37), we conclude that φ ^{ α } is submartingale timeconsistent. In particular, by Remark 8, φ ^{ α } is also weakly rejection time consistent.
On the other hand, as shown in (Artzner et al. 2007), φ ^{ α } is neither middle rejection time consistent nor weakly acceptance time consistent.
Example 5
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 semiweakly rejection time consistent but not semiweakly acceptance time consistent.
Example 6
Example 7
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 semiweakly acceptance and semiweakly rejection time consistent. For the sake of completeness, we will show here that dGLR is semiweakly 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
Using (44) we obtain
Example 8
where \(X\in \mathcal {X}=L^{\infty }\), \(t\in \mathbb {T}\). The parameter θ=−γ is commonly known as the riskaversion parameter. It can be proved that for γ≤0, the map \(\varphi ^{\gamma }_{t}\) is a dynamic concave utility measure, and that for any \(\gamma \in \mathbb {R}\), the map φ ^{ γ } is strongly time consistent (cf. (Kupper and Schachermayer 2009)). Since it is also cashadditive, strong time consistency implies both weak rejection and weak acceptance time consistency. Moreover (see (Bielecki et al. 2015a; Kupper and Schachermayer 2009) for details), \(\{\varphi ^{\gamma }_{t}\}_{t\in \mathbb {T}}\) is supermartingale time consistent if and only if γ≥0, and submartingale time consistent if and only if γ≤0.
Example 9
where \(\{\gamma _{t}\}_{t\in \mathbb {T}}\) is such that \(\gamma _{t}\in L^{\infty }_{t}\), \(t\in \mathbb {T}\). It has been shown in (Acciaio and Penner 2011) that \(\{\varphi _{t}^{\gamma _{t}}\}_{t\in \mathbb {T}}\) is strongly time consistent if and only if \(\{\gamma _{t}\}_{t\in \mathbb {T}}\) is a constant process, and that it is middle acceptance time consistent if and only if \(\{\gamma _{t}\}_{t\in \mathbb {T}}\) is a nonincreasing process, and that it is middle rejection time consistent if and only if \(\{\gamma _{t}\}_{t\in \mathbb {T}}\) is nondecreasing.
Example 10
It is easy to check that φ is a strongly time consistent dynamic LMmeasure. It belongs to the class of so called dynamic certainty equivalents Kupper and Schachermayer (2009). In Kupper and Schachermayer (2009), the authors showed that every dynamic LMmeasure, which is finite, normalized, strictly monotone, continuous, law invariant, admits The Fatou property, and is strongly time consistent, can be represented as (48) for some U. We also refer to Biagini and BionNadal (2014) for a more general approach to dynamic certainty equivalents (e.g., by using stochastic utility functions U), and to Bielecki et al. (2016) for the definition of certainty equivalents for processes.
Example 11
where γ is a fixed real number. It was proved in Bielecki et al. (2015a) that φ ^{ γ } is a dynamic measure of performance, and it is μacceptance time consistent with respect to \(\mu _{t}(m)=E[m\mathcal {F}_{t}], \ t\in \mathbb {T}\), if and only if γ>0, and μrejection time consistent, with respect to μ if and only if γ<0.
Taxonomy of examples
The following table is meant to help the reader to navigate through the examples presented above relative to various types of time consistency studied in this paper. We will use the following abbreviations for time consistency: WA  weak acceptance; WR  weak rejection; sWA  semiweak acceptance; sWR  semiweak rejection; MA  middle acceptance; MR  middle rejection; STR  strong; Sub  submartinagle; Sup  supermartinagle.
If a cell is marked with the check mark, that means that the corresponding property of time consistency is satisfied; otherwise the property is not satisfied in general.
We note that Example 11 is not represented in the table due to the distinct nature of the example. The dGLI evaluates a process V, but it does it through a limiting procedure, which really amounts to evaluating the process through its “values at T=∞.” We refer the reader to (Bielecki et al. 2015a) for a detailed discussion on various properties of this measure.
■■■
\(\mathcal {X}\)  WA  WR  sWA  sWR  MA  MR  STR  Sub  Sup  

Example 3  L ^{ p }  ✓  ✓  ✓  ✓  
Example 4  L ^{ p }  ✓  ✓  ✓  
Example 5  \(\mathbb {V}^{p}\)  ✓  
Example 6  \(\mathbb {V}^{p}\)  ✓  
Example 7  \(\mathbb {V}^{p}\)  ✓  ✓  
Example 8  γ≥0  L ^{ p }  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
γ≤0  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
Example 9  γ _{ t } ↓  L ^{ p }  ✓  ✓  ✓  \(\checkmark ^{*}\)  
γ _{ t } ↑  ✓  ✓  ✓  \(\checkmark ^{**}\)  
Example 10  L ^{ p }  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
Endnotes
^{1} An LMmeasure 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 LMmeasures” 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 “LMextensions” 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 semiweak 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 LMmeasures” 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 semicontinuous. 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 \(\mathcal {B}\).
Appendix
Here we provide a brief exposition of the three fundamental concepts used in the paper: the dynamic LMmeasures, the conditional essential suprema/infima, and the LMextensions.
Dynamic LMmeasures
Let \(\mathcal {X}\) denote the space of random variables or adapted stochastic processes as described in Section “Mathematical preliminaries”.

Superadditive if φ _{ t }(X+Y)≥φ _{ t }(X)+φ _{ t }(Y);

Normalized if φ _{ t }(0)=0;

Cashadditive if φ(X+m1_{{t}})=φ _{ t }(X)+m;

Quasiconcave if φ _{ t }(λ·_{ t } X+(1−λ)·_{ t } Y)≥φ _{ t }(X)∧φ _{ t }(Y);

Concave if φ _{ t }(λ·_{ t } X+(1−λ)·_{ t } Y)≥λ φ _{ t }(X)+(1−λ)φ _{ t }(Y);

Scale invariant if φ _{ t }(β·_{ t } X)=φ _{ t }(X);

Positively homogeneous if φ _{ t }(β·_{ t } X)=β φ _{ t }(X);

Lower semicontinuous with respect to the topology η, if \(\{Z\in \bar {L}_{t}^{0} \mid \varphi _{t}(X)\leq Z\}\) is ηclosed;^{16}

Upper semicontinuous with respect to the topology η, if \(\{Z\in \bar {L}_{t}^{0} \mid \varphi _{t}(X)\geq Z\}\) is ηclosed,

Independent of the past if φ _{ t }(X)=φ _{ t }(X−0·_{ t } X);

Translation invariant if φ _{ t }(X+m1_{{t}})=φ _{ t }(X+m1_{{s}}).
These last two properties are automatically satisfied for \(\mathcal {X}=L^{p}\).
Most of the above properties have a natural financial interpretation. For example, quasiconcavity, concavity, or superadditivity correspond to the positive effect of portfolio diversification. See (Cherny and Madan 2009; Föllmer and Schied 2010) for more details and for a financial interpretation of other properties listed above.

Fatou property, if \(\varphi _{t}(X)\geq \limsup _{n\to \infty }\varphi _{t}(X_{n})\);

Lebesgue property, if \(\varphi _{t}(X)={\lim }_{n\to \infty }\varphi (X_{n})\);

Lawinvariant property if φ _{ t }(X)=φ _{ t }(Y), whenever Law(X)=Law(Y);
for any \(t\in \mathbb {T}\), \(X,Y\in \mathcal {X}\) and any dominated sequence ^{17} \(\{X_{n}\}_{n\in \mathbb {N}}\) such that X _{ n }∈L ^{ p } and \(X_{n}\xrightarrow {a.s.}X\).
Classes of dynamic LMmeasures

Dynamic monetary utility measure, or just dynamic utility measure for short, if φ is translation invariant, independent of the past, normalized, monotone, and cashadditive;

Dynamic concave utility measure, if φ is a dynamic utility measure and concave;

Dynamic coherent utility measure, if φ is a dynamic utility measure, is positive homogeneous, and superadditive;

Dynamic performance measure, if φ is adapted, translation invariant, independent of the past, monotone increasing, and scale invariant;

Dynamic acceptability index, if φ is a dynamic performance measure, and it is quasiconcave.
It needs to be stressed that in the literature, typically, the negative of the dynamic (monetary, concave, or coherent) utility measure is used and referred to as dynamic (monetary, convex, or coherent) risk measure.
Robust representations for dynamic monetary utility measures
Robust representations have been studied for general dynamic LMmeasures, not only for dynamic monetary utility measures. However, in this paper we only use robust representations for dynamic monetary utility measures for random variables, and that is why our discussion here is limited to this case. Consequently, we take \(\mathcal {X}=L^{p}\) for a fixed p∈{0,1,∞}.

acceptance and rejection sets denoted by \(\mathcal {A}=\{\mathcal {A}_{t}\}_{t\in \mathbb {T}}\) and \(\mathcal {R}=\{\mathcal {R}_{t}\}_{t\in \mathbb {T}}\), respectively, where$$\begin{array}{ll} \mathcal{A}_{t} &:=\{X\in L^{p} \colon \varphi_{t}(X)\geq 0\},\\ \mathcal{R}_{t} &:=\{X\in L^{p} \colon \varphi_{t}(X)\leq 0\}. \end{array} $$

conditional acceptance and conditional rejection sets denoted by \(\{\mathcal {A}_{t,s}: t,s\in \mathbb {T},\, s>t\}\) and \(\{\mathcal {R}_{t,s}: t,s\in \mathbb {T},\, s>t\}\), respectively, where$$\begin{array}{ll} \mathcal{A}_{t,s} &:=\{X\in L^{p} \cap \bar{L}^{0}_{s} \colon \varphi_{t}(X)\geq 0\},\\ \mathcal{R}_{t,s} &:=\{X\in L^{p} \cap \bar{L}^{0}_{s} \colon \varphi_{t}(X)\leq 0\}. \end{array}$$

minimal penalty functions denoted by \(\alpha ^{\min }=\{\alpha ^{\min }_{t}\}_{t\in \mathbb {T}}\), where \(\alpha ^{\min }_{t}\colon \mathcal {M}(P)\to \bar {\mathbb {R}}\) is given by$$ \alpha^{\min}_{t}(Q):=\underset{X\in \mathcal{A}_{t}}{\mathrm{ess\,inf}}~E_{Q}[X\,\,\mathcal{F}_{t}]. $$

conditional minimal penalty functions denoted by \(\{\alpha ^{\min }_{t,s}: t,s\in \mathbb {T},\, s>t\}\), where \(\alpha ^{\min }_{t,s}\colon \mathcal {M}(P)\to \bar {L}^{0}_{t}\) is given by$$ \alpha^{\min}_{t,s}(Q):=\underset{X\in \mathcal{A}_{t,s}}{\mathrm{ess\,inf}}~E_{Q}[X\,\,\mathcal{F}_{t}]. $$
The following important definition is frequently used in this paper.
Definition 15
for any \(X\in \mathcal {X}\).
This type of representation is called robust or numerical representations. Moreover, such representation characterizes dynamic concave utility measures that admit the Fatou property.
Conditional expectation and conditional essential supremum/infimum
We present here some relevant properties of the generalized conditional expectation and conditional essential superemum and infimum, in the context of \(\bar {L}^{0}\).
Proposition 15
 1)
\(E[\lambda X\mathcal {F}_{t}]\leq \lambda E[X\mathcal {F}_{t}]\) for \(\lambda \in L^{0}_{t}\), and \(E[\lambda X\mathcal {F}_{t}]=\lambda E[X\mathcal {F}_{t}]\) for \(\lambda \in L^{0}_{t}\), λ≥0;
 2)
\(E[X\mathcal {F}_{t}]\leq E[E[X\mathcal {F}_{s}]\mathcal {F}_{t}]\), and \(E[X\mathcal {F}_{t}]=E[E[X\mathcal {F}_{s}]\mathcal {F}_{t}]\) for X≥0;
 3)
\(E[X\mathcal {F}_{t}]+E[Y\mathcal {F}_{t}]\leq E[X+Y\mathcal {F}_{t}]\), and \(E[X\mathcal {F}_{t}]+E[Y\mathcal {F}_{t}]=E[X+Y\mathcal {F}_{t}]\), if X,Y≥0.
For the proof, see (Bielecki et al. 2014a, Proposition A.1).
Remark 11
All inequalities in Proposition 15 can be strict. Assume that t=0 and \(k,s\in \mathbb {T}\), k>s>0, and let \(\xi \in L^{0}_{k}\) be such that ξ=±1, ξ is independent of \(\mathcal {F}_{s}\), and P(ξ=1)=P(ξ=−1)=1/2. We consider \(Z\in L_{s}^{0}\) such that Z≥0, and E[Z]=∞. By taking λ=−1, X=ξ Z and Y=−X, we get strict inequalities in 1), 2), and 3).
We call this random variable the \(\mathcal {F}_{t}\)conditional essential infimum of X. Accordingly, we define ess sup_{ t }(X):=−ess inf_{ t }(−X), the \(\mathcal {F}_{t}\)conditional essential supremum of X∈L ^{ ∞ }. The reader is referred to (Barron et al. 2003) for a proof of the existence and uniqueness of the conditional essential supremum/infimum.
Respectively, we put ess sup_{ t }(X):=−ess inf_{ t }(−X).
Proposition 16
 1)
ess inf_{ ω∈A } X=ess inf_{ ω∈A }(ess inf_{ t } X);
 2)
If ess inf_{ ω∈A } X=ess inf_{ ω∈A } U for some \(U\in \bar {L}^{0}_{t}\), then U=ess inf_{ t } X;
 3)
X≥ess inf_{ t } X;
 4)
If \(Z\in \bar {L}^{0}_{t}\), is such that X≥Z, then ess inf_{ t } X≥Z;
 5)
If X≥Y, then ess inf_{ t } X≥ess inf_{ t } Y;
 6)
 7)
ess inf_{ s } X≥ess inf_{ t } X;
Analogous results are true for \(\{\mathrm {ess\,sup}_{t}\}_{t\in \mathbb {T}}\).
The proof for the case X,Y∈L ^{ ∞ } can be found in (Barron et al. 2003). Since for any \(n\in \mathbb {N}\) and \(X,Y\in \bar {L}^{0}\), we get X ^{+}∧n∈L ^{ ∞ }, X ^{−}∧n∈L ^{ ∞ }, and X ^{+}∧X ^{−}=0, the extension of the proof to the case \(X,Y\in \bar {L}^{0}\) is straightforward.
It is worth mentioning that properties 3) and 4) from Proposition 16 imply that the conditional essential infimum ess inf_{ t }(X) can be defined as the largest \(\mathcal {F}_{t}\)measurable random variable, which is smaller than X (cf. (Barron et al. 2003)).
Note that, in view of (Karatzas and Shreve 1998, Appendix A), ess inf_{ i∈I } X _{ i }∧n and ess sup_{ i∈I } X _{ i }∧n are well defined, so that ess inf_{ i∈I } X _{ i } is well defined. It needs to be observed that the operations of the righthand side of (53) preserve measurability. In particular, if \(X_{i}\in \mathcal {F}_{t}\) for all i∈I, then \(\mathrm {ess\,inf}_{i\in I}X_{i}\in \mathcal {F}_{t}\).
Furthermore, if for any i,j∈I, there exists k∈I, such that X _{ k }≤X _{ i }∧X _{ j }, then there exists a sequence \(i_{n}\in I, n\in \mathbb {N}\), such that \(\{X_{i_{n}}\}_{n\in \mathbb {N}}\) is nonincreasing and \(\mathrm {ess\,inf}_{i\in I}X_{i}=\inf _{n\in \mathbb {N}}X_{i_{n}}={\lim }_{n\to \infty }X_{i_{n}}\). Analogous results hold true for ess sup_{ i∈I } X _{ i }.
LMextensions
In this part of the appendix, we introduce the concept of an LMextension of a dynamic LMmeasure for random variables.
Definition 16
Let φ be a dynamic LMmeasure 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 LMextension 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 LMmeasure. The collection of functions \(\varphi ^{+}=\{\varphi _{t}^{+}\}_{t\in \mathbb {T}}\), where \(\varphi ^{+}_{t}:\bar {L}^{0}\to \bar {L}^{0}_{t}\) is defined as^{19}
is called the upper LMextension 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 LMextension of φ.
The next result shows that φ ^{±} are two “extreme” extensions, and any other extension is sandwiched between them.
Proposition 17
Clearly, in general, the maps (54) and (55) are not equal, and thus the extensions of an LMmeasure 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 mapping^{20} \(f:\mathcal {B}\to \bar {L}_{t}^{0}\), 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 LMmeasures, 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 quasiconcave upper semicontinuous LMmeasures. Hence, an alternative construction of extensions can be obtained through the robust representations of LMmeasures, by considering conditional expectations defined on the extended real number line, etc.
Proofs
Proof of proposition 4
Proof
where \(X\in \mathcal {X}\) is such that X ^{′}=φ _{ s }(X). In view of the definition of \(\mathcal {X}_{\varphi _{s}}\) and strong time consistency of φ, the map ϕ _{ t,s } is welldefined.
Since there exists \(Z\in \mathcal {X}\), such that φ _{ s }(Z)=0 (see property (10)), using locality of φ, we get that for any \(X\in \mathcal {X}_{\varphi _{s}}, \ A\in \mathcal {F}_{t}\), there exists \(Y\in \mathcal {X}\), so that
Clearly, the family \(\mu =\{\mu _{t,s}:\, t,s\in \mathbb {T},\, s>t\}\) is an update rule, and using (57), we get that φ is both μacceptance and μrejection time consistent.
2)⇒3). Let \(s,t\in \mathbb {T}\) and \(X,Y\in \mathcal {X}\) be such that s>t and φ _{ s }(X)≥φ _{ s }(Y). From 2),(12), and by the monotonicity of μ, we have φ _{ t }(X)=μ _{ t,s }(φ _{ s }(X))≥μ _{ t,s }(φ _{ s }(Y))=φ _{ t }(Y).
3)⇒1), 4)⇔2), and 4)⇒5) are obvious.
The proof is complete. □
Proof of proposition 6
Proof
Let us consider \(\{\phi _{t}\}_{t\in \mathbb {T}}\) as given in (19).
and thus, ϕ _{ t }(m)=m, for any \(m\in \bar {L}^{0}_{t}\). Hence, \(\{\phi _{t}\}_{t\in \mathbb {T}}\) is projective.
Combining (59) and (60), ϕacceptance time consistency of \(\{g\circ \varphi _{t}\}_{t\in \mathbb {T}}\) follows. □
Proof of proposition 8
Proof
The first part follows immediately from the definition of LMextension. Clearly, projectivity of \(\widehat {\varphi }\) implies that φ _{ t }(X)=X, for \(X\in \mathcal {X}\cap \bar {L}^{0}_{t}\). To prove the opposite implication, it is enough to prove that φ ^{+} and φ ^{−} are projective. Assume that φ is such that φ _{ t }(X)=X, for \(t\in \mathbb {T}\) and \(X\in L^{p}\cap \bar {L}^{0}_{t}\). Let \(X\in \bar {L}^{0}_{t}\). For any \(n\in \mathbb {N}\), we get
Next, for any \(A\in \mathcal {F}_{t}\), such that A⊆{X=∞}, we get \(\mathcal {Y}^{+}_{A}(X)=\emptyset \), which implies Finally, for any \(n\in \mathbb {R}\), using locality of \(\varphi ^{+}_{t}\) and the fact that \(n\in \mathcal {X}\cap \bar {L}^{0}_{t}\), we get
which implies Hence, (61) holds true on entire space. The proof for φ ^{−} is analogous. □
Proof of proposition 14
Proof
Let φ be a dynamic LMmeasure, which is independent of the past.
Next, since there exists \(Z\in \mathcal {X}\), such that φ _{ t+1}(Z)=0, using the locality of φ, we get that for any \(X\in \mathcal {X}_{\varphi _{t+1}}, \ A\in \mathcal {F}_{t}\), there exist \(Y\in \mathcal {X}\), so that
The proof of the equivalence between 2) and 3) is straightforward and hence omitted here. □
Proof of Proposition 17
Proof
We show the proof for φ ^{+} only; the proof for φ ^{+} is similar. Consider a fixed \(t\in \mathbb {T}\).
As the above results are true for any \(t\in \mathbb {T}\), thus we have proved that φ ^{+} is an extension of φ. Let us now show (56) for φ ^{+}.
Declarations
Acknowledgments
Tomasz R. Bielecki and Igor Cialenco acknowledge support from the NSF grant DMS1211256. Part of the research was performed while Igor Cialenco was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. Marcin Pitera acknowledges the support by Project operated within the Foundation for Polish Science IPP Programme “Geometry and Topology in Physical Model” cofinanced by the EU European Regional Development Fund, Operational Program Innovative Economy 2007–2013. The authors would like to thank Marek Rutkowski for stimulating discussions and helpful remarks. We would also like to thank the anonymous referees, the associate editor and the editor for their helpful comments and suggestions which improved greatly the final manuscript.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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