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arXiv:1603.09030v1 [q-fin.MF] 30 Mar 2016
A survey of time consistency of dynamic risk measures and dynamic
performance measures in discrete time: LM-measure perspective
Tomasz R. Bielecki
a
, Igor Cialenco
a
, and Marcin Pitera
b
First Circulated: March 29, 2016
Abstract: In this work we give a comprehensive overview of the time consistency property of
dynamic risk and performance m easures, with focu s on discrete time setup. The two
key operational concepts used t hroughout are the notion of the LM-measure and t he
notion of the update rule that, we believe, are the key tools for studying the time
consistency in a unified framework.
Keywords: time consistency, update rule, dynamic LM-measure, dynamic risk measure, dynamic
acceptability index, measure of performance.
MSC2010: 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.
1 Introduction
The goal of this work is to give a comprehensive overview of the time consistency property of
dynamic risk and performance measur es. 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 m aking subject to
various uncertainties that evolve in time. Typically, the d ecisions made at a given p oint of time
have impact on the future resolution of uncertainty. Consequently, it appears to be a rational
modus operandi to postulate that the decisions are made through time in such a way that today ’s
decisions account on their temporal impact; this is one aspect of time consistency of any dynamic
decision making. Decisions are often made subject to decision maker’s preferences, which need to be
assessed as an integral part of the decision making process. Naturally, the assessment of preferen ces
should be done in such a way that th e future preferences are consistent with the present ones. This
is the second aspect of time consistency, which will be the main subject of this survey. We will
refer to this second aspect as to the time consistency of preferences.
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 rep re-
sentations, such as (dynamic) risk measures, (dynamic) performance measures, or, more generally,
a
Department of Applied Mathematics, Illinois Institute of Technology
10 W 32nd Str, Building E1, Room 208, Chicago, IL 60616, USA
Emails:
bielecki@iit.edu (T.R. Bielecki) and cialenco@iit.edu (I. Cialenco)
URLs:
http://math.iit.edu/
~
bielecki and http://math.iit.edu/
~
igor
b
Institute of Mathematics, Jagiellonian University, Cracow, Poland
Email:
marcin.pitera@im.uj.edu.pl, URL: http://www2.im.uj.edu.pl/MarcinPitera/
1
Time consistency of risk and performance measures: a survey 2
dynamic LM measures [
BCP15b]. Cons equently, the study of time consistency of preferen ces is con-
veniently done in terms of their numerical representations. This work is meant to survey various
approaches to modeling and analysis of time consistency of numerical representations of pr eferences.
As said above, the objects of our survey - the d y namic LM-measures - are meant to ‘put a
preference order’ on set of underlying entities. There exists a vast literature on the subject of
preference ordering, with various approaches towards establishing an order of choices, s uch as
decision theory or expected utility theory, that trace their origins to the mid 20th century. In this
study 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 Ar tzner et al. [
ADEH99], and has been going through a flourishing development since
then. The measures of risk introduced in [
ADEH99], called coherent risk measures, were m eant
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 term s 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 [
ADEH99] the authors considered the case of random variables, and the risk
measurement was don e 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 r isk 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 evolution 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 w ith the literature review relevant to the dynamic
risk and perform an ce measures, focusing on time consistency pr operty in discrete time setup. In
Section
3 we set the mathematical scene; in particular, we introduce the main notations used in the
paper and the notion of LM-measures. Section
4 is devoted to setting the scene regarding the time
consistency p roperty. We discuss there two generic approaches to time consistent assessm ent of
preferences, and we point out to several idiosyn cr atic approaches. We put forth in this section the
notion of update rule that, we believe, is the key tool for studying the time consistency in a unified
framework. Sections
5 and 6 survey concepts and results r egardin g time consistency in the case
of random variables and in the case of stochastic processes, respectively. Our s urvey is illustr ated
by numerous examples that are presented in Section 7. We end the survey with two appendices.
In Appendix A we provide a brief exposition of the three fundamental concepts used in the paper:
the dynamic LM-measures, the conditional essential suprema/infima, and LM-extensions. Finally,
in Appendix B we collect proofs of several results stated throughout our survey.
Time consistency of risk and performance measures: a survey 3
2 Literature review
The aim of this s ection is to give a chronological survey of the developments of theory of dynamic
risk and performance measures. Although it is not an obvious task to establish the exact order
of developments, we tried our best to account for the most relevant works according to adequate
ch ronological order.
We trace back the origins of the research regarding time consistency to Koopmans [
Koo60] who
put on the precise mathematical footing, in terms of the utility function, the notion of persistency
over time of the structure of preferences.
Subsequ ently, in the seminal paper, Kreps and Porteus [
KP78] treat the time consistency at
general level by axiomatising the “choice behavior” of an agent by taking into account h ow 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 to reviewing the works on dynamic risk an performance measures, it is worth
mentioning that the robust expected utility theory proposed by Gilboa and Schmeidler [
GS89]
can be viewed as a more comprehensive theory than that one discussed in [
ADEH99]; we refer to
[
RSE05] for the relevant discussion.
Starting with [
ADEH99], the axiomatic theory of r isk measures, understood as fu nctions map-
ping random variables into real numbers, was developing around the following main goals: a) to
define a set of properties (or axioms) that a risk measure should satisfy; b) to characterize all
functions that s atisfy th ese properties; c) provide particular examples of such functions. Each of
the imposed axioms should have a meaningful financial or actuarial interpretation. For example,
in [
ADEH99], a static coherent r isk measure is defi ned as a function ρ : L
∞
→ [−∞, ∞] that is
monotone decreasing (larger losses implies larger risk), cash-additive (the r isk is reduced by the
amount of cash added to the portfolio today), sub-additive (diversified portfolio has a smaller
risk) and positive h omogenous (the risk of a rescaled portfolio is rescaled correspondingly), w here
L
∞
is the space of (essentially) bounded random variables on some probability space
1
(Ω, F, P).
The descriptions or the r epresentations of these functions, also called robust representations, usu-
ally are derived via duality theory in convex analysis, and are necessary and sufficient in their
nature. Traditionally, among such representations we find: representations in terms of level or
acceptance sets; numerical representations in terms of dual pairings (e.g. expectations). For exam-
ple, the coherent risk measure ρ mentioned above can be described in terms of its acceptance set
A
ρ
= {X ∈ L
∞
| ρ(X) ≤ 0}. As it turns out, the acceptance set A
ρ
satisfies certain ch aracteristic
properties, and any set A with these properties generates a coherent risk measure via the represen-
tation ρ(X) = inf{m ∈ R | m + X ∈ A}. Alternatively, the function ρ is a coherent risk measure
if and only if there exists a nonempty set Q of probability measures, absolutely continuous with
respect to P , s uch that
ρ(X) = − inf
Q∈Q
E
Q
[X]. (2.1)
The set 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 even more general class called
the monetary risk measures. See, for instance, [
Sze02] for a sur vey of static risk measures, as well
as [
CL09, CL08]. On the other hand, axiomatic theory of performance measures was originated in
[
CM09]. A general th eory of risk preferences and their robust representations, based on only two
generic axioms, was stu died in [
Dra10, DK13].
1
In the original paper [
ADEH99] the aut hors considered finite probability spaces, but later the theory was elevated
to general space [
Del02].
Time consistency of risk and performance measures: a survey 4
Moving to the dynamic setup, we first intro duce an underlying filtered probability space
(Ω, F, {F
t
}
t≥0
, P), where the increasing collection of σ-algebras F
t
, t ≥ 0, models the flow of
information that is accumulated through time.
Artzner et al. [
ADE
+
02b] and [ADE
+
02a] stu dy an extension of the static models examined
in [
ADEH99] to the multiperiod case, assuming discrete time and discrete probability space. The
authors proposed a method of constructing dynamic risk measures {ρ
t
: L
∞
(F
T
) →
¯
L
0
(F
t
), t =
0, 1 . . . , T}, by a backward recursion, s tarting with ρ
T
(X) = −X, and letting
ρ
t
(X) = − inf
Q∈Q
E
Q
[−ρ
t+1
(X) | F
t
], 0 ≤ t < T, (2.2)
where, as before Q, is a set of probability measures. If, additionally, Q satisfies a property called
recursivity or consistency (cf. [
Rie04]),
inf
Q∈Q
E
Q
[Z | F
t
] = inf
Q∈Q
E
Q
[ inf
Q
1
∈Q
E
Q
1
[Z | F
t+1
] | F
t
], t = 0, 1, . . . , T − 1, Z ∈ L
∞
. (2.3)
In this case, one can show that (
2.2) is equivalent to
ρ
t
(X) = ρ
t
(−ρ
t+1
(X)), 0 ≤ t < T, X ∈ L
∞
(F
T
). (2.4)
The prop er ty (2.4) represents what has become known in the literature as the strong time consis-
tency property. For example, if Q = {P }, then the strong time consistency reduces to th e tower
property for conditional expectations. From the practical point of view, this property essentially
means that assessment of risks propagates in a consistent way through 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). Add itionally, the property (
2.4) is closely
related to Bellman principle of optimality or to dynamic programming principle (see, for instance,
[BD62, CCC
+
12]).
Delbaen [
Del06] studies the recursivity property in terms of m-stable sets of probability mea-
sures, and also d escribes the time consistency of dynamic coherent risk measur es in th e context of
martingale theory. The recursivity property is equivalent to prop er ties known as time consistency
and rectangularity in the multi-prior Bayesian decision theory. Epstein and Schneider [
ES03] study
time consistency and rectangularity property in the framework of “decision under ambiguity”.
It needs to be said that several authors refer to [Wan02] for an alternative axiomatic approach
to time consistency of dynamic risk measures.
The first study of dynamic risk measures for stochastic p rocesses (finite probability space and
discrete time) is attributed to Riedel [
Rie04], where the author introduced the (strong) time con-
sistency as one of the axioms. If ρ
t
, t = 0, . . . , T, is a dynamic coherent risk measure, acting on the
set of discounted terminal cash flows
2
, then ρ is strongly time consistent if the following implication
holds true:
ρ
t+1
(X) = ρ
t+1
(Y ) ⇒ ρ
t
(X) = ρ
t
(Y ). (2.5)
This means that if tomorrow we assess the riskiness of X and Y at the same level, th en 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, p roperty (
2.5) is equivalent to
(
2.4).
Motivated by results regarding the pricing procedure in incomplete markets, based on use of
risk measures, Roorda et al. [RSE05] study dynamic coherent risk measures (for the case of random
2
In [
Rie04] the auth or considered discounted dividend processes, but for simplicity here we write the time consis-
tency for random variables.
Time consistency of risk and performance measures: a survey 5
variables on finite probability space and discrete time) and introduce the notion of (strong) time
consistency; it needs to be noted that their work was similar and s imultaneous to [
Rie04]. They show
that the strong time consistency entails recursive computation of th e corresponding optimal hedging
strategies. Moreover, time consistency is also described in term s 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 r isk measures, by replacing sub-additivity and positive homogeneity properties with con-
vexity. In the continuous time setup, Rosazza Gianin [
RG02] links dynamic convex risk measures
to nonlinear expectations or g-expectations, and to backward s tochastic differential equations.
Strong time consistency plays a crucial role, and in view of (2.4) it is equivalent to the tower
property for conditional g-expectations. These results are further studied in a sequel of papers
[RG06, Pen04, FRG04], as well as in Coquet et al. [CHMP02].
A representation similar to (
2.1) holds tr ue for dy namic convex risk measure
ρ
t
(X) = − inf
Q∈M(P )
E
Q
[X | F
t
] + α
min
t
(Q)
, t = 0, 1, . . . , T, (2.6)
where M(P ) is the set of all probability measures abs olutely continuous with respect to P , and
α
min
is the min im al penalty function.
3
The natural question of describing (strong) time consistency
in terms of properties of the minimal penalty fun ctions was studied by Scandolo [
Sca03]. Also in
[
Sca03] the author discusses the importance in the dynamic setup of special p roperty called locality.
Similarly to previous studies, [
Sca03] finds a relationship between time consistency, the recur sive
construction of dynamic risk meassures and th e s upermartingale property. These results are further
investigated in Detlefsen and Scandolo [DS05]. Also in these works, it was shown that the dynamic
entropic risk measure is a strongly time consistent convex risk measure.
Weber [Web06] continues the study of dynamic convex risk measures for random variables in
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 characterises time
consistency in terms of acceptance indicator a
t
(X) =
ρ
t
(X)≤0
and in terms of acceptance sets of
the form N
t
= {X | ρ
t
(X) ≤ 0}. Along the same lines, F¨ollmer and Penner [
FP06] investigate the
dynamic convex risk measures, representation of strong time consistency as recursivity property,
and they r elate it to the Bellman principle of optimality. They also proved that the supermartingale
property of the penalty function corresponds to the weak or acceptance/rejection time consistency.
Moreover, the authors studied the co-cycle property of the penalty function for the dynamic convex
risk measures that admit robus t representation (see Definition
A.1).
Artzner et al. [
ADE
+
07] continue to study th e strong time consistency for dynamic risk mea-
sures, its equ ivalence with the stability property of test p robabilities and with the optimality
principle.
It is worth mentioning that Bion-Nadal [
BN04] studies dynamic monetary risk measures in
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 Shap iro [
RS06a] elevate
the concepts from [
RS06b] to the dynamic setting, with the main goal to establish conditions under
which the dynamic programming pr inciple holds.
Cheridito and Kupper [CK11] introduce the notion of aggregators and generators f or dynamic
convex risk measures, and give a thorough discussion about composition of time-consistent convex
3
See Appendix
A.1.2 for the definition of minimal penalty functions, up to a sign, and for the corresponding robust
representations.
Time consistency of risk and performance measures: a survey 6
risk measures in 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 [
CDK06,
CK09].
Jobert and Rogers [
JR08] take the valuation concept as the starting point, rather than the dy-
namics of acceptance sets, with valuation f unctional being the negative of a risk measure. To quote
the authors (strong) “time consistency is the heart of the matter.” Kloeppel and Schweizer [KS07]
use dynamic convex risk measures for valuation in incomplete markets, where the time consistency
plays the key role. Cher ny [Che07] uses dynamic coherent risk measure for pricing and hedging
European options; see also [
CM06].
Roorda and Schumacher [
RS07] study weak form of time consistency for dynamic convex risk
measure.
Bion-Nadal [
BN06] continues to study various properties of dynamic risk measures, both in
discrete and in continuous time, mainly focusing on the composition prop er ty mentioned above,
and thus on the strong time consistency. The composition property is characterized in terms of
stability of probability sets. T he author defines the co-cycle condition for the penalty function
and shows its equivalence to strong time consistency. In the follow up paper, [
BN08], the author
continues to study the characterization of time consistency in terms of the co-cycle condition for
minimal penalty function. For fu rther related developments in the continuous time framework see
[
BN09b].
Observing that Value at Risk (V@R) is not strongly time consistent, Boda an d Filar [
BF06],
and Cher idito and Stadje [
CS09] construct a strongly time consistent alternative to V@R by using
a recursive composition p rocedure.
Tutsch [Tut08] gives a different perspective on time consistency of convex risk measures by
introducing the update rules,
4
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 regulatory capital
requirement. For example, Cherny [
Che10] applies dynamic coherent risk measures to risk-reward
optimization problems and in [
Che09] to capital allocation; Bion-Nadal [BN09a] uses dynamic
risk measures f or time consistent pricing; Barrieu and El Karoui [
BEK04, BEK05, BEK07] study
optimal derivatives design under dynamic risk measures; Geman and Ohana [
GO08] explore the
time consistency in man aging a commodity portfolio via dynamic risk measures; Zariphopoulou
and Zitkovic [Zv10] investigate the maturity independent dynamic convex risk measures.
In Delbaen et al. [
DPRG10] the authors establish a representation of the penalty function of
dynamic convex risk measure usin g g-expectation and its relation to the strong time consistency.
There exists a significant literature on special class of risk measures that satisfy the law invari-
ance property. Kup per and Schachermayer [
KS09] proved 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 r efer
to [
BNK10b] and [BNK10a]. Acciaio et al. [AFP12] give a comprehensive study of various forms
of time consistency for dynamic convex risk measures in 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 [
AP11] of d iscr ete time dynamic convex risk measures. This work deals with
(essentially bounded) rand om variables and examis most of the papers mentioned above f rom the
perspective of the robust representation framework.
4
In the present manuscript we also use the name ‘update ru les’, although the concept used here is different from
that introd uced in [
Tut08].
Time consistency of risk and performance measures: a survey 7
Although the conn ection between BSDEs and the dynamic convex risk measures in continuous
time setting had been established for some time, it appears that Stadje [
Sta10] was the first author
to create a theoretical framework for stu dying dynamic risk measures in discrete time via the
Backward Stochastic Difference Equations (BS∆Es). Du e to the backward nature of BS∆Es, the
strong time consistency of risk measures p layed a critical role in characterizing the dynamic convex
risk measures as solutions of BS∆Es.
F¨ollmer and Penner [
FP11] 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 [NS12] for a study of
dynamic risk measures under volatility uncertainty and their connection to G-expectations.
From slightly different point of view, Ruszczynski [Rus10] stu dies Markov risk measures, that
enjoys the strong time consistency, in the framework of risk-averse preferences; see also [
Sha09,
Sha11, Sha12, FR14]. Some concepts from the theory of the dynamic risk measures are adopted to
the study of dynamic programming for Markov decision processes.
In the recent paper Mastrogiacomo and Rosazza Gianin [MRG15] provide several forms of time
consistency for sub-additive dynamic risk measures and th eir dual representations.
Finally, we want to mention that during the last decade significant advances were made towards
developing a general theory of set-valued risk measures [
HR08, HHR11, FR13, HRY13, FR15], in-
cluding the dynamic version of them where mostly the correspondin g 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 un its of the relevant
currency. Accordingly, the key property of any risk measure is cash additivity ρ(X−m) = ρ(X)+m.
Clearly, can think of the risk measures as of generalizations of V@R .
A concept that is, in a sense, complementary to the concept of risk measures, is that of per-
formance measures, which can be thou ght of generalizations of the well kn own Sharpe ratio. In
similarity with the theory of risk measures, the development of theory of performance measures fol-
lowed an axiomatic approach. This was initiated by Cherny and Madan [
CM09], where the authors
introduced the (static) notion of the coherent acceptability index - a fu nction on L
∞
with values
in R
+
that is monotone, quasi-concave and scale invariant. As a matter of fact, scale invariance is
the key property of acceptability indices that distinguishes them from risk m easures, and, typically,
acceptability ind ices are not cash additive. The dynamic version of coherent acceptability indices
was introduced by Bielecki et al. [
BCZ14], for the case of stochastic processes, finite probability
space and discrete time. From now on, we will us e as synony ms the terms measure s of performance,
performance measures and acceptability indices.
As it tur ns out, the time consistency for measures of performance is a delicate issue. None of
the forms of time consistency, that had been coined for dynamic risk measures, is appropriate for
dynamic performance measures. In [
BCZ14] the authors introduce a new form of time consistency
that is suitable for dynamic coherent acceptability indices. Let α
t
, t = 0, 1, . . . , T , be a dynamic
coherent acceptability index acting on L
∞
(i.e. discounted terminal cash flows). We say that α is
time consistent if the following implications hold true:
α
t+1
(X) ≥ m
t
⇒ α
t
(X) ≥ m
t
,
α
t+1
(X) ≤ n
t
⇒ α
t
(X) ≤ n
t
, (2.7)
where X ∈ L
∞
, and m
t
, n
t
are F
t
-measurable random variables. Biagini and Bion-Nadal [
BBN14]
study dynamic performance measures in a fairly general setup, that generalize the r esults of
[BCZ14]. Later, using the theory of dynamic coherent acceptability indices developed in [BCZ14],
Time consistency of risk and performance measures: a survey 8
Bielecki et al. [
BCIR13] propose a pricing f ramework, called d y namic conic finance, for dividend
pay ing securities in discrete time. Time consistency property was at the core of establishing th e
connection between the dynamic conic finance and the classical arbitrage pricing theory. The static
conic finance, that served as motivation for [
BCIR13], was introduced in [CM10]. Finally, in recent
papers [
BCC15, RGS13 ], the authors elevate the notion of dynamic coherent acceptability indices
to the case of sub-scale invariant performance measures. For that, BSDEs are used in [
RGS13] and
BS∆Es are used in [
BCC15].
For a general theory of robust representations of quasi-concave maps that covers both dynamic
risk measures and dynamic acceptability indices see [FM11, BCDK15, FM14, BN16]. Also in
[
BCDK15], the authors study the strong time consistency of quasi-concave maps via the concept
of certainty equivalence; see also [
FM10].
To our best knowledge, [
BCP15b] is the only paper that combines into a unifi ed framework the
time consistency for dynamic risk measures and dynamic performance measure. It uses the concept
of update rules that s er ve 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
our survey (not all of which are mentioned above though).
Dynamic Coherent Risk Measures
• r an dom variables, strong time consistency: [
ADE
+
02b], [ADE
+
02a], [RSE05].
• stochastic processes, strong time consistency: [
Rie04], [ADE
+
07].
Dynamic Convex Risk Measures,
• r an dom variables, strong time consistency: (discrete time) [Sca03], [DS05], [BF06], [FS06],
[
RS06a], [FP06], [CS09], [BN06], [GO08], [BN08], [CK09], [KS09], [AP11], [Sta10], [AFP12],
[
FS12], [BCDK15], [BCP15b], [IPS15], [MRG15], [RS15];
(continuous time) [
RG02], [RG06], [FRG04], [BEK04] [DPRG10], [KS 07 ], [BN06], [BEK07],
[
BN08], [Jia08], [BN09b], [BNK10b], [SS15], [NS12], [PR14].
• r an dom variables, supermartingale time consistency: [
Sca03], [DS05].
• r an dom variables, acceptance/rejection time consistency: [
Web06], [FP06], [AFP12], [RS07],
[
Tut08], [AFP12], [BCP15b], [RS15].
• stochastic processes, strong and su perm artin gale time consistency: (discrete time) [
Sca03],
[
BCP15b], (continuous time) [JR08]
Dynamic Monetary Risk Measures, stron g time consistency:
(discrete time) [
CK11], [CDK06]; (continuous time) [BN04], [FP11].
Dynamic Acceptability Indices: [
BCZ14], [BBN14], [BCIR13], [RGS13], [BCDK15], [FM14],
[
BCP15b], [BCC15].
3 Mathematical Preliminaries
Let (Ω, F, F = {F
t
}
t∈T
, P ) be a filtered probability space, with F
0
= {Ω, ∅}, and T = {0, 1, . . . , T },
where T ∈ N is a fixed and finite time horizon. We will also u se the notation T
′
= {0, 1, . . . , T − 1}.
For G ⊆ F we denote by L
0
(Ω, G, P ) and
¯
L
0
(Ω, G, P ) the sets of all G-measurab le random
variables with values in (−∞, ∞) and [−∞, ∞], respectively. In addition, we will use the notation
L
p
(G) := L
p
(Ω, G, P ), L
p
t
:= L
p
(F
t
) and L
p
:= L
p
T
, for p ∈ {0, 1, ∞}; analogously we define
¯
L
0
t
. We
will also use the notation V
p
:= {(V
t
)
t∈T
: V
t
∈ L
p
t
}, f or p ∈ {0, 1, ∞}.
5
Moreover, we use M(P ) to
5
Unless otherwise specified, it will be understood in the rest of the paper that p ∈ {0, 1, ∞}.
Time consistency of risk and performance measures: a survey 9
denote the set of all probability measures on (Ω, F) that are absolutely continuous with respect to
P , and we set M
t
(P ) := {Q ∈ M(P ) : Q|
F
t
= P |
F
t
}.
Throughout this paper, X will denote either the sp ace of random variables L
p
, or the space of
adapted p rocesses V
p
. If X = L
p
, then the elements X ∈ X are interpreted as discounted terminal
cash flow. On the other hand, if X = V
p
, then the elements of X , are interpreted as discounted
dividend processes. It needs to be remarked, th at all concepts developed for X = V
p
can be easily
adapted to the case of the cumulative discounted value processes. The case of random variables
can be viewed as a particular case of stochastic processes by considering cash flow with only the
terminal payoff, i.e. stochastic processes such that V = (0, . . . , 0, V
T
). Nevertheless, we treat this
case separately for transparency. In both cases we will consider the standard pointwise order,
understood in the almost sure sense. In what follows, we will also make use of the multiplication
operator denoted as ·
t
and defined by:
m ·
t
V := (V
0
, . . . , V
t−1
, mV
t
, mV
t+1
, . . .),
m ·
t
X := mX, (3.1)
for V ∈
(V
t
)
t∈T
| V
t
∈ L
0
t
, X ∈ L
0
, m ∈ L
∞
t
and t ∈ T. In order to ease the notation, if no
confusion arises, we will drop ·
t
from the above pr oduct, and we will simply write mV and mX
instead of m ·
t
V and m ·
t
X, respectively. For any t ∈ T we set
1
{t}
:=
(0, 0, . . . , 0
|
{z }
t
, 1, 0, 0, . . . , 0), if X = V
p
,
1 if X = L
p
.
For any m ∈
¯
L
0
t
, the value m1
{t}
corresponds to a cash fl ow of size m received at time t . We use this
notation for the case of random variables to present more unified definitions (see Appen dix
A.1).
Remark 3.1. We note that the space V
p
, endowed with the multiplication ·
t
does not define a
proper L
0
–module [
FKV09] (e.g., in general, 0 ·
t
V 6= 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
[BCDK15, BCP15a] for a th orou gh discussion on this matter.
We will use the convention ∞−∞ = −∞+∞ = −∞ and 0·±∞ = 0. Note that the distributive
law does not h old true in general: (− 1)(∞ − ∞) = ∞ 6= −∞ + ∞ = −∞. For t ∈ T and X ∈
¯
L
0
we define the (generalized) F
t
-conditional expectation of X by
E[X|F
t
] := E[X
+
|F
t
] − E[X
−
|F
t
],
where X
+
= (X ∨ 0) and X
−
= (−X ∨ 0). See Appendix
A.2 for some relevant properties of the
generalized expectation.
For X ∈
¯
L
0
and t ∈ T, we will denote by ess inf
t
X the unique (up to a set of probability zero),
F
t
-measurable random variable, su ch that
ess inf
ω∈A
X = ess inf
ω∈A
(ess inf
t
X) (3.2)
for any A ∈ F
t
. We call this random variable the F
t
-conditional essential infimum of X. Similarly,
we define ess sup
t
(X) := − ess inf
t
(−X), and we call it the F
t
-conditional essential supremum of X.
Again, see Appendix
A.2 for more details and some elementary properties of conditional essential
infimum and supremum .
The next definition introduces the main object of this work.
Time consistency of risk and performance measures: a survey 10
Definition 3.2. A family ϕ = {ϕ
t
}
t∈T
of maps ϕ
t
: X →
¯
L
0
t
is a Dynamic LM-measure if ϕ satisfies
1) (Locality)
A
ϕ
t
(X) =
A
ϕ
t
(
A
·
t
X);
2) (Monotonicity) X ≤ Y ⇒ ϕ
t
(X) ≤ ϕ
t
(Y );
for any t ∈ T, X, Y ∈ X and A ∈ 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 studies in the literature. Monotonicity property is natural for any
numerical representation of an order between elements of X . The locality property (also referred
in the literature as regularity) essentially means that the values of the LM-measure restricted to
a set A ∈ F remain invariant with respect to the values of the arguments outside of the same set
A ∈ F; in particular, the events that will not happen in the future do not affect the value of the
measure today.
Remark 3.3. 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 X = L
∞
, then monotonicity and cash additivity
imply locality (cf. [
Pit14, P roposition 2.2.4]). Similarly, any convex (or concave) map is also local
(cf. [
DS05]). I t 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. [KS09]).
In this paper we only consider dynamic LM-measures ϕ, such that
0 ∈ ϕ
t
[X ], (3.3)
for any t ∈ T. We impose this (technical) assumption to insure that the maps ϕ
t
that we consider
are not degenerate in the sense that they are not taking infinite values for all X ∈ X on some set
A
t
∈ F
t
of positive prob ab ility, for any t ∈ T; in the literature s ometimes such maps are referred
as proper [
KR09]. If this is the case, then there exists a family {Y
t
}
t∈T
, where Y
t
∈ X , such that
ϕ
t
(Y
t
) ∈ L
0
t
for any t ∈ T, an d so we can consider maps ˜ϕ given by ˜ϕ
t
(·) := ϕ
t
(·) − ϕ
t
(Y
t
), that
satisfy assumption (
3.3) and preserve the same order as the maps ϕ
t
do. Typically, in risk measure
framework, one assumes that ϕ
t
(0) = 0, which implies (
3.3). 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 th at traditionally in the literature the dynamic risk measures are mon otone
decreasing. On the other hand, the measures of performance are monotone increasing. In view of
condition 2) in Definition
3.2 whenever our LM-measure corresponds to a dynamic risk measure
it needs to be understood as the negative of that risk measure. In such cases, in order to avoid
confusion, we refer to th e respective LM-measure as to dynamic (monetary) utility measure rather
than as to dynamic (monetary) risk measure. See Ap pendix
A.1 for details.
4 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 d iscussed in the Intro-
duction time consistency is studied via numerical representations of preferences. Various numerical
representations will be surveyed below , and discus sed in the context of dy namic LM-measures.
Time consistency of risk and performance measures: a survey 11
To streamline the presentation we focus our attention on the case of random variables, that is
X = L
p
, for p ∈ {0, 1, ∞}.
6
Usually, the risk measures and 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 A.1), since for that a certain topological
structure is requir ed (cf. Remark
A.9). On the other hand, ‘time consistency’ refers only to
consistency of measurements in time, where no particular topological s tructure is needed, and thus
most of the results obtained here hold tru e for p = 0.
In Section
4.1 we outline two generic approaches to time consistent assessment of preferences:
an approach based on u pdate rules and an approach based on bench mark 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
4.2 are specific.
That is to say, those approaches are suited only for specific types of time consistency, specific classes
of dynamic LM-measures, specific spaces, etc.
4.1 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.
4.1.1 Update rule s
The approach to time consistency using up date rules was developed in [
BCP15b]. An u pdate rule
is a tool, that is applied to preference levels, and us ed for relating assessments of preferences d on e
using a dynamic LM-measure at different times.
Definition 4.1. A family µ = {µ
t,s
: t , s ∈ T, t < s} of maps µ
t,s
:
¯
L
0
s
→
¯
L
0
t
is called an update
rule if µ satisfies the following conditions:
1) (Locality)
A
µ
t,s
(m) =
A
µ
t,s
(
A
m);
2) (Monotonicity) if m ≥ m
′
, then µ
t,s
(m) ≥ µ
t,s
(m
′
);
for any s > t, A ∈ F
t
and m, m
′
∈
¯
L
0
s
.
Next, we give a definition of time consistency in terms of update rules.
Definition 4.2. Let µ be an update rule. We say that the dynamic LM-measure ϕ is µ-acceptance
(resp. µ -re jec tion) time consistent if
ϕ
s
(X) ≥ m
s
(resp. ≤) =⇒ ϕ
t
(X) ≥ µ
t,s
(m
s
) (resp. ≤), (4.1)
for all s > t, s, t ∈ T, X ∈ X , and m
s
∈
¯
L
0
s
. If property (
4.1) is satisfied for s = t + 1, t ∈ T
′
, then
we say that ϕ is one-step µ-acceptance (resp. one-step µ-rejection) time consistent.
We see that m
s
and µ
t,s
(m
s
) s er ve as benchmarks to which the measurements of ϕ
s
(X) and
ϕ
t
(X) are compared, r espectively. Thus, the interpretation of acceptance time consistency is
straightforward: if X ∈ X is accepted at some future time s ∈ T, at least at level m
s
, then
6
Most of th e concepts discussed in t his Section can be modified to deal with the case of stochastic processes, as
we will do in Section
6.
Time consistency of risk and performance measures: a survey 12
today, at time t ∈ T, it is accepted at least at level µ
t,s
(m
s
). Similarly for rejection time consis-
tency. Essentially, the update rule µ converts the preference levels at time s to preference levels at
time t.
We started our sur vey of time consistency with Definition
4.2 s ince, as we will demonstr ate
below, this concept of time consistency covers various cases of time consistency for risk and per-
formance measures that can be f ou nd in the existing literature. In particular it allows to establish
important connections between different types of time consistency. Time consistency property of
an LM-measure, in general, depends on th e choice of the updated rule; we refer to Section 5 for an
in-depth discu ssion.
It is useful to observe that the time consistency property given in terms of update rules can be
equivalently formulated as a version of the dynamic programming principle (see [
BCP15b, P ropo-
sition 3.6]): ϕ is µ-acceptance (resp. µ-rejection) time consistent if and only if
ϕ
t
(X) ≥ µ
t,s
(ϕ
s
(X)) (resp. ≤), (4.2)
for any X ∈ X and s, t ∈ T, such that s > t. The interpretation of (
4.2) is as follows: if the
numerical assessment of preferences about X is given in terms of a dynamic LM-measure ϕ, then
this measure is µ-acceptance time consistent if and only if the numerical assessment of preferences
about X done at time t is greater than the value of the measurement done at any futur e time s > t
and updated at time t via µ
t,s
. Analogous interpretation applies to rejection time consistency.
Next, we define two interesting and important classes of update rules.
Definition 4.3. Let µ be an update rule. We say that µ is
1) s-invariant, if there exists a family {µ
t
}
t∈T
of maps µ
t
:
¯
L
0
→
¯
L
0
t
, such that µ
t,s
(m
s
) = µ
t
(m
s
)
for any s, t ∈ T, s > t, and m
s
∈
¯
L
0
s
;
2) projective, if it is s-invariant and µ
t
(m
t
) = m
t
, for any t ∈ T, and m
t
∈
¯
L
0
t
.
Remark 4.4. If an update rule µ is s-invariant, then it is enough to consider only the corresponding
family {µ
t
}
t∈T
. Hence, with slight abu se of notation, we will write µ = {µ
t
}
t∈T
and call it an
update rule as well.
Example 4.5. The families µ
1
= {µ
1
t
}
t∈T
and µ
2
= {µ
2
t
}
t∈T
given by
µ
1
t
(m) = E[m|F
t
], and µ
2
t
(m) = ess inf
t
m, m ∈
¯
L
0
,
are projective update rules. It will be shown in Example
7.3 that there is a dynamic LM-measure
that is µ
2
–time consistent but not µ
1
–time consistent.
4.1.2 Benchmark families
The approach to time consistency based on families of benchmark sets was initiated by [
Tut08],
where the author applied this ap proach in the context of dyn amic risk measures. Essentially, a
benchmark f amily is a collection of subsets of X that contain r eference 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 4.6.
(i) A family Y = {Y
t
}
t∈T
of sets Y
t
⊆ X is a benchmark family if
0 ∈ Y
t
and Y
t
+ R = Y
t
,
Time consistency of risk and performance measures: a survey 13
for any t ∈ T.
(ii) A dynamic LM-measure ϕ is acceptance (resp. rejection) time consistent with respect to the
benchmark family Y, if
ϕ
s
(X) ≥ ϕ
s
(Y ) (resp. ≤) =⇒ ϕ
t
(X) ≥ ϕ
t
(Y ) (resp. ≤), (4.3)
for all s ≥ t, X ∈ X and Y ∈ Y
s
.
Informally, the “degree” of time consistency with respect to Y is measured by the size of Y.
Thus, th e larger the sets Y
s
are, for each s ∈ T, the stronger is the degree of time consistency of ϕ.
Example 4.7. The families of sets Y
1
= {Y
1
t
}
t∈T
and Y
2
= {Y
2
t
}
t∈T
given by
Y
1
t
= R and Y
2
t
= X ,
are benchmark families. They relate to weak and strong types of time consistency, as will be
discussed later on.
For future reference we recall fr om [
BCP15b, Proof of Proposition 3.9] that ϕ is acceptance
(resp. rejection) time consistent with respect to Y, if and only if ϕ is acceptance (r esp. rejection)
time consistent with respect to the benchmark family
b
Y given by
b
Y
t
:= {Y ∈ X : Y =
A
Y
1
+
A
c
Y
2
, for some Y
1
, Y
2
∈ Y
t
and A ∈ F
t
}. (4.4)
4.1.3 Relation between update rule approach and the benchmark a pproach
The difference between the update rule approach and the benchmark family approach is that pref-
erence levels are chosen in different ways in these approaches. Specifically, in the former ap proach,
the preference level at time s is chosen as any m
s
∈
¯
L
0
s
, and then updated to 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 ∈ Y
s
, where Y
s
is an element of
the benchmark family Y.
These two approaches are strongly related to each other. Indeed, for any LM-measure ϕ and
for any benchmark family Y, one can construct an update rule µ such that ϕ is time consistent
with respect to Y if and only if it is µ-time consistent.
For example, in case of acceptance time consistency of ϕ with respect to Y, us ing locality of ϕ,
it is easy to note that (
4.3) is equivalent to
ϕ
t
(X) ≥ ess sup
A∈F
t
h
A
ess sup
Y ∈Y
−
A,s
(ϕ
s
(X))
ϕ
t
(Y ) +
A
c
(−∞)
i
,
where Y
−
A,s
(m
s
) := {Y ∈
b
Y
s
:
A
m
s
≥
A
ϕ
s
(Y )} and
b
Y = {
b
Y
s
}
s∈T
is defined in (
4.4). Consequently,
setting
eµ
t,s
(m
s
) := ess sup
A∈F
t
h
A
ess sup
Y ∈Y
−
A,s
(m
s
)
ϕ
t
(Y ) +
A
c
(−∞)
i
,
and using (
4.2), we deduce that ϕ satisfies (4.3) if and only if ϕ is time consistent with respect
to the up date rule eµ
t,s
(see [
BCP15b, Proposition 3.9] for details). 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 d ynamic coherent
acceptability index cannot be expressed in terms of a single benchmark family.
Time consistency of risk and performance measures: a survey 14
4.2 Idiosyncratic Approaches
Each such approach to time consistency of a given LM-measure exploits the idiosyncratic properties
of this LM-measure, which are not necessarily sh ared by other LM measures, and typically it is
suited only for a s pecific subclass of dynamic LM-measures. For example, in case of dynamic convex
or monetary risk measures time consistency can be characterized in terms of relevant properties
of associated acceptance sets and/or dynamics of the penalty functions and/or the rectangular
property of the families of probab ility m easures. These idiosyncratic approaches, and the relevant
references, were mentioned and briefly discussed in Section
2. Detailed analysis of each of th ese
approaches is beyond the scope of this survey.
5 Time consistency for random variables
In this Section we survey time consistency of LM-measures applied to random variables. Accord-
ingly, we assume here that X = L
p
, for a fixed p ∈ {0, 1, ∞}. We pro ceed with discussion of various
related types of time consistency, without much reference to the existing literature. Such references
are provided in Section 2.
5.1 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 smaller than
or equal to ϕ
s
(X), adjus ted by the information F
t
available at time t. Similarly, if tomorrow we
reject X at level ϕ
s
(X), then to day, we should also reject X at any level b igger than or equal to
ϕ
s
(X), adapted to the information F
t
.
Definition 5.1. A dynamic LM-measure ϕ is weakly acceptance (resp. weakly rejection) time
consistent if
ϕ
t
(X) ≥ ess inf
t
ϕ
s
(X), (resp. ϕ
t
(X) ≤ ess sup
t
ϕ
s
(X) )
for any X ∈ L
p
and s, t ∈ T, such that s > t.
Propositions
5.2 and 5.3 provide some characterizations of weak acceptance time consistency.
Proposition 5.2. Let ϕ be a dynamic LM-measure on L
p
. The following prope rties are equivalent:
1) ϕ is weakly acceptance time consistent.
2) ϕ is µ-acceptance time consistent, where µ is a projective update rule, given by
µ
t
(m) = ess inf
t
m.
3) The following inequality is satisfied
ϕ
t
(X) ≥ ess inf
Q∈M
t
(P )
E
Q
[ϕ
s
(X)|F
t
], (5.1)
for any X ∈ L
p
, s, t ∈ T, s > t.
4) For any X ∈ L
p
, s, t ∈ T, s > t, and m
t
∈
¯
L
0
t
, it holds that
ϕ
s
(X) ≥ m
t
⇒ ϕ
t
(X) ≥ m
t
.
Time consistency of risk and performance measures: a survey 15
Similar results hold true for weak rejection time consistency.
For the proof of equivalence between 1), 2) and 4), see [
BCP15b, Proposition 4.3]. Regarding
3), note that any measure Q ∈ M
t
(P ) may be expressed in terms of a Radon-Nikodym derivative
with respect to measure P. In other words, instead of (6.4), we may write
ϕ
t
(X) ≥ ess inf
Z∈P
t
E[Zϕ
s
(X)|F
t
],
where P
t
:= {Z ∈ L
1
| Z ≥ 0, E[Z|F
t
] = 1}. Thus, one can show equivalence between 1) and 3)
noting that for any m ∈
¯
L
0
we get ess inf
t
m = ess inf
Z∈P
t
E[Zm|F
t
]. See [
BCP15b, Proposition
4.4] for the proof.
It is worth mentioning that Property 4) in Proposition 5.2 was suggested as the notion of (weak)
acceptance and (weak) rejection time consistency in the context of scale invariant measures, called
acceptability indices (cf. [BBN14, BCZ14]).
Usually, the weak time consistency is considered f or dynamic monetary r isk measures on L
∞
(cf. [
AP11] and references therein). This case lends itself to even more characterizations of this
property.
Proposition 5.3. Let ϕ be a representable dynamic monetary u tility measure
7
on L
∞
. The fol-
lowing properties are equivalent
1) ϕ is weakly acceptance time consistent.
2) ϕ is acceptance time consistent with respect to {Y
t
}
t∈T
, where Y
t
= R.
3) For any X ∈ L
p
and s, t ∈ T, s > t,
ϕ
s
(X) ≥ 0 ⇒ ϕ
t
(X) ≥ 0. (5.2)
4) A
t+1
⊆ A
t
, for any t ∈ T, such that t < T .
5) For any Q ∈ M(P ) and t ∈ T, such that t < T ,
α
min
t
(Q) ≥ E
Q
[α
min
t+1
(Q) | F
t
],
where α
min
is the minimal penalty function in the robust representation of ϕ.
Analogous results are obtained for weak rejection time consistency.
We note that equivalence of p roperties 1), 2) an d 3) also holds true in the case of X = L
0
,
and not only for r epresentable but for any dynamic monetary utility measure; for the proof, see
[
BCP15b, 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 pr oof of equivalence of 3), 4) and 5), see [AP11, Proposition 33].
Next result shows that the weak time consistency is indeed one of the weakest forms of time
consistency, in the sense that the weak time consistency if implied by any time consistency generated
by a projective update rule; we refer to [
BCP15b, Proposition 4.5] for the proof.
Proposition 5.4. Let ϕ be a dynamic LM -measure on L
p
, and let µ be a projective update rule.
If ϕ is µ-acceptance (resp. µ-rej ection) time consistent, then ϕ is weakly acceptance (resp. weakly
rejection) time consistent.
7
See section
A.1 for details.
Time consistency of risk and performance measures: a survey 16
Remark 5.5. An important feature of the weak time consistency is its invariance with respect to
monotone transformations. Specifically, let g :
¯
R →
¯
R be a strictly increasing function an d let ϕ
be a weakly acceptance/rejection time consistent dynamic LM-measure. Then, {g ◦ ϕ
t
}
t∈T
is also
a weakly acceptance/rejection time consistent dynamic LM-measure.
Remark 5.6. In the case of general LM-measures the weak time consistency may not be characterized
as in 2) of Proposition
5.3. For example, if ϕ is a (normalized) acceptability index, then ϕ
t
(R) =
{0, ∞}, for t ∈ T, which does not agree with 4) in Proposition
5.2.
5.2 Strong time consistency
As already said in the Introduction the origins of the strong form of time consistency can be
traced to [
Koo60]. Historically, this is the first, and the most extensively studied form of the time
consistency in the dynamic risk measures literature.
We start with the definition of strong time consistency.
Definition 5.7. Let ϕ be a dynamic LM-measure on L
p
. Then, ϕ is said to be strongly time
consistent if
ϕ
s
(X) = ϕ
s
(Y ) =⇒ ϕ
t
(X) = ϕ
t
(Y ), (5.3)
for any X, Y ∈ L
p
and s, t ∈ T, such that s > t.
The strong time consistency gains its popularity and importance due to its equivalence to the
dynamic programming principe. This equivalence, as well as other characterisations of strong time
consistency, are the subject of the follow ing two propositions.
Proposition 5.8. Let ϕ be a dynamic LM-measure on L
p
. The following prope rties are equivalent:
1) ϕ is strongly time consistent.
2) There exist an u pdate rule µ such that ϕ is both µ-acceptance and µ-rejection time consistent.
3) ϕ is acceptance time consistent with respect to {Y
t
}
t∈T
, where Y
t
= L
p
.
4) There exists an update rule µ such that for any X ∈ L
p
, s, t ∈ T, s > t,
µ
t,s
(ϕ
s
(X)) = ϕ
t
(X). (5.4)
5) There exists a one-step update rule µ such that for any X ∈ L
p
,