Time Consistency in Decision Making

Transcription

1 Time Consistency in Decision Making Igor Cialenco Department of Applied Mathematics Illinois Institute of Technology Joint work with Tomasz R. Bielecki and Marcin Pitera Mathematical Finance Colloquium University of Southern California November 13, 2017

2 Introduction Preference Order Let X be a non-empty set, and assume that at each time t {0, 1,..., T } we have a total preorder t on X, that admits a numerical representation ϕ t X [, ], i.e. x t y ϕ t (x) ϕ t (y), x, y X. Main Question: If the decisions are made through time using the preorder t or its numerical representation ϕ t, t = 0, 1, 2,..., T, how to insure that the decisions are made consistently in time? A total preorder is a binary relation on X such that 1 (reflexive) x x, for any x X ; 2 (transitive) if x y and y z, then x z; 3 (total) for any x, y X, x y or y x, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 2

3 Introduction Preference Order through time The progressive assessment of preferences should be an integral part of the decision making process. The assessment of preferences should be done in such a way that the future preferences are assessed consistently with the present ones. Main Goal: To develop a unified theory for studying time consistency for dynamic risk and dynamic performance measures. We will take a top-down approach and define time consistency for functions ϕ that are only local and monotone. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 3

4 Introduction Preference Order through time The progressive assessment of preferences should be an integral part of the decision making process. The assessment of preferences should be done in such a way that the future preferences are assessed consistently with the present ones. Main Goal: To develop a unified theory for studying time consistency for dynamic risk and dynamic performance measures. We will take a top-down approach and define time consistency for functions ϕ that are only local and monotone. 1 T.R. Bielecki, I. Cialenco, M. Pitera, A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time, Forthcoming in Mathematics of Operations Research (28 pages), T.R. Bielecki, I. Cialenco, M. Pitera, A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective, Probability, Uncertainty and Quantitative Risk, 2:3, pp.1-52, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 3

5 T.R. Bielecki, I. Cialenco and T. Chen, Dynamic Conic Finance via Backward Stochastic Difference Equations, SIAM Journal of Financial Mathematics, 6(1), pp , T.R. Bielecki, I. Cialenco, S. Drapeau and M. Karliczek, Dynamic Assessment Indices, Stochastics, vol. 88, No 1, pp. 1-44, T.R. Bielecki, I. Cialenco, I. Iyigunler and R. Rodriguez, Dynamic Conic Finance: Pricing and Hedging in Market Models with Transaction Costs via Dynamic Coherent Acceptability Indices, IJTAF, vol. 16, No 1, T.R. Bielecki, I. Cialenco and Z. Zhang, Dynamic Coherent Acceptability Indices and their Applications to Finance, Math. Fin., Vol. 24(3), pp , July 2014 (accepted 2011). T.R. Bielecki, I. Cialenco and R. Rodriguez, No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs, Math. Fin. 25(4), pp , October 2015 (accepted 2012).

6 Literature review Before the theory of risk measures: Koopmans [Koo60] - Stationary ordinal utility and impatience Kreps and Porteus [KP78] - Temporal resolution of uncertainty and dynamic choice theory Duffie and Epstein [DE92] - Stochastic differential utility Sarin and Wakker [SW98] - dynamic consistency for non-expected utilities in a decision theoretic framework There exists a significant literature on dynamic risk and performance measures, with time consistency axiom playing a crucial role. 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, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 5

7 Notations Notations T -fixed time horizon; Discrete time setup T = {0, 1,..., T }; (Ω, F T, F = (F t ) t T, P) - the underlying filtered probability space; L p (F t ) = L p (Ω, F t, P; R), p {0, 1, }, L p t = Lp (Ω, F t, P; R), = and 0 = 0, E[X F t ] = lim n E[(X + n) F t ] lim n E[(X n) F t ]. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 6

8 Notations Case of Random Variables: X = L p (F T ) X L p (F T ) is interpreted as terminal payoff, or portfolio s value at T. In this talk, we focus on this case. Throughout we will assume zero interest rates. Case of Stochastic Processes was studied too. X = V p = {(V t ) t T V t L p t }. V V p is interpreted as a cash flow with dividend payments V t at t T. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 7

9 Dynamic LM-measure Preference orders on X Dynamic LM-measures approach Definition A dynamic LM-measures is a family {ϕ t } t T of maps ϕ t X L 0 t, such that, for any t T, ϕ t is Local: 1 A ϕ t (X) = 1 A ϕ t (1 A X), for any X X, A F t Monotone: X Y ϕ t (X) ϕ t (Y ), for any X, Y X. Two minimal properties, with clear financial interpretation, to define a preference order X. Locality is also known as regularity, zero-one law, or relevance property. With some additional properties, an LM-measure becomes a monetary utility measure, or a risk measure, or a performance measure, or assessment index, etc. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 8

10 Examples of LM measures Conditional essential infimum ϕ t (X) = ess inf t X; Dynamic LM-measure ϕ t (X) = E[X F t ]; Value At Risk ϕ t (X) = V@R α (X F t ) = ess sup{s L 0 (G ) P (X < s G ) α}; Average Value at Risk ϕ t (X) = α 0 V@R β(x F t )dβ; Gain to Loss Ratio E[X Ft] ϕ t (X) = E[X F, if E[X F t] t] > 0 and zero otherwise. Sortino Ratio ϕ t (X) = E[X F t] E[(X ) 2 F t] ; Any dynamic risk measure; any dynamic assessment index. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 9

11 Approaches to Time Consistency Existing approaches to time consistency Generic approach: families of benchmark sets Tutsch [Tut08]. It is implied by the proposed methodology, but not equivalent. Idiosyncratic approaches: define time consistency in terms of objects specific to a certain class of measures Conditional acceptance sets Dynamic of the minimal penalty functions Cocycle condition g-expectation Recursive construction rectangular property, prudence, etc All these approaches are particular cases of the proposed method; see the survey [BCP16] Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 10

12 Update Rules Update Rules Approach to Time Consistency Definition 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) (Local) 1 A µ t,s (m) = 1 A µ t,s (1 A m); 2) (Monotone) 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. An update rule is meant to relate the assessment level of preferences between different times. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 11

13 Update Rules Examples Definition The update rule µ is called Example s-invariant if µ t,s (m) = µ t (m), for s t, m L 0 s. projective if it is s-invariant and µ t (m t ) = m t, for t T, m t L 0 t. 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. For a fixed ε (0, 1), consider the update rule (not s-invariant) µ 3 t,s(m, X) = { εs t E[m F t ], on {E[m F t ] 0}, ε t s E[m F t ], on {E[m F t ] < 0}.. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 12

14 Update Rules Examples Definition Dynamic LM-measure ϕ is µ-acceptance time consistent if ϕ s (X) m s ϕ t (X) µ t,s (m s ), for all s > t, s, t T, X X, and m s L 0 s. Similarly, ϕ is µ-rejection time consistent if ϕ s (X) m s ϕ t (X) µ t,s (m s ), for all s > t, s, t T, X X, and m s L 0 s. In this talk we focus on acceptance time consistency. ϕ t (X) t µ t,s (m s) F t ϕ s(x) s m s F s X T Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 13

15 Update Rules Examples An analog of Dynamic Programming Principle Proposition The LM-measure ϕ is µ-acceptance time consistent if and only if ϕ t (X) µ t,s (ϕ s (X)), for any X X and s > t. Proof. ( ) Take in the definition of acceptance time consistency m s = ϕ s (X). ( ) Let m s L 0 s be such that ϕ s (X) m s. By monotonicity of µ, ϕ t (X) µ t,s (ϕ s (X)) µ t,s (m s ). Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 14

16 Update Rules Strong Time Consistency Strong Time Consistency Definition A dynamic LM-measure ϕ is strongly time consistent if there exist an update rule µ such that ϕ is both µ-acceptance and µ-rejection time consistent. Proposition The following statements are equivalent: 1) ϕ is strongly time consistent. 2) There exists an update rule µ such that µ t,s (ϕ s (X)) = ϕ t (X), X L p T, s > t. 3) There exists a one-step update rule µ such that µ t,t+1 (ϕ t+1 (X)) = ϕ t (X), X L p, t < T. 4) For any X, Y L p and s > t, ϕ s (X) = ϕ s (Y ) ϕ t (X) = ϕ t (Y ). Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 15

17 Update Rules Strong Time Consistency Strong time consistency is one of the most popular forms of time consistency. Suitable, and traditionally used, for dynamic convex risk measures. A dynamic monetary utility measure is an LM-measure that is also normalized ϕ t (0) = 0, and cash-additive ϕ t (X + m t ) = ϕ t (X) + m, m L p t. A monetary risk measure is the negative of a monetary utility measure. A dynamic monetary utility measure ϕ is representable, if where α min t ϕ t (X) = ess inf (E Q[X F t ] + αt min (Q)), X X, Q P is a minimal penalty function. This type of representation is called robust or numerical representations. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 16

18 Update Rules Strong Time Consistency Proposition Let ϕ be a representable dynamic monetary utility measure on L. The following properties are equivalent: 1) ϕ is strongly time consistent. 2) ϕ is recursive, i.e. for any X L p, s, t T, s > t, ϕ t (X) = ϕ t (ϕ s (X)). 3) A t = A t,s + A s, for all t, s T, s > t, A t,s = {X L p L 0 s ϕ t(x) 0}. 4) For any Q P, t, s T, s > t, 5) For any X L p, Q P, s, t T, s > t, α min t (Q) = α min t,s (Q) + E Q [α min s (Q) F t]. ϕ t(x) α min t (Q) E Q [ϕ s(x) α min (Q) F t]. Recursivity plays a critical role in pricing by using risk measures. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 17 s

19 Update Rules Strong Time Consistency Example (Dynamic Entropic Utility/Risk Measure) ϕ γ t (X) = { 1 γ ln E[exp(γX) F t] if γ 0, E[X F t ] if γ = 0, where X X = L, t T; θ = γ is the risk-aversion parameter For γ 0, the map ϕ γ t is a dynamic concave utility measure. For any γ R, the map ϕ γ is strongly time consistent. More generally, for U R R a strictly increasing, continuous function, a dynamic certainty equivalent is defined as ϕ t (X) = U 1 (E[U(X) F t ]), X X, t T. (5.1) ϕ is a strongly time consistent dynamic LM-measure. Every dynamic LM-measure, which is finite, normalized, strictly monotone, continuous, law invariant, admits the Fatou property, and is strongly time consistent, can be represented as (5.1) for some U [KS09]. See also [BBN14, BCDK16]. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 18

20 Update Rules Strong Time Consistency α is not strongly time consistent [CS09]. Dynamic Gain Loss Ratio (dglr) defined as ϕ t (X) = is not strongly time consistent. E[X F t] E[(X) F, t] if E[X F t] > 0, 0, otherwise, An LM measure is scale invariant if ϕ t (λx) = ϕ t (X), λ F + t. No scale invariant LM measure is strongly time consistent. Let A F s such that P(A) = 1/2, and X n = n1 A 1 A C, n N. By locality and scale invariance ϕ s (X n ) = ϕ s (X 1 ), n N. If ϕ is strongly time consistent, then ϕ 0 (X n ) = ϕ 0 (X 1 ). On the other hand, any reasonable performance measure should assess X n at higher level than X 1. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 19

21 Update Rules Weak Time Consistency Weak Time Consistency: Update Rule µ t (m) = ess inf t m Proposition Let ϕ be a dynamic LM-measure on L p T. The following are equivalent: 1) ϕ is is weakly acceptance time consistent, i.e. for any X L p T, s > t, ϕ s (X) m s ϕ t (X) ess inf t m s. 2) For any X L p T, s > t, ϕ t(x) ess inf t ϕ s (X). 3) For any X L p T, s > t, with M t(p) = {Q M(P) Q Ft = P Ft }, ϕ t (X) ess inf Q M t(p) E Q[ϕ s (X) F t ]. 4) For any X L p T, s > t, and m t L 0 t, ϕ s (X) m t ϕ t (X) m t. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 20

22 Update Rules Weak Time Consistency Most of the known examples are weakly acceptance or/and rejection time consistent. Proposition Let ϕ be a dynamic LM-measure on L p T, and let µ be any projective update rule. If ϕ is µ-acceptance time consistent, then ϕ is weakly acceptance time consistent. Proof. It is easy to check that ess inf s X ess inf t X. Then, for any t, s T, s > t, and any X L p, we get ϕ t (X) µ t (ϕ s (X)) µ t (ess inf s (ϕ s (X))) µ t (ess inf t (ϕ s (X))) = ess inf t (ϕ s (X)). Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 21

23 Performance Preference Order Dynamic Accessability Indices Performance Based Preference Orders Definition (Dynamic Performance Measure) A Dynamic Acceptability Indices is a family {f t } t T of maps f t X L t (F t ), such that, for any X, Y X, (P1) Local: 1 A f t (X) = 1 A f t (1 A X), for any A F t (P2) Monotone: X Y f t (X) f t (Y ) (P3) Scale Invariant: f t (λx) = f t (X), for any λ L p t, λ 0. (P4) Quasi-Concave: f t (λx + (1 λ)y ) min{f t (X), f t (Y )}, for any 0 λ 1, λ L p t. (P5) Time Consistency: For any m t, n t L 0 t (F t ), (acceptance) f t+1 (X) m t f t (X) m t, (rejection) f t+1 (X) n t f t (X) n t. For general theory see [BCZ14, BCC15, RGS13, BBN14] Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 22

24 Performance Preference Order Examples of Performance Measures Dynamic Accessability Indices Dynamic Gain-Loss Ratio dglr is Weakly time consistent. Sortino Ratio dglr t (X) = E[X F t] E[X F t ]. SR t (X) = E[X F t ] E[(X ) 2 F t ], Risk Adjusted Return on Capital RAROC t (X) = E[X F t ]/ρ t (X), where ρ is a dynamic risk measure. Dynamic TV@R Acceptability Index - performance measure constructed using TV@R dynamic risk measure via dual representations. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 23

25 Performance Preference Order Dynamic Accessability Indices Proposition Let ϕ be a representable dynamic monetary utility measure on L. The following properties are equivalent: 1) ϕ is weakly acceptance time consistent. 2) For any X L p and s, t T, s > t, 3) A t+1 A t, for any t T, such that t < T. ϕ s (X) 0 ϕ t (X) 0. 4) 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. Note: V@R α is weakly acceptance and rejection time consistent. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 24

26 Performance Preference Order Dynamic Accessability Indices Proposition Let {ϕ x } x R+ be a decreasing family of weakly acceptance/rejection time consistent dynamic LM-measures. Then, α t (X) = sup{x R + ϕ x t (X) 0}. is a weakly acceptance/rejection time consistent dynamic LM-measure. Let {α t } t T be a weakly acceptance/rejection time consistent dynamic LM-measure Then, for any x R +, ϕ x t (X) = ess inf {c1 {α c R t(x c) x}}, is a weakly acceptance/rejection time consistent dynamic LM-measure. Remark: Similar dual results hold true for processes, where the weak time consistency of α is replaced with semi-weak time consistency. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 25

27 Performance Preference Order Robust Expectations Time consistency wrt Robust Expectations Proposition Let D be a determining family of sets, and let ϕ be a dynamic LM-measure. Consider the family of maps φ = {φ t } t T, φ t L 0 L 0 t, given by the following robust expectations φ t (m) = ess inf Z D t E[Zm F t ]. (6.1) Then, φ is a projective update rule. Moreover, if ϕ is φ-acceptance time consistent, then {g ϕ t } t T is also φ-acceptance time consistent, for any increasing, concave g R R. Remark: Dynamic Coherent Risk Measures are good examples to generate update rules. D = {D t} t T is a determining family if for any t T, the set D t satisfies the following properties: D t, D t P t, it is L 1 -closed, F t-convex 1, and uniformly integrable, with P t = {Z L 1 Z 0, E[Z F t] = 1}. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 26

28 Performance Preference Order Robust Expectations Super(sub) martingale time consistency For the trivial determining sets D t = {1}, the projective update rule is µ t (m) = E[m F t ], m L 0. Definition Let ϕ be a dynamic LM-measure on L p. We say that ϕ is supermartingale (resp. submartingale) time consistent if for any X L p and t, s T, s > t. ϕ t (X) E[ϕ s (X) F t ], (resp. ) Any super(sub)martingale time consistent LM-measure is also weakly acceptance(rejection) time consistent. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 27

29 Performance Preference Order Robust Expectations Example Dynamic Entropic Risk Measure with non-constant risk aversion 1 t (X) = { γ t ln E[exp(γ t X) F t ] if γ t 0, E[X F t ] if γ t = 0, ϕ γt where {γ t } t T is such that γ t L t, t T. {ϕ γt t } t T is strongly time consistent if and only if {γ t } t T is a constant process [AP11]; it is middle acceptance time consistent if and only if {γ t } t T is a non-increasing process; is middle rejection time consistent if and only if {γ t } t T is non-decreasing. Other Examples: Dynamic Risk Sensitive Criterion [BCP15], and Conditional Weighted V@R. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 28

30 Performance Preference Order Time consistency induced by LM-measures Definition Let ϕ be a dynamic LM-measure on L p. A family ϕ = { ϕ t } t T of maps ϕ t L 0 L 0 t is an LM-extension of ϕ, if for any t T, ϕ t X ϕ t, and ϕ t is local and monotone on L 0. Define the collection of functions ϕ ± = {ϕ ± t } t T, where ϕ ± t L 0 L 0 t and ϕ + t (X) = ess inf A F t [1 A ess inf Y Y + A (X) ϕ t(y ) + 1 A c(+ )], ϕ t (X) = ess sup [1 A ess sup ϕ t (Y ) + 1 A c( )], A F t Y YA (X) where Y + A (X) = {Y X 1 AY 1 A X}, Y A (X) = {Y X 1 AY 1 A X}, is called the upper/lower LM-extension of ϕ, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 29

31 Performance Preference Order Proposition Let ϕ be a dynamic LM-measure on L p. 1 The functions ϕ and ϕ + are LM-extensions of ϕ. 2 For any ϕ LM-extension of ϕ ϕ t (X) ϕ t (X) ϕ + t (X), X L 0. 3 ϕ is an s-invariant update rule. 4 ϕ is projective if and only if ϕ t (X) = X, for t T and X L p L 0 t. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 30

32 Performance Preference Order Definition ϕ is middle acceptance time consistent, if it is ϕ acceptance time consistent. For representable risk measures on L, continuous from above, this is equivalent to and, also equivalent to ϕ s (X) ϕ s (Y ) ϕ t (X) ϕ t (Y ), ϕ t (X) ϕ t (ϕ s (X)). Clearly, strong time consistent implies middle acceptance time consistent. The converse implication is not true; for counterexample see [AP11, Proposition 37] Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 31

33 Performance Preference Order Similarly, ϕ is middle rejection time consistency, if it is ϕ + rejection time consistent. For representable risk measures on L, continuous from above, this is equivalent to or ϕ t (X) ϕ t (ϕ s (X)). ϕ s (X) ϕ s (Y ) ϕ t (X) ϕ t (Y ), ϕ is strongly time consistent, if there exists an LM-extension ˆϕ, of ϕ, such that ϕ is both ˆϕ-acceptance and ˆϕ-rejection time consistent, or equivalently ϕ t (X) = ˆϕ t (ϕ s (X)), X X, s, t T, s > t. Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 32

34 Figure: Summary of results for acceptance time consistency for random variables ϕ s (X) 0 ϕ t (X) 0 if ϕ is a monetary utility measure ϕ s (X) m t ϕ t (X) m t Dynamic LM-Measure ϕ is Weakly Accept Consist if ϕ t(x) ess inf t(ϕ s(x)) Dynamic LM-Measure ϕ is Supermartingale Consist if ϕ t(x) E[ϕ s(x) F t] if µ is projective ϕ is µ - accept consist and µ is projective ϕ s (X) = ϕ t (ϕ s (X)) for X = L, where ϕ is a monetary utility measure 6 7 ϕ is scale invariant ϕ is both µ - accept and µ - reject consist 8 Dynamic LM-Measure ϕ is Middle Accept Consist ϕ s(x) ϕ s(y ) ϕ t(x) ϕ t(y ) for Y X L 0 s 5 Dynamic LM-Measure ϕ is Strongly Consist ϕ s(x) = ϕ s(y ) ϕ t(x) = ϕ t(y ) 9

35 Examples Performance Preference Order X WA WR swa swr MA MR STR Sub Sup Cond. WV@R L p TV@R AI V p RAROC V p dglr V p dent γ 0 L p γ 0 dent+ γ t L p γ t RSC L p RSC = Risk Sensitive Criterion if γ t 0, if γ t 0 Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 34

36 Thank You! The end of the talk... but not of the story

37 Bibliography I [AP11] B. Acciaio and I. Penner. Dynamic risk measures. In G. Di Nunno and B. Øksendal (Eds.), Advanced Mathematical Methods for Finance, Springer, pages 1 34, [BBN14] S. Biagini and J. Bion-Nadal. Dynamic quasi-concave performance measures. Journal of Mathematical Economics, 55: , December [BCC15] T.R. Bielecki, I. Cialenco, and T. Chen. Dynamic conic finance via Backward Stochastic Difference Equations. SIAM J. Finan. Math., 6(1): , [BCDK16] T.R. Bielecki, I. Cialenco, S. Drapeau, and M. Karliczek. Dynamic assessment indices. Stochastics: An International Journal of Probability and Stochastic Processes, 88(1):1 44, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 36

38 Bibliography II [BCP15] T.R. Bielecki, I. Cialenco, and M. Pitera. Dynamic limit growth indices in discrete time. Stochastic Models, 31: , [BCP16] T.R. Bielecki, I. Cialenco, and M. Pitera. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Forthcoming in Probability, Uncertainty and Quantitative Risk, [BCZ14] T.R. Bielecki, I. Cialenco, and Z. Zhang. Dynamic coherent acceptability indices and their applications to finance. Mathematical Finance, 24(3): , [CS09] P. Cheridito and M. Stadje. Time-inconsistency of VaR and time-consistent alternatives. Finance Research Letters, 6(1):40 46, Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 37

39 Bibliography III [DE92] D. Duffie and L. G. Epstein. Stochastic differential utility. Econometrica, 60(2): , With an appendix by the authors and C. Skiadas. [Koo60] T. C. Koopmans. Stationary ordinal utility and impatience. Econometrica, 28: , [KP78] D. M. Kreps and E. L. Porteus. Temporal resolution of uncertainty and dynamic choice theory. Econometrica, 46(1): , [KS09] M. Kupper and W. Schachermayer. Representation results for law invariant time consistent functions. Mathematics and Financial Economics, 2(3): , [RGS13] E. Rosazza Gianin and E. Sgarra. Acceptability indexes via g-expectations: an application to liquidity risk. Mathematical Finance (2013) 7: , Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 38

40 Bibliography IV [SW98] R. Sarin and P. Wakker. Dynamic choice and nonexpected utility. Journal of Risk and Uncertainty, 17(2):87 120, [Tut08] S. Tutsch. Update rules for convex risk measures. Quant. Finance, 8(8): , Ig. Cialenco Illinois Tech Nov 13, 2017 Slide 39