Michel De Lara

Vincent Leclère
email: vincent.leclere@cermics.enpc.fr

Building up Time-Consistency for Risk Measures and Dynamic Optimization

Keywords: dynamic programming, time-consistency, dynamic risk measures

In stochastic optimal control, one deals with sequential decision-making under uncertainty; with dynamic risk measures, one assesses stochastic processes (costs) as time goes on and information accumulates. Under the same vocable of time-consistency (or dynamic-consistency), both theories coin two different notions: the latter is consistency between successive evaluations of a stochastic processes by a dynamic risk measure (a form of monotonicity); the former is consistency between solutions to intertemporal stochastic optimization problems. Interestingly, both notions meet in their use of dynamic programming, or nested, equations.

We provide a theoretical framework that offers i) basic ingredients to jointly define dynamic risk measures and corresponding intertemporal stochastic optimization problems ii) common sets of assumptions that lead to time-consistency for both. We highlight the role of time and risk preferences -materialized in one-step aggregators -in time-consistency. Depending on how one moves from one-step time and risk preferences to intertemporal time and risk preferences, and depending on their compatibility (commutation), one will or will not observe time-consistency. We also shed light on the relevance of information structure by giving an explicit role to a state control dynamical system, with a state that parameterizes risk measures and is the input to optimal policies.

Introduction

You come accross time-consistency in two different mathematical fields. You are time-consistent if, as time goes on and information accumulates, you do not question the original assessment of stochastic processes (dynamic risk measures) or planning of policies (stochastic optimal control).

We propose a general mechanism to build up time-consistent dynamic risk measures, that serve as criteria for optimal control problems under uncertainty, which henceforth inherit time-consistency. We show how in a few words.

Consider two sets T 1 and T 2 , representing sets of time periods (T 1 = {1, 2, 3}, T 2 = {4, 5} for instance). Consider two sets W 1 and W 2 , representing possible values of uncertainties. For any set S, denote by L(S) the set of functions S → R ∪ {+∞}, and by G S : L(S) → R ∪ {+∞} a mapping. You can assess any function A :

T 1 × T 2 × W 1 × W 2 → R ∪ {+∞},
• either by block-aggregation: start by aggregating by time, yielding G T2 G T1 :

W 1 × W 2 → R ∪ {+∞}, then by uncertainty, yielding G W2 G W1 G T2 G T1 A ∈ R ∪ {+∞},
• or by nested-aggregation, yielding

G W2 G T2 G W1 G T1 A ∈ R ∪ {+∞}.
We will show that nested-aggregation produces both time-consistent dynamic risk measures and optimal control problems, and that so does block-aggregation when a commutation property holds true. For example sum and integral are commuting operators and a block-aggregation is equivalent to a nested-aggregation as shown in the following equality Now, let us be more specific.

In stochastic optimal control, one deals with sequential decision-making under uncertainty; with dynamic risk measures, one assesses stochastic processes (costs) as time goes on and information accumulates. We discuss the definition of time-consistency in each setting one after the other (see also [START_REF] Rudloff | Time consistency and risk averse dynamic decision models: Definition, interpretation and practical conse-510 quences[END_REF] for another analysis of links between both notion).

In optimal control problems, we consider a dynamical process that can be influenced by exogenous noises as well as decisions made at every time step.

The decision-maker (DM) wants to optimize a criterion (for instance, minimize a net present value) over a given time horizon. As time goes on and the system evolves, the DM makes observations. Naturally, it is generally more profitable for the DM to adapt his decisions to the observations. He is hence looking for policies (strategies, decision rules) rather than simple decisions: a policy is a function that maps every possible history of the observations to corresponding decisions.

The notion of "consistent course of action" (see [START_REF] Peleg | On the existence of a consistent course of action when tastes are changing[END_REF]) is well-known in the field of economics, with the seminal work of [START_REF] Strotz | Myopia and inconsistency in dynamic utility maximization[END_REF]: an individual having planned his consumption trajectory is consistent if, reevaluating his plans later on, he does not deviate from the originally chosen plan. This idea of consistency as "sticking to one's plan" may be extended to the uncertain case where plans are replaced by decision rules ("Do thus-and-thus if you find yourself in this portion of state space with this amount of time left", Richard Bellman cited in [START_REF] Dreyfus | Richard Bellman on the birth of dynamic programming[END_REF]): [START_REF] Hammond | Changing tastes and coherent dynamic choice[END_REF] addresses "consistency" and "coherent dynamic choice", [START_REF] Kreps | Temporal resolution of uncertainty and dynamic choice theory[END_REF] refers to "temporal consistency".

In this context, we loosely state the property of time-consistency in optimal control problems as follows [START_REF] Carpentier | Dynamic consistency for stochastic optimal control problems[END_REF]. The decision maker formulates an optimization problem at time t 0 that yields a sequence (planning) of optimal decision rules for t 0 and for the following increasing time steps t 1 , . . . , t N = T . Then, at the next time step t 1 , he formulates a new problem starting at t 1 , that yields a new sequence of optimal decision rules from time steps t 1 to T . Suppose the process continues until time T is reached. The sequence of optimization problems is said to be time-consistent if the optimal strategies obtained when solving the original problem at time t 0 remain optimal for all subsequent problems. In other words, time consistency means that strategies obtained by solving the problem at the very first stage do not have to be questioned later on. Now, we turn to dynamic risk measures. At time t 0 , you assess, by means of a risk measure ρ t0,T , the "risk" of a stochastic process A t t N t=t0 , that represents a stream of costs indexed by the increasing time steps t 0 , t 1 , . . . , t N = T . Then, at the next time step t 1 , you assess the risk of the tail A t t N t=t1 of the stochastic process knowing the information obtained and materialized by a σ-field F t1 . For this, you use a conditional risk measure ρ t1,T with values in F t1 -measurable random variables. Suppose the process continues until time T is reached. The sequence ρ t,T t N t=t0 of conditional risk measures is called a dynamic risk measure.

Dynamic or time-consistency has been introduced in the context of risk measures (see [START_REF] Riedel | Dynamic coherent risk measures[END_REF][START_REF] Detlefsen | Conditional and dynamic convex risk measures[END_REF][START_REF] Cheridito | Dynamic monetary risk measures for bounded discrete-time processes[END_REF][START_REF] Artzner | Coherent multiperiod risk adjusted values and Bellmans principle[END_REF][START_REF] Cheridito | Composition of time-consistent dynamic monetary risk measures in discrete time[END_REF] for definitions and properties of coherent and consistent dynamic risk measures). The dynamic risk measure ρ t,T t N t=t0 is said to be time-consistent when the following property holds. Suppose that two streams of costs, A t t N t=t0 and A t t N t=t0 , are such that they coincide from time t i up to time t j > t i and that, from that last time t j , the risk of the tail stream

A t t N t=tj is more than that of A t t N t=tj (i.e. ρ tj ,T ( A t t N t=tj ) ≥ ρ tj ,T ( A t t N t=tj )). Then, the whole stream A t t N t=ti has higher risk than A t t N t=ti (i.e. ρ ti,T ( A t t N t=ti ) ≥ ρ ti,T ( A t t N t=ti )).
We observe that both notions of time-consistency look quite different: the latter is consistency between successive risk assessments of a stochastic process by a dynamic risk measure (a form of monotonicity); the former is consistency between solutions to intertemporal stochastic optimization problems. We now stress the role of information accumulation in both notions of time-consistency, because it highlights how the two notions can be connected. For dynamic risk measures, the flow of information is materialized by a filtration F t t N t=t1 . In stochastic optimal control, an amount of information more modest than the past of exogenous noises is often sufficient to make an optimal decision. In the seminal work of [START_REF] Bellman | Dynamic Programming[END_REF], the minimal information necessary to make optimal decisions is captured in a state variable (see [START_REF] Whittle | Optimization over time[END_REF] for a more formal definition). Moreover, the famous Bellman or Dynamic Programming Equation (DPE) provides a theoretical way to find optimal strategies (see [START_REF] Bertsekas | Dynamic Programming and Optimal Control, 2nd Edition[END_REF] for a broad overview on Dynamic Programming (DP)).

Interestingly, time-consistency in optimal control problems and time-consistency for dynamic risk measures meet in their use of DPEs. On the one hand, in optimal control problems, it is well known that the existence of a DPE with state x for a sequence of optimization problems implies time-consistency when solutions are looked after as feedback policies that are functions of the state x. On the other hand, proving time-consistency for a dynamic risk measure appears rather easy when the corresponding conditional risk measures can be expressed by a nested formulation. In both contexts, such nested formulations are possible only for proper information structures. In optimal control problems, a sequence of optimization problems may be consistent for some information structure while inconsistent for a different one (see [START_REF] Carpentier | Dynamic consistency for stochastic optimal control problems[END_REF]). For dynamic risk measures, timeconsistency appears to be strongly dependent on the underlying information structure (filtration or scenario tree). Moreover, in both contexts, nested formulations and the existence of a DPE are established under various forms of decomposability of operators that display monotonicity and commutation properties.

Our objective is to provide a theoretical framework that offers i) basic ingredients to jointly define dynamic risk measures and corresponding intertemporal optimization problems under uncertainty ii) common sets of assumptions that lead to time-consistency for both. We wish to highlight the role of time and risk preferences, materialized in one-step aggregators, in time-consistency. Depending on how you move from one-step time and risk preferences to intertemporal time and risk preferences, and depending on their compatibility (commutation), you will or will not observe time-consistency. We also shed light on the relevance of information structure by giving an explicit role to a dynamical system with state x.

The paper is organized as follows. In §2, we define dynamic uncertainty criteria ("cousins" of dynamic risk measures) and their time-consistency. Then, we introduce the notions of time and uncertainty-aggregators, define their composition, and show two ways to craft a dynamic uncertainty criterion from onestep aggregators: in the nested-aggregation case, we prove time-consistency; in the block-aggregation case, we have to add a commutation property for this. In §3, we introduce the basic material to formulate intertemporal optimization problems under uncertainty from dynamic uncertainty criteria, and define their time-consistency. In the nested-aggregation case, we prove time-consistency by displaying a DPE; in the block-aggregation case, we have to add a commutation property for this. We end with applications in §4, before concluding.

Notations

We fix notations used throughout the paper:

• [[a, b]] is the set of integers between a and b (included);

• F E, F is the set of functions mapping E into F ;

• u t T 0 is the sequence u 0 , . . . , u T ;

• R = R ∪ {+∞}; • W [0:s] is the Cartesian product W 0 × • • • × W s ;
• G is used to refer to an aggregator with respect to uncertainty;

• φ is used to refer to an aggregator with respect to time.

Furthermore, the superscript notation indicates that the domain of the mapping

G [t:s] is F(W [t:s] ; R) (not to be confused with G [t:s] = G r s r=t ).

Building up Time-Consistent Risk Measures

In §2.1, we lay out adapted uncertainty processes and dynamic uncertainty criterion, cousins of adapted processes and dynamic risk measures, then propose a definition of time-consistency. Then, in §2.2, we introduce the notions of time and uncertainty-aggregators, define their composition, and outline general ways of building a dynamic uncertainty criterion from one-step aggregators. Further, in §2.3 we define nested criterion and give a time-consistency result relying on monotonicity of one-step aggregators. Finally, in §2.4 we define a commutation property between time and uncertainty aggregators. This commutation property allow to give a time-consistency result for a block-aggregated dynamic uncertainty criterion.

Time-Consistent Dynamic Uncertainty Criterion

Inspired by the definitions of risk measures and dynamic risk measures in Mathematical Finance, and motivated by intertemporal optimization, we introduce the following definitions of dynamic uncertainty criterion, and Markov dynamic uncertainty criterion. Such criteria are not restricted to assess measurable mappings (as stochastic processes would be), and we pay the price by letting +∞ be a possible assessment (for instance, the "mathematical expectation" of a non-measurable function is +∞).

Mimicking the definition of adapted processes in probability theory, we first introduce the following definition of adapted uncertainty processes. Definition 1. We say that a sequence

A [0:T ] = A s T 0 is an adapted uncertainty process if A s ∈ F W [0:s] ; R (that is, A s : W [0:s] → R), for all s ∈ [[0, T ]]. In other words, F(W [0:s] ; R) T s=0
is the set of adapted uncertainty processes. Definition 2. A dynamic uncertainty criterion is a sequence { t,T } T t=0 , such that, for all t ∈ [[0, T ]],

• t,T is a mapping

t,T : F(W [0:s] ; R) T s=t → F(W [0:t] ; R) , (1a)
• the restriction of t,T to the domain1 F(W [t:s] ; R)

T s=t yields constant functions, that is,

t,T : F(W [t:s] ; R) T s=t → R . (1b)
A Markov dynamic uncertainty criterion is a dynamic uncertainty criterion parametrized at each step t by a state x t belonging to a state space X t .

We establish a parallel between uncertainty criteria and risk measures. For this purpose, when needed, we implicitely suppose that each uncertainty set W t is endowed with a σ-algebra W t , so that the set W [0:T ] of scenarios is naturally equipped with the filtration

F t = W 0 ⊗ • • • ⊗ W t ⊗ {∅, W t+1 } ⊗ • • • ⊗ {∅, W T } , ∀t ∈ [[0, T ]] . (2)
Notice that, when the σ-algebra W t is the complete σ-algebra made of all subsets of W t , F W [0:t] ; R is exactly the space of random variables that are F tmeasurable. We provide a definition of time-consistency for Markov dynamic uncertainty criteria, inspired by the definition for dynamic risk measures. If two streams of random costs coincide up to time t, but that the tail streams can be ranked pointwise, so will their risk measure.

Definition 3. The Markov dynamic uncertainty criterion {{ xt t,T } xt∈Xt } T t=0 is said to be time-consistent if, for any couple of times 0 ≤ t < t ≤ T , the following property holds true.

If two adapted uncertainty processes {A s } T 0 and {A s } T 0 , satisfy

A s = A s , ∀s ∈ [[t, t]] , (3a)
ρ x t,T {A s } T t ≤ ρ x t,T {A s } T t , ∀x ∈ X t , (3b)
then we have:

ρ x t,T {A s } T t ≤ ρ x t,T {A s } T t , ∀x ∈ X t . (3c)

Aggregators and their Composition

We introduce the notions of time and uncertainty-aggregators, define their composition, and outline general ways of constructing a dynamic uncertainty criterion from one-step aggregators.

One-Step Time-Aggregators and their Composition

For notational clarity the argument of time-aggregators are written between curly braces ({•}) whereas the argument of uncertainty aggregators are written between straight brackets [•]. Definition 4. A multiple-step time-aggregator is a function mapping Rk into R, where k ≥ 2. When k = 2, we call one-step time-aggregator a function mapping R2 into R. A one-step time-aggregator is said to be non-decreasing if it is non-decreasing in its second variable.

Let Φ 1 : R2 → R be a one-step time-aggregator and Φ k : Rk → R be a multiple-step time-aggregator. We define the composition

Φ 1 Φ k : Rk+1 → R by Φ 1 Φ k c 1 , c 2 , . . . , c k+1 = Φ 1 c 1 , Φ k c 2 , . . . , c k+1 . ( 4
)
The composition of multiple one-step time-aggregator is defined recursively by

T -1 s=t Φ s c [t:T ] = Φ t c t , T -1 s=t+1 Φ s c [t+1:T ] . (5)
Example. If each one-step time-aggregators is simply the sum of two terms (Φ t c t , c t+1 = c t + c t+1 ), their composition is just the total sum:

T -1 s=t Φ s c [t:T ] = T s=t c s .
More generally, consider the sequence Φ t T -1 t=0 of one-step time-aggregators given by

Φ t c t , c t+1 = α t (c t ) + β t (c t )c t+1 , ∀t ∈ [[0, T -1]] , (6)
where

(α t ) t∈[[0,T -1]] and (β t ) t∈[[0,T -1]] are sequences of functions, each mapping R into R. With the convention that α T (c T ) = c T , we have T -1 s=t Φ s c s T t = T s=t α s c s s-1 r=t β r c r , ∀t ∈ [[0, T -1]] . (7)

One-Step Uncertainty-Aggregators and their Composition

Definition 5. Let t ∈ [[0, T ]] and s ∈ [[t, T ]]. A [t : s]-multiple-step uncertainty- aggregator is a mapping G [t:s] from F(W [t:s] ; R) into R. When t = s, we call G [t:t] a t-one-step uncertainty-aggregator. A [t : s]-multiple-step uncertainty-
aggregator is said to be non-decreasing if it is monotonous with respect to the pointwise partial order of functions. To a [t : s]-multiple-step uncertainty-aggregator G [t:s] , we attach a mapping

G [t:s] : F(W [0:s] ; R) → F(W [0:t-1] ; R) , (8)
obtained by freezing the first variables (seen as parameters).

We define the notion of chained sequence of uncertainty-aggregators and their composition as follows.

Definition 6. Let t ∈ [[0, T ]] and s ∈ [[t + 1, T ]]. Let G [t:t] : F(W t ; R) → R be a t-one-step uncertainty-aggregator, and G [t+1:s] : F(W [t+1:s] ; R) → R be a [t + 1 : s]-multiple-step uncertainty-aggregator. We define the [t : s]-multiple-step uncertainty-aggregator G [t:t] G [t:s] by, for all function A t ∈ F W [t:s] ; R , G [t:t] G [t:s] A t = G [t:t] w t → G [t+1:s] w [t+1:s] → A t w t , w [t+1:s] . (9)
We say that a sequence

G t T t=0 of one-step uncertainty-aggregators is a chained sequence if G t is a t-one-step uncertainty-aggregator, for all t ∈ [[0, T ]].
Quite naturally, we define the composition of chained sequences by

T s=T G s = G T and T s=t G s = G t T s=t+1 G s . (10)

Time-Consistency for Nested Dynamic Uncertainty Criteria

After having introduced the ingredients of one-step aggregators and their composition, we now cook up nested dynamic uncertainty criterion and prove that they are time-consistent. From a monotonous pair Φ, G , we build a dynamic uncertainty criterion by successive compositions. Definition 8. We inductively define a nested dynamic uncertainty criterion

N t,T T t=0 by N T A T = G T A T , ( 11a
) N t,T A s T s=t = G t Φ t A t , N t+1,T A s T s=t+1 , ∀t ∈ [[0, T -1]] , (11b)
for any adapted uncertainty process A s T s=0 . The following Theorem 9 states that monotonicity is enough to ensure the time-consistency of nested dynamic uncertainty criteria.

A s =A s , ∀s ∈ [[t, t]] , (12a)
N t,T {A s } T t ≤ N t,T {A s } T t , (12b)
We show by backward induction that, for all t ∈ [[t, t]], the following statement (H t ) holds true:

(H t ) N t,T {A s } T t ≤ N t,T {A s } T t . (13)
First, we observe that (H t ) holds true by assumption (12b). Second, by (H t ), and as

A t-1 = A t-1 monotonicity of Φ t-1 yields Φ t-1 A t-1 , N t,T {A s } T t ≤ Φ t-1 A t-1 , N t,T {A s } T t .
Monotonicity of G t-1 then gives

G t-1 Φ t-1 A t-1 , N t,T {A s } T t ≤ G t-1 Φ t-1 A t-1 , N t,T {A s } T t .
By definition of N t-1,T in [START_REF] Artzner | Coherent multiperiod risk adjusted values and Bellmans principle[END_REF], we obtain (H t-1 ).

Remark 10. In Definition 8, we build nested aggregators, first starting by aggregating with respect to time, second with respect to uncertainty. If we aggregate first with respect to uncertainty, second with respect to time, we obtain the dynamic uncertainty criterion given by

N T A T = G T A T , N t,T A s T s=t = Φ t G t A t , G t N t+1,T A s T s=t+1 , ∀t ∈ [[0, T -1]] .
It is shown in [START_REF] Leclère | Contribution to decomposition methods in stochastic optimisation[END_REF]Chapter 2] that monotonicity is also sufficient to yield timeconsistency.

Time-Consistency for Block-Aggregated Criteria 210

Nested uncertainty criteria carry the time-consistency property in the very manner they are built. However, nested uncertainty criterion are not the most natural candidates to assess risk and their economic interpretation is delicate. In practice, uncertainty criterion are more often given in block-aggregation form -integrate with respect to all times, then with respect to all uncertainties -

G [t:s] w [t:s] → Φ c, D t w [t:s] = Φ c, G [t:s] w [t:s] → D t w [t:s] , (14)
for any function D t ∈ F(W [t:s] ; R) and any extended scalar c ∈ R. We say that a (Φ, G) is a commuting pair of sequence of aggregators if G t commutes with Φ s , for any 0 ≤ s < t ≤ T .

Example. If (W t , F t , P t ) is a probability space and if Φ c, c t = α(c) + β(c)c t , (15)
where α : R → R and β : R → R + , then the extended2 expectation G [t:t] = E Pt commutes with Φ.

The following lemma shows how commutation of time-step aggregators leads to commutation of multi-step aggregators.

Lemma 12. Consider a commuting pair (Φ, G) of aggregators. Then,

T s=t G s commutes with Φ r , for any 0 ≤ r < t ≤ T , that is, T s=t G s Φ r c r , A = Φ r c, T s=t G s A , ∀ 0 ≤ r < t ≤ T , (16)
for any extended scalar c ∈ R and any function A ∈ F W [0:T ] ; R .

Beware that, for a given time t, we require the commutation of all subsequent uncertainty-aggregators G t+1 , . . . , G T with the time aggregator Φ t (see Definition 11). The proof of Lemma 12, by induction, is detailed in Appendix B.1.

Time-Consistency for Block-Aggregated Criteria

We now define block dynamic uncertainty criteria and prove their timeconsistency under a commutation property. Definition 13. We define the block dynamic uncertainty criterion

{ B t,T } T t=0 by B t,T = T s=t G s • T -1 s=t Φ s , ∀t ∈ [[0, T -1]] . (17)
The following Theorem 14 is our main result on time-consistency in the block-aggregation case. Proof. Since, for any 0 ≤ s < t ≤ T , G t commutes with Φ s , the blockdynamic uncertainty criterion { B t,T } T t=0 , given by Definition 17, coincides with { N t,T } T t=0 , given by Definition 8. Indeed, we prove that { B t,T } T t=0 satisfies the backward induction [START_REF] Artzner | Coherent multiperiod risk adjusted values and Bellmans principle[END_REF].

With the convention that

T -1 r=T Φ r is the identity mapping, we have B T = G T , that is, (11a).
Then, let t be fixed. For any

A s T t ∈ F(W [0:s] ; R)
T s=t , we have:

B t A s T s=t = s r=t G r T -1 r=t Φ r A s T s=t by (17), = G t s r=t+1 G r T -1 r=t Φ r A s T s=t by (10), = G t s r=t+1 G r Φ t A t , T -1 r=t+1 Φ r A s T s=t+1 by (5), = G t Φ t A t , s r=t+1 G r T -1 r=t+1 Φ r A s T s=t+1
by ( 16)

= G t Φ t A t , B t+1 A s T s=t+1
by [START_REF] Ruszczyski | Risk-averse dynamic programming for Markov decision processes[END_REF].

Thus, { B t,T } T t=0 satisfies the backward induction [START_REF] Artzner | Coherent multiperiod risk adjusted values and Bellmans principle[END_REF]. Remark 15. In Definition 13, we build block-aggregators, first starting by aggregating with respect to time, second with respect to uncertainty. If we aggregate first with respect to uncertainty, second with respect to time, we obtain another dynamic uncertainty criterion. However, to show time-consistency would require monotonicity and another notion of commutation, less widespread than Definition 11. See [START_REF] Leclère | Contribution to decomposition methods in stochastic optimisation[END_REF]Chapter 2] for more details on the subject.

In [START_REF] Leclère | Contribution to decomposition methods in stochastic optimisation[END_REF]Chapter 2], the construction of this §2 is extended to Markovian aggregators. Roughly speaking, the time-step time and uncertainty aggregators are parametrized by a state, that follows a control dynamical system. All results remain true if the commutation property holds true for every possible value of the underlying state.

Building up Time-Consistent Intertemporal Optimization Problems

In §2, we considered dynamic uncertainty criterion that value (the risk of) a stream of costs. We now use such criteria to formulate intertemporal opti-mization problems under uncertainty. In §3.1, we lay out the basic material to formulate intertemporal optimization problems. Then, we provide definition of time-consistency in §3.2, and time-consistency results in §3.3.

Ingredients for Optimal Control Problems

In §3.1.1, we recall the formalism of Control Theory, with dynamical system, state, control and costs. In §3.1.2, we show how to produce adapted uncertainty processes of costs, that will be the inputs to dynamic uncertainty criteria.

Dynamical System, State, Control and Costs

Let T ≥ 2 be an integer. We define a control T -stage dynamical system as follows. We consider sequences of sets of states ( X t T 0 ), controls

( U t T -1 0
), and uncertainties ( W t T 0 ). We also consider sequences of functions representing dynamics

( f t T -1 0 , where f t : X t × U t × W t → X t+1 ) con- straints ( U t T -1 0
, where U t : X t ⇒ U t is a set-valued function), and costs

( J t T -1 0 with J t : X t × U t × W t → R being instantaneous cost functions) and J T : X T × W T → R the final cost function. 3 A policy π = (π t ) T -1
0 is a sequence of functions π t : X t → U t , and we denote by Π the set of all policies. More generally, for all t ∈ [[0, T ]], we call (tail) policy a sequence π = (π s ) T -1 t and we denote by Π t the set of all such policies. We restrict our search of optimal solutions to so-called admissible policies belonging to a subset Π ad ⊂ Π. An admissible policy π ∈ Π ad always satisfies:

π t (x) ∈ U t (x) , ∀x ∈ X t , ∀t ∈ [[0, T -1]] .
We can express in Π ad other types of constraints, such as measurability or integrability ones when we are in a stochastic setting. Naturally, we set Π ad t = Π t ∩ Π ad . Definition 16. For any time t ∈ [[0, T ]], state x ∈ X t and policy π ∈ Π, the flow {X x,π t,s } T s=t of the dynamics

f t T -1 0
is defined by the forward induction:

∀w ∈ W [0:T ] ,    X x,π t,t (w) = x , X x,π t,s+1 (w) = f s X x,π t,s (w), π s (X x,π t,s (w)), w s , ∀s ∈ [[t, T ]] . (18)
The expression X x,π t,s (w) is the state Remark 17. For 0 ≤ t ≤ s ≤ T , the flow X x,π t,s is a function that maps the set W [0:T ] of scenarios into the state space X s :

x s ∈ X s reached at time s ∈ [[0, T ]],
X x,π t,s : W [0:T ] → X s . (19)
By (18),

• when t > 0, the expression X x,π t,s (w) depends only on the inner part w [t:s-1] of the scenario w = w [0:T ] , hence depends neither on the head w [0:t-1] , nor on the tail w [s:T ] ,

• when t = 0, the expression X x,π 0,s (w) in (18) depends only on the head w [0:s-1] of the scenario w = w [0:T ] , hence does not depend on the tail w [s:T ] . This is why we often consider that the flow X x,π t,s is a function that maps the set W [t:s-1] of scenarios into the state space X s :

X x,π t,s : W [t:s-1] → X s , ∀s ∈ [[1, T ]] , ∀t ∈ [[0, s -1]] . (20)
A state trajectory is a realization of the flow X x,π 0,s (w)

T s=0 for a given scenario w ∈ W [0:T ] . The flow property X x,π t,s ≡ X X x,π t,s ,π s ,s , ∀t, s, s , t < s < s , ∀x ∈ X t [START_REF] Clark | Mathematical Bioeconomics[END_REF] expresses the fact that we can stop anywhere along a state trajectory and start again.

Producing Streams of Costs

With a policy and a scenario, we obtain state and control trajectories that we plug into the instantaneous cost functions J s to deliver streams of costs. Definition 18. For a given policy π ∈ Π, and for all times t ∈ [[0, T ]] and s ∈ [[t, T ]], we define the uncertain costs evaluated along the state trajectories by:

J x,π t,s : w ∈ W [0:T ] -→ J s X x,π t,s (w), π X x,π t,s (w) , w s . ( 22
)
Remark 19. By Remark 17, we consider that J x,π t,s is a function that maps the set W [t:s] of scenarios into R:

J x,π t,s : W [t:s] → R , ∀s ∈ [[0, T ]] , ∀t ∈ [[0, s]] . ( 23
)
As a consequence, the stream J x,π 0,s T s=0 of costs is an adapted uncertainty process (see Definition 1). By ( 22) and ( 18), we have, for all t ∈ [[0, T ]] and s ∈

[[t + 1, T ]], ∀w [t:T ] ∈ W [t:T ] ,      J x,π t,t (w t ) = J t x, π t (x), w t , J x,π t,s (w t , {w r } T t+1 ) = J ft(x,πt(x),wt),π t+1,s ({w r } T t+1 ) . (24)
We relate dynamic uncertainty criteria and optimization problems as follows.

Definition of Time-Consistency for Markov Optimization Problems

With the formalism of §2.1, we can now give a definition of time-consistency for Markov optimization problems.

For the clarity of exposition, suppose for a moment that any optimization problem (P t )(x) in (25) has a unique solution, a policy that we denote π t,x = {π s t,x } T -1 s=t ∈ Π ad t to stress that it parametrically depends on t, x as (P t )(x) does. Consider 0 ≤ t < t ≤ T . Suppose that, starting from the state x at time t, the flow (18) drives you to

x = X x,π t,t (w) , π = π t,x (26)
at time t, along the scenario w ∈ W [0:T ] and adopting the optimal policy π t,x ∈ Π ad t . Arrived at x, you solve (P t )(x) and get the optimal policy π t,x = {π s t,x } T -1 s=t ∈ Π ad t . Time-consistency holds true when

π s t,x = π s t,x , ∀s ≥ t , (27)
that is, when the "new" optimal policy, obtained by solving (P t )(x), coincides, after time t, with the "old" optimal policy, obtained by solving (P t )(x). In other words, you "stick to your plans" (here, a plan is a policy) and do not reconsider your policy whenever you stop along an optimal path and optimize ahead from this stop point.

To account for non-uniqueness of optimal policies, we propose the following formal definition.

Definition 21. We say that the Markov optimization problem (25) of Definition 20 is time-consistent if, for any couple of times t ≤ t in [[0, T ]] and any state x ∈ X t , the following property holds: there exists a policy π = {π s } T -1 s=t ∈ Π ad t such that

• {π s } T -1
s=t is optimal for Problem P t (x);

• the tail policy {π s } T -1 s=t is optimal for Problem P t (x), where x ∈ X t is any state achieved by the flow X

x,π t,t in [START_REF] Shapiro | Minimax and risk averse multistage stochastic programming[END_REF].

We stress that the above definition of time-consistency of a sequence of families of optimization problems is contingent on the state x and on the dy-

namics f t T -1 0
by the flow [START_REF] Shapiro | Minimax and risk averse multistage stochastic programming[END_REF]. In particular, we assume that, at each time step, the control is taken only in function of the state: this defines the class of solutions as policies that are feedbacks of the state x (such restriction is justified in the Markovian case, for example).

Time-Consistency for Optimal Control Problems

We now provide time-consistency results, differing whether in the nestedaggregation or in the block-aggregation case.

Time-Consistency for Nested Criteria

We define the nested Markov optimization problem formally by

(P N t )(x) V N t (x) = min π∈Π ad t N t,T J x,π t,s T s=t , ∀t ∈ [[0, T ]] , ∀x ∈ X t , ( 28
)
where the functions J x,π t,s are defined by [START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF], and the nested uncertainty criterion N t,T T t=0 by [START_REF] Artzner | Coherent multiperiod risk adjusted values and Bellmans principle[END_REF].

The following Proposition 22 expresses sufficient conditions under which any Problem (P N t )(x), for any time t ∈ [[0, T -1]] and any state x ∈ X t , can be solved by means of a Dynamic Programming Equation (DPE).

Proposition 22. Let Φ, G be a monotonous pair of sequences of aggregators. Assume that there exists an admissible policy π ∈ Π ad such that

π t (x) ∈ arg min u∈Ut(x) G t Φ t J t (x, u, •),V N t+1 • f t (x, u, •) , ∀t ∈ [[0, T -1]] , ∀x ∈ X t . ( 29
)
Then, π is an optimal policy for any Problem (P N t )(x), for all t ∈ [[0, T ]] and for all x ∈ X t , and the value functions V t satisfy the DPE

V N T (x) = G T J T (x, •) , ∀x ∈ X T , (30a)
V N t (x) = min u∈Ut(x) G t Φ t J t (x, u, •), V N t+1 • f t (x, u, •) , ( 30b
) ∀t ∈ [[0, T -1]] , ∀x ∈ X t .
Remark 23. It may be difficult to prove the existence of a measurable selection among the solutions of (29). Since it is not our intent to consider such issues, we make the assumption that an admissible policy π ∈ Π ad exists, where the definition of the set Π ad is supposed to include all proper measurability conditions.

The proof of Proposition 22 is detailed in the appendix.

The following Theorem 9 is our main result on time-consistency in the nested-aggregation case. defined in (28) is time-consistent, as soon as there exists an admissible policy π ∈ Π ad such that (29) holds true.

Time-Consistency for Block-Aggregated Criteria

We define the block Markov optimization problem formally by

(P B t )(x) min π∈Π ad t B t,T J x,π t,s T s=t , ∀t ∈ [[0, T ]] , ∀x ∈ X t , (31)
where the functions J x,π t,s are defined by [START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF], and the block uncertainty criterion B t,T T t=0 by [START_REF] Ruszczyski | Risk-averse dynamic programming for Markov decision processes[END_REF]. The following Theorem 25 is our main result on time-consistency in the block-aggregation case. We do not give the proof since it directly follows from Theorem 9 and Theorem 14.

Theorem 25. Let Φ, G be a monotonous and commuting pair of sequences of aggregators. Then In [START_REF] Leclère | Contribution to decomposition methods in stochastic optimisation[END_REF]Chapter 2], the time consitency result is extended to Markovian aggregators, made of one-step time and uncertainty aggregators parameterized by the state. All results remain true if the commutation property holds true for every possible value of the state.

Applications

We end by providing classes of dynamic uncertainty criteria and corresponding intertemporal optimization problems that display time-consistency, as well as examples of applications.

Coherent Risk Measures

We introduce a class of dynamic uncertainty criteria, that are related to coherent risk measures, and we show that they display time-consistency. We thus extend, to more general one-step time-aggregators, results known for the sum (see e.g. [START_REF] Ruszczyski | Risk-averse dynamic programming for Markov decision processes[END_REF][START_REF] Shapiro | Minimax and risk averse multistage stochastic programming[END_REF]).

We denote by P(W t ) the set of probabilities over the set W t endowed with the σ-algebra W t . Let P 0 ⊂ P(W 0 ), . . . , P T ⊂ P(W T ). If A and B are sets of probabilities, then A ⊗ B is defined as

A ⊗ B = {P A ⊗ P B |P A ∈ A , P B ∈ B} . (32) Let (α t ) t∈[[0,T -1]] and (β t ) t∈[[0,T -1]
] be sequences of functions, each mapping R into R, with the additional property that

β t ≥ 0, for all t ∈ [[0, T -1]]. We set, for all t ∈ [[0, T ]], co t,T A s T t = sup Pt∈Pt E Pt • • • sup P T ∈P T E P T T s=t α s A s s-1 r=t β r A r • • • , (33)
min π∈Π ad co t,T J x,π t,s T s=t , ∀t ∈ [[0, T ]] , ∀x ∈ X t , (34)
where J x,π t,s (w) is defined by [START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF], as soon as there exists an admissible policy π ∈ Π ad such that, for all

t ∈ [[0, T -1]], for all x ∈ X t , π t (x) ∈ arg min u∈Ut(x) sup Pt∈Pt E Pt α t J t (x, u, •) +β t J t (x, u, •) V t+1 •f t (x, u, •) ,
where the value functions are given by the following DPE

V T (x) = sup P T ∈P T E P T J T (x, •) , (35a)
V t (x) = min u∈Ut(x) sup Pt∈Pt E Pt α t J t (x, u, •) (35b)
+ β t J t (x, u, •) V t+1 • f t (x, u, •) .
Proof. The setting is that of Theorem 14 and Proposition 22, where the onestep time-aggregators are defined by

Φ t c t , c t+1 = α t (c t ) + β t (c t )c t+1 , ∀t ∈ [[0, T -1]] , ∀ c t , c t+1 ∈ R2 , (36a)
and the one-step uncertainty-aggregators are defined by

G t C t = sup Pt∈Pt E Pt C t , ∀t ∈ [[0, T -1]] , ∀C t ∈ F(W t ; R) . (36b)
The DPE (35) is the DPE (30), which holds true as soon as the assumptions of Theorem 14 hold true (mainly that we have a commuting monotonous pair of operators). First, we prove that, for any 0 ≤ t < s ≤ T , G s commutes with Φ t (this is a special case of Proposition 30 shown in appendix). Indeed, letting c t be an extended real number in R and C s a function in F(W s ; R), we have

G s Φ t {c t , C s } = sup Ps∈Ps E Ps α(c t ) + β(c t )C s by (36b) and (36a), = α t (c t ) + β t (c t ) sup Ps∈Ps E Ps [C s ] as β t ≥ 0 , = α t (c t ) + β t (c t )G s [C s ] by (36b), = Φ t {c t , G s [C s ]} by (36a).
Second, we observe that G t is non-decreasing (see Definition 5), and that c t+1 ∈ R → Φ t c t , c t+1 = α t (c t ) + β t (c t )c t+1 is non-decreasing, for any c t ∈ R.

The one-step uncertainty-aggregators G t in (36b) correspond to a coherent risk measure (see [START_REF] Artzner | Coherent measures of risk[END_REF]). In fact, our result extends to aggregators of the form

θ sup P∈P E P +(1-θ) inf Q∈Q E Q , with θ ∈ [0, 1]
, and even to more complex convex combinations of infima and suprema as shown in Proposition 31 in Appendix A.

Our result extends to a Markovian setting where one-step aggregators G t and Φ t are indexed by the state x. More precisely, the coefficient α t and β t and the set of probability P t can depend on the state x.

Our result differs from the work of A. Ruszczyński [START_REF] Ruszczyski | Risk-averse dynamic programming for Markov decision processes[END_REF]Theorem 2] in two ways. On the one hand, Ruszczyński provides arguments to show that there exists an optimal Markovian policy among the set of adapted policies (that is, having a policy taking as argument the whole past uncertainties would not give a better cost than a policy taking as argument the current value of the state). We do not tackle this issue since we directly deal with policies as functions of the state. Where we suppose that there exists an admissible policy π ∈ Π ad such that (29) holds true, Ruszczyński gives conditions ensuring this property. On the other hand, where Ruszczyński restricts to the sum to aggregate instantaneous costs, we consider more general one-step time-aggregators Φ t . For instance, our results applies to the product of costs.

Convex Risk Measures

We introduce a class of dynamic uncertainty criteria, that are related to convex risk measures (see [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]), and we show that they display time-consistency.

We consider the same setting as in §4.1, with the restriction that β t ≡ 1 in (33) and an additional data (Υ t ) t∈[[0,T ]] .

Let P 0 ⊂ P(W 0 ), . . . , P T ⊂ P(W T ), and (Υ t ) t∈[[0,T ]] be sequence of functions, each mapping P(W t ) into R. Let (α t ) t∈[[0,T ]] be sequence of functions, each mapping R into R. We set, for all t ∈ [[0, T ]],

cx t,T ( A s T t ) = sup Pt∈Pt E Pt • • • sup P T ∈P T E P T T s=t α s (A s ) -Υ s (P s ) • • • , (37)
for any adapted uncertain process A t T 0 , with the convention that α T (c T ) = c T .

Proposition 28. Time-consistency holds true for 420

• the dynamic uncertainty criterion { cx t,T } T t=0 given by (37),

• the Markov optimization problem

min π∈Π ad cx t,T ( J x,π t,s T s=t ) , ∀t ∈ [[0, T ]] , ∀x ∈ X t , (38)
where J x,π t,s (w) is defined by [START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF], as soon as there exists an admissible policy π ∈ Π ad such that, for all t ∈ [[0, T -1]], for all x ∈ X t ,

π t (x) ∈ arg min u∈Ut(x) sup Pt∈Pt E Pt α t J t (x, u, •) + V t+1 • f t (x, u, •) -Υ t (P t ) ,
where the value functions are given by the following DPE

V T (x) = sup P T ∈P T E P T J T (x, •) -Υ T (P T ) , (39a)
V t (x) = min u∈Ut(x) sup Pt∈Pt E Pt α t J t (x, u, •) + V t+1 • f t (x, u, •) -Υ t (P t ) . (39b)
Proof. We follow the proof of Proposition 27, where the one-step time-aggregators are defined by

Φ t c t , c t+1 = α t (c t ) + c t+1 , ∀t ∈ [[0, T -1]] , ∀ c t , c t+1 ∈ R2 , (40a)
and the one-step uncertainty-aggregators are defined by

G t C t = sup Pt∈Pt E Pt C t -Υ t (P t ) , ∀t ∈ [[0, T -1]] , ∀C t ∈ F(W t ; R) . ( 40b
= α t (c t ) + sup Ps∈Ps E Ps [C s ] -Υ s (P s ) = α t (c t ) + G s [C s ] by (40b) = Φ t {c t , G s [C s ]} by (40a).
This ends the proof.

The one-step uncertainty-aggregators G t in (40b) correspond to a convex risk measure (see [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]). Moreover, Proposition 31 shows that we could also consider any positive linear combination of suprema and infima of expectation. A special case of coherent risk measures consists of the worst case scenario operators, also called "fear operators". For this subclass of coherent risk measures, we show that time-consistency holds for a larger class of time-aggregators than the ones above.

430

For any t ∈ [[0, T -1]], let W t be a non empty subset of W t , and let Φ t : R2 → R be a function which is continuous and non-decreasing in its second variable. We set, for all t ∈ [[0, T ]],

wc t,T A s T t = sup ws T t ∈ Wtו••× W T Φ t A t ( w s T t ), Φ t+1 • • • , Φ T -1 A T -1 (w T -1 , w T ), A T (w T ) , (41)
where J x,π t,s (w) is defined by [START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF], as soon as there exists an admissible policy π ∈ Π ad such that, for all t ∈ [[0, T -1]], for all x ∈ X t ,

π t (x) ∈ arg min u∈Ut(x) sup wt∈ Wt Φ t J t x, u, w t , V t+1 • f t x, u, w t ,
where the value functions are given by the following DPE

V T (x) = sup w T ∈ W T J T (x, w T ) , (43a)
V t (x) = min u∈Ut(x) sup wt∈ Wt Φ t J t x, u, w t , V t+1 • f t x, u, w t . (43b)
Proof. We follow the proof of Proposition 27, where This ends the proof.

G t C t = sup wt∈ Wt C t (w t ) , ∀t ∈ [[0, T -1]] , ∀C t ∈ F(W t ; R) . (44)

Examples

State-dependent discounting and beliefs

We consider the following long term investment problem. Let J t x t , u t , w t be the cost incurred at time t in the state x t , under decision u t and uncertainty w t . The state x t includes economic indicators, one of them affecting the discount factor e -rt (xt) . Hence, the time-aggregation of the cost process is given by

T -1 t=0 e -rt(I t ) J t X t , U t , W t . (45)
We suppose that the one-step uncertainty aggregators are coherent risk measures

G xt t • = sup Q∈P(xt) E Q • , (46)
where the probability set P(x t ) of beliefs is allowed to depend on the economic indicators in x t . Such an optimization problem, where both discounting and beliefs depend on

Non-additive time preferences

In environmental economics literature, when time spans across generations (like with climate change issues), scholars discuss the use of additive time preferences. Indeed, additivity implies possible compensations between distant generations, and discounting can lead to myopic decisions [START_REF] Clark | Mathematical Bioeconomics[END_REF][START_REF] Weitzman | A review of the Stern review on the economics of climate change[END_REF]. The so-called Rawls or maximin criterion [START_REF] Lara | Sustainable management of natural resources[END_REF] min

J 0 x 0 , u 0 , w 0 , • • • , J T -1 x T -1 , u T -1 , w T -1 (47)
is a possible alternative, which can be obtained by aggregation of one-step timeaggregators Φ t c t , c t+1 = max c t , c t+1 . Now, when the uncertainty aggregator is the worst-case operator over the Cartesian product

W t × • • • × W T -1 ,
the resulting optimization problem falls into the framework developped in §4.3.

Risk-sensitive optimization

We consider a family of risk-sensitive optimization problems

min log E exp T -1 t=t0 J t X t , U t , W t (48)
X t+1 = f t X t , U t , W t , X t0 = x 0 U t ∈ U t (X t ) .
This family is time-consistent, after an equivalent reformulation as follows.

First, as the function log is increasing, it is equivalent to minimize its argument. Second, denoting Jt X t , U t , W t = exp J t X t , U t , W t , we arrive at the following optimization problem

min E T -1 t=t0 Jt X t , U t , W t X t+1 = f t X t , U t , W t , X t0 = x 0 U t ∈ U t (X t ) .
Hence, by changing costs, we are falling into the setting of Proposition 27, and we can write a time-consistent Markov decision problem equivalent to Problem 48.

Conclusion

We have provided basic ingredients -one-step time and uncertainty aggregators -to make up dynamic uncertainty criteria and corresponding intertemporal optimization problems under uncertainty. Nested criteria carry the time-consistency property in the very manner they are built, only relying on monotonicity property. Block-aggregated are more natural candidates than nested ones as dynamic uncertainty criteria. However, we prove that they display time-consistency at the additional price of commutation between one-step time and uncertainty aggregators.

Thus equipped, we have tools to cook up time-consistent dynamic uncertainty criteria and intertemporal optimization problems under uncertainty. Moreover, our framework extends to the Markovian setting, where the one-step time and uncertainty aggregators are parametrized by the state, that follows a control dynamical system. This is how our results cover a large class of applications. However, our results also point to the fact that, in practice, time-consistency is mostly related to linearity and expectation (see [START_REF] Shapiro | Minimax and risk averse multistage stochastic programming[END_REF] for some comments on the subject). Other ways to obtain time-consistent formulation consist in using either a worst case approach, or a nested formulation from the beginning.

We think that an interesting question for further research is the following. Given a dynamic risk measure or an intertemporal optimization problem under uncertainty, can we identify a state or a new state such that time-consistency holds true with this new information setting? A first set of answers can be found in [START_REF] Pflug | On dynamic decomposition of multistage stochastic programs[END_REF]. We think that our analysis provides insight to look after the properties expected from such a state.

Appendix A. Constructing new commuting aggregators

We show that we can construct new commuting one-step uncertainty aggregators either as linear combination of suprema or infima of one-step commuting aggregators.

Proposition 30. Let Φ be a one-step time-aggregator and (G i t ) i∈I be a family of one-step uncertainty aggregators. Suppose that G i t commutes with Φ, for all i ∈ I, and that

• either, for all c ∈ R, Φ c, • is continuous and non-decreasing;

• or, for all C t ∈ F(W t , R), sup i∈I G i t [C t ] is attained (always true for I finite).

Then, the one-step uncertainty-aggregator sup i∈I G i t commutes with Φ, and so does inf i∈I G i t , provided inf i∈I G i t never takes the value -∞.

Proof. We consider the supremum case, the infimum being similar. For any (c, C t ) ∈ R × F(W t , R), we have The last equality being obtained either because the supremum is attained or by continuity.

The following Proposition allows to build uncertainty aggregators commuting with affine or linear (in the second variable) time-aggregator.

Proposition 31. Let Φ be a one-step time-aggregator and (G i t ) i∈I be a family of one-step uncertainty aggregators. Suppose that for all i ∈ I, the timestep uncertainty aggregator G i commutes with the time-step time aggregator Φ.

G i t C t + n j=1 θ j sup i∈Ij G i t C t = Φ c, Ḡ C t ,
where the last equality is obtained either because the coefficient θ j sum to one 495 (affine case) or because α is equal to zero (linear case). This expression is well defined because J x,π t,s : W [t:s] → R, for s ∈ [[t, T ]] by [START_REF] Lara | Sustainable management of natural resources[END_REF]. First, we show that the functions {V π t } T t=0 satisfy a backward equation "à la Bellman": We also have, for t ∈ [[0, T -1]],

V π t (x) = G t Φ t J t (x, π t (x), •), V π t+1 •f t (x,
V π t (x) = N t,T J x,
= G t Φ t J t (x, π t (x), •), V π t+1 • f t (x, π t (x), •) .

Second, we show that V t (x), as defined in (30) is lower than the value of the optimization problem P N t (x) in (28). For this purpose, we denote by (H t ) the following assertion

(H t ) : ∀x ∈ X t , ∀π ∈ Π ad , V t (x) ≤ V π t (x) .
By definition of V π T (x) in (B.3) and of V T (x) in (30a), assertion (H T ) is true.

505

Now, assume that (H t+1 ) holds true. Let x be an element of X t . Then, by definition of V t (x) in (30b), we obtain V t (x) ≤ inf π∈Π ad G t Φ t J t x, π t (x), • , V t+1 • f t x, π t (x), • , (B.5)

X×Y ×Z [c 1

 1 (x) + c 2 (x, y) + c 3 (x, y, z)]dxdydz = X c 1 (x) + Y c 2 (x, y) + Z c 3 (x, y, z)dx dy dz .

Definition 7 .

 7 A monotonous pair Φ, G of sequences of aggregators consists in • a sequence Φ t T -1 t=0 of non-decreasing one-step time-aggregators, • a chained sequence G t T t=0 of non-decreasing one-step uncertainty-aggregators.

Theorem 9 .

 9 Let Φ, G be a monotonous pair of sequences of aggregators.Then, the nested dynamic uncertainty criterion N t,T T t=0 defined by (11) is time-consistent (in the sense of Definition 3).

  Proof. Let t < t be both in [[0, T ]]. Consider two adapted uncertainty processes {A s } T 0 and {A s } T 0 , where A s and A s maps W [0:T ] into R, such that

2. 4 . 1 .

 41 Commutation of Aggregators Definition 11. Let t ∈ [[0, T ]] and s ∈ [[t+1, T ]]. A [t : s]-multiple-step uncertaintyaggregator G [t:s] is said to commute with a one-step time-aggregator Φ if

Theorem 14 .

 14 Let (Φ, G) be a commuting and monotonous pair of sequences of aggregators. Then the block dynamic uncertainty criterion B t,T T t=0 defined by (17) is time-consistent.

  when starting at time t ∈ [[0, s]] from state x ∈ X t and following the dynamics f r T -1 0 with the policy π ∈ Π along the scenario w ∈ W [0:T ] .

Definition 20 .

 20 Given a Markov dynamic uncertainty criterion Markov optimization problem as the following sequence of families of optimization problems, indexed by t ∈ [[0, T ]], and x ∈ X t :(P t )(x)min 25) is indeed well defined by (1b), because J x,π

  for any adapted uncertain process A t T 0 , with the convention that α T (c T ) = 385 c T . Proposition 27. Time-consistency holds true for • the dynamic uncertainty criterion { co t,T } T t=0 given by (33), • the Markov optimization problem

425 4 . 3 .

 43 Worst-Case Risk Measures (Fear Operator)

  Ḡt Φ c, C t = sup i∈I G i t Φ c, C t by definition of Ḡt , = sup i∈I Φ c, G i t C t by commutation of G i and Φ, = Φ c, sup i∈I G i t C t .

•

  For j ∈ [[1, n]], let I j ⊂ I, and I j ⊂ I, be families of subsets of I, and({θ j } j∈[[1,N ]] , {θ j } j∈[[1,N ]]) be non-negative scalars. We define If Φ is affine in its second variable, that is, ifΦ c, d = α(c) + β(c)d , (A.2)and if({θ j } j∈[[1,N ]] , {θ j } j∈[[1,N ]]) sum to one, then the convex combina-490 tion Ḡ of infima and suprema of subfamilies of {G i t } i∈I commutes with Φ, provided inf i∈I j G i t never takes the value -∞.• If Φ is linear in its second variable, that is, if Φ c, d = β(c)d , (A.3)then the linear combination Ḡ of infima and suprema of subfamilies of {G i t } i∈I commutes with Φ.Proof. Assume that Φ is given by (A.2)

  cost from time t to time T when following policy π starting from state x: [0, T ]] , ∀x ∈ X t .(B.3)

  π t (x), •) , ∀t ∈ [[0, T -1]] , ∀x ∈ X t . (B.[START_REF] Dreyfus | Richard Bellman on the birth of dynamic programming[END_REF]) Indeed, we have,V π T (x) = N T,T J x,π T,T by the definition (B.3) of V π T (x), = N T,T J T (x, •) by (22) that defines J x,π T,T , = G T J T (x, •) by the definition (11a) of N T , = G T J T (x, •)by Definition of G T .

  Theorem 24. Let Φ, G be a monotonous pair of sequences of aggregators.

	Then
	1. the nested dynamic uncertainty criterion N t,T consistent;	T t=0 defined by (11) is time-
	2. the Markov optimization problem	(P N t )(x) x∈Xt	T t=0

)

  We show that, for any t ∈ [[0, T -1]] and s ∈ [[t + 1, T ]], G s commutes with Φ t (we could also apply Proposition 31). Letting c t be an extended real number in R and C s a function in F(W s ; R), we have G s Φ t {c t , C s } = sup Ps α(c t ) + C s -Υ s (P s ) by (40a) and (40b)

Ps∈Ps

E

  We prove that, for any t ∈ [[0, T -1]] and s ∈ [[t + 1, T ]], G s commutes with Φ t . Letting c t be an extended real number in R and C s a function in F(W s ; R), we haveG s Φ t {c t , C s } = sup

	Φ t c t , C s (w s )	by (44),
	ws∈ Ws
	= Φ t c t , sup	C s (w s )	by continuity of Φ t {c t , •} ,
	w∈ Ws
	= Φ t c t , G s [C s ]	by (44).

Where F (W [t:s] ; R) is naturally identified as a subset of F (W [0:s] ; R).

than in nested form.We first propose a commutation property that will allow to go from one formulation to the other, and that will stand as one of the key ingredients for a DPE and lead to the time-consistency result in Theorem 14.

We set β ≥ 0, so that, when Ct ∈ F (Wt; R) is not integrable with respect to Pt, the equality (14) still holds true.

For notational consistency with the Jt for t = [[0, T -1]], we will often write J T (x, u, w) to mean J T (x, w).

the state, falls into the framework developped in §4.1 in its Markovian version.

Appendix B.1. Proof of Lemma 12 Proof. We prove by induction that

for any extended scalar c ∈ R and any function

let (H t ) be the following assertion

The assertion (H T ) coincides with the property that, for any 0 ≤ r < T , G T commutes with Φ r (apply [START_REF] Whittle | Optimization over time[END_REF] where t = T , Φ = Φ r ), and hence holds true. Now, suppose that (H t+1 ) holds true. Let r < t, c ∈ R and

and where, for all w t , D t+1 :

by commutation property ( 14) of G t with Φ = Φ r , since 0 ≤ r < t ≤ T , and where

This ends the induction, hence the proof of (B.1) which leads to [START_REF] Leclère | Contribution to decomposition methods in stochastic optimisation[END_REF].

Appendix B.2. Proof of Proposition 22

Proof. In the proof, we denote by V t the sequence of function given by (30). Let π ∈ Π ad be a policy. For any t ∈ [[0, T ]], we define V π t (x) as the intertemporal since, for all π ∈ Π ad we have π t (x) ∈ U t (x). By (H t+1 ) we have, for any

• . From monotonicity of Φ t and monotonicity of G t , we deduce:

We obtain:

Hence, assertion (H t ) holds true.

Third, we show that the lower bound V t (x) for the value of the optimization problem P N t (x) is achieved for the policy π in (29). For this purpose, we consider the following assertion

By definition of V π T (x) in (B.3) and of V T (x) in (30a), (H T ) holds true. For t ∈ [[0, T -1]], assume that (H t+1 ) holds true. Let x be in X t . We have

by definition of π in (29), = G t Φ t J t x, π t (x), • , V π t+1 • f t (x, π t (x), •) by (H t+1 ) = V π t (x) by (B.3).

Hence (H t ) holds true, and the proof is complete by induction.


Géraldine Féraud

Natalia Esteves-Lopez

Claude Dedonder

Christophe Jouvet
email: christophe.jouvet@univ-amu.fr

Natalia Esteves-López

Claude Dedonder-Lardeux

UV spectroscopy of cold ions as a probe of the protonation site

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Protonated aromatic hydrocarbon molecules, AH+, constitute a fundamental class of organic molecules occurring as short-lived intermediates in various environments, from astrochemistry, jet engine gas exhaust, organic chemistry to biophysics. For example, AH+ are widely accepted as intermediates in electrophilic aromatic substitution reactions (σ complexes), an important reaction mechanism of aromatic molecules, and thus the protonation site governs the stereochemistry.

In the past, NMR and IR spectroscopy were employed to characterize AH+ ions in the condensed phase [START_REF]Carbonium ions[END_REF][START_REF]Carbocation Chemistry[END_REF][START_REF] Olah | [END_REF] . In solution, intrinsic properties of AH+ are difficult to characterize, due to the strong interaction with surrounding solvent molecules and counter ions. Gas-phase studies of AH+ are required to elucidate the properties of the ions. Structural and energetic information have been obtained from mass spectrometry experiments, but the results are not unambiguous. In particular, these techniques provide indirect structural determination of the protonation site, since the information is basically obtained from systems heated either by collisions leading to dissociation or by collision with a proton acceptor/donor. In both cases, the collisional energies involved are larger than the energy for proton isomerization (in the order of 1eV) and the proton is not necessarily well localized (mobile proton) when the reactive event occurs.

Recently spectroscopic techniques, mainly IR spectroscopy, have been extensively used to derive the structure of the ions. [4][5][6][7][8][9][10][11][12] With the development of mass spectrometry experiments near the free electron laser facility [13][14][15] the infrared spectra are obtained from multiphoton absorption leading to dissociation, which is not a linear process, thus there is no guaranties that all the vibrations can be observed. Additionally, the excitation energies being high and the experiments being performed at room temperature, vibrational bands are not very narrow and more than one tautomer can coexist in the ion trap. This makes the interpretation difficult, and as shown for example in the case of Ala-Ala 16 the position of the proton may not be clearly defined (mobile proton).

The electronic spectroscopy of cold protonated ions has made considerable progress these last past ten years with the development of cold ion traps [17][18][19][20][21] and the question we would like to address in this paper is the following: can we identify the protonation site by analyzing the electronic spectra? To answer this question, we studied the different isomers of a simple protonated aromatic molecule, aminophenol i.e., o-aminophenolH+ (or 2-aminophenolH + ), m-aminophenol (or 3-aminophenolH + ), paminophenol (or 4-aminophenolH + ). Indeed, the experimental results presented here show that the electronic spectroscopy of cold ions, in combination with excited-state ab initio calculations, is an alternative tool to get information on the protonation site.

Moreover, there is an active discussion on the relationship between the most stable ions in solution and the ions obtained in the gas phase after the electrospray process 22,23 . For aminophenol the solvent stabilization of one conformer versus the other ones is quite large, changing the most stable tautomer between the gas phase and the liquid phase (vide infra). This is then an ideal system to discuss this issue.

Methodology Experimental

The experimental setup has been presented previously 19 and is just outlined here. The electronic spectra of the protonated aromatic ions were obtained via parent ion photo-fragment spectroscopy in a cryogenically-cooled Paul trap, the protonated ions being produced in an electrospray source. The ions are injected in the trap and thermalized at a temperature between 20 and 50 K through collisions with the cold Helium buffer gas. The ions are kept in the trap for several tens of ms before the photo-dissociation laser is triggered. After laser excitation, the ions are extracted to a TOF mass spectrometer and the fragment(s) ion signal is recorded as the laser wavelength is scanned. The photo-dissociation laser is an OPO (NT342B) laser from EKSPLA. The absence of vibrational hot bands indicates that the temperature of the ions is less than 50K (see the case of protonated acridine in Figure SI.2 of the supplementary information).

Calculations

Ab initio calculations have been performed with the TURBOMOLE program package 24 , making use of the resolution-of-the-identity (RI) approximation 25 for the evaluation of the electronrepulsion integrals. The equilibrium geometry of the protonated species in their ground states (S 0 ) has been determined at the MP2 and DFT/B3LYP levels. Vertical excitation energies of the lowest excited singlet states have been determined at the RI-ADC(2) (second order Algebraic Diagrammatic Construction) 26 level since TD-DFT is suffering from the bad representation of charge transfer states. Calculations were performed with the correlation-consistent polarized valence double-zeta (cc-pVDZ) basis set, and some values were checked with the triple-zeta cc-pVTZbasis set. To get an idea on the relative stability of the proton site in the liquid phase, we used the Conductor-like Screening Model (COSMO) 27 as implemented in Turbomole.

Results and discussion

The experimental photofragmentation spectra obtained for the three isomers are presented in Figure 1. These spectra are the sum of the fragment ions as a function of the laser wavelength. For p-aminophenolH + , the two main fragmentation channels are the loss of NH 3 and loss of hydrogen atom, the loss of H 2 O being a minor channel. The H loss channel is not seen in CID experiments, it is specific of the electronic excitation and has been explained in the framework of the ππ * /πσ * interaction model 28 in a previous paper 29 .This channel is not observed in the photo-dissociation of mand o-aminophenolH + . For meta and ortho aminophenolH + , the fragments are m/z 92 and 93, corresponding to the loss of NH

para-aminophenolH +

The p-aminophenolH + has already been reported previously 29 . It presents a very intense 0-0 transition and narrow vibronic bands. This electronic spectrum is quite similar to the spectrum of cold neutral p-aminophenol 30 . This similarity and the comparison with ground and excited states ab-initio calculations clearly indicate that the proton is localized on the amino group.

ortho-aminophenolH +

The photofragmentation spectrum of o-aminophenolH + is as a whole similar to that of p-aminophenolH + , but also presents vibrational progressions on a low frequency mode (96 ± 1 cm -1 ) as in its neutral homologue 31 or fluorophenol 32 . This is due to the steric hindrance between the amino group (or the F atom) and the hydroxyl group, which imposes weak out of plane deformation. The band origin of o-aminophenolH + at 276.82 nm (4.48 eV) is very close to that of the p-aminophenolH + isomer (277.36 nm).

meta-aminophenolH +

The photofragmentation spectrum of m-aminophenolH + is very much red-shifted as compared to the o-and p-aminophenolH + spectra. Indeed, the origin of its first electronic excited state is located at 365.4 nm (3.39 eV) i.e. more than 1 eV lower in energy than the band origin of the other isomers. Two other band systems are observed starting at 321.8 nm (3.85 eV) and 255.3 nm (4.86 eV). Moreover, all the vibrational bands are very broad (FWHM ≈ 200 cm -1 instead of 15 cm -1 for the p-aminophenolH + bands) and separated by about 500 cm -1 . At this point it is not possible to state whether these band systems belong to the different electronic states of the same tautomer of m-aminophenolH + or to different tautomers. It is quite clear that the electronic spectrum is drastically different from those of the other isomers and thus the proton is not located on the amino group: the spectrum of the tautomer protonated on the amino group (NH 3 + ) should exhibit a strong transition around 277 nm, which is not observed, indicating that this tautomer is not present (or in very low abundance) in the ion trap at low temperature (50 K). Further assignment of the tautomers possibly observed requires ab initio calculations.

Calculations para and ortho-aminophenolH +

The assignment of the different tautomers that can be involved in the photofragmentation spectra relies on two calculated values, the ground state energy and the S 1 <-S 0 transition energy. For protonated o-and p-aminophenol, the assignment is very simple and without ambiguity. The ground state calculations (with both DFT and MP2 methods) predict that the protonation site on the ammonium group is more stable by more than 0.2 eV 15 for p-aminophenolH + and more than 0.3 eV for o-aminophenoH + as compared to other positions of the proton on carbon atoms of the aromatic ring (see SI). For o-aminophenoH + , there could be two orientations (cis and trans) of the hydroxyl group with respect to the ammonium group, however a ground state optimization starting from the cis geometry leads without barrier to the trans geometry. In addition, the electronic transition of the trans o-aminophenoH + tautomer protonated on the amino group (trans ortho NH 3 + isomer: isomers protonated on the amino group are denoted NH 3 + in the next paragraphs) is calculated within 0.10 eV of the experimental band origin (i.e. less than 5% error), as shown in Table 2.

meta-aminophenolH +

The situation is more complex for m-aminophenolH + : for each proton location, there are cis and trans conformations depending on the orientation of the OH group versus the amino group in meta position.

The ground state energies of the different m-aminophenolH+ isomers have been calculated at two levels of theory (MP2 and DFT) and are presented in Table 1, where the isomers protonated on the amino group are labeled cis and trans NH 3 + and the tautomers protonated on carbon atoms (see Figure 1 for atom numbering) are labeled cis or trans CnH + . As it can be seen there is a strong discrepancy between MP2 and DFT methods mostly for the NH 3 + tautomer. At the MP2/cc-pVDZ level of calculation, the C6H + and NH 3 + tautomers (both cis and trans) have their ground state energy within less than 0.07 eV, which can be considered as within the calculation error. But at the DFT/B3LYP level, the C6H + tautomer is more stable than the NH 3 + tautomer by 0.3/0.4 eV, which is usually far from the usual uncertainty in such calculations. Calculations with extended basis set (cc-pVTZ) give similar results. So for the NH 3 + tautomer of m-aminophenolH + , ground state calculations seem to be in trouble.

For the other CnH + (n=2,4,5) tautomers, DFT and MP2 methods agree quite well. The cis and trans C4H + and C2H + may be considered as potentially involved in the m-aminophenolH + photofragmentation spectrum since and their ground state energies are not very high as compared to that of C6H + (see Table 1).

Calculations at the DFT/B3LYP+COSMO level were performed to investigate the relationship between the protonation sites in solution (before the electrospray evaporation) and in the gas phase. These calculations indicate that protonation on the amino site is clearly more stable than on the C6 site in the liquid phase (MP2+COSMO calculations give the same result).

Excited states

From the excited state calculations and comparison with the para-and ortho-isomers in which the proton is located on the amino group, it is clear that the cis and trans NH 3 + meta tautomers should be characterized by sharp transitions in the 277 nm region, which are not observed. The vertical transitions for cis and trans C6H + meta-aminophenolH + tautomers are calculated at 4.51 eV and 4.41 eV, respectively, i.e. red shifted by 0.4 eV compared to the transition of the NH 3 + meta tautomers. Upon excited state optimization, there is a strong change of geometry for both species leading to a stabilization energy of around 1 eV. The transition energies have also been calculated for the cis and trans C4H + and C2H + meta tautomers, which have their ground state energies less than 0.4 eV above the ground state energy of the C6H + tautomer.

Table 2: Transition energies calculated for the first electronic states of the o-, m-and p-aminophenol protonated isomers. For o-and p-aminophenolH + , only the most stable NH

Discussion

Assignment

The electronic spectra of the three isomers are very different, considering the positions and widths of the bands. The most stable isomers of o-and p-aminophenolH + are the ones with the proton on the amino substituent, and they show electronic transitions in the UV (~ 277 nm) and their spectroscopy is similar to the anilinium one. 29 On the contrary, m-aminophenolH + has a characteristic spectrum in the near UV (origin at 365.4 nm) i.e., red-shifted as compared to the spectra of the other two isomers. These transitions to the red part of the spectrum are assigned to tautomers of m-aminophenolH + with the proton on a carbon atom of the cycle and not on the amino group. Moreover, the vibrational progressions based on modes at ~ 500 cm -1 are not the same as observed for the neutral isomer, showing the effect of the proton on the cycle (contrary to the spectra of molecules protonated on the amino group, which look like that of the neutral). 29 Previous calculations on protonated benzene 33 or phenol 34 , have predicted that the proton localization on a carbon of the aromatic skeleton should lead to broad quasi structure-less spectra due to a barrier-less internal conversion process, which seems similar to what is observed for m-aminophenolH + .

The ab initio calculations necessary for a more complete assignment are problematic since there is a discrepancy between the MP2 and DFT calculations for the ground state of m-aminophenolH + . With MP2 calculations, both the NH 3 + and the C6H + tautomers have very similar ground state energies and then should both be present in the ion trap. At the opposite, the DFT calculations predict that the C6H + is more stable by 0.3 eV (cis isomer) or 0.4 eV (trans isomer) as compared to the NH 3 + tautomer. The oscillator strengths have been calculated for the S 1 <-S 0 transitions of C6H + and NH 3 + tautomers and they are within a factor of 2. Excited state optimization indicates a large geometry change between the ground and excited state for the C6H + tautomer, so that the FC factors should be smaller as compared to those of the NH 3 + tautomer. Thus the excitation efficiency of the NH 3 + tautomer should be greater than that of C6H + . However the NH 3 + tautomer is not observed, thus one can deduce that it is not present (or in very small amount) in the cold ion trap, and consequently less stable than the C6H + tautomer. This implies also that, for ground state calculations, DFT is more reliable than MP2, which seems to fail rather badly in the present case. At the opposite, the TD-DFT method does not give correct results for excited state calculations since the presence of a calculated charge transfer (πσ*) state forbids the excited state optimization in the case of the NH 3 + tautomers. As shown also previously, the CC2 or ADC(2) methods are quite good for excited state calculations [35][36]21 and in the present case, the ADC(2) method gives adiabatic energies in quite good agreement with the experimental values for ortho and para-aminophenolH + (NH 3 + tautomers). For m-aminophenolH+, the C6H + adiabatic energies are calculated at 3.59 eV and 3.54 eV (cis and trans). The C4H + tautomer, which may be populated in the trap since its ground state energy is only 0.1 eV higher than that of C6H + , has a calculated vertical transition energy around 0.5 eV lower than that of the C6H + tautomer. The adiabatic transition energy for the cis meta C4H + tautomer is calculated at 3.34 eV, not far from the origin of the first band system, while for the trans meta C4H + tautomer there seems to be a conical intersection between the excited and the ground state.

To sum up, a reasonable assignment for the m-aminophenolH+ photofragmentation spectrum would be to assign the band system starting at 3.39 eV to the S 1 <-S 0 transition of the C4H+ tautomer, and the second band system starting at 3.85 eV to the S 1 <-S 0 transition of the C6H+ tautomer. The third band system starting at 4.85 eV may be assigned to the S 2 <-S 0 transition of the C6H + tautomer calculated at 5.6 eV vertically.

The most stable proton site determination

For protonated species there are not so many experiments which are able to determine the most abundant tautomer in well-defined conditions. Usually the temperature of the ions is not known nor measured and the observation method is not necessarily unambiguous. Besides, it has been shown that the proton is not necessary localized on one structure at high temperature 16 . Determination of the protonation site by collision processes relies on the assumption that the tautomerization rate is slower that other reaction/fragmentation processes and this assumption is not so obvious owing the small barrier to the tautomerisation process, around 1ev in an aromatic ring 15 .

Detection of tautomers through IR spectroscopy is another approach, often realized through IRMPD on hot ions and the spectroscopy is not unambiguous due to the highly nonlinear multiphoton absorption process leading to the absence of some vibrational bands.

The electronic photo-fragmentation of cold ions approach can be a powerful alternative but one has to be aware that some assumptions are also behind the measurement. In particular quantitative relationship between the ion intensity and the stability of the tautomer/conformer may not be so easy to do.

The fragmentation process might depend on the tautomer. However the excitation energy is such (4 eV) that, when the excited state undergoes internal conversion, the randomization between tautomers (barrier around 1 eV) 15 is realized, thus the fragmentation rate should be the same for all tautomers.

The excitation efficiency, which depends on the oscillator strength of the electronic transition and on the Franck-Condon (FC) factors is more difficult to evaluate. The oscillator strength can be calculated and in the present case it has the same order of magnitude for the C6H+ and NH3+ tautomers of m-aminophenolH+ (within a factor 2). When the ground and excited state optimized geometries are similar, the FC factors can be calculated using calculated ground and excited vibrational frequencies. When there is a large change in the optimized geometry between ground and excited states, the density of states in the Franck-Condon window is such that Franck Condon analysis becomes impossible and the Franck-Condon factors are very weak. In the case of the m-aminophenolH+ isomers, the distortion in geometry of the C6H+ tautomer (this holds also for the C4H+) in S1 is larger than for the NH3+ tautomer, which can be seen easily deduced from the difference between vertical and the adiabatic transition energies: a change of less than 0.5 eV for the NH3+ tautomer, as in protonated aniline or in p-aminophenolH+, indicates a relatively small geometry change associated with a strong 0-0 transition and a well resolved spectrum. On the other hand, a difference of 1 eV or more between vertical and adiabatic energies indicates a large geometry change that leads to weak Franck-Condon factors. These considerations imply that if the NH3+ tautomer was present, it would have been observed quite easily as in the case of ortho and para-aminophenolH+. Thus we can conclude that the C6H+ and C4H+ are by far the most abundant conformers in the experiment and, if thermal equilibrium at 50 K is assumed, these tautomers should be the most stable ones. The absence of the NH3+ tautomer is not compatible with MP2 calculations, which predict that the NH3+ and C6H+ tautomers have the same ground state energy, but it matches with DFT calculations.

We are very surprised by the inconsistency between the two methods, which usually gives similar results, but in this case are in strong disagreement as far as the NH3+ tautomer is concerned, while for the C2H+, C4H+, C5H+ tautomers both methods give very similar results. These results show that there is a problem in the calculation methods but it is beyond our competence and is certainly a problem that theoreticians have to solve2 .

The stronger proton affinity of the meta-isomer on the C6 and C4 carbon atoms can be explained by a resonance effect. It is known that electron donating groups with lone pairs (OH, -NH2) on the atoms adjacent to the π system activate the aromatic ring by increasing the electron density on the ring through a resonance donating effect that allows electron density to be positioned at the orthoand para-positions. For the para-and ortho-isomers, the resonance effect due to OH and NH2 acts on different carbon atoms, whereas in meta-aminophenolH+ both substituents induce electron density on the same carbon atoms increasing their negative charge thus their proton affinity, which becomes larger than that of the amino group and indeed, both C4H+ and C6H+ tautomers seem to be observed.

Electrospray source and produced ions.

There is an active discussion on the relationship between the most stable ions in solution and the ions obtained in the gas phase after the electrospray process 22,23 . We have calculated the stability of the NH3+ and the C6H+ tautomers of meta-aminophenolH+ in solution using the COSMO model implemented in Turbomole at the DFT/B3LYP level. Interestingly, there is a complete reversal of the stability between the gas and solution phase for these two tautomers. As seen in the SI (Fig. SI.3), the absorption spectrum of the meta amino phenol in solution (H20/methanol) at two different pHs shows a blue shift of the absorption at low pH (at 270 nm) only compatible with protonation on the amino group. The NH3+ form is 0.3 eV more stable than the C6H+ in solution and this structure is less stable by 0.3 eV in the gas phase. The absence of the NH3+ conformer in the ion trap clearly shows that the liquid equilibrium is not conserved in the ESI process, at least in our experimental setup...

Conclusions

In this paper it is shown that one can use electronic spectroscopy to identify the most abundant protonated tautomers at low temperature, which are probably the most stable. In particular it shows that in the para-and ortho-isomers of protonated aminophenol, the proton is attached to the amino group while it is linked to a carbon atom of the aromatic ring for the meta-isomer. This experiment also shows that the most stable tautomers cannot be always predicted by ab-initio calculations as long as the large discrepancies between MP2 and DFT methods are not understood. It seems also that the most stable tautomeric structure is not conserved from the liquid phase to the gas phase.

Figure 1 Figure 1 :

 11 Figure 1

Table 1 :

 1 Ground state energies of the m-aminophenolH + isomers (all the values are in eV): The energies are referenced to the energy of cis C6H + tautomer (i.e. the proton is on C6 carbon atom of the aromatic ring as numbered in Figure1).

	m-aminophenolH +	Relative energy	Relative energy	Relative energy
	isomers	DFT/B3LYP/cc-pVDZ	MP2/cc-pVDZ	DFT/B3LYP+COSMO/cc-pVDZ
	cis C6H +	0.00	0.00	0.00
	trans C6H +	0.06	0.05	0.03
	cis C4H +	0.09	0.10	0.11
	trans C4H +	0.10	0.12	0.09
	trans C2H +	0.19	0.25	0.18
	cis C2H +	0.33	0.41	0.27
	trans NH 3 +	0.34	-0.01	-0.21
	cis NH 3 +	0.42	0.07	-0.19
	trans C5H +	1.45	1.47	1.47
	cis C5H +	1.50	1.52	1.51

Similar calculations have done with other software with which it is forbidden to compare and similar results are obtained

Acknowledgements

This works was supported the ANR Research Grant (ANR2010BLANC040501). We acknowledge the use of the computing facility cluster GMPCS of the LUMAT federation (FR LUMAT 2764). We thank Dr. Anne Zehnacker and Pr. Ernesto Marceca for fruitful discussions.