1 Introduction and notation
Although our results are valid in more general filtrations, we start with a two-period model. In this setting, we work with a probability space equipped with three sigma-algebras, \((\Omega ,{\mathcal{F}}_{0}\subseteq {\mathcal{F}}_{1}\subseteq { \mathcal{F}}_{2},{\mathbb{P}})\). The sigma-algebra \({\mathcal{F}}_{0}\) is supposed to be trivial, i.e., every \(A\in {\mathcal{F}}_{0}\) satisfies \({\mathbb{P}}[A]=0\text{ or } 1\), whereas \({\mathcal{F}}_{2}\) is supposed to express innovations with respect to \({\mathcal{F}}_{1}\). Since we do not put topological properties on the set \(\Omega \), we make precise definitions later that do not use conditional probability kernels. But essentially, we could say that we suppose that conditionally on \({\mathcal{F}}_{1}\), the probability ℙ is atomless on \({\mathcal{F}}_{2}\). We shall show that such a hypothesis implies that there is an atomless sigma-algebra \({\mathcal{B}}\subseteq {\mathcal{F}}_{2}\) which is independent of \({\mathcal{F}}_{1}\). The space \(L^{\infty }({\mathcal{F}}_{i})\) is the space of bounded \({\mathcal{F}}_{i}\)-measurable random variables modulo equality almost surely (a.s.). We say that two random variables \(\xi ,\eta \) are commonotonic1 if there are two nondecreasing functions \(f,g\colon {\mathbb{R}}\rightarrow {\mathbb{R}}\) and a random variable \(\zeta \) such that \(\xi =f(\zeta ), \eta =g(\zeta )\). Commonotonicity can be seen as the opposite of diversification. If \(\zeta \) increases, then both \(\xi \) and \(\eta \) increase (or, better, do not decrease). By the way, if \(\xi \) and \(\eta \) are commonotonic, then one can choose \(\zeta =\xi +\eta \); see Delbaen [7, Chap. 2.4]. It can be shown that in this case one can choose representatives — still denoted by \((\xi ,\eta )\) — such that \((\xi (\omega )-\xi (\omega '))(\eta (\omega )-\eta (\omega '))\ge 0\) for all \(\omega ,\omega '\). Since we do not need this result, we do not include a proof. We say that a set \(E\subseteq {\mathbb{R}}^{2}\) is commonotonic if \((x,y),(x',y')\in E\) implies \((x-x')(y-y')\ge 0\). In convex function theory, such sets are also called monotone or monotonic sets. Random variables \(\xi ,\eta \) are commonotonic if and only if the support of the image measure of \((\xi ,\eta )\) is a commonotonic set.
The present paper deals with time-consistent utility functions. This means that for \(0\le i< j\le 2\), there are functions \(u_{i,j}\colon L^{\infty }({\mathcal{F}}_{j})\rightarrow L^{\infty }({ \mathcal{F}}_{i})\) such that we have \(u_{0,2}=u_{0,1}\circ u_{1,2}\). These utility functions satisfy the following properties; see [7, Chap. 11] for more information on the relation between these properties:
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1) \(u_{i,j}\colon L^{\infty }({\mathcal{F}}_{j})\rightarrow L^{\infty }({ \mathcal{F}}_{i})\), and if \(\xi \ge 0\), then also \(u_{i,j}(\xi )\ge 0\), and \(u_{i,j}(0)=0\).
2) For \(\xi ,\eta \in L^{\infty }({\mathcal{F}}_{j})\) and \(0\le \lambda \le 1\) and \({\mathcal{F}}_{i}\)-measurable, we have
$$ u_{i,j}\big(\lambda \xi +(1-\lambda )\eta \big)\ge \lambda u_{i,j}( \xi )+(1-\lambda ) u_{i,j}(\eta ). $$
3) Since commonotonicity implies (as easily seen) positive homogeneity, we use a stronger property and suppose coherence. For \(\xi \in L^{\infty }({\mathcal{F}}_{j})\) and \(\lambda \geq 0\) and \({\mathcal{F}}_{i}\)-measurable, we have
$$ u_{i,j}(\lambda \xi )=\lambda u_{i,j}(\xi ). $$
4) For \(\xi \in L^{\infty }({\mathcal{F}}_{j})\) and \(a\in L^{\infty }({\mathcal{F}}_{i})\), we have
$$ u_{i,j}(\xi +a)=u_{i,j}(\xi ) + a. $$
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5) We need Lebesgue-continuity which means that if \((\xi _{n}) \subseteq L^{\infty }({\mathcal{F}}_{j})\) is a uniformly bounded sequence such that \(\xi _{n}\rightarrow \eta \) in probability, then \(u_{i,j}(\xi _{n})\) tends to \(u_{i,j}(\eta )\) in probability.
6) The Lebesgue property is stronger than the Fatou property which says that for a sequence \((\xi _{n}) \subseteq L^{\infty }\) such that a.s. \(\xi _{n}\downarrow \eta \in L^{\infty }\), we have \(u_{ij}(\xi _{n})\rightarrow u_{ij}(\eta )\) a.s.
The utility functions we need are coherent and hence we can use their dual representation; see Delbaen [6, end of the proof of Theorem 6]. This means that there is a uniquely defined convex closed set \({\mathcal{S}}\subseteq L^{1}\) of probability measures, absolutely continuous with respect to ℙ, such that The set \({\mathcal{S}}\) is viewed as a subset of \(L^{1}\) via the Radon–Nikodým theorem. The Lebesgue-continuity is equivalent to the weak compactness of \({\mathcal{S}}\). We suppose that our utility functions are relevant, i.e., for each \(A\) with \({\mathbb{P}}[A]>0\), we have \(u(-\mathbf {1}_{A})<0\); see [7, Chap. 4.14]. By the Halmos–Savage theorem, this means that \({\mathcal{S}}\) contains an equivalent probability measure. We need this property in order to avoid some problems with negligible sets appearing in the definition and with comparisons of conditional expectations.
$$ u_{0,2}(\xi )=\inf _{{\mathbb{Q}}\in {\mathcal{S}}}{\mathbb{E}}_{ \mathbb{Q}}[\xi ]. $$
Without further notice, we always assume that our utility functions are relevant and Lebesgue-continuous. These assumptions are not always needed; sometimes Fatou-continuity is sufficient. Since we want to put more emphasis on the methods of proof, we do not aim for the most general results.
One may ask in which way the utility functions \(u_{i,j}\) can be constructed from the utility function \(u_{0,2}\). The construction is easier when \(u_{0,2}\) is relevant. The Fatou or Lebesgue property is less important for this development. As shown in [7, Chap. 11], there is a way to check whether the utility function \(u_{0,2}\) can be embedded in a time-consistent family of utility functions. To do this, we introduce the acceptability cones The necessary and sufficient condition for the existence of a time-consistent extension is \({\mathcal{A}}_{0,2}={\mathcal{A}}_{0,1}+{\mathcal{A}}_{1,2}\). If this is fulfilled, we put and \(u_{0,1}\) is simply the restriction of \(u_{0,2}\) to \(L^{\infty }({\mathcal{F}}_{1})\). This gives sense to expressions such as “\(u_{0,2}\) is time-consistent”.
$$\begin{aligned} {\mathcal{A}}_{0,2} &=\{\xi \in L^{\infty }({\mathcal{F}}_{2}) \colon u_{0,2}(\xi )\ge 0\}, \\ {\mathcal{A}}_{0,1}&=\{\xi \in L^{\infty }({\mathcal{F}}_{1})\colon u_{0,2}( \xi )\ge 0\}, \\ {\mathcal{A}}_{1,2}&=\{\xi \in L^{\infty }({\mathcal{F}}_{2})\colon \text{for all } A\in {\mathcal{F}}_{1} , u_{0,2}(\xi \mathbf {1}_{A})\ge 0 \}. \end{aligned}$$
$$ u_{1,2}(\xi )=\mathop{\mathrm{ess\,inf}}\{\eta \in L^{\infty }({ \mathcal{F}}_{1})\colon \xi -\eta \in {\mathcal{A}}_{1,2}\}, $$
Already in the case where the utility functions are expected value and conditional expectations, the main theorem leads to the following result. (The notion “conditionally atomless” will be explained and analysed in the next section.)
Theorem 1.1
If \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\), then for any couple \((f,g)\) of \({ \mathcal{F}}_{1}\)-measurable finite-valued random variables, there is a commonotonic couple \((\xi ,\eta )\) of \({\mathcal{F}}_{2}\)-measurable random variables such that (in an extended sense, made precise later) \(f={\mathbb{E}}[\xi \, | \, {\mathcal{F}}_{1}],g={\mathbb{E}}[\eta \, | \, {\mathcal{F}}_{1}]\). Furthermore, for every norm on \({\mathbb{R}}^{2}\), there is a constant \(C\) such that \(\Vert (\xi ,\eta )\Vert \le C \Vert (f,g)\Vert \) almost surely.
Both concepts, time-consistency and commonotonicity, are important in the theory of risk evaluation. The concept of time-consistency (and -inconsistency) was introduced and investigated by Koopmans [12]. The role of commonotonicity found its way into insurance and is present in several papers. The use of Choquet integration as premium principle was emphasised by Denneberg [9] who was inspired by the pioneering work of Yaari [21]. Schmeidler proved the relation between commonotonic principles, convex games and Choquet integration [14]. Modern uses can be found for instance in Wang et al. [17] and Wang [18]. For more references and different proofs of these results, we refer to [7, Chap. 7]. Although commonotonicity seems to be a desirable property, there might be some difficulties when insurance contracts are priced in this way; see Castagnoli et al. [5] for some unexpected consequences.
The concept of risk measures (up to sign changes monetary utility functions) was introduced in Artzner et al. [1, 2].
Using the general version of Theorem 1.1, we shall show that except in very restrictive cases, a utility function \(u_{0,2}\) cannot be time-consistent and commonotonic at the same time. It seems that time-consistency is a strong property that excludes some other desirable properties. For instance in Kupper and Schachermayer [11], it is shown that in a filtration with innovations (comparable to the requirement of being conditionally atomless), utility functions that are time-consistent and law-determined are necessarily of entropic type. We refer to [11] for the details and the precise form of the innovations. The present paper studies time-consistent utility functions that might depend on past history and are not necessarily law-determined. The methods we use are different from the approaches used for law-determined or law-invariant utility functions. Among the many papers on these utility functions, we could refer the reader to the cited papers and to e.g. Bellini et al. [3], Bellini et al. [4], Wang and Ziegel [19], Weber [20] and Ziegel [22].
2 Atomless extension of sigma-algebras
In this section, we work with a probability space \((\Omega ,{\mathcal{F}}_{2},{\mathbb{P}})\) equipped with the filtration \({\mathcal{F}}_{0}\subseteq {\mathcal{F}}_{1}\subseteq { \mathcal{F}}_{2}\).
Definition 2.1
We say that \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\) if for every \(A\in {\mathcal{F}}_{2}\), there exists a set \(B\subseteq A\), \(B\in {\mathcal{F}}_{2}\), such that \(0< { \mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]<{\mathbb{E}}[\mathbf {1}_{A} \, | \, {\mathcal{F}}_{1}]\) on the set \(\{{\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]>0\}\).
If the conditional expectation can be calculated with a – under extra topological conditions – regular probability kernel, say \(K(\omega , A)\), then the above definition is a measure-theoretic way of saying that the probability measure \(K(\omega , \cdot )\) is atomless for almost every \(\omega \in \Omega \). The precise relation between these two notions is not the topic of this paper. See Delbaen [8] for the details.
Theorem 2.2
\({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\) if for every \(A\in {\mathcal{F}}_{2}\) with \({\mathbb{P}}[A]>0\), there is \(B\subseteq A\), \(B\in {\mathcal{F}}_{2}\), such that
$$ {\mathbb{P}}\big[0< {\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]< { \mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\big]>0. $$
Proof
The proof is a standard exhaustion argument. For completeness, we give the details. Let \({\mathcal{D}}\) be the collection of \({\mathcal{F}}_{1}\)-measurable sets given by We show that there is a biggest set in \({\mathcal{D}}\) and this must then equal \(\{{\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]>0\}\). To show that there is a biggest set in \({\mathcal{D}}\), it is sufficient to show that \({\mathcal{D}}\) is stable for countable unions. Let \((D_{n})\) be a sequence in \({\mathcal{D}}\) and suppose that for each \(n\), we have a set \(B_{n}\subseteq A\), \(B_{n}\in {\mathcal{F}}_{2}\), such that \(D_{n}=\{ 0<{\mathbb{E}}[\mathbf {1}_{B_{n}}\, | \, {\mathcal{F}}_{1}]<{ \mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\}\). Now take It is easy to check that \(\{ 0<{\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]<{\mathbb{E}}[ \mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\}=\bigcup _{n\in {\mathbb{N}}} D_{n}\) and therefore \(\bigcup _{n\in {\mathbb{N}}} D_{n}\in {\mathcal{D}}\). Let now \(D=\{ 0<{\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]<{\mathbb{E}}[ \mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\}\) be a maximum in \({\mathcal{D}}\). Suppose that \({\mathbb{P}}[\{{\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]>0\} \setminus D ]>0\). This implies that \({\mathbb{P}}[A\setminus D]>0\). According to the hypothesis of the theorem, there will be a set \(B'\subseteq A\setminus D\), \(B'\in {\mathcal{F}}_{2}\), with \(D' = \{ 0<{\mathbb{E}}[\mathbf {1}_{B'}\, | \, {\mathcal{F}}_{1}]<{ \mathbb{E}}[\mathbf {1}_{A\setminus D}\, | \, {\mathcal{F}}_{1}] \} \) having nonzero probability. Since \(D\cup D'\in {\mathcal{D}}\) and \(D\cap D'=\emptyset \), the element \(D\) is not a maximum, which is a contradiction. □
$$ {\mathcal{D}}=\big\{ \{0< {\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]< { \mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\}\colon B\subseteq A, B \in {\mathcal{F}}_{2} \big\} . $$
$$ B=\bigcup _{n\in {\mathbb{N}}} \bigg( B_{n}\cap \Big(D_{n}\setminus \big(\bigcup _{k=1}^{n-1}D_{k}\big)\Big) \bigg). $$
The main result of this section is the following.
Theorem 2.3
\({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\) if and only if there exists an atomless sigma-algebra \({\mathcal{B}}\subseteq {\mathcal{F}}_{2}\) that is independent of \({\mathcal{F}}_{1}\).
The “if” part is easy, but requires some continuity argument. Because ℬ is atomless, there is a ℬ-measurable random variable \(U\) uniformly distributed on \([0,1]\). The sets \(B_{t}=\{U\le t\}, 0\le t \le 1\), form an increasing family of sets with \({\mathbb{P}}[B_{t}]=t\). Fix \(A\in {\mathcal{F}}_{2}\) and let \(F=\{ 0 < {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\}\). We may suppose that \({\mathbb{P}}[F]>0\) since otherwise there is nothing to prove. We now show that there is \(t\in (0,1)\) with \({\mathbb{P}}[ 0 < {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, { \mathcal{F}}_{1}]< {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}] ] > 0\). According to Theorem 2.2, \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\). Obviously for \(0\le s\le t \le 1\), we have by independence of ℬ and \({\mathcal{F}}_{1}\) that It follows that there is a set of measure 1, say \(\Omega '\), such that for all \(s\le t\), \(s,t\) rational, and all \(\omega \in \Omega '\), \({\mathbb{E}}[\mathbf {1}_{A\cap B_{t}}\, | \, {\mathcal{F}}_{1}](\omega )\) can be taken to satisfy For each \(\omega \in \Omega '\), we can extend the function to a continuous function on \([0,1]\). The resulting continuous extension then represents the equivalence classes of random variables \(({\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}])_{t \in [0,1]}\). For \(t=0\), we have zero, and for \(t=1\), we find \({\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\). Because the trajectories are continuous for \(\omega \in \Omega '\), a simple application of Fubini’s theorem shows that the real valued function becomes strictly positive for some \(t\). With some extra work – done later –, one can even show that there is \(G\subseteq A\) such that \({\mathbb{E}}[\mathbf {1}_{G}\, | \, {\mathcal{F}}_{1}]= (1/2){\mathbb{E}}[ \mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]\).
$$ \Vert {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}] - { \mathbb{E}}[\mathbf {1}_{A\cap B_{s}} \, | \, {\mathcal{F}}_{1}]\Vert _{\infty }\le \Vert {\mathbb{E}}[\mathbf {1}_{B_{t}\setminus B_{s}} \, | \, { \mathcal{F}}_{1}]\Vert _{\infty }= t-s. $$
$$ | {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}](\omega ) - {\mathbb{E}}[\mathbf {1}_{A\cap B_{s}} \, | \, {\mathcal{F}}_{1}](\omega ) | \le t-s. $$
$$ [0,1] \cap {\mathbb{Q}}\ni q \mapsto {\mathbb{E}}[\mathbf {1}_{A\cap B_{q}}\, | \, {\mathcal{F}}_{1}](\omega ) $$
$$ t\mapsto {\mathbb{P}}\left [ 0 < {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}]< {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}] \right ] $$
For completeness, let us now give the details of the application of Fubini’s theorem. Suppose to the contrary that for all \(t\in [0,1]\), we have Then on the product space \([0,1]\times \Omega '\), we find that the (clearly measurable) set has \((m\times {\mathbb{P}})\)-measure zero (\(m\) denotes Lebesgue measure). By Fubini’s theorem, we have that for almost all \(\omega \in \Omega '\), the set must have Lebesgue measure zero. However, for \(\omega \in \Omega '\), this contradicts the continuity of the mapping
$$ {\mathbb{P}}\left [ 0 < {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, { \mathcal{F}}_{1}]< {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}] \right ] =0. $$
$$ \{(t,\omega )\colon 0 < {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, { \mathcal{F}}_{1}](\omega )< {\mathbb{E}}[\mathbf {1}_{A}\, | \, { \mathcal{F}}_{1}] (\omega )\} $$
$$ \{t \colon 0 < {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}]( \omega )< {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}] (\omega ) \} $$
$$ t\mapsto {\mathbb{E}}[\mathbf {1}_{A\cap B_{t}} \, | \, {\mathcal{F}}_{1}]( \omega ). $$
The proof of the “only if” part is broken down into several steps stated in the lemmas that follow. Without further notice, we always suppose that \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\).
Lemma 2.4
Suppose \(A\in {\mathcal{F}}_{1}\) and \(C\subseteq A\), \(C\in {\mathcal{F}}_{2}\), is such that \({\mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]>0\) on \(A\). Then we can construct a decreasing sequence \((B_{n})_{n\ge 0}\) of sets \(B_{n}\subseteq C\), \(B_{n}\in {\mathcal{F}}_{2}\), such that \(0<{ \mathbb{E}}[\mathbf {1}_{B_{n}}\, | \, {\mathcal{F}}_{1}]\le 2^{-n}\) on \(A\).
Proof
The statement is obviously true for \(n=0\) since we can take \(B_{0}=C\). We now proceed by induction and suppose the statement holds for \(n\). So the set \(B_{n}\subseteq A\) satisfies \(0<{\mathbb{E}}[\mathbf {1}_{B_{n}}\, | \, {\mathcal{F}}_{1}]\le 2^{-n}\) on \(A\). Clearly, \(A\subseteq \{{\mathbb{E}}[\mathbf {1}_{B_{n}}\, | \, {\mathcal{F}}_{1}]>0 \}\). By assumption, there is a set \(D\subseteq B_{n}\), \(D\in {\mathcal{F}}_{2}\), such that on \(A\subseteq \{ {\mathbb{E}}[\mathbf {1}_{A}\, | \, {\mathcal{F}}_{1}]>0\}\), we have We now take The set \(B_{n+1}\) satisfies the requirements. □
$$ 0< {\mathbb{E}}[\mathbf {1}_{D}\, | \, {\mathcal{F}}_{1}]< {\mathbb{E}}[\mathbf {1}_{B_{n}} \, | \, {\mathcal{F}}_{1}]. $$
$$\begin{aligned} B_{n+1}& = \bigg(D\cap \bigg\{ {\mathbb{E}}[\mathbf {1}_{D}\, | \, { \mathcal{F}}_{1}]\le \frac{1}{2}{\mathbb{E}}[\mathbf {1}_{B_{n}} | { \mathcal{F}}_{1}]\bigg\} \bigg) \\ & \phantom{=:}\cup \bigg((B_{n}\setminus D)\cap \bigg\{ {\mathbb{E}}[\mathbf {1}_{D}\, | \, { \mathcal{F}}_{1}]> \frac{1}{2}{\mathbb{E}}[\mathbf {1}_{B_{n}} | { \mathcal{F}}_{1}]\bigg\} \bigg). \end{aligned}$$
Lemma 2.5
Let \(C\in {\mathcal{F}}_{2}\) and let \(h\colon \Omega \rightarrow [0,1]\) be \({\mathcal{F}}_{1}\)-measurable. Then there is a set \(B\subseteq C\), \(B\in {\mathcal{F}}_{2}\), such that \({\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]=h\,{\mathbb{E}}[ \mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]\).
Proof
Let \(B_{0}=\emptyset \). Inductively, we define for \(n\ge 1\) classes \({\mathcal{B}}_{n}\) and sets \(B_{n}\in {\mathcal{B}}_{n}\). For \(n\ge 1\), let Let \(\beta _{n}=\sup \{ {\mathbb{P}}[B]\colon B\in {\mathcal{B}}_{n}\}\) and take \(B_{n}\in {\mathcal{B}}_{n}\) such that \({\mathbb{P}}[B_{n}]\ge (1-2^{-n})\beta _{n}\). Clearly, \((B_{n})\) is nondecreasing, and we set \(B_{\infty }=\bigcup _{n\geq 0} B_{n}\). Obviously, We claim that \({\mathbb{E}}[\mathbf {1}_{B_{\infty }}\, | \, {\mathcal{F}}_{1}]=h\,{ \mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]\). We have \({\mathbb{E}}[\mathbf {1}_{B_{\infty }}\, | \, {\mathcal{F}}_{1}]\le h\,{ \mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]\) by construction. If \({\mathbb{P}}[ {\mathbb{E}}[\mathbf {1}_{B_{\infty }}\, | \, {\mathcal{F}}_{1}] < h\,{\mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}] ]>0\), then \({\mathbb{P}}[B_{\infty }]<{\mathbb{P}}[C]\) and there must be \(m\ge 1\) such that \({\mathbb{P}}[ {\mathbb{E}}[\mathbf {1}_{B_{\infty }}\, | \, {\mathcal{F}}_{1}] < h\,{\mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}] -2^{-m}]>0\). Lemma 2.4 allows us to find \(D\subseteq C\setminus B_{\infty }\), \(D \in {\mathcal{F}}_{2}\), \({\mathbb{P}}[D]=\eta >0\), with \(0<{\mathbb{E}}[\mathbf {1}_{D}\, | \, {\mathcal{F}}_{1}]\le 2^{-m}\) on the set \(\{{\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}] < h\,{\mathbb{E}}[ \mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}] - 2^{-m}\}\) and zero elsewhere. The set \(D\cup B_{\infty }\) is in all classes \({\mathcal{B}}_{n}\), and for \(n\) big enough, we have yielding a contradiction. So we must have \({\mathbb{E}}[\mathbf {1}_{B_{\infty }}\, | \, {\mathcal{F}}_{1}]=h\,{ \mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]\). □
$$ {\mathcal{B}}_{n}=\{ B_{n-1}\subseteq B\subseteq C \colon B\in { \mathcal{F}}_{2},\,{\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}] \le h\,{\mathbb{E}}[\mathbf {1}_{C}\, | \, {\mathcal{F}}_{1}]\}. $$
$$ {\mathbb{P}}[B_{\infty }]\ge \limsup _{n\to \infty } \beta _{n}\ge \liminf _{n\to \infty }\beta _{n}\ge \lim _{n\to \infty } {\mathbb{P}}[B_{n}]={ \mathbb{P}}[B_{\infty }]. $$
$$ \beta _{n}\ge {\mathbb{P}}[D\cup B_{\infty }] \ge {\mathbb{P}}[B_{n}]+ \eta \ge (1-2^{-n})\beta _{n} +\eta \ge \beta _{n}+\eta -2^{-n}> \beta _{n}, $$
Remark 2.6
Lemma 2.5 is a variant of Sierpiński’s theorem [15]. This theorem states that in an atomless probability space \((\Omega ,{\mathcal{E}},{\mathbb{P}})\), for every set \(A\in {\mathcal{E}}\) and every \(0< t<1\), there is a set \(B\subseteq A\), \(B \in {\mathcal{E}}\), with \({\mathbb{P}}[B]=t{\mathbb{P}}[A]\). The usual proof – presented in many probability courses – uses the axiom of choice (AC). A referee pointed out that for many people AC – or Zorn’s lemma – is an extra assumption. To prove Sierpiński’s theorem, we only need the axiom of countable dependent choice, which is a countable form of the axiom of choice. In analysis, this is the axiom that is usually needed and used. The proof above follows the approach given by Lorenc and Witula [13].
Lemma 2.7
There is an increasing family \((B_{t})_{t\in [0,1]}\) of sets such that \({\mathbb{E}}[\mathbf {1}_{B_{t}}\, | \, {\mathcal{F}}_{1}]=t\). The sigma-algebra ℬ generated by the family \((B_{t})\) is independent of \({\mathcal{F}}_{1}\). The system \((B_{t})\) can also be described as \(B_{t}=\{U\le t\}\), where \(U\) is a random variable that is independent of \({\mathcal{F}}_{1}\) and uniformly distributed on \([0,1]\).
Proof
The proof is a repeated use of Lemma 2.5 where we take \(h=1/2\). We start with \(B_{0}=\emptyset , B_{1}=\Omega \). Suppose that for the dyadic numbers \(k 2^{-n}\), \(k=0,\ldots , 2^{n}\), the sets are already defined. Then we consider the set \(B_{(k+1)2^{-n}}\setminus B_{k2^{-n}}\) and apply Lemma 2.5 with \(h=1/2\). We get a set \(D\subseteq B_{(k+1)2^{-n}}\setminus B_{k2^{-n}}\), \(D \in {\mathcal{F}}_{2}\), with \({\mathbb{E}}[\mathbf {1}_{D}\, | \, { \mathcal{F}}_{1}]=2^{-(n+1)}\). We then define \(B_{(2k+1)2^{-(n+1)}}=B_{k2^{-n}}\cup D\). For non-dyadic numbers \(t\), we find a sequence \((d_{n})\) of dyadic numbers such that \(d_{n}\uparrow t\). Then we define \(B_{t}=\bigcup _{n\in {\mathbb{N}}} B_{d_{n}}\). This completes the construction. Since the system \((B_{t})\) is trivially stable under intersections, the relation \({\mathbb{E}}[\mathbf {1}_{B_{t}}\, | \, {\mathcal{F}}_{1}]=t\) shows that the sigma-algebra ℬ generated by \((B_{t})\) is independent of \({\mathcal{F}}_{1}\). The construction of \(U\) is standard. At level \(n\), we put \(U_{n}=\sum _{k=1}^{2^{n}} k2^{-n}\mathbf {1}_{B_{k2^{-n} }\setminus B_{(k-1)2^{-n}}}\). Then \((U_{n})\) decreases to a random variable \(U\) that satisfies the needed properties. The proof of Theorem 2.3 is now completed. □
Remark 2.8
Suppose that for the probability ℙ, there is an atomless sigma-algebra \({\mathcal{B}}\subseteq {\mathcal{F}}_{2}\) that is independent of \({\mathcal{F}}_{1}\). Suppose now that \({\mathbb{Q}}\approx {\mathbb{P}}\) is an equivalent probability measure. Clearly, the definition of being conditionally atomless is invariant for equivalent measure changes. Hence there is an atomless sigma-algebra \({\mathcal{B}}'\subseteq {\mathcal{F}}_{2}\) that is independent of \({\mathcal{F}}_{1}\) for the probability ℚ. Proving this directly does not seem easy.
The following proposition is Lemma 2.5 where we take \(C=\Omega \). For didactic reasons, we give another proof that directly uses the existence of an independent sigma-algebra. We use the same assumptions and notations as in Theorem 2.3.
Proposition 2.9
For every \({\mathcal{F}}_{1}\)-measurable function \(h\colon \Omega \rightarrow [0,1]\), there is a set \(B_{h}\in {\mathcal{F}}_{2}\) such that \({\mathbb{E}}[\mathbf {1}_{B_{h}}\, | \, {\mathcal{F}}_{1}]=h\).
Proof
The idea is to use the set \(B_{t}\) on the set \(\{h=t\}\), i.e., \(B=\bigcup _{t}(\{h=t\}\cap B_{t})\). However, because the set of real numbers is uncountable, this definition is not good enough to obtain a set in \({\mathcal{F}}_{2}\). So we need a trick. Let \(\phi \) be the mapping This mapping is obviously measurable and the image measure is because of independence the product measure. We also define \(h_{1}(\omega ,\omega ')=h(\omega )\) and \(U_{2}(\omega ,\omega ')=U( \omega ')\). For \(A\in {\mathcal{F}}_{1}\), we set \(A_{1}=A\times \Omega \). We define \(B_{h}=\{U\le h\}=\phi ^{-1}\{U_{2}\le h_{1}\}\). We now verify that \({\mathbb{E}}[\mathbf {1}_{B_{h}}\, | \, {\mathcal{F}}_{1}]=h\). To do this, we calculate for a set \(A\in {\mathcal{F}}_{1}\) the probability showing \({\mathbb{E}}[\mathbf {1}_{B_{h}}\, | \, {\mathcal{F}}_{1}]=h\). □
$$ \phi \colon (\Omega ,{\mathcal{F}}_{2})\rightarrow (\Omega ,{ \mathcal{F}}_{1})\times (\Omega ,{\mathcal{B}}), \qquad \phi ( \omega )=(\omega ,\omega ). $$
$$\begin{aligned} {\mathbb{P}}[B_{h}\cap A] =& ({\mathbb{P}}\times {\mathbb{P}})[\{U_{2} \le h_{1}\}\cap A_{1}] \\ =& \int {\mathbb{P}}[d\omega ']\int {\mathbb{P}}[d\omega ] \, \mathbf {1}_{ \{U_{2}\le h_{1}\} }(\omega ,\omega ')\mathbf {1}_{A_{1}}(\omega ,\omega ') \\ =& \int {\mathbb{P}}[d\omega ']\,{\mathbb{P}}[\{h\ge U(\omega ')\} \cap A] \\ =& \int _{0}^{1} dt\, {\mathbb{P}}[\{h\ge t\}\cap A] \\ =& {\mathbb{E}}[h\mathbf {1}_{A}], \end{aligned}$$
Remark 2.10
Proposition 2.9 is not actually needed. We need the stronger version where the conditional expectation is replaced by the utility function \(u_{1,2}\). To prove this stronger version, we use a slightly different approach. However, if we are only interested in conditional expectations, the above proof might be of some didactic interest.
Remark 2.11
After the first version of this paper was made available, we got the remark that the paper of Shen et al. [16] contains similar concepts and results.2 In their notation, they work with a measurable space \((\Omega ,{\mathcal{A}})\) on which they have a finite number of probability measures \({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n}\). Their paper also considers an infinite number of measures, but to clarify the relation between their paper and our approach, we only consider a finite number of measures. They introduce
Definition 2.12
The set \(({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n})\) is conditionally atomless if there exist a dominating measure ℚ (i.e., \({\mathbb{Q}}_{k}\ll {\mathbb{Q}}\) for each \(k\le n\)) as well as a continuously distributed random variable \(X\) (for the measure ℚ) such that the vector of Radon–Nikodým derivatives \((\frac{d{\mathbb{Q}}_{k}}{d{\mathbb{Q}}})_{k=1,\dots ,n}\) is independent of \(X\).
They then prove the following result.
Proposition 2.13
The following are equivalent:
1) \(({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n})\) is conditionally atomless.
2) In the definition, we can take \({\mathbb{Q}}=\frac{1}{n}({\mathbb{Q}}_{1}+\cdots +{\mathbb{Q}}_{n})\).
3) \(X\) can be taken as uniformly distributed over \([0,1]\).
There are several differences with our approach. There is the technical difference that [16] suppose the existence of a continuously distributed random variable \(X\). In doing so, they avoid the technical points between the more conceptual definition using conditional expectations and the construction of a suitable sigma-algebra with a uniformly distributed random variable. A further difference is that they use a dominating measure that later can be taken as the mean of \(({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n})\). Of course, their result together with the results here show that the definition of \(({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n})\) being conditionally atomless is equivalent to the statement that for the measure \({\mathbb{Q}}_{0}=\frac{1}{n}({\mathbb{Q}}_{1}+\cdots +{\mathbb{Q}}_{n})\), the sigma-algebra \({\mathcal{A}}\) is atomless conditionally to the sigma-algebra generated by the Radon–Nikodým derivatives \((\frac{d{\mathbb{Q}}_{k}}{d{\mathbb{Q}}_{0}})_{k=1,\dots ,n}\). In [16], it is also shown that one can take any strictly positive convex combination of the measures \(({\mathbb{Q}}_{1},\ldots ,{\mathbb{Q}}_{n})\). Below we show that the sigma-algebra \({\mathcal{A}}\) in some sense has a minimality property, a result that clarifies the relation between the two approaches. Before doing so, let us recall two easy results from introductory probability theory.
Exercise 2.14
For a probability space \((\Omega ,{\mathcal{A}},{\mathbb{Q}})\), set \({\mathcal{N}}=\{N\in {\mathcal{A}}: {\mathbb{Q}}[N]=0\}\). Suppose that a sub-sigma-algebra \({\mathcal{F}}\subseteq {\mathcal{A}}\) is given and that \({\mathcal{G}}\) with \({\mathcal{F}}\subseteq {\mathcal{G}}\), is another sub-sigma-algebra which is included in the sigma-algebra generated by ℱ and \({\mathcal{N}}\). Then for each \(\xi \in L^{1}(\Omega ,{\mathcal{A}},{\mathbb{Q}})\),
$$ {\mathbb{E}}_{\mathbb{Q}}[\xi \, | \, {\mathcal{F}}]={\mathbb{E}}_{ \mathbb{Q}}[\xi \, | \, {\mathcal{G}}] \qquad \mbox{a.s.} $$
Exercise 2.15
With the notation in Exercise 2.14, let \(F\colon \Omega \rightarrow {\mathbb{R}}^{n}\) and \(F'\colon \Omega \rightarrow {\mathbb{R}}^{n}\) be two random vectors that are equal a.s. Let ℱ be generated by \(F\) and \({\mathcal{G}}\) by \(F'\). Then ℱ and \({\mathcal{G}}\) are equal up to sets in \({\mathcal{N}}\). More precisely, \({\mathcal{G}}\) is contained in the sigma-algebra generated by ℱ and \({\mathcal{N}}\) (and of course vice versa), i.e., \(\sigma ({\mathcal{F}},{\mathcal{N}})=\sigma ({\mathcal{G}},{ \mathcal{N}})\).
Proposition 2.16
Let \({\mathbb{Q}}_{1},\dots ,{\mathbb{Q}}_{n}\) be probability measures on a measurable space \((\Omega ,{\mathcal{A}})\). Let \({\mathbb{Q}}_{0}=\sum _{k=1}^{n} \lambda _{k} {\mathbb{Q}}_{k}\) be a convex combination of these measures with each \(\lambda _{k} > 0\). Let \(f_{k}\) denote an \({\mathcal{A}}\)-measurable version of \(\frac{d{\mathbb{Q}}_{k}}{{\mathbb{Q}}_{0}}\). Let ℚ be another dominating measure with \(g_{k}\) an \({\mathcal{A}}\)-measurable version of \(\frac{d{\mathbb{Q}}_{k}}{d{\mathbb{Q}}}\). Let \({\mathcal{N}}= \{N\in {\mathcal{A}}\colon {\mathbb{Q}}_{0}[N]=0\}\). Let ℱ be generated by \(f_{k},k=1,\dots , n\), and let \({\mathcal{G}}\) be generated by \(g_{k},k=1,\ldots , n\). Then \({\mathcal{F}}\subseteq \sigma ({\mathcal{G}},{\mathcal{N}})\).
Proof
Clearly, \({\mathbb{Q}}_{0}\ll {\mathbb{Q}}\); so let \(h=\frac{d{\mathbb{Q}}_{0}}{d{\mathbb{Q}}}\). It is now immediate that \(g_{k}= f_{k} h\) ℚ-a.s. To see this, observe that the values of \(f_{k}\) on \(\{h=0\}\) do not matter. The functions \(g_{k}\) and \(h\) are \({\mathcal{G}}\)-measurable since \(h\) can be taken as \(h=\sum _{k=1}^{n} \lambda _{k} g_{k}\). Then we define \(f_{k}'= \frac{g_{k}}{h}\) on \(\{h>0\}\) and \(f_{k}'=0\) on \(\{h=0\}\). This choice shows that the \(f_{k}'\) are \({\mathcal{G}}\)-measurable. It is immediate that \(f_{k}=f_{k}'\) \({\mathbb{Q}}_{0}\)-a.s. The result now follows. □
From Proposition 2.16, it follows that the sigma-algebra augmented with the class \({\mathcal{N}}\) is the same for all strictly positive convex combinations. This shows that in the definition of atomless conditionally to ℱ, we can also add the nullsets \({\mathcal{N}}\) to ℱ. To check that \({\mathcal{A}}\) is atomless conditionally to a sigma-algebra ℱ, it is clear that the smaller ℱ, the easier it is to satisfy the condition. In our opinion, the above clarifies the relation between this paper and [16].
3 A continuity result
Let us recall the standing assumptions: \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\), and \(U\) is independent of \({\mathcal{F}}_{1}\) and uniformly distributed on \([0,1]\). Further, the utility function \(u_{1,2}\colon L^{\infty }({\mathcal{F}}_{2})\rightarrow L^{\infty }({ \mathcal{F}}_{1})\) is coherent and Lebesgue-continuous. For each mapping \(h\colon \Omega \rightarrow [0,1]\) that is \({\mathcal{F}}_{1}\)-measurable, we put \(\phi (h)=u_{1,2}(\mathbf {1}_{\{U\le h\}})\). Clearly, \(\phi \) takes values in the space \(L^{\infty }({\mathcal{F}}_{1})\). We have the following continuity result.
Proposition 3.1
If \(h_{n}\downarrow h\) or \(h_{n}\uparrow h\), then \(\phi (h_{n})\rightarrow \phi (h)\).
Proof
If \(h_{n}\downarrow h\), then \(\mathbf {1}_{\{U\le h_{n}\}}\downarrow \mathbf {1}_{\{U\le h\}}\) and the Fatou property gives the desired result. For the upward convergence, we must be more careful. Because \(U\) has a continuous distribution function and is independent of \({\mathcal{F}}_{1}\), we conclude that \({\mathbb{P}}[U=h]=0\) and hence \(\mathbf {1}_{\{U\le h_{n}\}}\uparrow \mathbf {1}_{\{U\le h\}}\) a.s. The Lebesgue property then allows to conclude. □
Theorem 3.2
If \(h\colon \Omega \rightarrow [0,1]\) is \({\mathcal{F}}_{1}\)-measurable, there is an \({\mathcal{F}}_{1}\)-measurable function \(g\colon \Omega \rightarrow [0,1]\) such that the set \(B_{g}=\{U\le g\}\) satisfies \(u_{1,2}(\mathbf {1}_{B_{g}})=h\).
Proof
The statement can be rewritten as \(\phi (g)=h\). Let us introduce the class Then \({\mathcal{G}}\) is nonempty since \(1\in {\mathcal{G}}\). Furthermore, \({\mathcal{G}}\) is stable under taking minima. Indeed, take \(g_{1},g_{2}\in {\mathcal{G}}\) and put \(g=g_{1}\mathbf {1}_{A}+g_{2}\mathbf {1}_{A^{c}}\), where \(A=\{g_{1}< g_{2}\}\). Since \(u_{1,2}(\mathbf {1}_{B_{g}})=\mathbf {1}_{A} u_{1,2}(\mathbf {1}_{B_{g_{1}}}) + \mathbf {1}_{A^{c}}u_{1,2}(\mathbf {1}_{B_{g_{2}}})\ge h\), we have \(g\in {\mathcal{G}}\). Let now \(g_{n}\downarrow g\), where \((g_{n}) \subseteq {\mathcal{G}}\) and \({\mathbb{E}}[g_{n}]\downarrow \inf \{{\mathbb{E}}[g'] : g'\in { \mathcal{G}}\}\). The continuity for decreasing sequences then shows that \(g\in {\mathcal{G}}\). The previous lines are enough to show that \({\mathcal{G}}\) has a minimum. Let \(g\) be the smallest function in \({\mathcal{G}}\). We claim that the continuity for increasing sequences (the Lebesgue property) implies that actually \(u_{1,2}(\mathbf {1}_{B_{g}})=h\). Indeed, suppose to the contrary that the set \(\{u_{1,2}(\mathbf {1}_{B_{g}})>h\}\) has nonzero measure. This assumption trivially implies that \({\mathbb{P}}[g>0]>0\). Take now a sequence \(g_{n}\uparrow g\) such that on \(\{g>0\}\), we have \(g_{n}< g\). By Proposition 3.1, \(u_{1,2}(\mathbf {1}_{B_{g_{n}}})\uparrow u_{1,2}(\mathbf {1}_{B_{g}})\). Hence there must exist \(n\) such that \(A_{n}=\{u_{1,2}(\mathbf {1}_{B_{g_{n}}})>h\}\) has nonzero measure. On \(A_{n}\), we have \(g_{n}>0\), hence also \(g>0\), and therefore also \(g_{n}< g\). Put now \(g'=g_{n}\mathbf {1}_{A_{n}}+g\mathbf {1}_{A_{n}^{c}}\). We have \({\mathbb{E}}[g']<{\mathbb{E}}[g]\), but also \(g'\in {\mathcal{G}}\), which is a contradiction to the minimality of \(g\). □
$$ {\mathcal{G}}=\{ g\colon g\text{ is } {\mathcal{F}}_{1} \text{-measurable and }u_{1,2}(\mathbf {1}_{B_{g}})=\phi (g)\ge h \}. $$
Remark 3.3
Although “intuitively clear”, the continuity of the process \(t \mapsto u_{1,2}(\mathbf {1}_{B_{t}})\) is not an easy result. First of all, we are working with random variables identified under the equivalence a.s. That means that we must first select or construct measurable functions instead of classes of measurable functions. Then we must show that with respect to \(t\), these outcomes are continuous. The general theory of stochastic processes gives us the necessary tools to achieve this goal. We do not really need these finer results so that if you do not belong to the amateurs of the general theory of stochastic processes à la Dellacherie and Meyer [10], the remark can be skipped. First we construct a process \(\alpha (t,\omega )\). For each rational point \(q\in [0,1]\), we select an \({\mathcal{F}}_{1}\)-measurable function \(\alpha '(q)\) that represents \(u_{1,2}(\mathbf {1}_{B_{q}})\). Because of monotonicity we can – if needed – change these selections on a set of zero measure to make sure that a.s., the mapping \({\mathbb{Q}}\cap [0,1]\rightarrow {\mathbb{R}}, q\mapsto \alpha '(q)\) is increasing. For each \(t\in [0,1]\), we now define \(\alpha (t)= \inf _{q\ \text{rational,}\ q\ge t}\alpha '(q)\). The functions \(\alpha (t)\) are of course \({\mathcal{F}}_{1}\)-measurable and represent \(u_{1,2}(\mathbf {1}_{B_{t}})\) by the Fatou property. We may also suppose that \(\alpha (0)=0,\alpha (1)=1\) a.s. It is clear that \(\alpha \) is a.s. nondecreasing in \(t\) and right-continuous. This means there is a set (independent of \(t\)) such that on this set, \(t\mapsto \alpha (t,\omega )\) is right-continuous and nondecreasing.
We claim that the function \(\alpha \) also satisfies \(\alpha (h)=u_{1,2}(\mathbf {1}_{\{U\le h\}})=\phi (h)\) for each \({\mathcal{F}}_{1}\)-measurable function \(h\colon \Omega \rightarrow [0,1]\). To avoid misunderstandings, the random variable \(\alpha (h)\) is defined as \(\alpha (h)(\omega )=\alpha (h(\omega ),\omega )\). Such a notation is common in stochastic process theory. The above property of \(\alpha \) is easy to verify for elementary functions \(h\), and the general statement trivially follows by approximating \(h\) from above by elementary functions. Let us give the details. For an elementary function \(h=\sum _{k=1}^{K} t_{k}\mathbf {1}_{A_{k}}\) (the sets \(A_{k}\) are disjoint and in \({\mathcal{F}}_{1}\)), we have As indicated above, the Fatou property then completes the proof by using right-continuity. Indeed, let \(h\colon \Omega \rightarrow [0,1]\) be \({\mathcal{F}}_{1}\)-measurable and \(h_{n} \downarrow h\) a sequence of elementary functions that are \({\mathcal{F}}_{1}\)-measurable. Since \(\mathbf {1}_{\{U\le h_{n}\}} \downarrow \mathbf {1}_{\{U\le h\}}\), the Fatou property and the right-continuity of \(t \mapsto \alpha (t)\) give us \(\phi (h)=u_{1,2}(\mathbf {1}_{\{U\le h\}})\).
$$\begin{aligned} \alpha (h) =& \sum _{k=1}^{K} \alpha (t_{k})\mathbf {1}_{A_{k}} \\ =&\sum _{k=1}^{K} u_{1,2}(\mathbf {1}_{B_{t_{k}}})\mathbf {1}_{A_{k}} \\ =&\sum _{k=1}^{K} u_{1,2}(\mathbf {1}_{B_{t_{k}}}\mathbf {1}_{A_{k}})\mathbf {1}_{A_{k}} \\ =&\sum _{k=1}^{K} u_{1,2} ( \mathbf {1}_{B_{t_{k}}\cap A_{k}})\mathbf {1}_{A_{k}} \\ =&\sum _{k=1}^{K} u_{1,2}\bigg(\Big(\sum _{\ell =1}^{K} \mathbf {1}_{B_{t_{\ell }}\cap A_{\ell }}\Big)\mathbf {1}_{A_{k}} \bigg)\mathbf {1}_{A_{k}} \\ =&\sum _{k=1}^{K} u_{1,2}(\mathbf {1}_{\{U\le h\}}\mathbf {1}_{A_{k}})\mathbf {1}_{A_{k}} \\ =& u_{1,2}(\mathbf {1}_{\{U\le h\}})=\phi (h). \end{aligned}$$
The proof of the left-continuity can be done by using ideas from the general theory of stochastic processes. For \(\varepsilon >0\), we define Observe that \(r>0\) by construction. Suppose now that at the point \(r\), the probability that \(\alpha \) has a jump of size at least \(\varepsilon \) is nonzero. Take \(r_{n}\uparrow r\), \(r_{n}< r\). The continuity result in Proposition 3.1 gives us that \(\alpha (r_{n})\uparrow \alpha (r)\) which is a contradiction to \(\alpha \) having a jump. So for almost every \(\omega \in \Omega \), \(\alpha (\cdot ,\omega )\) has no jumps of size at least \(\varepsilon \). Since the latter was arbitrary, the a.s. continuity of the process \(\alpha \) is proved.
$$ r=\inf \Big\{ t \colon \lim _{s\rightarrow t, \, s< t}\alpha (s)\le \alpha (t)-\varepsilon \Big\} \wedge 1. $$
4 Some special commonotonic set
In this section, we define a special norm on \({\mathbb{R}}^{2}\). Part of its unit sphere will then be used as a commonotonic set. The reader could make some drawings to help visualise the constructions. The construction is done in several steps. The first step consists in taking the curve obtained as the concatenation of the convex intervals that join the points The convex hull of this set is a parallelogram \(P_{0}\), with parallel vertical sides given by the line segments The set \(P_{0}\) will be used as the unit ball of a norm on \({\mathbb{R}}^{2}\). More precisely, we use the Minkowski functional Note that every point of \(P_{0}\) is the convex combination of points taken on the vertical sides. An easy and continuous way to obtain such convex combination goes as follows. Through a point in \(P_{0}\), take a line parallel to the “skew” sides of \(P_{0}\) and see where it intersects the vertical sides. Elementary calculations give us that for \((x,y)\in P_{0}\), we may write \((x,y)=(1-\lambda _{0})(u^{0}_{1},u^{0}_{2})+\lambda _{0}(v^{0}_{1},v^{0}_{2})\) with \(u^{0}, v^{0} \in P\) and \(0 \leq \lambda _{0} \leq 1\), or more explicitly For each \(n\in {\mathbb{Z}}\), we now define \(P_{n}=2^{n}P_{0}\) and similarly as for \(n=0\), we define \(\lambda _{n}\), \((u^{n}_{1},u^{n}_{2})\), \((v^{n}_{1},v^{n}_{2})\). These functions are obviously continuous. The set \(E\) consists of all the vertical segments with the origin added. It forms a commonotonic set. This follows from the equality We now construct functions \(\Lambda , U, V\) on \({\mathbb{R}}^{2}\) as follows. For \((x,y)\in P_{n}\setminus P_{n-1}\), we define \(\Lambda (x,y)= \lambda _{n}(x,y)\), \(U(x,y)=u^{n}(x,y)\), \(V(x,y)=v^{n}(x,y)\). At \((0,0)\), we put \(\Lambda (0,0)=1\), \(U(0,0)=(0,0)=V(0,0)\). These functions are no longer continuous, but are certainly Borel-measurable. They satisfy the following properties:
$$ (-4,-4)\rightarrow (-4,-2)\rightarrow (0,0)\rightarrow (4,2) \rightarrow (4,4). $$
$$ (-4,-4)\rightarrow (-4,-2)\qquad \text{and}\qquad (4,2) \rightarrow (4,4). $$
$$ \Vert (x,y)\Vert := \inf \{ \alpha >0: (x,y)\in \alpha P_{0}\}. $$
$$ (x,y)=\frac{4-x}{8}\left (-4,y-3-\frac{3x}{4}\right )+\frac{4+x}{8} \left (4,y+3-\frac{3x}{4}\right ). $$
$$ E=\{(0,0)\}\cup \bigcup _{n\in {\mathbb{Z}}}\Big( 2^{n}\big( [(-4,-4),(-4,-2)] \cup [(2,4),(4,4)] \big) \Big). $$
1) \(\Lambda \colon {\mathbb{R}}^{2}\rightarrow [0,1]\).
2) \(U \colon {\mathbb{R}}^{2}\rightarrow E\), \(V\colon {\mathbb{R}}^{2}\rightarrow E\).
3) We have \(\Vert U(x,y)\Vert \le 2 \Vert (x,y)\Vert \) and \(\Vert V(x,y)\Vert \le 2 \Vert (x,y)\Vert \). Indeed, for \((x,y)\in P_{n}\setminus P_{n-1}\), we have \(2^{n}=\Vert U(x,y)\Vert \ge \Vert (x,y)\Vert \ge 2^{n-1}\), and the same holds for \(V\).
4) For all \((x,y)\in {\mathbb{R}}^{2}\), \((x,y)=(1-\Lambda (x,y))U(x,y)+\Lambda (x,y)V(x,y)\).
5) The coordinates \(V_{1}(x,y)-U_{1}(x,y)\) and \(V_{2}(x,y)-U_{2}(x,y)\) of \(V-U\) are nonnegative.
5 The main result
We start by giving an extension of the usual definition of conditional expectation.
Definition 5.1
We say that an \({\mathcal{F}}_{2}\)-measurable random variable \(\xi \) has an extended conditional expectation with respect to \({\mathcal{F}}_{1}\) if there is a countable \({\mathcal{F}}_{1}\)-measurable partition \((A_{n})\) such that each \(\mathbf {1}_{A_{n}}\xi \) is integrable. The conditional expectation is then defined as \(\sum _{n} {\mathbb{E}}[\mathbf {1}_{A_{n}}\xi \, | \, {\mathcal{F}}_{1}]\).
The reader can check that the existence and definition of an extended conditional expectation are independent of the choice of the \({\mathcal{F}}_{1}\)-measurable partition. We sometimes drop the word “extended”.
Again we suppose that \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\). The utility function \(u_{1,2}\) is Lebesgue-continuous.
Before giving the main result of the paper, we first prove a special case.
Theorem 5.2
For every couple \((f,g)\) of \({\mathcal{F}}_{1}\)-measurable finite-valued random variables, there is a commonotonic couple \((\xi ,\eta )\) of \({\mathcal{F}}_{2}\)-measurable random variables such that \(f={\mathbb{E}}[\xi \, | \, {\mathcal{F}}_{1}],g={\mathbb{E}}[\eta \, | \, {\mathcal{F}}_{1}]\). Furthermore, \(\Vert (\xi ,\eta )\Vert \le 2 \Vert (f,g)\Vert \) almost surely.
Proof
The proof is almost given in the previous sections. Let \((f,g)\colon \Omega \rightarrow {\mathbb{R}}^{2}\) be \({ \mathcal{F}}_{1}\)-measurable. Using the functions \(\Lambda ,U,V\) of Sect. 4, we can then write Because \(\Lambda (f,g):\Omega \rightarrow [0,1]\) is \({\mathcal{F}}_{1}\)-measurable and \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\), there is a set \(B \in {\mathcal{F}}_{2}\) such that \({\mathbb{E}}[\mathbf {1}_{B}\, | \, {\mathcal{F}}_{1}]=\Lambda (f,g)\). The random variables \((\xi ,\eta )\) are now defined as or in other words Both random variables have extended conditional expectations, and because \(U(f,g)\), \(V(f,g)\) are \({\mathcal{F}}_{1}\)-measurable, we get \((f,g)={\mathbb{E}}[(\xi ,\eta )\, | \, {\mathcal{F}}_{1}]\). Because \((\xi ,\eta )\) takes its values in the commonotonic set \(E\) from Sect. 4, we get that \(\xi \) and \(\eta \) are commonotonic. The estimate of the norms follows from the estimates for \(U\) and \(V\). □
$$ (f,g)=\Lambda (f,g) V(f,g)+\big(1-\Lambda (f,g)\big)U(f,g). $$
$$ \xi =\mathbf {1}_{B} V_{1}(f,g)+\mathbf {1}_{B^{c}}U_{1}(f,g), \qquad \eta = \mathbf {1}_{B} V_{2}(f,g)+\mathbf {1}_{B^{c}}U_{2}(f,g), $$
$$ (\xi ,\eta )=\mathbf {1}_{B} V(f,g) + \mathbf {1}_{B^{c}}U(f,g). $$
Corollary 5.3
The random variable \((\xi ,\eta )\) has the same integrability properties as the couple \((f,g)\). In particular, if \((f,g)\) is bounded, the couple \((\xi ,\eta )\) is bounded.
Remark 5.4
If one wants to use another norm than the Minkowski functional of \(P_{0}\), one must adapt the constant. Because all norms on \({\mathbb{R}}^{2}\) are equivalent, this is an exercise in linear algebra. We did not try to find the best estimates for e.g. the Euclidean norm, where a rough calculation gave \(10\sqrt{2}\). This problem would require to find a better commonotonic set than the one used above.
The next theorem is an improvement of the preceding result in the sense that we replace the conditional expectation by a more general utility function. The proof follows the same lines.
Theorem 5.5
For every couple \((f,g)\) of \({\mathcal{F}}_{1}\)-measurable bounded random variables, there is a commonotonic couple \((\xi ,\eta )\) of \({\mathcal{F}}_{2}\)-measurable random variables such that \(f=u_{1,2}(\xi ),g=u_{1,2}(\eta )\). Furthermore, \(\Vert (\xi ,\eta )\Vert \le 2 \Vert (f,g)\Vert \) almost surely.
Proof
We use the same notation \((\Lambda ,U,V)\) as in the previous proof. But this time we take a set \(B\) such that \(u_{1,2}(\mathbf {1}_{B})=\Lambda \). Again we define We then have and similarly for \(g\) and the second coordinate. Note that we can apply the positive homogeneity of \(u_{1,2}\) because \(V_{1}(f,g)-U_{1}(f,g)\ge 0\). □
$$ (\xi ,\eta )=\mathbf {1}_{B} V(f,g) + \mathbf {1}_{B^{c}}U(f,g)=U(f,g)+\mathbf {1}_{B} \big(V(f,g)-U(f,g)\big). $$
$$\begin{aligned} u_{1,2}(\xi ) =&u_{1,2}\Big(U_{1}(f,g)+\mathbf {1}_{B}\big(V_{1}(f,g)-U_{1}(f,g) \big)\Big) \\ =&U_{1}(f,g)+u_{1,2}(\mathbf {1}_{B})\big(V_{1}(f,g)-U_{1}(f,g)\big) \\ =&U_{1}(f,g)+\Lambda (f,g)\big(V_{1}(f,g)-U_{1}(f,g)\big)=f, \end{aligned}$$
Remark 5.6
If \((f,g)\) is only finite-valued, we can write and this is a sum of bounded random variables. For each \(n\), we can define \(\xi _{n},\eta _{n}\) as in Theorem 5.5. These random variables are zero outside \(\{(f,g)\in P_{n}\setminus P_{n-1}\}\), and hence the sum \((\xi ,\eta )=\sum _{n\in {\mathbb{Z}}}(\xi _{n},\eta _{n})\) is well defined. We could then extend \(u_{1,2}\) as we did for conditional expectations. Finally, we get \(u_{1,2}(\xi )=f,u_{1,2}(\eta )=g\). This extension is important when the utility functions are defined on e.g. Orlicz or Riesz spaces. Important for such extensions is the pointwise (almost sure) estimate \(\Vert (\xi ,\eta )\Vert \le 2\Vert (f,g)\Vert \).
$$ (f,g)=\mathbf {1}_{\{(f,g)=(0,0)\}}(f,g)+\sum _{n\in {\mathbb{Z}}}\mathbf {1}_{\{(f,g) \in P_{n}\setminus P_{n-1}\}}(f,g), $$
6 Commonotonicity and time-consistency
In this section, we use the same hypothesis on the filtration \(({\mathcal{F}}_{0},{\mathcal{F}}_{1},{\mathcal{F}}_{2})\). In particular, we suppose that \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\). We start with a monetary coherent utility function \(u_{0,2}\colon L^{\infty }({\mathcal{F}}_{2})\rightarrow {\mathbb{R}}\). We suppose – as in the rest of the paper – that \(u_{0,2}\) is relevant.
Theorem 6.1
Suppose that
1) \({\mathcal{F}}_{2}\) is atomless conditionally to \({\mathcal{F}}_{1}\);
2) \(u_{0,2}\) is coherent and relevant;
3) \(u_{0,2}\) is time-consistent;
4) \(u_{0,2}\) is commonotonic, i.e., if the random variables \(\xi ,\eta \in L^{\infty }({\mathcal{F}}_{2})\) are commonotonic, then \(u_{0,2}(\xi +\eta )=u_{0,2}(\xi )+u_{0,2}(\eta )\);
5) \(u_{0,2}\) is Lebesgue-continuous.
Then there is a probability \({\mathbb{Q}}\approx {\mathbb{P}}\) such that \(u_{0,1}(f)={\mathbb{E}}_{\mathbb{Q}}[f]\) for all \(f\in L^{\infty }({\mathcal{F}}_{1})\).
Proof
According to Theorem 5.5, for each \(f,g\in L^{\infty }({\mathcal{F}}_{1})\), there are commonotonic \(\xi ,\eta \in L^{\infty }({\mathcal{F}}_{2})\) with \(u_{1,2}(\xi )=f,\, u_{1,2}(\eta )=g\) and \(u_{1,2}(\xi +\eta )=f+g\). We then have \(u_{0,1}(f)=u_{0,1}(u_{1,2}(\xi ))=u_{0,2}(\xi )\) and similarly for \(g\). The combination with commonotonicity then gives This shows that \(u_{0,1}\) is additive (therefore linear) and hence given by a finitely additive probability measure. But Lebesgue-continuity implies that this measure, say ℚ, must be sigma-additive and absolutely continuous with respect to ℙ. Because \(u_{0,2}\) and hence \(u_{0,1}\) are relevant, we must have \({\mathbb{Q}}\approx {\mathbb{P}}\). □
$$\begin{aligned} u_{0,1}(f+g) =&u_{0,1}\big(u_{1,2}(\xi +\eta )\big) \\ =&u_{0,2}(\xi +\eta ) \\ =&u_{0,2}(\xi )+u_{0,2}(\eta ) \\ =&u_{0,1}\big(u_{1,2}(\xi )\big)+u_{0,1}\big(u_{1,2}(\eta )\big) \\ =&u_{0,1}(f)+u_{0,1}(g). \end{aligned}$$
Remark 6.2
For general commonotonic \(\xi ,\eta \) (not just for those used in the proof of Theorem 6.1), we can now prove that \(u_{1,2}(\xi +\eta )=u_{1,2}(\xi )+u_{1,2}(\eta )\). We already know that \(u_{1,2}(\xi +\eta )\ge u_{1,2}(\xi )+u_{1,2}(\eta )\). If \({\mathbb{Q}}[u_{1,2}(\xi +\eta )> u_{1,2}(\xi )+u_{1,2}(\eta )] >0\), then we get which is a contradiction to \(u_{0,2}(\xi +\eta )=u_{0,2}(\xi )+u_{0,2}(\eta )\). The strict inequality in the third line follows from the fact that \(u_{0,1}\) is the expectation with respect to the equivalent probability measure ℚ.
$$\begin{aligned} u_{0,2}(\xi +\eta ) =&u_{0,1}\big(u_{1,2}(\xi +\eta )\big) \\ =&{\mathbb{E}}_{\mathbb{Q}}[u_{1,2}(\xi +\eta ) ] \\ >& {\mathbb{E}}_{\mathbb{Q}}[u_{1,2}(\xi )] +{\mathbb{E}}_{\mathbb{Q}}[u_{1,2}( \eta ) ] \\ =& u_{0,1}\big(u_{1,2}(\xi )\big)+u_{0,1}\big(u_{1,2}(\eta )\big) \\ =& u_{0,2}(\xi )+u_{0,2}(\eta ), \end{aligned}$$
Remark 6.3
If the assumption of relevance is dropped, we must start with a time-consistent system of utility functions \(u_{0,2},u_{0,1},u_{1,2}\). In that case, we only obtain \({\mathbb{Q}}\ll {\mathbb{P}}\), and the result of Remark 6.2 only holds ℚ-a.s.
Remark 6.4
There is no reason that \(u_{0,2}\) is additive on \(L^{\infty }({\mathcal{F}}_{2})\) as the following example shows. We take \(\Omega =[0,1]\times [0,1]\), \({\mathcal{F}}_{2}\) is the product sigma-algebra of the Borel sigma-algebras on \([0,1]\), and the measure ℙ is the product measure of the usual Lebesgue measures. \({\mathcal{F}}_{0}\) is the trivial sigma-algebra and \({\mathcal{F}}_{1}\) is generated by the first coordinate mapping. For \(\xi \in L^{\infty }({\mathcal{F}}_{2}),\xi \ge 0\), we define For \(0\le \xi \in L^{\infty }({\mathcal{F}}_{2})\), the utility function \(u_{1,2}\) is then given by Such expressions are known as distortions or Choquet integrals. They are standard examples of commonotonic utility functions; see [7, Chap. 7]. We need a bit less than commonotonicity; in fact, we only need for \(\xi ,\eta \) that \(u_{1,2}(\xi +\eta )=u_{1,2}( \xi )+u_{1,2}(\eta )\) as soon as for each \(\alpha \), the random variables \(\xi (\alpha ,\cdot ),\eta (\alpha ,\cdot )\) are commonotonic. To see that \(u_{0,2}\) is not linear, let us calculate the outcomes for \(\xi (\alpha ,y)=\mathbf {1}_{[0,1/2]}(y)\) and \(\eta (\alpha ,y)=\mathbf {1}_{[1/2,1]}(y)\). For both random variables, we find \(\frac{1}{4\log 2}\) which do not sum up to \(u_{0,2}(\xi +\eta )=u_{0,2}(1)=1\).
$$ u_{0,2}(\xi )=\int _{0}^{1}d\alpha \int _{0}^{\infty }dx \big({ \mathbb{P}}[\xi (\alpha ,\cdot )\ge x]\big)^{1+\alpha }. $$
$$ u_{1,2}(\xi )(\alpha )=\int _{0}^{\infty }\big({\mathbb{P}}[\xi (\alpha , \cdot )>x]\big)^{1+\alpha }\,dx. $$
7 A continuous-time result
In this section, we use a filtration indexed by the time interval \([0,T]\). This filtration \(\left ({\mathcal{F}}_{t}\right )_{0\le t\le T}\) does not necessarily fulfil the usual assumptions. The only assumption is that \({\mathcal{F}}_{T}\) is generated by \(\bigcup _{0\le t< T}{\mathcal{F}}_{t}\). We also suppose that we are given a family \(u_{t,s},0\le t\le s\le T\), \(u_{t,s}\colon L^{\infty }({\mathcal{F}}_{s})\rightarrow L^{\infty }({ \mathcal{F}}_{t})\), of coherent utility functions. We assume the following time-consistency: for \(t\le s\le v\), we have \(u_{t,v}=u_{t,s}\circ u_{s,v}\).
Theorem 7.1
With the notation introduced in this section, we suppose that for all \(0\le t < T\), the sigma-algebra \({\mathcal{F}}_{T}\) is atomless conditionally to \({\mathcal{F}}_{t}\). If \(u_{0,T}\) is relevant, Lebesgue-continuous and commonotonic, there is a probability \({\mathbb{Q}}\approx {\mathbb{P}}\) such that \(u_{0,T}(\xi )={ \mathbb{E}}_{\mathbb{Q}}[\xi ]\) for all \(\xi \in L^{\infty }({\mathcal{F}}_{T})\).
Proof
The results of Sect. 6 show that on each \(L^{\infty }({\mathcal{F}}_{t})\), the utility function \(u_{0,T}\) is linear. The utility function \(u_{0,T}\) is therefore linear on the vector space \(\bigcup _{t< T}L^{\infty }({\mathcal{F}}_{t})\). This space is sequentially dense in \(L^{\infty }({\mathcal{F}}_{T})\) for the Mackey topology (simply use the martingale convergence theorem). Because of Lebesgue-continuity, the utility function \(u_{0,T}\) is therefore linear on \(L^{\infty }({\mathcal{F}}_{T})\). It is thus given by a probability measure \({\mathbb{Q}}\ll {\mathbb{P}}\). But since the utility function is relevant, we find that \({\mathbb{Q}}\approx {\mathbb{P}}\). □
Remark 7.2
The previous results can be applied for most filtrations used in finance and insurance. This is for instance true for filtrations coming from a Brownian motion in one or several dimensions, filtrations generated by most Lévy processes, and so on. In other words, commonotonicity and time-consistency are not good friends.
Acknowledgements
This research was done while the author was visiting Tokyo Metropolitan University in October and November 2018. We thank the staff of TMU for the many fruitful discussions, and in particular we thank Prof. Adachi for many critical remarks. We also thank Prof. T. Yamada and Prof. K. Takaoka for fruitful discussions while the author was visiting Hitotsubashi University, Kunitachi, Tokyo, in November and December 2019.
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