We preserve the same names for various types of time consistency for both the random variables and the stochastic processes. However, we stress that the nature of time consistency for stochastic processes is usually much more intricate. If φ is an LMmeasure, and \(V\in \mathbb {V}^{p}\), then in order to compare φ
_{
t
}(V) and φ
_{
s
}(V), for s>t, one also needs to take into account the cash flows between times t and s.
In order to account for the intermediate cash flows, we modify appropriately the concept of the update rule.
Definition 10
The family \(\mu =\{\mu _{t,s}:\, t,s\in \mathbb {T},\, t<s\}\) of maps \(\mu _{t,s}:\bar {L}^{0}_{s}\times \mathcal {X}\to \bar {L}^{0}_{t}\) is called a generalized update rule if for any \(X\in \mathcal {X}\) the family \(\mu (\cdot,X)=\{\mu _{t,s}(\cdot,X):\, t,s\in \mathbb {T},\, t<s\}\) is an update rule.
Note that the update rule introduced in Definition 3 may be considered as the generalized update rule, which is constant with respect to X, i.e., μ(·,X)=μ(·,Y) for any \(X,Y\in \mathcal {X}\). In what follows, if there is no ambiguity, we drop the term generalized.
As before, we say that the update rule μ is sinvariant, if there exists a family \(\{\mu _{t}\}_{t\in \mathbb {T}}\) of maps \(\mu _{t}:\bar {L}^{0}\times \mathcal {X} \to \bar {L}^{0}_{t}\), such that μ
_{
t,s
}(m
_{
s
},X)=μ
_{
t
}(m
_{
s
},X) for any \(s,t\in \mathbb {T}\), s>t, \(X\in \mathcal {X}\), and \(m_{s}\in \bar {L}^{0}_{s}\).
We now arrive at the corresponding definition of timeconsistency.
Definition 11
Let μ be a generalized update rule. We say that the dynamic LMmeasure φ is μacceptance (resp.
μrejection) time consistent if
$$ \varphi_{s}(X)\geq m_{s} \quad(\mathrm{resp.} \leq)\quad \Longrightarrow\quad \varphi_{t}(X)\geq \mu_{t,s}(m_{s},X)\quad(\mathrm{resp.} \leq), $$
(23)
for all \(s,t\in \mathbb {T}\), s>t, \(X\in \mathcal {X}\), and \(m_{s}\in \bar {L}^{0}_{s}\). In particular, if property (23) is satisfied for s=t+1, t=0,…,T, then we say that φis onestep
μacceptance (resp. onestep
μrejection) time consistent.
Throughout this section, we assume that \(\mathcal {X}=\mathbb {V}^{p}\).^{12} We will focus our attention on onestep update rules μ such that
$$ \mu_{t,t+1}(m,V)=\tilde \mu_{t,t+1}(m)+f(V_{t}), \quad t=0,\ldots,T1, $$
(24)
where \(\tilde \mu \) is the onestep update rule for random variables, and \(f:\bar {\mathbb {R}}\to \bar {\mathbb {R}}\) is a Borel measurable function such that f(0)=0. Property (24) is postulated primarily to allow establishing a direct connection between our results and the existing literature. Moreover, when using onestep update rules of form (24), the onestep time consistency for random variables is a particular case of onestep time consistency for stochastic processes by considering cash flows with only terminal payoff, namely stochastic processes such that V=(0,…,0,V
_{
T
}).
Finally, we note that for update rules, which admit the so called nested composition property (cf. (Ruszczyński 2010; Ruszczyński and Shapiro 2006b)),
$$ {\mu_{t,s}(m,V)=\mu_{t,t+1}(\mu_{t+1,t+2}(\ldots\mu_{s2,s1}(\mu_{s1,s}(m,V),V)\ldots V),V),} $$
(25)
we have that μacceptance (resp. μrejection) time consistency is equivalent to one step μacceptance (resp. μrejection) time consistency. This is another reason why we consider only one step update rules for stochastic processes.
Weak time consistency
We start with the following definition.
Definition 12
A dynamic LMmeasure φ on \(\mathbb {V}^{p}\) is weakly acceptance (resp. weakly rejection) time consistent if
$$\varphi_{t}(V)\geq \mathrm{ess\,inf}_{t}\varphi_{t+1}(V)+V_{t},\quad (\mathrm{resp.}\quad \varphi_{t}(V)\leq \mathrm{ess\,sup}_{t}\varphi_{t+1}(V)+V_{t}\,) $$
for any \(V\in \mathbb {V}^{p}\) and \(t\in \mathbb {T}\), such that t<T.
The next result is the counterpart of Proposition 1 and Proposition 2.
Proposition 9
Let φ be a dynamic LMmeasure on \(\mathbb {V}^{p}\). The following properties are equivalent:

1)
φ is weakly acceptance time consistent.

2)
φ is μacceptance time consistent, where μ is an sinvariant update rule, given by
$$\mu_{t}(m,V)=ess\,{inf}_{t}m+V_{t}. $$

3)
For any \(V\in \mathbb {V}^{p}\) and t<T
$$ \varphi_{t}(V)\geq \underset{Q\in\mathcal{M}_{t}(P)}{\mathrm{ess\,inf}}~E_{Q}[\varphi_{t+1}(V)\mathcal{F}_{t}]+V_{t}. $$
(26)

4)
For any \(V\in \mathbb {V}^{p}\), t<T, and \(m_{t}\in \bar {L}^{0}_{t}\),
$$\varphi_{t+1}(V)\geq m_{t} \Rightarrow \varphi_{t}(V)\geq m_{t}+V_{t}. $$
Additionally, if φ is a dynamic monetary risk measure, then the above properties are equivalent to

5)
For any \(V\in \mathbb {V}^{p}\) and t<T,
$$\varphi_{t+1}(V)\geq 0 \Rightarrow \varphi_{t}(V)\geq V_{t}. $$
Analogous equivalences are true for weak rejection time consistency.
The proof of Proposition 9 is analogous to the proofs of Proposition 1 and Proposition 2, and we omit it.
As mentioned earlier, the update rule, and consequently time consistency for stochastic processes, depends also on the value of the process (the dividend paid) at time t. In the case of weak time consistency this feature is interpreted as follows: if tomorrow, at time t+1, we accept \(V\in \mathbb {V}^{p}\) at the level greater than \(m_{t+1}\in \mathcal {F}_{t+1}\), then today at time t, we will accept V at least at the level ess inf_{
t
}
m
_{
t+1} (i.e., the worst level of m
_{
t+1} adapted to the information \(\mathcal {F}_{t}\)) plus the dividend V
_{
t
} received today.
Finally, we present the counterpart of Proposition 3 for the case of stochastic processes.
Proposition 10
Let ϕ be a projective update rule for random variables and let the update rule μ for stochastic processes be given by
$$ \mu_{t,t+1}(m,V)=\phi_{t}(m)+V_{t},\quad m\in\bar{L}^{0}_{t+1},\ V\in\mathbb{V}^{p}. $$
(27)
If φis a dynamic onestep LMmeasure on \(\mathbb {V}^{p}\), which is μacceptance (resp. μrejection) time consistent, then φ is weakly acceptance (resp. weakly rejection) time consistent.
Proposition 10 can be proved in a way analogous to the proof of Proposition 3.
Remark 9
The statement of Proposition 10 remains true if we replace (27) with
$$\mu_{t,t+1}(m,V)=\phi_{t}(m+V_{t}),\quad m\in\bar{L}^{0}_{t+1},\ V\in\mathbb{V}^{p}. $$
Indeed, it is enough to note that, for any \(V\in \mathbb {V}^{p}\) and t<T,
$$\begin{array}{@{}rcl@{}} \varphi_{t}(V) &\geq& \mu_{t,t+1}(\varphi_{t+1}(V),V)=\phi_{t}(\varphi_{t+1}(V)+V_{t})\\ & \geq & \phi_{t}(\mathrm{ess\,inf}_{t}[\varphi_{t+1}(V)\,+\,V_{t}])\,=\,\mathrm{ess\,inf}_{t}[\varphi_{t+1}(V)\,+\,V_{t}]\geq \mathrm{ess\,inf}_{t}\varphi_{t+1}(V)\,+\,V_{t}. \end{array} $$
Semiweak time consistency
In this section, we introduce the concept of semiweak time consistency for stochastic processes. We have not discussed semiweak time consistency in the case of random variables, since, in that case, semiweak time consistency coincides with the weak time consistency.
As it was shown, (Bielecki et al. 2014b), none of the forms of time consistency existing in the literature at the time when that paper was written were suitable for scaleinvariant maps such as acceptability indices. In fact, even the weak acceptance and the weak rejection time consistency for stochastic processes (as defined in the present paper) are too strong in the case of scale invariant maps. This is a reason why we introduce yet a weaker notion of time consistency, which we will refer to as semiweak acceptance and semiweak rejection time consistency. The notion of semiweak time consistency for stochastic processes, introduced next, is well suited for scaleinvariant maps; we refer the reader to (Bielecki et al. 2014b) for a detailed discussion on time consistency for such maps and their dual representations.^{13}
Definition 13
Let φ be a dynamic LMmeasure on \(\mathbb {V}^{p}\). Then, φ is semiweakly acceptance time consistent if
and it is semiweakly rejection time consistent if
Clearly, weak acceptance/rejection time consistency for stochastic processes implies semiweak acceptance/rejection time consistency.
Next, we will show that the definition of semiweak time consistency is indeed equivalent to the time consistency introduced in (Bielecki et al. 2014b).
Proposition 11
Let φ be a dynamic LMmeasure on \(\mathbb {V}^{p}\). The following properties are equivalent

1)
φ is semiweakly acceptance time consistent.

2)
φ is one step μacceptance time consistent, where the (generalized) update rule is given by
$$\mu_{t,t+1}(m,V) =1_{\{V_{t}\geq 0\}}\mathrm{ess\,inf}_{t} m+1_{\{V_{t}< 0\}}(\infty). $$

3)
For all \(V\in \mathbb {V}^{p}\), \(t\in \mathbb {T}\), t<T, and \(m_{t}\in \bar {L}^{0}_{t}\), such that V
_{
t
}≥0
$$\varphi_{t+1}(V)\geq m_{t}\quad \Longrightarrow\quad\varphi_{t}(V)\geq m_{t}. $$
A similar result is true for semiweak rejection time consistency.
For the proof, see (Bielecki et al. 2014a, Proposition 4.8).
Property 3) in Proposition 11, which is the definition of the (acceptance) time consistency given in (Bielecki et al. 2014b), best illustrates the financial meaning of semiweak acceptance time consistency: if tomorrow we accept the dividend stream \(V\in \mathbb {V}^{p}\) at level m
_{
t
}, and if we get a positive dividend V
_{
t
} paid today at time t, then today we accept the cash flow V at least at level m
_{
t
} as well. A similar interpretation is valid for semiweak rejection time consistency.
The next two results are important. In particular, they generalize the work done in (Bielecki et al. 2014b) regarding duality between cashadditive risk measures and acceptability indices.
Proposition 12
Let \(\{\varphi ^{x}\}_{x\in \mathbb {R}_{+}}\) be a decreasing family^{14} of dynamic LMmeasures on \(\mathbb {V}^{p}\). Assume that for each \(x\in \mathbb {R}_{+}\), φ
^{x} is weakly acceptance (resp. weakly rejection) time consistent. Then, the family \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) of maps \(\alpha _{t}:\mathbb {V}^{p}\to \bar {L}^{0}_{t}\) defined by
is a semiweakly acceptance (resp. semiweakly rejection) time consistent dynamic LMmeasure.
For the proof, see (Bielecki et al. 2014a, Proposition 4.9). It will be useful to note that α
_{
t
}(V) defined in (28) can also be written as
$$ \alpha_{t}(V)=\sup\{x \in \mathbb{R}^{+} \mid \varphi_{t}^{x}(V)\geq 0\}. $$
(29)
Proposition 13
Let \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) be a dynamic LMmeasure, which is independent of the past and translation invariant.^{15} Assume that \(\{\alpha _{t}\}_{t\in \mathbb {T}}\) is semiweakly acceptance (resp. semiweakly rejection) time consistent. Then, for any \(x\in \mathbb {R}_{+}\), the family \(\varphi ^{x}=\{\varphi _{t}^{x}\}_{t\in \mathbb {T}}\) of maps \(\varphi ^{x}_{t}:\mathbb {V}^{p}\to \bar {L}^{0}_{t}\) defined by
is a weakly acceptance (resp. weakly rejection) time consistent dynamic LMmeasure.
For the proof, see (Bielecki et al. 2014a, Proposition 4.10). In what follows, we will use the fact that \(\varphi ^{x}_{t}(V)\) defined in (30) can also be written as
This type of dual representation, i.e., (28) and (30), or, equivalently, (29) and (31), first appeared in (Cherny and Madan 2009) where the authors studied the static (one period of time) case. Subsequently, in (Bielecki et al. 2014b), the authors extended these results to the case of stochastic processes with special emphasis on the time consistency property. In contrast to the results of (Bielecki et al. 2014b), Propositions 12 and 13 consider an arbitrary probability space, not just a finite one.
Strong time consistency
Let us start with the definition of strong time consistency.
Definition 14
Let φbe a dynamic LMmeasure on \(\mathbb {V}^{p}\). Then φ is said to be strongly time consistent if
$$V_{t}=V'_{t}\ \text{and} \ \varphi_{t+1}(V)= \varphi_{t+1}(V') \quad\Longrightarrow \quad\varphi_{t}(V)= \varphi_{t}(V'), $$
for any \(V,V'\in \mathbb {V}^{p}\) and \(t\in \mathbb {T}\), such that t<T.
Now, let us present the counterpart of Proposition 4.
Proposition 14
Let φ be a dynamic LMmeasure on \(\mathbb {V}^{p}\), which is independent of the past. The following properties are equivalent:

1)
φ is strongly time consistent.

2)
There exists an update rule μ such that: for any \(t\in \mathbb {T}'\), \(m\in \bar {L}^{0}_{t}\), and \(V,V'\in \mathbb {V}^{p}\), satisfying \({V}_{t}={V}^{\prime }_{t}\), we have μ
_{
t,t+1}(m,V)=μ
_{
t,t+1}(m,V
^{′}); the family φ is both onestep μacceptance and onestep μrejection time consistent.

3)
There exists an update rule μ such that for any t<T and \(V\in \mathbb {V}^{p}\)
$$\varphi_{t}(V)=\mu_{t,t+1}(\varphi_{t+1}(V),1_{\{t\}}V_{t}). $$
As in the case of random variables, strong time consistency is usually considered for dynamic monetary risk measures on \(\mathbb {V}^{\infty }\). In this case, additional equivalent properties can be established. For brevity, we skip the details, and only show the general idea for deriving a litany of equivalent properties. This idea is rooted in a specific construction of strongly time consistent dynamic LMmeasures.
Corollary 1
Let μbe a update rule for random variables. Let \(\widetilde {\varphi }\) be a dynamic LMmeasure on \(\mathbb {V}^{\infty }\) given by
$$\left\{ \begin{array}{ll} \tilde{\varphi}_{T}(V) & =V_{T}\\ \tilde{\varphi}_{t}(V) & =\mu_{t,t+1}(\tilde\varphi_{t+1}(V))+V_{t}, \end{array}\right. $$
Then, \(\tilde \varphi \) is a strongly time consistent dynamic LMmeasure on \(\mathbb {V}^{\infty }\).
For a more detailed explanation of this idea and other equivalent properties see, e.g., (Cheridito and Kupper 2011) or (Ruszczyński and Shapiro 2006b).
Other types of time consistency
Other types of time consistency for stochastic processes may be defined in analogy to what is done in Section “Other types of time consistency” for the case of random variables. For brevity, we limit our discussion here to the update rules derived from dynamic LMmeasures.
First, given a dynamic LMmeasure φ on \(\mathbb {V}^{p}\), we denote by \(\widetilde {\varphi }\) the family of maps \(\widetilde {\varphi }_{t}: L_{t+1}^{p}\to \bar {L}^{0}_{t}\) given by
$$ \widetilde{\varphi}_{t}(X):=\varphi_{t}(1_{\{t+1\}}X),\ \text{for}\ t\in\mathbb{T}'. $$
(32)
Since φ is monotone and local on \(\mathbb {V}^{p}\), then, clearly, \(\widetilde {\varphi }_{t}\) is local and monotone on \(L_{t+1}^{p}\).
Next, for any \(t\in \mathbb {T}'\), we extend \(\widetilde {\varphi }_{t}\) to \(\bar {L}_{t+1}^{0}\), preserving locality and monotonicity (see Remark 12), and this extension produces a onestep update rule.
For example, the middle acceptance time consistency is obtained by taking the update rule μ given as
$$\mu_{t,t+1}(m,V)=\widetilde{\varphi}^{}_{t}(m+V_{t}),\quad t\in\mathbb{T}', $$
where \(\widetilde {\varphi }_{t}^{}:\bar {L}^{0}_{t+1}\to \bar {L}^{0}_{t}\) is defined as in (55), with the sets \(\mathcal {Y}^{}_{A}(X)\) replaced by
Taxonomy of results
In Fig. 2, we summarize the results surveyed in Section “Time consistency for stochastic processes”. We label each arrow (implication or equivalence) in the flowchart with numbers in squares and we relate the labels to the relevant results. Additionally, we provide comments on converse implications whenever appropriate.

■■■
Proposition 9, 5)

■■■
Proposition 9, 4)

■■■
Proposition 11, 3)

■■■
Proposition 10

■■■
Proposition 10, and see also .

■■■
Proposition 14.
Remark 10
The converse of implications and in Flowchart 2 do not hold true in general; one can use the same counterexamples as in the case of random variables. For a counterexample showing that the converse of does not hold true in general, see Example 5.