Dynamically consistent investment under model uncertainty: the robust forward criteria

Despite the popularity of the above model, there has been a considerable amount of criticism of the model fundamentals \((\mathbb {P},T, U)\), for these inputs might be ambiguous, inflexible, not very amenable to applications, and difficult to specify. First, there are numerous issues regarding elucidation and choice of the utility function \(U\). Some authors argue that the concept of utility per se is elusive and that one should look for different, more pragmatic criteria to use in order to quantify the risk preferences of an investor. We refer the reader to an old note of F. Black [8] where the criterion is the choice of the optimal portfolio, see also He and Huang [29] and Cox et al. [13], and to Monin [50] where the criterion is a targeted wealth distribution. Another line of research accepts the utility as an appropriate device to rank outcomes but challenges the classical EUM, for empirical evidence shows that investors feel differently with respect to gains and losses. Among others, see Hershey and Schoemaker [33] and Kahneman and Tversky [35] which then led to the development of the area of behavioural finance (see e.g. Barberis and Thaler [4] and Jin and Zhou [34]). Yet others generalise the concept of utility and move away from terminal-horizon deterministic utilities, as \(U(\, \cdot\,)\) above, by allowing state- and path-dependence which can alleviate several drawbacks of the classical setting. One of the best known paradigms are recursive utilities; see e.g. Duffie and Epstein [18], El Karoui et al. [22], Skiadas [67]. State-dependent utilities have also been considered in static frameworks; see e.g. Drèze [17] and Karni [41].

Second, the investment horizon \(T\) might not be fixed or a priori known. Such situations arise, for example, in investment problems with rolling horizons or in problems in which the horizon needs to be modified due to an inflow of new funds, new market opportunities, or new investment options and obligations. In this context, it is natural to study under which model conditions and preference structures one could extend the standard investment problem beyond a pre-specified horizon in a time-consistent manner; see e.g. Källblad [36, Sect. 2.2] and [37]. It is also interesting to study utilities that are not biased by the horizon choice, like the horizon-unbiased utilities introduced by Henderson and Hobson [30]; see also Choulli et al. [12].

Last but not least, an investor frequently faces significant ambiguity as to which market model to use; specifically, how to determine the probability measure ℙ. This is often referred to as Knightian uncertainty in reference to the original contribution of Knight [44]. In the seminal work by Gilboa and Schmeidler [28], motivated by the Ellsberg [23] paradox, the independence axiom was weakened to account for ambiguity aversion which led to a generalised robust EUM paradigm. It built on earlier contributions, including Anscombe and Aumann [2] and Schmeidler [66], and has since been followed and extended in a large number of works; we refer the reader to Maccheroni et al. [47], Schied [64] and to Föllmer et al. [27] and the references therein.

Our work here was motivated by the above considerations on the triplet of model inputs \((\mathbb{P},T, U)\). We propose a framework that alleviates some of the above shortcomings in a unified manner, combining elements from classical robustness theory and the recently developed forward investment performance approach. We now briefly introduce the latter before describing our main contributions.

In this paper, we build an analogous decision framework for an agent who faces model ambiguity. As in the classical robust EUM, we consider an investor in a stochastic market environment for which she does not know the “true” model. Instead, she describes the market reality through relative weighting of stochastic models with some models being more likely than others, some being excluded altogether, etc. These views are expressed by a penalty function and are updated dynamically in time. The investor’s personal evaluation of wealth is expressed through her preferences. When considering a given investment horizon, say \(T\), the investor aims to maximise the robust expected utility (max-min) functional, similarly to Maccheroni et al. [47] and Schied [64]. However, we generalise their criterion by considering stochastic preferences. These preferences evolve forward in time, taking into account the model ambiguity, and are defined for all investment horizons. Accordingly, we call them robust forward criteria. They are encoded by pairs of utility fields and penalty functions which are dynamically consistent.

Our theoretical focus is on defining and further characterising the new investment criteria. We consider their duals and establish an appropriate duality theory. Similarly to Schied [64], as well as Quenez [60] and Schied and Wu [65], the proof of duality proceeds by using an appropriate minimax theorem and then applying a model-specific duality result to the inner maximisation. However, unlike [64] which relied on results of Kramkov and Schachermayer [45], we view the inner maximisation problems under the fixed reference measure ℙ but featuring stochastic utility functions and apply the duality in Žitković [69]. Our proofs involve a number of technical and conceptual novelties. In particular, we prove relevant conjugacy relations and the existence of a dual optimiser for a class of utility functions which are allowed to be stochastic and finite on the entire real line. Notably, the dynamic consistency conditions are imposed jointly on the penalty function and the utility random field. Unlike for convex risk measures or the classical EUM, the dynamic aspects of robust portfolio optimisation seem to have been studied only for specific examples; see e.g. Laeven and Stadje [46] and Müller [51, Chap. 7]. We provide general results which in particular highlight the necessity of a conditional stability property of the penalty functions, see property (2.11) below, in the past only considered for dynamic risk measures. Further, we also obtain the equivalence between dynamic consistency in the primal and dual domain and characterise the latter via a suitable submartingale property. While these are natural properties which are well understood in other contexts, e.g. classical EUM, they appear to be novel in the context of robust portfolio optimisation. We use the dual formulation to study the question of time-consistency of the optimal strategies. We show that in general, both in our framework as well as in the classical robust EUM, the optimal strategies may fail to be time-consistent. This is caused by possibly arbitrary dynamics of the penalty functions. We show that time-consistency of the optimal strategies is guaranteed under suitable assumptions of dynamic consistency of the penalty functions.

Apart from the theoretical contribution, we also construct and solve explicitly some practically relevant examples which showcase the advantages of our approach. Most notably, we consider an investor who starts with a logarithmic utility and applies a quadratic penalty function. The investor then builds a dynamic estimate of the market price of risk, say \(\hat{\lambda}\), and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with \(\hat{\lambda}\). The leverage is a function of the investor’s confidence in the estimate \(\hat{\lambda}\). This solution is both intuitive and relevant since it corresponds to strategies often followed by large investors in practice. In the classical robust EUM approach, for a fixed time interval \([0,T]\), such behaviour is consistent with the simplest setting of a complete market and constant penalty weighting and is essentially the only explicit example available with the classical approach; see Hernández-Hernández and Schied [32]. In a more complex setting – e.g. incomplete market or general adapted penalty weights –, this structure is lost; the solution is described via PDE or BSDE methods and the optimal investment strategies may depend on the setting and on the investment horizon \(T\). This complexity is due to the entangled nature of solving the problem backwards and having a deterministic boundary constraint at \(T\). Our approach, in contrast, does not suffer from such drawbacks and offers a solution which holds in great generality. We discuss this in detail in Sect. 2.1. A further example of an investor initially endowed with an exponential utility is studied in Sect. 2.2. In Sect. 5, we discuss the structure of forward criteria and identify some particular classes or robust forward criteria – this provides us with yet some further examples.

The rest of the paper is organised as follows. In Sect. 2, the market assumptions are specified, the robust forward criteria are introduced and motivating examples are studied. In Sect. 3, equivalent dual characterisations of robust forward criteria are established. Then, in Sect. 4, we study the link between dynamic consistency of penalty functions and time-consistency of optimal investment strategies. In particular, we discuss a simple example of criteria leading to time-inconsistent optimal investment strategies. Section 5 is devoted to a mostly formal discussion of various classes of criteria. Our aim is to illustrate the flexibility of the notion and the fact that interesting preferences might be identified under additional evolutionary requirements. In particular, time-monotone criteria are linked to a specific PDE. We also argue that for each robust forward criterion, there exists a specific (standard) forward criterion in the reference market producing the same optimal behaviour. The proofs are deferred to Sect. 6.

In order to motivate and illustrate the upcoming definition, we first consider two examples. In Sect. 2.1, we build a robust forward criterion which combines logarithmic preferences with a quadratic penalty structure for model ambiguity. The example is of particular interest as it gives theoretical justification for fractional Kelly strategies which are often used in practice. Subsequently, in Sect. 2.2, we consider an example with initial exponential preferences. In Sect. 2.3, we then introduce the general setup and definition.

2.3 Definition of robust forward performance criteria

We now turn to a general market setup and define the robust forward criteria.

2.3.1 The underlying market assumptions

The market consists of \(d+1\) securities whose prices \((S^{0}_{t};S_{t})=(S^{0}_{t},S^{1}_{t},\dots,S^{d}_{t})\), \(t\ge0\), are modelled by a \((d+1)\)-dimensional càdlàg semimartingale on a filtered probability space \(({\Omega},\mathcal{F},\mathbb{F},\mathbb{P})\), where the filtration \(\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,\infty)}\) satisfies the usual conditions. We let \(S^{0}\equiv1\) and assume \(S\) to be locally bounded. A portfolio process \(\pi=(\pi_{t})_{t\in[0,\infty)}\) is an \(\mathbb {F}\)-predictable process which is \(S\)-integrable on \([0,T]\) for each \(T>0\) and denotes the number of shares held in the risky asset. The associated wealth process \(X^{\pi}\) is given by

$$ X^{\pi}_{t}=\int_{0}^{t}\pi_{u}dS_{u}, \qquad t\ge0. $$

The set of admissible portfolio processes available to the investor is denoted by \(\mathcal{A}\) and is typically a subset of all portfolio processes.

For each \(T>0\), \(\mathcal{M}^{e}_{T}\) denotes the set of equivalent local martingale measures, that is, the set of measures ℚ on \(\mathcal{F}_{T}\) such that \(\mathbb{Q}\sim\mathbb{P}|_{\mathcal{F}_{T}}\) and each component of \(S\) is a ℚ-local martingale. Similarly, \(\mathcal{M}^{a}_{T}\) denotes the set of absolutely continuous local martingale measures. The corresponding sets of density processes are denoted, respectively, by \(\mathcal{Z}^{e}_{T}\) and \(\mathcal {Z}^{a}_{T}\). Put differently,

$$ \mathcal{Z}^{e}_{T}=\bigg\{ \bigg(\mathrm{E}\bigg[\frac{\mathrm {d}\mathbb{Q}}{\mathrm {d}\mathbb{P}|_{\mathcal{F} _{T}}}\bigg|\mathcal{F}_{t}\bigg]\bigg)_{0 \leq t \leq T}: \mathbb {Q}\in\mathcal {M}^{e}_{T}\bigg\} , $$

and similarly for \(\mathcal{Z}^{a}_{T}\). For any nonnegative martingale \(Z_{t}\), \(t\le T\), and in particular for density processes in \(\mathcal{Z}^{a}_{T}\), we use the notation \(Z_{s,t}:=\frac{Z_{t}}{Z_{s}}\) for \(0\le s\le t\le T\), with the convention \(Z_{s,t}\equiv1\) on \(\{Z_{s}=0\}\).

We impose the following assumption throughout:

Assumption 2.3

The set \(\mathcal{M}^{e}_{T}\) is nonempty for each \(T>0\).

This assumption is referred to as the absence of arbitrage (NFLVR) on finite horizons; see [69, Sect. 2] for further discussion. Note that while

$$ \mathcal{M}^{e}_{T_{1}}=\{\mathbb{Q}|_{\mathcal{F}_{T_{1}}}:\mathbb{Q}\in \mathcal {M}^{e}_{T_{2}}\} \qquad \textrm{for all }0\le T_{1}\le T_{2}, $$

there need not exist a set \(\mathcal{M}^{e}\) of probability measures equivalent to ℙ such that \(\mathcal {M}^{e}_{T}=\{ \mathbb{Q} |_{\mathcal{F}_{T}}:\mathbb{Q}\in\mathcal{M}^{e}\}\) for all \(T>0\). As argued in [69], the condition NFLVR on finite horizons implies that any density process \(Z\in\mathcal{Z}^{e}_{T}\) can be extended to a strictly positive martingale \((Z_{t})_{t\in[0,\infty)}\) such that \(Z_{0}=1\) and \(ZS\) is a local martingale. The set of all such processes \(Z\) is denoted by \(\mathcal{Z}^{e}\). In particular, NFLVR on finite horizons holds if and only if \(\mathcal{Z}^{e}\) is nonempty. If the condition of strict positivity is replaced by the one of nonnegativity, the obtained family is denoted by \(\mathcal{Z}^{a}\).

2.3.2 Utility random fields and penalty functions

The robust forward criteria which we introduce below combine two elements: a utility random field \(U(\omega,x,t)\), \(t\ge0\), and a family of penalty functions \(\gamma_{t,T}(\mathbb{Q})\), \(0\le t\le T<\infty\). The component \(U(\omega, \, \cdot\, ,t)\) models the preferences at time \(t\) and may depend on past observations. In addition, the investor faces ambiguity about the “true model” for the dynamics of the financial assets and forms a view about the relative plausibility of different probability measures; this is reflected in \(\gamma_{t,T}(\mathbb{Q})(\omega)\) which gives the weighting of measures ℚ on \(\mathcal{F}_{T}\). From now on, we focus on the case of \(U\) defined on ℝ; this simplifies some aspects of the duality theory, as explained in Sect. 3 below. Alterations of our abstract definitions to the case of \(U\) on \(\mathbb{R}_{+}\) are immediate.

Definition 2.4

A random field is a mapping \(U:{\Omega}\times\mathbb {R}\times [0,\infty)\to\mathbb{R}\) which is measurable with respect to the product of the optional \(\sigma\)-algebra on \({\Omega}\times[0,\infty)\) and \(\mathcal{B}(\mathbb{R})\). A utility random field is a random field which satisfies the following conditions:

1. (i)
   For all \(t\in[0,\infty)\), the mapping \(x\mapsto U(\omega ,x,t)\) is \(\mathbb{P}(d\omega)\)-a.s. a strictly concave and strictly increasing \(C^{1}(\mathbb{R})\)-function which satisfies the Inada conditions

   $$ \lim_{x\to-\infty}\frac{\partial}{\partial x}U(\omega,x,t)=\infty ,\qquad \lim_{x\to\infty}\frac{\partial}{\partial x}U(\omega,x,t)=0. $$
    
2. (ii)

   \(\mathbb{P}(d\omega)\)-a.s., the mapping \(t\mapsto U(\omega,x,t)\) is càdlàg on \([0,\infty)\) for all \(x\in\mathbb{R}\).

    
3. (iii)

   For each \(x\in\mathbb{R}\) and \(T\in[0,\infty)\), \(U(\, \cdot\, ,x,T)\in L^{1}(\mathcal{F}_{T})\).

    

In what follows, we suppress \(\omega\) from the notation and simply write \(U(x,t)\).

The penalty function \(\gamma_{t,T}(\, \cdot\,)\) should not distinguish between two probability measures \(\mathbb{Q}_{1},\mathbb{Q}_{2}\sim \mathbb{P}|_{\mathcal{F}_{T}}\) which agree at time \(t\) when considering the horizon \([t,T]\), i.e.,

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(2.11)

This means that \(\gamma_{t,T}(\mathbb{Q})\) is a function of the conditional density \(Z^{\mathbb{Q}}_{t,T}\), where for \(\mathbb{Q}\sim\mathbb{P}|_{\mathcal{F}_{T}}\), we set \(Z^{\mathbb{Q}}_{t}:=\mathrm{E}[\frac{\mathrm{d}\mathbb{Q}}{\mathrm {d}\mathbb{P}|_{\mathcal{F}_{T}}}|\mathcal{F} _{t}]\) and \(Z^{\mathbb{Q}}_{t,T}= Z^{\mathbb{Q}}_{T}/Z^{\mathbb{Q}}_{t}\), \(t\in[0,T]\), and we consider \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\sim\mathbb{P}|_{\mathcal{F}_{T}}\}\) as a subset of \(L^{1}_{+}(\mathcal{F}_{T})\). This justifies the following definition.

Definition 2.5

For \(0\le t\le T<\infty\), we call a penalty function any mapping \(\gamma_{t,T}\) from \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\sim \mathbb{P}|_{\mathcal{F}_{T}}\} \) to \(\mathcal{F}_{t}\)-measurable \([0,+\infty]\)-valued random variables which satisfies the following conditions:

1. (i)

   (convexity) \(\forall\lambda\in L^{0}(\mathcal{F}_{t})\), \(0\leq \lambda \leq1\), and \(Z^{1},Z^{2}\in\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\sim\mathbb {P}|_{\mathcal{F}_{T}}\}\), we have \(\gamma_{t,T}(\lambda Z^{1}+(1-\lambda)Z^{2})\leq\lambda\gamma _{t,T}(Z^{1})+(1-\lambda)\gamma_{t,T}(Z^{2})\) a.s.;

    
2. (ii)

   for \(\kappa\in L^{\infty}_{+}(\mathcal{F}_{t})\), \(Z\mapsto \mathrm{E}[\kappa \gamma _{t,T}(Z)]\) is \(\sigma(L^{1},L^{\infty})\)-lower semicontinuous.

    

We write \(\gamma_{t,T}(\mathbb{Q}):=\gamma_{t,T}(Z^{\mathbb{Q}}_{t,T})\) for \(\mathbb{Q} \sim \mathbb{P}|_{\mathcal{F}_{T}}\). Moreover, for a given utility random field \(U(x,t)\) and a set of admissible strategies \(\mathcal{A}\), we say that \((\gamma_{t,T})\), \(0\le t\le T<\infty\), is an admissible family of penalty functions if for all \(T>0\) and \(\pi\in\mathcal{A}\), \(\mathrm{E}^{\mathbb{Q}}[U(X_{T}^{\pi},T)]\) is well defined in \(\mathbb{R} \cup\{\infty\}\) for all \(\mathbb{Q}\in\mathcal{Q}_{t,T}\), \(t\le T\), where \(\mathcal{Q}_{t,T}\) is the set of measures on \(\mathcal{F}_{T}\) given by

$$ \mathcal{Q}_{t,T}:=\{\mathbb{Q}: \mathbb{Q}\sim\mathbb {P}|_{\mathcal{F}_{T}}\textrm{ and }\gamma _{t,T}(\mathbb{Q} )< \infty\textrm{ a.s.}\}. $$

We note that conditional convexity (i) above readily implies (2.11). Conversely, (2.11) together with convexity only for deterministic \(\lambda\) implies conditional convexity for simple \(\lambda\in L^{0}(\mathcal{F}_{t})\), which then yields (i) by using the continuity in (ii).

Condition (2.11) above simply says that if at time \(t\) an investor considering \([t,T]\) cannot tell apart \(\mathbb{Q}_{1}\) from \(\mathbb{Q}_{2}\), then she assigns them the same penalty. To the best of our knowledge, such a condition has previously not been invoked in the context of robust portfolio optimisation, but it is required here since unlike previous works, we consider a dynamic problem and prove conditional conjugacy relations. Analogous conditions have appeared before in the context of dynamic risk measures; see Definition 3.11 of the local property of penalty functions in Cheridito et al. [11] or the pasting property in Lemma 3.3 in Klöppel and Schweizer [43]. Its importance here becomes apparent in the proof of Lemma 6.3.

In the above definition, \(\mathcal{Q}_{t,T}\) is the set of feasible measures considered at time \(t\) when investing over \([t,T]\). It may depend on \(t\) and \(T\) but is non-random. Both larger and smaller sets could be used, e.g. the (random) set of measures ℚ with \(\gamma _{t,T}(\mathbb{Q} )(\omega)<\infty\) or the set of measures ℚ with \(\mathrm {E}[\gamma _{t,T}(\mathbb{Q})]<\infty\). However, for many natural penalty functions, these different choices lead to the same value function. Finally, note that we do not impose any regularity or consistency assumptions on \(\gamma_{t,T}(\mathbb{Q})\) in the time variables. These are not necessary for the abstract results in Sect. 3 and will be introduced later when they appear naturally; see Assumption 4.1.

2.3.3 Robust forward performance criteria

We are now ready to introduce the robust forward criteria. As highlighted above, these are pairs \((U,\gamma)\) which exhibit a dynamic consistency akin to the dynamic programming principle.

Definition 2.6

Let \(U\) be a utility random field, \(\mathcal{A}\) a set of admissible strategies and \(\gamma\) an admissible family of penalty functions. We say that \((U,\gamma)\) is a robust forward criterion if for all \(0\le t\le T<\infty\) and all \(\xi\in L^{\infty}(\mathcal{F}_{t})\),

$$\begin{aligned} U(\xi,t)= \mathop{\mathrm{ess}\;\mathrm{sup}}_{\pi\in\mathcal{A}}\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,T}} \bigg(\mathrm{E}^{\mathbb{Q}}\bigg[U\bigg(\xi+\int_{t}^{T}\pi _{s}dS_{s},T\bigg)\bigg|\mathcal{F}_{t}\bigg] +\gamma_{t,T}(\mathbb{Q})\bigg)\ \ \textrm{a.s.}\qquad\ \end{aligned}$$

(2.12)

We note that the above definition is well posed. Indeed, given the assumptions on \(U\) and \(\gamma\), the conditional expectations in (2.12) are well-defined (extended-valued) random variables. Since all \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) are equivalent to ℙ, for each \(\pi\in\mathcal{A}\), the essential infimum is also well defined (extended-valued) with respect to the reference measure ℙ. The set of admissible strategies \(\mathcal{A}\) which we consider is specified below; more generally, and in particular if \(U\) were defined on \(\mathbb{R}_{+}\), one might need to take an \(\mathcal{A}\) which depends on \((\xi,t)\). Naturally, we call an admissible strategy optimal if it attains the supremum in (2.12). However, our definition of robust forward criteria does not require the existence of optimal investment strategies. In that aspect, we follow the approach in [69] rather than the original definition in [53, 54]. This is particularly helpful for the duality theory developed in Sect. 3.

Example 2.7

An example of a robust forward criterion as in Definition 2.6 is given by the pair \((U,\gamma)\) considered in Proposition 2.2. The pair \((U,\gamma)\) in Proposition 2.1 is an example corresponding to an analogous definition, but for the case of random fields defined on \(\mathbb{R}_{+}\). We discuss further examples below; see in particular Sect. 5.

The optimisation in (2.12) fits within the robust EUM paradigm as discussed in the introduction. The crucial difference is that we require (2.12) to hold for all time pairs \(t\leq T\). We refer to (2.12) as the dynamic consistency property of \((U,\gamma)\); allowing model ambiguity, it provides a direct extension of the notion of self-generating utility fields studied in [69] and, consequently, of the notion of forward performance criteria; see the introduction and Sect. 5.

To relate (2.12) to the more classical dynamic programming principle, it is useful to introduce the family of value functions \(\{u(\,\cdot\,;t,T):0\le t\le T<\infty\}\) with \(u(\,\cdot\, ;t,T):L^{\infty}(\mathcal{F}_{t})\to L^{0}(\mathcal{F}_{t};\mathbb{R}\cup\{ \infty\})\) given by

$$ u(\xi;t,T):= \mathop{\mathrm{ess}\;\mathrm{sup}}_{\pi\in\mathcal{A}}\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,T}} \bigg(\mathrm{E}^{\mathbb{Q}}\bigg[U\bigg(\xi+\int_{t}^{T}\pi _{s}dS_{s},T\bigg)\bigg|\mathcal{F}_{t}\bigg] +\gamma_{t,T}(\mathbb{Q})\bigg). $$

(2.13)

Then \((U,\gamma)\) is a robust forward criterion if and only if for all \(0\le t\le T<\infty\) and all \(\xi\in L^{\infty}(\mathcal{F}_{t})\),

$$ U(\xi,t)= u(\xi;t,T)\qquad \textrm{a.s.} $$

This then implies a familiar DPP (or martingale optimality principle), namely

$$\begin{aligned} u(\xi;t,T)&=U(\xi,t)=u(\xi;t,r) \\ & = \mathop{\mathrm{ess}\;\mathrm{sup}}_{\pi\in\mathcal{A}}\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,r}} \bigg(\mathrm{E}^{\mathbb{Q}}\bigg[U\bigg(\xi+\int_{t}^{r}\pi _{s}dS_{s},r\bigg)\bigg|\mathcal{F}_{t}\bigg] +\gamma_{t,r}(\mathbb{Q})\bigg) \\ & = \mathop{\mathrm{ess}\;\mathrm{sup}}_{\pi\in\mathcal{A}}\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,r}} \bigg(\mathrm{E}^{\mathbb{Q}}\bigg[u\bigg(\xi+\int_{t}^{r}\pi _{s}dS_{s};r,T\bigg)\bigg|\mathcal{F} _{t}\bigg] +\gamma_{t,r}(\mathbb{Q})\bigg), \end{aligned}$$

(2.14)

for \(0\leq t\leq r\leq T\) and \(\xi\in L^{\infty}(\mathcal{F}_{t})\).

The setting of (2.13) corresponds to a very general robust EUM, but we note that it also has its limitations. For example, the penalty \(\gamma_{t,T}(\mathbb{Q})\) associated to a given measure is fixed and independent of wealth. This has important implications for the time-consistency of optimal investment strategies. Indeed, as we show in Proposition 4.4, when the \((\gamma_{t,T})\) are dynamically consistent and if we have saddle points \((\pi^{t,T},\mathbb{Q}^{t,T})\) solving (2.13), then \(\mathbb{Q}^{t,r}=\mathbb{Q}^{t,T}|_{\mathcal{F}_{r}}\), \(t\leq r\leq T\), and also the optimal investment strategies are time-consistent. However, in all generality, we could have (dynamically consistent) robust forward criteria which lead to time-inconsistent optimal strategies – an example is given in Sect. 4. Independence of \(\gamma_{t,T}(\mathbb{Q})\) from the investor’s wealth is also contrary to the empirical evidence as discussed in behavioural finance, see e.g. Kahneman and Tversky [35], which points to the importance of the investor’s reference point for judging scenarios. In consequence, we believe that it might be interesting to study generalisations of the problem in (2.13). Within the framework of robust EUM, these are possible using quasi-concave utility functionals introduced in Cerreia-Vioglio et al. [10]. Their use for the (classical) optimal investment problem has recently been investigated by Källblad [38].

Dual methods have proved useful for the study of optimal investment problems and this applies also within our setup. In particular, while the primal problem features a saddle point, the dual problem amounts to the search for a pure infimum, and robust forward criteria are therefore easier to characterise in the dual rather than the primal domain. The aim of this section is to establish the equivalence between dynamic consistency in the primal and the dual domain.

We focus on utility random fields which are finite on the entire real line. The reasons are twofold. First, we complement the work of Schied [64] where only utilities defined on the positive half-line were studied. Second, this simplifies certain technical aspects, see also e.g. [25], and allows us to focus on the novelty of our setting. We note that allowing negative wealth usually complicates the choice of an appropriate set of admissible strategies yielding the existence of an optimiser; cf. [59, 62]. This is not a concern for us since we do not require the existence of a primal optimiser, and hence, without loss of generality, we can restrict to the set of bounded wealth processes.⁵ Accordingly, we set in Definitions 2.4 and 2.6\(\mathcal{A} =\mathcal{A}_{\mathrm{bd}}\), the set of all portfolios producing bounded wealth processes. Specifically, \(\mathcal{A}_{\mathrm{bd}} =\bar{\mathcal{A}}\cap(-\bar{\mathcal{A}})\), where \(\bar{\mathcal{A}}\) is the set of all admissible portfolio processes for which for any \(T>0\), there exists a constant \(c>0\) such that \(X^{\pi}_{t}\ge-c\), \(0\le t\le T\), a.s.

3.1 Equivalence between primal and dual dynamic consistency

We first introduce the following technical assumption.

Assumption 3.2

For any \(0\le t\le T<\infty\), the set of densities \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\} \) is \(\sigma (L^{1},L^{\infty})\)-compact, and the family \(\{Z^{\mathbb{Q}}_{t,T}U^{-}(x,T):\mathbb{Q}\in\mathcal {Q}_{t,T}\}\) is uniformly integrable for any \(x\in\mathbb{R}\). In addition, for any \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and any nonincreasing sequence \((D_{n})_{n\in\mathbb{N}}\) in \(\mathcal{F}_{T}\) with \(\bigcap _{n}D_{n}=\emptyset\), there exists a sequence \((a_{n})_{n\in\mathbb{N}}\) in \((0,\infty)\) such that

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For measures \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) such that \(Z^{\mathbb{Q}}_{t,T}U(x,T)\in L^{1}\) for some and hence for all \(x\in\mathbb{R}\), although seemingly weaker, the second part of the above assumption is equivalent to the fact that the stochastic utility function \(Z^{\mathbb{Q}}_{t,T}U(\, \cdot\,,T)\) satisfies the non-singularity condition in Definition 3.3 in [69]. This is a mild technical assumption which precludes pathological appearances of non-countably additive measures in the dual treatment. In particular, it is satisfied whenever the utility field is \((x,\omega )\)-uniformly bounded from below by a deterministic utility function; see [69, Remark 3.4]. We also note that since \(\{Z^{\mathbb{Q}} _{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\}\) is convex, weak compactness is equivalent to closedness in \(L^{0}\); cf. [65, Lemma 3.2].

Next, we present the first main result, which yields the conjugacy relations between the functions \(u(x;t,T)\) and \(v(y;t,T)\). We stress that even for \(t=0\), Theorem 3.3 differs from Theorem 2.4 in [64] in that \(U(\, \cdot\,,T)\) is defined on the entire real line and allowed to be stochastic, and, moreover, we do not impose any finiteness assumptions. The proof is reported in Sect. 6.1.

Theorem 3.3

Let\((U,\gamma)\)be a pair of a utility random field and an admissible family of penalty functions, suppose that Assumption 3.2holds, and let\(V\)be the associated dual random field. Then for all\(0\le t\le T<\infty\), \(\xi\in L^{\infty}(\mathcal{F}_{t})\)and\(\eta \in L^{0}_{+}(\mathcal{F}_{t})\), the associated value fields satisfy

$$\begin{aligned} u(\xi;t,T)& =\mathop{\mathrm{ess}\;\mathrm{inf}}_{\eta\in L^{0}_{+}(\mathcal{F}_{t})}\big(v(\eta;t,T)+\xi\eta \big)\qquad \textrm{a.s.,} \end{aligned}$$

(3.4)

$$\begin{aligned} v(\eta;t,T) &=\mathop{\mathrm{ess}\;\mathrm{sup}}_{\xi\in L^{\infty}(\mathcal{F}_{t})}\big(u(\xi;t,T)-\xi \eta\big) \qquad \textrm{a.s.} \end{aligned}$$

(3.5)

In consequence, the combination of a utility random field\(U(x,t)\)and a family of penalty functions\(\gamma_{t,T}\)is dynamically consistent if and only if the combination of the dual random field\(V(y,t)\)and\(\gamma_{t,T}\)is dynamically consistent.

The next result shows that the dual problem admits a solution even though the primal problem need not (since we have restricted to the use of bounded wealth strategies).

Proposition 3.4

Let\((U,\gamma)\)be a pair of a utility random field and an admissible family of penalty functions, and let\(V\)be the associated dual random field. Suppose that Assumption 3.2holds. Then for any\(t\le T<\infty\)and\(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), there exist\(\mathbb{Q}\in\mathcal{Q}_{t,T}\)and\(Z\in\mathcal{Z}^{a}_{T}\)attaining the infimum in (3.2).

We provide the proof in Sect. 6.1, but remark that the fact that the second component of the optimiser lies in \(\mathcal {M}^{a}_{T}\) (as opposed to the larger set of finitely additive measures) is a consequence of the utility function being finite on the entire real line (see [69] and also [5, 62]).

We work here under the assumption that the measures in \(\mathcal {Q}_{t,T}\) are equivalent to the reference measure. However, under the convention that \(Z^{\mathbb{Q}}_{t,T}V(\eta Z_{t,T}/Z^{\mathbb{Q}}_{t,T})=\infty\) on \(\{Z^{\mathbb{Q}}_{t,T}=0\}\), our proofs go through with straightforward modifications also when allowing \(\mathcal{Q}_{t,T}\) to include all measures absolutely continuous with respect to the reference measure with finite penalty a.s.; cf. [38] for similar results in the case of (static) utility functions defined on the positive half-line. We also expect that our proofs might be further developed so as to rely on weak compactness of level sets of the form \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\ll \mathbb{P}|_{\mathcal{F}_{T}}\textrm{ and }\gamma_{t,T}(\mathbb {Q})\le\xi\textrm{ a.s.}\}\), \(\xi\in L^{\infty}(\mathcal{F}_{t})\), rather than of \(\mathcal {Q}_{t,T}\). We leave this topic for future research.

The definition of robust forward criteria requires the combined criterion consisting of \(U(x,t)\) and \(\gamma_{t,T}\) to be dynamically consistent (cf. Definition 2.6). In this section, we further investigate this assumption and relate it to dynamic consistency of the penalty functions and time-consistency of the optimal investment strategies. The corresponding proofs are reported in Sect. 6.2.

For any penalty function satisfying (4.1), \(\mathcal {Q}_{t,T}\subseteq \tilde{\mathcal{Q}}_{t,T}\). However, in general, stability under pasting (4.2) may fail. It may be recovered if different definitions of \(\mathcal{Q} _{t,T}\) are used, e.g. with measures satisfying \(\mathrm{E}[\gamma _{t,T}(\mathbb{Q} )]<\infty\); see the remarks below on penalty functions associated with risk measures.

The additional structure resulting from Assumption 4.1 allows us to consider the question of whether for a fixed \(T>0\), the value field \(u(x;t,T)\) associated with a general utility field satisfies itself the dynamic programming principle (2.14) for \(t\le T\). We show that under suitable assumptions on the penalty function, this is the case. For particular choices of preferences, this property has been used to address the ambiguity-averse problem by stochastic control methods in [31, 32, 51]. The proof proceeds by first establishing appropriate consistency in the dual domain and then applying Theorem 3.3.

For the case of standard (non-robust) utility maximisation and deterministic utility functions, it is well known that the value process satisfies the DPP, also referred to as the martingale optimality principle; see [19, Chap. I]. Proposition 4.2 shows that a similar consistency property holds for certain ambiguity-averse criteria. However, the value field associated with a general penalty function may fail to be dynamically consistent; see [64] for counterexamples. Hence, while the standard forward criteria are effectively a generalisation (to all positive times) of the value functions associated with (stochastic) utility functions, within the robust setting, our Definition 2.6 enforces additional structure by imposing the dynamic consistency requirement (2.12) on the pair \((U,\gamma)\). In general, however, this is weaker than the assumption of dynamic consistency of \(\gamma\). Indeed, as illustrated by the next example, there are dynamically consistent pairs \((U,\gamma)\) where the penalty function \(\gamma\) itself is not dynamically consistent. Such robust forward criteria may lead to time-inconsistent optimal investment strategies.

Observe that in the above example, property (4.1) is violated in a rather simplistic way. Indeed, at any time \(t\), looking to invest on \([t,T]\), the investor believes that only one model is feasible. This is a degenerate case since the choice of this model changes arbitrarily with \(t\) and \(T\) and there is no consistency requirement. Consider, for example, the extreme situation when all \(\lambda^{t,T}\) are constant and \(T\) is fixed. Then at time zero, the investor picks possibly different models which she will choose to believe in when making investment decisions at \(t\) for the period \([t,T]\); it is not surprising that this might lead to time-inconsistent investment strategies. However, the flexibility of fixing the penalty \(\gamma _{t,T}\) implies that the dynamic consistency of the value functions, i.e., (2.14) on \([0,T]\) or (2.12) in general, may nevertheless be preserved.

In Example 4.3, the lack of time-consistency of optimal strategies is inherited from the lack of dynamic consistency of the penalty functions, i.e., from the violation of (4.1). In contrast, when the penalty functions are consistent, we recover the time-consistency of the optimisers.

The above result, combined with Example 4.3, shows that dynamic consistency of the penalty functions, i.e., (4.1), is a necessary and sufficient condition for time consistency of the optimal investment strategies to hold for any corresponding criterion. This applies both to the robust forward criteria studied here as well as to classical robust expected utility maximisation on a fixed horizon. It leads to interesting open questions. First, the economic and empirical justification for (4.1) remains unclear. In fact, it is a non-trivial requirement, and, for example, penalty functions associated to convex risk measures do not satisfy (4.1) in general; see also Remark 3.5 in Schied [64]. Second, are there generalisations of the optimisation problem in (2.13) which would preserve time-consistency of optimal strategies while still violating (4.1)?

Next, we show that the dynamic consistency property of penalty functions leads to a characterisation of robust forward criteria in terms of a certain “weighted submartingale” property of the dual field. This is used in Sect. 5 to derive an equation allowing us to investigate particular classes and examples of robust forward criteria.

We conclude this section with brief remarks on the penalty functions \(\gamma_{t,T}\) associated with (dynamic) convex risk measures (see [1, 7, 11, 43]). Such penalty functions, under minimal regularity/continuity assumptions, satisfy the properties of Definition 2.5. However, the weak compactness condition in Assumption 3.2 usually requires stronger assumptions. Recall that for static risk measures, it is obtained for risk measures continuous from below, see [64, Lemma 4.1], and in particular by a coherent risk measure which only assigns zero penalty to equivalent measures; see [31] for an example. Regarding Assumption 4.1, the time-consistency of convex risk measures is characterised by property (4.1), and any time-consistent coherent risk measure⁶ admits the pasting property (4.2) (cf. [1, Corr. 1.26]). However, in general, (4.1) does not imply (4.2). Nevertheless, any convex risk measure admits a robust representation where \(\mathcal {Q}_{t,T}\) is replaced by the set \(\{\mathbb{Q}\sim\mathbb{P}|_{\mathcal {F}_{T}}:\mathrm{E}[\gamma_{t,T}(\mathbb{Q} )]<\infty\}\), which in turn satisfies (4.2). This property is crucial for proving the equivalence between time-consistency of the risk measure and property (4.1) (see e.g. [1, Thm. 1.20] or [7]). Assuming (4.2) is therefore consistent with the use of time-consistent penalty functions associated with risk measures.

In this section, we study the structure of robust forward criteria and subsequently discuss specific cases. Throughout, we consider the Brownian setup of Sect.  2.1 , and the discussion is mostly formal. We start with the structure of forward criteria and focus on the non-uniqueness of robust forward criteria for given initial preferences. Then we study examples of classes where the uniqueness may be recovered. These classes are obtained by generalising, in various ways, the main example studied in Sect. 2.1. First, in Sect. 5.2, we consider fields which exhibit logarithmic dependence on wealth. Then, in Sect. 5.3, we focus on robust forward criteria with no volatility (cf. (5.6) below). Such criteria are characterised by a specific evolutionary property and linked to a certain PDE (Eq. (5.7) below). For both examples, the discussion is in terms of dual fields. Finally, in Sect. 5.4, we show that for each robust forward criterion, there exists a (standard) forward criterion in the fixed reference market producing the same optimal behaviour.