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Optimal control with delayed information flow of systems driven by GBrownian motion
Probability, Uncertainty and Quantitative Risk volume 3, Article number: 8 (2018)
Abstract
In this paper, we study strongly robust optimal control problems under volatility uncertainty. In the Gframework, we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.
Introduction
One of the motivations for this paper is to study the problem of optimal consumption and optimal portfolio allocation in finance under model uncertainty. In particular, we focus on volatility uncertainty, i.e., a situation where the volatility affecting the asset price dynamics is unknown and we need to consider a family of different volatility processes instead of just one fixed process (and hence also a family of models related to them).
Volatility uncertainty has been investigated in the literature by following two approaches, i.e., by introducing an abstract sublinear expectation space with a special process called GBrownian motion (see (Peng 2007), (Peng 2010)), or by capacity theory (see (Denis et al. 2011)). In (Denis et al. 2011), it is proven that these two methods are strongly related. The link between these two approaches is the representation of the sublinear expectation \(\hat {\mathbb {E}}\) associated with the GBrownian motion as a supremum of ordinary expectations over a tight family of probability measures \(\mathcal {P}\), whose elements are mutually singular:
see (4) and Theorem 1 for more details.
In this paper, we work in a GBrownian motion setting as in (Peng 2007) and use the related stochastic calculus, including the Itô formula, GSDEs, martingale representation and GBSDEs, as developed in (Peng 2007), (Peng 2010), (Soner et al. 2011a), (Song 2011), (Soner et al. 2011b), (Peng et al. 2014), (Hu et al. 2014c), (Hu et al. 2014a). It is important for understanding the nature of GBrownian motion to note that its quadratic variation 〈B〉 is not deterministic, but it is absolutely continuous with the density taking value in a fixed set (for example, \([\underline {\sigma }^{2},\bar \sigma ^{2}]\) for d=1). Each \(\mathbb {P}\in \mathcal {P}\) can then be seen as a model with a different scenario for the quadratic variation. That justifies why GBrownian motion is a good framework for investigating model uncertainty.
In a GBrownian motion setting one considers the following stochastic optimal control problem: to find the control \(\hat {u}\in \mathcal {A}\) such that
with
where X^{u} is a controlled GSDE, see (8). This problem has been studied in (Matoussi et al. 2013), (Hu et al. 2014b). In (Hu et al. 2014b), they show that the value function associated with such an optimal control problem satisfies the dynamic programming principle and is a viscosity solution of some HJB equation.^{1} (Matoussi et al. 2013) investigates the robust investment problem for geometric GBrownian motion, where 2BSDEs (which are closely related to GBSDEs) are used to find an optimal solution. In both papers the optimal control is robust in a worstcase scenario sense.
It is interesting to note that in the simplest example of the optimal portfolio problem, which is the Merton problem with the logarithmic utility, one can easily prove that there exists a portfolio which is optimal not only in the worstcase scenario, but also for all probability measures \(\mathbb {P}\) (with the optimality criterion \(J^{\mathbb {P}}\)). We call this a strongly robust control. This strongly robust control is thus optimal in a much more robust sense than the worstcase scenario optimality. The new strongly robust optimality uses the fact that probability measures \(\mathbb {P}\) are mutually singular. Informally speaking, one can therefore modify the \(\mathbb {P}\)optimal control \(\hat {u}^{\mathbb {P}}\) outside the support of a probability measure \(\mathbb {P}\) without losing the \(\mathbb {P}\)optimality. As a consequence, if the family \(\{\hat {u}^{\mathbb {P}}\}_{\mathbb {P} \in \mathcal {P}}\) satisfies some consistency conditions, under a suitable choice of the underlying filtration the controls can be aggregated into a unique control \(\hat {u}\), which is optimal under every probability measure \(\mathbb {P}\). See (Soner et al. 2011b) for more details on aggregation.
In this paper, we study strongly robust optimal control problems. However, instead of checking the consistency condition for the family of controls and using the aggregation theory established in (Soner et al. 2011b), we adapt the stochastic maximum principle to the Gframework to find necessary and sufficient conditions for the existence of a strongly robust optimal control. We stress that this method has the clear advantage that we solve only one GBSDE to produce the strongly robust optimal control instead of considering the optimal control problem for all \(\mathbb {P}\in \mathcal {P}\) (which are usually not Markovian laws) and checking the consistency condition. Another advantage is that we work with the raw filtration instead of enlarging it.
In the recent paper (Hu and Ji 2016), they also study a stochastic maximum principle for stochastic recursive optimal control problems in the Gsetting, but still using the worstcase approach. They use the Minimax Theorem to obtain the variational inequality under a reference probability P^{∗}: the stochastic maximum principle holds then under such a P^{∗}a.s, which is the main difference with respect to our approach. They prove that this stochastic maximum principle is also a sufficient condition under some convex assumptions, but our control problem is different from the one in (Hu and Ji 2016) and considers delayed information.
The notion of the strongly robust optimal control also has a better financial interpretation than the standard robust optimality mentioned above. The main drawback of the classical robust optimal control is that it is a differential game from a mathematical point of view, where one player chooses the optimal control \(\hat {u}\) and the other chooses the optimal volatility represented by the law \(\hat {\mathbb {P}}\):
Therefore, the optimal pair \((\hat {u},\hat {\mathbb {P}})\) has the Nash equilibrium interpretation^{2}. The problem is that in real life there is no reason to why we should assume that the worst case is true, as there are no players who try to maximize gains from choosing \(\mathbb {P}\).
However, this is not the only problem with the standard robust optimality. Since the optimal probability measure \(\hat {\mathbb {P}}\) is mutually singular to any other measure \(\mathbb {Q}\in \mathcal {P}\), we can modify the control \(\hat {u}\) outside the support of \(\hat {\mathbb {P}}\) without losing the (classical) robust optimality. Since, as we noted above, usually the true probability will be different than \(\hat {\mathbb {P}}\), the classical robust optimal control may have little sense for \(\mathbb {Q}\). Moreover, in the standard robust optimality, the measure \(\hat {\mathbb {P}}\) is chosen to be static and does not change with time. As a result, not all available information is taken into consideration, as shown for the Merton problem with logarithmic utility in Section 5.
The paper is structured in the following way. In Section 2, we give a quick overview of the Gframework. Section 3 is devoted to a sufficient maximum principle in the partialinformation case. In Section 4, we investigate the necessary maximum principle for the fullinformation case. In Section 5, we give four examples, including the Merton problem with logarithmic utility, mentioned earlier and an LQproblem. In Section 6, we provide a counterexample and show that it is not possible to relax the crucial assumption of the sufficient maximum principle without losing the strongly robust sense of optimality.
Preliminaries
Let Ω be a given set and \(\mathcal {H}\) be a vector lattice of real functions defined on Ω, i.e. a linear space containing 1 such that \(X\in \mathcal {H}\) implies \(X\in \mathcal {H}\). We will treat elements of \(\mathcal {H}\) as random variables.
Definition 1
A sublinear expectation\(\mathbb {E}\) is a functional \(\mathbb {E}\colon \mathcal {H}\to \mathbb {R}\) satisfying the following properties

1.
Monotonicity: If \(X,Y\in \mathcal {H}\) and X≥Y, then \(\mathbb {E} [X]\geq \mathbb {E} [Y]\).

2.
Constant preserving: For all \(c\in \mathbb {R}\), we have \(\mathbb {E} [c]=c\).

3.
Subadditivity: For all \(X,Y\in \mathcal {H}\), we have \(\mathbb {E} [X]  \mathbb {E}[Y]\leq \mathbb {E} [XY]\).

4.
Positive homogeneity: For all \(X\in \mathcal {H}\), we have \(\mathbb {E} [\lambda X]=\lambda \mathbb {E} [X]\) for all λ≥0.
The triple \((\Omega,\mathcal {H},\mathbb {E})\) is called a sublinear expectation space.
We will consider a space \(\mathcal {H}\) of random variables having the following property: if \(X_{i}\in \mathcal {H},\ i=1,\ldots n\), then
where \({C_{b,Lip}(\mathbb {R}^n)}\) is the space of all bounded Lipschitz continuous functions on \(\mathbb {R}^{n}\).
Definition 2
An mdimensional random vector Y=(Y_{1},…,Y_{m}) is said to be independent of an ndimensional random vector X=(X_{1},…,X_{n}) if for every ϕ∈C_{b,Lip}(R^{n}×R^{m})
Let X_{1} and X_{2} be ndimensional random vectors defined on sublinear random spaces \((\Omega _{1},\mathcal {H}_{1},\mathbb {E}_{1})\) and \((\Omega _{2},\mathcal {H}_{2},\mathbb {E}_{2})\), respectively. We say that X_{1} and X_{2} are identically distributed and denote it by X_{1}∼X_{2}, if for each \(\phi \in {C_{b,Lip}(\mathbb {R}^n)}\) one has
Definition 3
A ddimensional random vector X=(X_{1},…,X_{d}) on a sublinear expectation space \((\Omega,\mathcal {H},\mathbb {E})\) is said to be Gnormally distributed if for each a,b≥0 and each \(Y\in \mathcal {H}\) such that X∼Y and Y is independent of X, one has
The letter G denotes a function defined as
where §_{d} is the space of all d×d symmetric matrices. We assume that G is nondegenerate, i.e., G(A)−G(B)≥βtr[A−B] for some β>0.
It can be checked that G might be represented as
where Θ is a nonempty bounded and closed subset of \(\mathbb {R}^{d\times d}\).
Definition 4
Let \(G\colon \S _{d}\to \mathbb {R}\) be a given monotonic and sublinear function. A stochastic process B=(B_{t})_{t≥0} on a sublinear expectation space \((\Omega,\mathcal {H},\mathbb {E})\) is called a GBrownian motion if it satisfies the following conditions:

1.
B_{0}=0;

2.
\(B_{t}\in \mathcal {H}\) for each t≥0;

3.
For each t,s≥0 the increment B_{t+s}−B_{t} is independent of \((B_{t_{1}},\ldots, B_{t_{n}})\) for each \(n\in \mathbb {N}\) and 0≤t_{1}<…<t_{n}≤t. Moreover, (B_{t+s}−B_{t})s^{−1/2} is Gnormally distributed.
Definition 5
Let \(\Omega =C_{0}(\mathbb {R}_{+},\mathbb {R}^{d})\), i.e., the space of all \(\mathbb {R}^{d}\)valued continuous functions starting at 0. We equip this space with the uniform convergence on compact intervals topology and denote by \(\mathcal {B}(\Omega)\) the Borel σalgebra of Ω. Let
A Gexpectation \(\hat {\mathbb {E}}\) is a sublinear expectation on \((\Omega,\mathcal {H})\)defined as follows: for X∈Lip(Ω)of the form
we set
where ξ_{1},…ξ_{n} are ddimensional random variables on sublinear expectation space \((\tilde {\Omega },\tilde {\mathcal {H}},\mathbb {E})\) such that for each i=1,…,n, ξ_{i} is Gnormally distributed and independent of (ξ_{1},…,ξ_{i−1}). We denote by \({L^{p}_{G}(\Omega)}\)the completion of Lip(Ω) under the norm \(\X\_{p}:=\hat {\mathbb {E}}[X^{p}]^{1/p}\), p≥1. Then it is easy to check that \(\hat {\mathbb {E}}\) is also a sublinear expectation on the space \((\Omega,{L^{p}_{G}(\Omega)})\), \({L^{p}_{G}(\Omega)}\) is a Banach space, and the canonical process B_{t}(ω):=ω_{t} is a GBrownian motion.
Following (Peng 2010) and (Denis et al. 2011), we introduce the notation: for each t∈[0,∞)

1.
Ω_{t}:={w_{.∧t}:ω∈Ω}, \(\mathcal {F}_{t}:=\mathcal {B}(\Omega _{t})\),

2.
L^{0}(Ω): the space of all \(\mathcal {B}(\Omega)\)measurable real functions,

3.
L^{0}(Ω_{t}): the space of all \(\mathcal {B}(\Omega _{t})\)measurable real functions,

4.
Lip(Ω_{t}):=Lip(Ω)∩L^{0}(Ω_{t}), \(L^{p}_{G}(\Omega _{t}):=L^{p}_{G}(\Omega)\cap L^{0}(\Omega _{t})\),

5.
\(M^{2}_{G}(0,T)\) is the completion of the set of elementary processes of the form
$$ \eta(t)=\sum_{i=1}^{n1}\xi_{i}\mathbbm{1}_{[t_{i},t_{i+1})}(s), $$where 0≤t_{1}<t_{2}<…<t_{n}≤T, n≥1 and \(\phantom {\dot {i}\!}\xi _{i}\in Lip(\Omega _{t_{i}})\). The completion is taken under the norm
$$ \\eta\^{2}_{M^{2}_{G}(0,T)}:=\hat{\mathbb{E}}\left[\int_{0}^{T}\eta(t)^{2}ds\right]. $$
Definition 6
Let X∈Lip(Ω)have the representation
We define the conditional Gexpectation under \(\mathcal {F}_{t_{j}}\) as
where
Similarly to the Gexpectation, the conditional Gexpectation might also be extended to the sublinear operator \(\hat {\mathbb {E}}[.\mathcal {F}_{t}]\colon L^{p}_{G}(\Omega)\to L^{p}_{G}(\Omega _{t})\) using the continuity argument. For more properties of the conditional Gexpectation, see (Peng 2010).
G(conditional) expectation plays a crucial role in the stochastic calculus for GBrownian motion. In (Denis et al. 2011), it was shown that the analysis of the Gexpectation might be embedded in the theory of upperexpectations and capacities.
Theorem 1
((Denis et al. 2011), Theorem 52 and 54) Let \((\tilde {\Omega }, \mathcal {G}, \mathbb {P}_{0})\) be a probability space carrying a standard ddimensional Brownian motion W with respect to its natural filtration \(\mathbb {G}\). Let Θ be a representation set defined as in (3) and denote by \({\mathcal {A}^{\Theta }_{0,\infty }}\) the set of all Θvalued \(\mathbb {G}\)adapted processes on an interval [0,∞). For each \(\theta \in {\mathcal {A}^{\Theta }_{0,\infty }}\) define \(\mathbb {P}^{\theta }\) as the law of a stochastic integral \(\int _{0}^.\,\theta _{s}dW_{s}\) on the canonical space \(\Omega =C_{0}(\mathbb {R}_{+},\mathbb {R}^{d})\). We introduce the sets
where the closure is taken in the weak topology. \(\mathcal {P}_{1}\) is tight, so \(\mathcal {P}\) is weakly compact. Moreover, one has the representation
For convenience, we always consider only a Brownian motion on the canonical space Ω with the Wiener measure \(\mathbb {P}_{0}\). Similarly, an analogous representation holds for the Gconditional expectation.
Proposition 1
((Soner et al. 2011a), Proposition 3.4) Let \(\mathcal {Q}(t,P):=\left \{\mathbb {P}'\in \mathcal {Q}\colon \mathbb {P}=\mathbb {P}'\ \text {on}\ \mathcal {F}_{t}\right \}\), where \(\mathcal {Q}=\mathcal {P}\) or \(\mathcal {P}_{1}\). Then, for any \(X\in L^{1}_{G}(\Omega)\) and \(\mathbb {P}\in \mathcal {Q}\), \(\mathcal {Q}=\mathcal {P}\) or \(\mathcal {P}_{1}\), one has
We now introduce the Choquet capacity (see (Denis et al. 2011)) related to \(\mathcal {P}\)
Definition 7

1.
A set A is said to be polar, if c(A)=0. Let \(\mathcal {N}\) be a collection of all polar sets. A property is said to hold quasisurely (abbreviated to q.s.) if it holds outside a polar set.

2.
We say that a random variable Y is a version of X if X=Y q.s.

3.
A random variable X is said to be quasicontinuous (q.c. in short), if for every ε>0 there exists an open set O such that c(O)<ε and \(\phantom {\dot {i}\!}X_{O^{c}}\) is continuous.
We have the following characterization of spaces \(L^{p}_{G}(\Omega)\). This characterization shows that \(L^{p}_{G}(\Omega)\) is a rather small space.
Theorem 2
(Theorem 18 and 25 in (Denis et al. 2011)) For each p≥1 one has
Gexpectation turns out to be a good framework to develop stochastic calculus of the Itô type. We can also use GSDEs and a version of the backward SDEs. As backward equations are a key tool to consider the maximum principle, we now give a short introduction to GBSDEs and their properties (for simplicity in a onedimensional case).
Fix two functions \(f,g:\Omega \times [0,T]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) and \(\xi \in L^{p}_{G}(\Omega _{T}),\ p>2\). We say that the triple (p^{G},q^{G},K) is a solution of the GBSDE with drivers f,g and terminal condition ξ if
where K is a nonincreasing Gmartingale starting at 0. In (Hu et al. 2014c), the existence and uniqueness of such a GBSDE are proved under some Lipschitz and regularity conditions on the driver.
Furthermore, under any \(\mathbb {P}\in \mathcal {P}_{1}\), the process p^{G} is a supersolution of a classical BSDE with drivers f and g and terminal condition ξ on the probability space \((\Omega,\mathcal {F},\mathbb {P})\) (we will call such a BSDE a \(\mathbb {P}\)BSDE). Hence, by the comparison theorem for supersolutions and solutions, we get
where \(p^{\mathbb {P}}\) is a solution of a \(\mathbb {P}\)BSDE. It can also be checked that p^{G} is minimal in the sense that
see (Soner et al. 2011c) for this representation. From now on we drop the superscript G in the notation for GBSDEs whenever this doesn’t lead to confusion.
A sufficient maximum principle
Let B(t) be a GBrownian motion with associated sublinear expectation operator \(\hat {\mathbb {E}}\). We consider controls u taking values in a closed convex set \(U\subset \mathbb {R}\). Let X(t)=X^{u}(t) be a controlled process of the form
We assume that the coefficients b, μ, σ are Lipschitz continuous w.r.t. the space variable uniformly in (t,u). Moreover, if the coefficients are not deterministic, they must belong to the space \(M^{2}_{G}(0,T)\) for each \((x,u)\in \mathbb {R}\times U\).
Let \(f:[0,T]\times \mathbb {R}\times U\to \mathbb {R}\) and \(g:\mathbb {R}\to \mathbb {R}\) be two measurable functions such that f is \(\mathcal {C}^{1}\) w.r.t. the second variable and g is a lowerbounded, differentiable function with quadratic growth such that there exists a constant C>0 and ε>0 s.t.
We let \(\mathcal {A}\) denote the set of all admissible controls. For u to be in \(\mathcal {A}\) we require that u(t) is quasicontinuous for all t∈[0,T] and adapted to \((\mathcal {F}_{(t\delta)^{+}})_{t\geq \delta }\), where δ≥0 is a given constant. This means that our control u has access only to a delayed information flow. Moreover, we assume that for each \(u\in \mathcal {A}\) the following integrability condition is satisfied
Then, for each \(\mathbb {P}\in \mathcal {P}\), the performance functional associated with \(u\in \mathcal {A}\) is assumed to be of the form
We study the following strongly robust optimal control problem: find \(\hat {u}\in \mathcal {A}\) such that
where the set \(\mathcal {P}\) is introduced in (4). To this end we define the Hamiltonian
and the associated GBSDE with adjoint processes p(t),q(t),K(t) by
Note that the solution of such a GBSDE exists thanks to the assumption on the functions f and g and on the definition of the admissible control (see (Hu et al. 2014c) for details).
Theorem 3
Let \(\hat {u}\in \mathcal {A}\)with corresponding solution \(\hat {X}(t), \hat p(t),\hat q(t), \hat K(t)\)of (8) And (12) such that \(\hat K\equiv 0\). Assume that:
and
for all t q.s. Then \(\hat {u}=u\) is a strongly robust optimal control for the problem (10).
Proof
For the sake of simplicity, in the following we adopt the concise notation f(t):=f(t,X^{u}(t),u(t)), \(\hat f(t)=f(t,X^{\hat {u}}(t),\hat {u}(t))\), X(T)=X^{u}(T), \(\hat {X}(T)=X^{\hat {u}}(T)\). Let \(u\in \mathcal {A}\) be arbitrary and consider
where J is introduced in (2) and
By the definition of H, we can write
By the concavity of g, (12), and the Itô formula, we have
Adding (16) and (17) and using the concavity of H we get, by the sublinearity of the Gexpectation and by (15), that
since \(u=\hat {u}\) is a critical point of the Hamiltonian. This proves that \(\hat {u}:=\hat {u}\) is optimal. □
Remark 1
Note that if δ=0, we can slightly relax the assumption in Eq. (14) by only requiring that
A necessary maximum principle for the fullinformation case
A drawback of the previous result is that the concavity conditions are not satisfied in many applications. Therefore, it is of interest to have a maximum principle, which does not need this condition. Moreover, the requirement that the nonincreasing Gmartingale \(\hat {K}\) disappears from the adjoint equation for the optimal control \(\hat {u}\) is a very strong assumption, which, however, is crucial in the proof. In this section, we prove a result which doesn’t depend on the concavity of the Hamiltonian. Moreover, in the Merton problem we show that the necessary maximum principle might be obtained without the assumption on the process \(\hat K\). We make the following assumptions.

A1.
For all \(u,\beta \in \mathcal {A}\) with β bounded, there exists δ>0 such that
$$u+a\beta\in \mathcal{A} \quad \text{for all} \ a\in (\delta, \delta). $$ 
A2.
For all t,h such that 0≤t<t+h≤T and all bounded random variables \(\alpha \in L^{1}_{G}(\Omega _{t})\)^{3}, the control
$$\beta(s):= \alpha \mathbbm{1}_{[t,t+h]} (s) $$belongs to \(\mathcal {A}\).
Remark 2
Note that given \(u,\beta \in \mathcal {A}\) with β bounded, the derivative process
exists, Y(0)=0, and
This follows by the smoothness (\(\mathcal {C}^{1}\)) assumptions on the coefficients b,μ,σ and the Itô formula.
Before we give the necessary maximum principle for this problem, we will state the following remark showing that it is sufficient to consider just a control which is optimal for all \(\mathbb {P}\in \mathcal {P}_{1}\).
Remark 3
Note that if \(\hat {u}\in \mathcal {A}\) is a strongly robust optimal control, it is of course the optimal control for the following problem:
However, we have also the opposite, thanks to the conditions on the set of admissible controls \(\mathcal {A}\). Namely, if \(\hat {u}\) satisfies (18), then we have that for any fixed \(u\in \mathcal {A}\) and any \(\mathbb {P}\in \mathcal {P}_{1}\)
We use again the shortened notation \(\hat f(t):=f\left (t,X^{\hat {u}}(t), \hat {u}(t)\right)\) and f(t):=f(t,X^{u}(t),u(t))and conclude that
Note that due to the conditions on the admissible controls we know that the random variables: \(\int _{0}^{T}f(t)dt,\ \int _{0}^{T}\hat f(t)dt,\ g(X^{u}(T))\), and \(g\left (X^{\hat {u}}(T)\right)\) belong to \(L^{1}_{G}(\Omega)\), hence by the representation of the Gexpectation, we have
and that implies by Proposition 1 that \(\hat {u}\) is a strongly robust optimal control.
Lemma 1
Assume that A1, A2 hold and that \(\hat {u}\) is an optimal control for the performance functional
for some probability measure \(\mathbb {P}\in \mathcal {P}_{1}\). Consider the adjoint equation as a BSDE under the probability measure \(\mathbb {P}\):
Then
Proof
Consider
By the Itô formula,
If \(\hat {u}\) is an optimal control, then the above gives
for all bounded \(\beta \in \mathcal {A}\). Applying this to both βand −β, we conclude that
By A2 together with the footnote about the denseness, we can then proceed to deduce that
□
Using the lemma, we can easily get the following necessary maximum principle.
Theorem 4
Assume that A1, A2 hold and that \(\hat {u}\) is a strongly robust optimal control for the performance functional
for every probability measure \(\mathbb {P}\in \mathcal {P}_{1}\). Consider the adjoint equation as a GBSDE:
If \(\hat K\equiv 0\ q.s.\), then
Proof
We now prove that the relation in (23) holds for every \(\mathbb {P}\in \mathcal {P}_{1}\). Fix \(\mathbb {P}\in \mathcal {P}_{1}\). If \(\hat K\equiv 0\ q.s.\), then by the uniqueness of the solution of \(\mathbb {P}\)BSDE, we get that \(\hat p^{G}\equiv \hat p^{\mathbb {P}}\ \mathbb {P}a.s.\) and \(\hat q^{G}\equiv \hat q^{\mathbb {P}}\ \mathbb {P}a.s\). But by Lemma 1, we know that \(\hat {u}\) is a \(\mathbb {P}a.s.\) critical point of \(\hat H^{\mathbb {P}}(t)\) hence also \(\hat H^{G}(t)\). By the arbitrariness of \(\mathbb {P}\in \mathcal {P}_{1}\), we get
We get the assertion of the theorem by stating a general fact that if \(\xi,\eta \in L^{1}_{G}(\Omega)\) and \(\xi =\eta \ \mathbb {P}a.s.\) for all \(\mathbb {P}\in \mathcal {P}_{1}\), then ξ=η q.s. □
As we mentioned at the beginning of this section, the assumption on the process \(\hat K\) is a big disadvantage. However, if we limit our considerations to Mertontype problems, we are able to show the necessary maximum principle without this assumption.
Theorem 5
Assume that

1.
A1, A2 hold,

2.
b≡0, μ(t,x,u)=ψ(x)l(u)m(t) and σ(t,x,u)=ψ(x)h(u)s(t) for \(\psi, l, h \in \mathcal {C}^{1}(\mathbb {R})\) and some bounded processes m and s such that for each t∈[0,T]m(t) and s(t) are quasicontinuous. Moreover, let c(s(t)=0)=0 for all t∈[0,T],

3.
f≡0,

4.
X(0)=x≠0.
Let \(\hat {u}\) be a strongly robust optimal control for the performance functional
for every probability measure \(\mathbb {P}\in \mathcal {P}_{1}\). If \(c(l(\hat {u}(t))=0)=0\), \(c(h(\hat {u}(t))=0)=0\), \(c(h'(\hat {u}(t))=0)=0\), \(c(\psi (\hat {X}(t))=0)=0\), \(c(\psi ^{\prime }(\hat {X}(t))=0)=0\), and \(c(l(\hat {u}(t))h'(\hat {u}(t))l'(\hat {u}(t))h(\hat {u}(t))\neq 0)=0\)^{4} for all t∈[0,T], then
Proof
Fix a probability measure \(\mathbb {P}\in \mathcal {P}_{1}\). By Lemma 1, we know that \(\hat {u}\) is a critical point (\(\mathbb {P}\)a.s.) of the Hamiltonian
Using this fact, we get
By the assumption on process s and h^{′}, we compute that
We have that
since \(c(l(\hat {u})h'(\hat {u})l'(\hat {u})h(\hat {u})\neq 0)=0\) by hypothesis. But then we see that \(\hat p^{\mathbb {P}}\) has dynamics
Hence,
We also remember that
Thus, by the characterization of the conditional Gexpectation in (6), we obtain that \(\hat p^{G}(t)\) is a Gmartingale with representation
and consequently it has dynamics
But in that case we know that for almost all t∈[0,T], we must have that
q.s. By assumption on \(\psi ^{\prime }(\hat {X})\), we conclude that
Hence,
since \(c(h(\hat {u}(t))=0)=0\) for all t∈[0,T], and we can easily check then that
□
Examples
We now consider some examples to illustrate the previous results. In the following, we assume to work with a onedimensional GBrownian motion with operator G of the form
i.e., with quadratic variation 〈B〉(t) lying within the bounds \(\underline {\sigma }^{2}t\) and t.
Example I
Consider
where c(t), t∈[0,T], is stochastic process such that \(c(t)\in L^{1}_{G}(\Omega _{t})\) for all t∈[0,T]. We wish to solve the optimal control problem for every \(\mathbb {P}\in \mathcal {P}\) under the performance criterion
In the notation of Section 3, we have chosen here f(t,x,c)= lnc and g(x)=x, i.e., g^{′}(x)=1. Then, the Hamiltonian is given by
and by (12), we obtain
i.e., q=0,p=1. Furthermore, by (29), we have
i.e., \(\hat c(t)=1\), t∈[0,T], is a strongly robust optimal control by Theorem 3.
Note that by the proof we could choose a general utility function instead of a logarithmic utility function without losing the existence of the strongly robust optimal control.
Example II
Consider
and problem (28). Here, b(t) is a deterministic measurable function. Then, the Hamiltonian is given by
Here,
Let q=0, then
i.e., \(p(t)=\exp {\int _{t}^{T} b(s) ds}\) and \(\hat c(t)=\frac {1 }{p(t)}\) is a strongly robust optimal control by Theorem 3.
Example III
Consider the Mertontype problem with the logarithmic utility, i.e., let
where \(u(t)\in L^{2}_{G}(\Omega _{t})\) for all t∈[0,T] and m and s are two deterministic functions. Assume that s(t)≠0 for all t∈[0.T]. We are interested in finding a strongly robust optimal control problem for the family of probability measures \(\mathcal {P}\) with the performance criterion given by
The Hamiltonian associated with this problem is given by
and for each admissible control u we consider adjoint GBSDE of the form
Note that the adjoint equation is linear, hence by Remark 3.3 in (Hu et al. 2014a), we obtain the representation formula for the solution
Moreover, by the dynamics of X^{−1}, we deduce that
Plugging this solution into the Hamiltonian (34), we get that
hence the critical point of the Hamiltonian must satisfy
and this is our strongly robust optimal control.
Note that we can also solve this problem directly by omegawise maximization, without using the maximum principle and GBSDEs. In fact, we may consider more general dynamics in X
and by direct computation it might be checked that the strongly robust optimal control takes the form
However, it is important to note that this control is no longer quasicontinuous (see (Song 2012)) and it doesn’t make sense to consider GBSDEs associated with such a control.
Finally, note that the classical robust optimal control for this problem would be \(u^{*}(t)=m(t)/\underline {\sigma }^{2}\). It is clear that this control ignores the flow of information about the volatility path and instead it just sticks to its worstcase scenario assumption. It makes sense to assume the worstcase scenario at time 0, but later one should rather update its view about the past volatility, which is not done for the classical robust optimal controls.
Example IV
As another example of a problem which admits a strongly robust optimal control, consider an LQproblem, in which the state equation has the linear dynamics
for \(u\in \mathcal {A}\) as described in the beginning of Section 3. The performance functional is quadratic
Here, F,G,μ,σ,Q,R are continuous deterministic functions on [0,T], Q(t)>0, R(t)>0 and L>0 is a constant.
We want to find \(\hat {u}\in \mathcal {A}\) (as described in Section 3) which maximizes \(J^{\mathbb {P}}(u)\) over all \(u\in \mathcal {A}\) for all \(\mathbb {P}\in \mathcal {P}\).
In this case, the Hamiltonian in (11) gets the form
and the adjoint BSDE (12) becomes
We intend to apply Theorem 3 and note that
which is 0 when
where \(\hat p(t)\) refers to the solution of (38) when \(u=\hat {u}\) is applied to the BSDE.
Let us guess that (38) with \(u=\hat {u}\) admits the solution of the form
for some deterministic functions \(S, Z\in \mathcal {C}^{1}(\mathbb {R}_{+})\) to be determined.
We apply the Itô formula to the equation for \(\hat p\) in (41) and plug in the candidate for optimal control from (40). By comparison with (38), after some simple computations we get that
i.e., (41) is indeed the solution of the adjoint Eq. 38 if S satisfies the Riccati equation
and Z satisfies the linear differential equation
By Theorem 3, we conclude that
is a strongly optimal control with S and Z given by (42) and (43), respectively.
Counterexample: the Merton problem with the power utility
In this example, we consider the Merton problem with the power utility and show that generally we cannot drop the assumption \(\hat K\equiv 0\) without losing the strong sense of the optimality. First, we solve the classical robust utility maximization problem and then we prove that the optimal control for that problem is optimal usually only in a weaker sense, i.e., there exists a probability measure \(\mathbb P\in \mathcal {P}\) such that the control is not optimal under \(\mathbb P\), even though the control satisfies all the conditions of the sufficient maximum principle with the exception of \(\hat K\equiv 0\).
Consider first the classical robust utility maximization problem
where X has dynamics for any \(u\in \mathcal {A}\)
Then,
We assume that m and s are bounded and deterministic and s≠0. Put f≡0 and \(g(x)=\frac {1}{\alpha }x^{\alpha },\ \alpha \in ]0,1[\). Hence,
We now use the Girsanov theorem for Gexpectation and the Gmartingale
see Section 5.2 in (Hu et al. 2014a). We get the sublinear expectation \(\hat {\mathbb {E}}^{u}\) under which the process \({B}^{u}(t):=B(t)\int _{0}^{t}s(r)u(r)d\langle B\rangle (r)\) is a GBrownian motion. Note that
q.s. Moreover, it is easy to check that the deterministic control
is a maximizer of the following function
Hence, we get that
The last equalities are a consequence of (45) and of the fact that the integrand is deterministic and that B^{u} and \(B^{\hat {u}}\) are GBrownian motions under \(\mathbb {E}^{u}\) and \(\mathbb {E}^{\hat {u}}\), respectively. Eq. 46 shows then that \(\hat {u}\) is an optimal control for this weaker optimization problem.
Now, consider the adjoint equation related to \(\hat {u}\) in terms of a GBSDE. The backward equation is linear due to linearity of the Hamiltonian, hence we may use the conditional expectation representation of linear GBSDEs (compare with Remark 3.3 in (Hu et al. 2014a)):
Applying the Girsanov theorem and the same reasoning as in (46), we easily get that
Furthermore, we also know that the integrand is always positive by the assumption α∈]0,1[, hence we get by the representation of the conditional Gexpectation (6) that for every \(\mathbb {P}\in \mathcal {P}\) and by (26) that
Hence,
By integration by parts for \(\hat {X}^{1}\) and Z, one can compute that
By comparing Eq. 47 with the adjoint Eq. 12, we first obtain that
and hence that \(\hat {u}\) is a maximizer of the function \(u\mapsto H\left (t,\hat {X}(t),u,\hat p^{G}(t), \hat q^{G}(t)\right)\). Secondly, we get that the process \(\hat K\) has the explicit form
and consequently is a nontrivial process.
To summarize the example so far: we have shown that \(\hat {u}\) is optimal in a weaker sense. We also showed that it satisfies the assumption for the necessary maximum principle for strongly robust optimality and that all assumptions of the sufficient maximum principle are satisfied, with the exception of the vanishing of process \(\hat K\). Now, we prove that \(\hat {u}\) is not optimal in the stronger sense, hence the assumption on the process \(\hat K\) is really crucial for our result and cannot be dropped.
Fix \(\mathbb {P} \in \mathcal {P}_{1}\) and assume that \(\hat {u}\) is optimal under \(\mathbb {P}\). By Lemma 1, we know that \(\hat {u}\) is a critical point of the Hamiltonian evaluated in \(\hat p^{\mathbb {P}}\) and \(\hat q^{\mathbb {P}}\). Hence, by the same analysis as in Theorem 5, we see that
therefore
However, we know by the dynamics of \(\hat {X}\) and the terminal condition of \(\mathbb {P}\)BSDE that
Dividing (48) by (49), we get that
The equalities here are \(\mathbb {P}\)a.s. so we get that the integral \(\int _{0}^{T}\frac {\alpha m^{2}(t)}{2s^{2}(t)(\alpha 1)}d\langle B\rangle (t)\) must be equal \(\mathbb {P}\)a.s. to a constant. However, the quadratic variation of the canonical process under \(\mathbb {P}\) is generally a nondeterministic stochastic process, hence also the integral is a random variable, in general nonconstant. This shows that \(\hat {u}\) is optimal under \(\mathbb {P}\) only for very specific probability measures such as the Wiener measure.
To conclude, \(\hat {u}\) is not optimal for every probability measure \(\mathbb {P}\in \mathcal {P}\) even though it is a maximizer of the Hamiltonian related to \(\hat {u}\). This example shows that the new strong notion of optimality is rather restricted and we may expect it only in very special cases when the process \(\hat K\) vanishes.
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Acknowledgments
The authors would like to thank three anonymous referees and the editor for a careful reading of the paper, and the Journal for providing English editing of the paper.
Funding
The research leading to these results received funding from the European Research Council under the European Community’s Seventh Framework Program (FP7/20072013) / ERC grant agreement 228087.
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Biagini, F., MeyerBrandis, T., Øksendal, B. et al. Optimal control with delayed information flow of systems driven by GBrownian motion. Probab Uncertain Quant Risk 3, 8 (2018). https://doi.org/10.1186/s415460180033z
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Keywords
 GBrownian motion
 optimal control problem
 stochastic maximum principle
Mathematics Subject Classification (2010):
 60H99
 93E20