Optimal control with delayed information flow of systems driven by $G$-Brownian motion

In this paper we study strongly robust optimal control problems under volatility uncertainty. In the $G$-framework 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 here 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 G-Brownian motion (see [? ], [? ]), or by quasi-sure analysis (see [? ]). In [? ] it is proven that these two methods are strongly related. The link between these two approaches is the representation of the sublinear expectationÊ associated with the G-Brownian motion as a supremum of ordinary expectations over a tight family of probability measures P, whose elements are mutually singular: see (2.2) and Theorem 2.7 for more details.
In this paper we work in a G-Brownian motion setting as in [? ] and use the related stochastic calculus, including the Itô formula, G-SDE's, martingale representation and G-BSDE's, as developed in [? ], [? ], [? ], [? ], [? ], [? ], [? ], [? ]. It is important for understanding the nature of the G-Brownian 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 [σ 2 ,σ 2 ] for d = 1). Each P ∈ P can be seen then as a model with a different scenario for the quadratic variation. That justifies why G-Brownian motion is a good framework for investigating model uncertainty.
In a G-Brownian motion setting one considers the following stochastic optimal control problem: to find the controlû ∈ A such that J(û) = sup f (t, X u (t), u(t))dt + g(X u (T ))] =: sup where X u is a controlled G-SDE, see (3.1). This problem has been studied in [? ], [? ]. In [? ] 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 [? ] investigates the robust investment problem for geometric G-Brownian motion and 2BSDE's (which is a version of G-BSDE's) are used to find an optimal solution. In both papers the optimal control is robust in the worst case 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 worst case scenario, but also for all probability measures P (with the optimality criterion J P ). We call this a strongly robust control. This strongly robust control is thus optimal in a much more robust sense than the worst case scenario optimality. The new strongly robust optimality uses the fact that probability measures P are mutually singular, hence one can modify the P-optimal controlû P outside the support of a probability measure P without losing the P-optimality. As a consequence, if the family {û P } P∈P satisfies some consistency conditions, the controls can be aggregated into a unique controlû, which is optimal under every probability measure P. See [? ] 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 [? ], we adapt the stochastic maximum principle to the G-framework to find necessary and sufficient conditions for the existence of a strongly robust optimal control. The paper is structured in the following way. In Section 2 we give a quick overview on the G-framework. Section 3 is devoted to a sufficient maximum principle in the partial information case. In Section 4 we investigate the necessary maximum principle for the full-information case. In Section 5 we give three examples, including the Merton problem with the logarithmic utility, already mentioned earlier. In Section 6 we provide a counter-example 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 H be a vector lattice of real functions defined on Ω, ie. a linear space containing 1 such that X ∈ H implies |X| ∈ H. We will treat elements of H as random variables.

Positive homogeneity: For all
The triple (Ω, H, E) is called a sublinear expectation space.
We will consider a space H of random variables having the following property: Let X 1 and X 2 be n-dimensional random vectors defined on sublinear random spaces (Ω 1 , H 1 , E 1 ) and (Ω 2 , H 2 , E 2 ) respectively. We say that X 1 and X 2 are identically distributed and denote it by The letter G denotes a function defined as It can be checked that G might be represented as where Θ is a non-empty bounded and closed subset of R d×d .
Definition 2.4. Let G : S d → R be a given monotonic and sublinear function. A stochastic process B = (B t ) t≥0 on a sublinear expectation space (Ω, H, E) is called a G-Brownian motion if it satisfies following conditions 3. For each t, s ≥ 0 the increment B t+s − B t is G-normally distributed and independent of (B t 1 , . . . , B tn ) for each n ∈ N and 0 ≤ t 1 < . . . < t n ≤ t. A G-expectationÊ is a sublinear expectation on (Ω, H) defined as follows: for X ∈ Lip(Ω) of the form where ξ 1 , . . . ξ n are d-dimensional random variables on sublinear expectation space (Ω,H, E) such that for each i = 1, . . . , n ξ i , is G-normally distributed and independent of (ξ 1 , . . . , ξ i−1 ). We denote by L p G (Ω) the completion of Lip(Ω) under the norm X p :=Ê[|X| p ] 1/p , p ≥ 1. Then it is easy to check thatÊ is also a sublinear expectation on the space (Ω, L p G (Ω)), L p G (Ω) is a Banach space and the canonical process B t (ω) := ω t is a G-Brownian motion.
is the completion of the set of elementary processes of the form where 0 ≤ t 1 < t 2 < . . . < t n ≤ T, n ≥ 1 and ξ i ∈ Lip(Ω t i ). The completion is taken under the norm Definition 2.6. Let X ∈ Lip(Ω) have the representation Similarly to the G-expectation, the conditional G-expectation might be also extended to the sublinear operatorÊ[.|F t ] : L p G (Ω) → L p G (Ω t ) using the continuity argument. For more properties of the conditional G-expectation, see [? ].
G-(conditional) expectation plays a crucial role in the stochastic calculus for G-Brownian motion. In [? ] it was shown that the analysis of the G-expectation might be embedded in the theory of upper-expectations and capacities.
Theorem 2.7 ([? ], Theorem 52 and 54). Let (Ω, G, P 0 ) be a probability space carrying a standard d-dimensional Brownian motion W with respect to its natural filtration G. Let Θ be a representation set defined as in eq. (2.1) and denote by We introduce the sets where the closure is taken in the weak topology. P 1 is tight, so P is weakly compact. Moreover, one has the representation For convenience we will always consider only a Brownian motion on the canonical space Ω with the Wiener measure P 0 . Similarly an analogous representation holds for the G-conditional expectation.
We now introduce the Choquet capacity (see [? ]) related to P Definition 2.9.

A set
A is said to be polar, if c(A) = 0. Let N be a collection of all polar sets. A property is said to hold quasi-surely (abbreviated to q.s.) if it holds outside a polar set.

We say that a random variable Y is a version of
We have the following characterization of spaces L p G (Ω). This characterization shows that L p G (Ω) is a rather small space. The G-expectation turns out to be a good framework to develop stochastic calculus of the Itô type. We can have also G-SDE's and a version of the backward SDE's. As backward equations are a key tool to consider the maximum principle, we now give some short introduction to G-BSDE's and their properties (for simplicity in a one-dimensional case).
Fix two functions f, g : where K is a non-increasing G-martingale starting at 0. In [? ] the existence and uniqueness of such a G-BSDE are proved under some Lipschitz and regularity conditions on the driver. Furthermore under any P ∈ P the process p G is a supersolution of a classical BSDE with drivers f and g and terminal condition ξ on the probability space (Ω, F, P) (we will call such a BSDE a P-BSDE). Hence, by comparison theorem for supersolutions and solutions we get where p P is a solution of P-BSDE. It might be also checked that p G is minimal in the sense that p G (t) = ess sup P Q∈P(t,P) see [? ] for this representation. From now on we drop the superscript G in the notation for G-BSDE's whenever this doesn't lead to confusion.

A sufficient maximum principle
Let B(t) be a G-Brownian motion with associated sublinear expectation operator E. We consider controls u taking values in a closed convex set U ⊂ 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 Let f : [0, T ]×R×U → R and g : R → R be two measurable functions such that f is continuous w.r.t the second variable and g is a lower-bounded, differentiable function with quadratic growth s.t. there exists a constant C > 0 and ǫ > 0 s.t We let A denote the set of all admissible controls. For u to be in A we require that u is quasi-continuous and adapted to (F (t−δ) + ) t≥δ , where δ ≥ 0 is a given constant. This means that our control u has only access to a delayed information flow. Moreover, we assume that for each u ∈ A the following integrability condition is satisfiedÊ Then for each P ∈ P, the performance functional associated to u ∈ A is assumed to be of the form ( 3.2) We study the following strongly robust optimal control problem: findû ∈ A such that where the set P is introduced in (2.2). To this end we define the Hamiltonian (3.4) and the associated G-BSDE with adjoint processes p(t), q(t), K(t) by Note that the solution of such G-BSDE exists thanks to the assumption on the functions f and g and on the definition of the admissible control (see [? ] for details).
Proof. For the sake of simplicity , in the sequel we adopt the concise notation Let u ∈ A be arbitrary and consider where J is introduced in (1.2) and By definition of H we can write (3.9) By concavity of g, (3.5) and the Itô formula we have (3.11) Adding (3.9) and (3.11) and using concavity of H we get, by the sublinearity of the G-expectation and by (3.8), that since u =û is a critical point of the Hamiltonian. This proves thatû :=û is optimal.

A necessary maximum principle for full-information case
It is a drawback of the previous result 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 non-increasing G-martingaleK disappears from the adjoint equation for the optimal controlû is a very strong assumption, which is however 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 processK. We make the following assumptions: A1. for all u, β ∈ A with β bounded, there exists δ > 0 such that u + aβ ∈ A, for all a ∈ (−δ, δ).

A2. For all t, h such that 0 ≤ t < t + h ≤ T and all bounded random variables
A3. Given u, β ∈ A with β bounded, the derivative process exists, Y (0) = 0 and Lemma 4.1. Assume that A1, A2, A3 hold and thatû is an optimal control for the performance functional u → J P (u) for some probability measure P ∈ P. Consider the adjoint equation as a BSDE under probability measure P: p P (T ) = g ′ (X(T )) P − a.s.

(4.2)
2 It is easy to see that for a fixed P ∈ P the set of all bounded random variables from the space L 1 G (Ω) is dense in the space L p P (Ωt) under the norm (E P [|.| p ]) 1/p for any p ≥ 1.

By the Itô formula
Adding (4.2) and (4.3) we get Ifû is an optimal control, then the above gives for all bounded β ∈ A. Applying this to both β and −β, we conclude that By A2 together with the footnote about the denseness we can then proceed to deduce that ∂Ĥ P ∂u (t) = 0 P − a.s.
Using the lemma we can easily get the following necessary maximum principle.
Theorem 4.2. Assume that A1, A2, A3 hold and thatû is a strongly robust optimal control for the performance functional for every probability measure P ∈ P. Consider the adjoint equation as a G-BSDE: p G (T ) = g ′ (X(T )) q.s.
Proof. We now prove that the relation in (4.5) holds for every P ∈ P. Fix P ∈ P.
IfK ≡ 0 q.s. then by the uniqueness of the solution of P-BSDE we get that p G ≡p P P − a.s. andq G ≡q P P − a.s. But by Lemma 4.1 we know thatû is a P − a.s. critical point ofĤ P (t) hence alsoĤ G (t). By the arbitrariness of P ∈ P we get the assertion of the theorem.
Just as we mentioned at the beginning of this section, the assumption on the processK is a big disadvantage. However, if we limit our considerations to the Merton-type problem, we are able to show the necessary maximum principle without this assumption.

f ≡ 0.
Letû is a strongly robust optimal control for the performance functional for every probability measure P ∈ P. Then ∂Ĥ G ∂u (t) := ∂ ∂u H(t,X(t), u,p G (t),q G (t)) = 0, q.s. (4.6) Proof. Fix a probability measure P ∈ P. By Lemma 4.1 we know thatû is a critical point (P-a.s.) of the Hamiltonian Using this fact we get By the assumption on the process s we compute that But then we see thatp P has dynamics Hencep We also remember that p G (t) = ess sup P Q∈P(t,P)p Thus by the characterization of the conditional G-expectation in (2.4) we obtain thatp G (t) is a G-martingale with representation and consequently it has dynamics dp G (t) =q G (t)dB(t) + dK(t).
But in that case we know that for almost all t ∈ [0, T ] we must have that By assumption onû we conclude that m(t)p G (t) + s(t)q G (t) = 0 q.s.
and we can easily check then that ∂ ∂u H(t,û,p G (t),q G (t)) = 0.

Examples
We now consider some examples to illustrate the previous results. In the sequel we assume to work with a one-dimensional G-Brownian motion with operator G of the form G(a) := 1 2 (a + − σ 2 a − ), σ 2 > 0, (5.1) i.e. with quadratic variation B (t) lying within the bounds σ 2 t and t.

Example I
Consider where c(t), t ∈ [0, T ], is stochastic process such that c(t) ∈ L 1 G (Ω t ) for all t ∈ [0, T ]. We wish to solve the optimal control problem for every P ∈ P under the performance criterion (5.3) In the notation of Section 3, we have chosen here f (t, x, c) = ln c and g(x) = x, i.e. g ′ (x) = 1. Then the Hamiltonian is given by and by (3.5) we obtain i.e. q = 0, p = 1. Furthermore by (5.4) we have i.e.ĉ(t) = 1, t ∈ [0, T ], is strongly robust optimal by Theorem 3.1. Note that by the proof we could choose a general utility function instead of logarithmic utility without losing the existence of the strongly robust optimal control.

Example II
Consider and Problem (5.3). Here b(t) is a deterministic measurable function. Then the Hamiltonian is given by Here (5.8) p(T ) = g ′ (X(T )) = 1.

Example III
Consider the Merton-type problem with the logarithmic utility: let where u(t) ∈ L 2 G (Ω 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 to find a strongly robust optimal control problem for the family of probability measures 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 G-BSDE of the form Note that the adjoint equation is linear, hence by Remark 3.3 in [? ] we obtain the representation formula for the solution Moreover, by the dynamics of X −1 we deduce that Plugging this solution into the Hamiltonian (5.9) we get that hence the critical point of the Hamiltonian must satisfŷ and this is our strongly robust optimal control. Note that we can also solve this problem directly by omega-wise maximization, without using the maximum principle and G-BSDE's. 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 formû .
However it is important to note that this control is not quasi-continuous any more (see [? ]) and it doesn't have sense to consider G-BSDE's associated with such a control.
6 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 assumptionK ≡ 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 P ∈ P such that the control is not optimal under P, even though the control satisfies all the conditions of the sufficient maximum principle with the exception ofK ≡ 0. Consider first the classical robust utility maximization problem where X has dynamics for any u ∈ A dX(t) = m(t)X(t)u(t)d B (t) + s(t)X(t)u(t)dB(t). Then We assume that m and s are bounded and deterministic and s = 0. Put f ≡ 0 and g(x) = 1 α x α , α ∈]0, 1[. Hencê We now use the Girsanov theorem for G-expectation and the G-martingale see Section 5.2. in [? ]. We get the sublinear expectationÊ u under which the process B u (t) : 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 consequence of (6.1) and of the fact that the integrand is deterministic and that B u and Bû are G-Brownian motions under E u and Eû (respectively). Equation (6.2) shows then thatû is an optimal control for this weaker optimization problem. Now consider the adjoint equation related toû in terms of a G-BSDE. The backward equation is linear due to linearity of the Hamiltonian, hence we may use the conditional expectation representation of a linear G-BSDE's (compare with Remark 3.3 in [? ]): Applying the Girsanov theorem and the same reasoning as in (6.2) 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 G-expectation (2.4) that for every P ∈ P and by (5.1) that E exp{ Hencep G (t) = (X(t)) α−1 exp{ T t α 2(1 − α) m 2 (r) s 2 (r) dr} =: (X(t)) α−1 · Z(t).
To summarize the example so far: we have shown thatû 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 the procesŝ K. Now we prove thatû is not optimal in the stronger sense, hence the assumption on the processK is really crucial for our result and cannot be dropped.
Fix P ∈ P and assume thatû is optimal under P. By Lemma 4.1 we know that u is a critical point of the Hamiltonian evaluated inp P andq P . Hence, by the same analysis as in Theorem 4.3 we see that dp P (t) = − m(t) s(t)p P (t)dB(t), Dividing (6.4) by (6.5) we get that 1 =p P (0) x α−1 exp T 0 αm 2 (t) 2s 2 (t)(α − 1) d B (t) .
The equalities here are P-a.s. so we get that the integral T 0 αm 2 (t) 2s 2 (t)(α−1) d B (t) must be equal P-a.s. to a constant. However the quadratic variation of the canonical process under P is generally a non-deterministic stochastic process, hence also the integral is a random variable, in general non-constant. This shows thatû is optimal under P only for very specific probability measures such as the Wiener measure.
To conclude,û is not optimal for every probability measure P ∈ P even though it is a maximizer of the Hamiltonian related toû. 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 processK vanishes.