Proofs of Theorem 3.3 and Proposition 3.4
Throughout Sect.
6.1
, we consider a pair \((U,\gamma )\) of a utility random field and an admissible family of penalty functions and the associated dual field \(V\) given in (3.1). Further, we consider the arbitrary but fixed time points \(0\le t\le T<\infty\). We start by introducing relevant notation from Zitković [69] since we then apply the duality from there in our proofs; see (6.6) below. Then, in Sect. 6.1.1, we prove conjugacy relations and existence of a dual optimiser for a specific auxiliary problem. In Sect. 6.1.2, Theorem 3.3 and Proposition 3.4 are proved via a reduction to this auxiliary problem.
The spaces \(L^{p}\), \(p\in[0,\infty]\), are defined with respect to \(({\Omega},\mathcal{F}_{T},\mathbb{P}|_{\mathcal{F}_{T}})\); the space \(L^{1}\) is identified with its image in \((L^{\infty})^{*}\) under the isometric embedding of a Banach space into its bidual.
Let \(\mathcal{K}_{t,T}:=\{\int_{t}^{T}\pi_{s}dS_{s}:\pi\in\mathcal{A}_{\mathrm{bd}}\}\) and \(\mathcal{C}_{t,T}:=(\mathcal{K}_{t,T} -L^{0}_{+})\cap L^{\infty}\). The optimisation over \(\mathcal{K}_{t,T}\) in (2.13) can then be replaced by optimisation over \(\mathcal{C}_{t,T}\). Given \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and a random variable \(\kappa \in L^{\infty}_{+}(\mathcal{F} _{t})\) – we typically consider 
, \(A\in\mathcal {F}_{t}\), and use it to localise arguments to a set –, we then introduce the function
$$ u^{\mathbb{Q}}_{\kappa}(\xi)=\sup_{g\in\mathcal{C}_{t,T}}\mathrm {E}[\kappa Z^{\mathbb{Q}}_{t,T}U(\xi +g,T)],\qquad \xi\in L^{\infty}(\mathcal{F}_{t}). $$
Next, let \(\mathcal{D}_{t,T}:=\{\zeta^{*}\in(L^{\infty})^{*}:\langle \zeta ^{*},\zeta\rangle\le0\textrm{ for all }\zeta\in\mathcal{C}_{t,T}\} \), and for \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), let \(\mathcal{D}^{\eta}_{t,T}:=\{\zeta^{*}\in \mathcal{D}_{t,T}:\langle \zeta ^{*},\xi\rangle=\langle\eta,\xi\rangle\textrm{ for all }\xi\in L^{\infty}(\mathcal{F}_{t})\}\). Recall that according to Lemma A.4 in [69],
$$ \zeta^{*}\in\mathcal{D}_{t,T}\cap L^{1}_{+} \qquad \textrm{if and only if}\qquad \zeta^{*}=\eta Z_{t,T}, $$
(6.1)
for some \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\) and \(Z\in\mathcal{Z}^{a}_{T}\). Note that the proof of this result uses that the market satisfies NFLVR on finite horizons. Define the function \(\mathbb{V}^{\mathbb{Q}}_{\kappa}:\mathcal{D}_{t,T}\to(-\infty,\infty]\) by
$$\begin{aligned} \mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*}) := \textstyle\begin{cases}\mathrm{E}[\kappa Z^{\mathbb{Q}}_{t,T}V(\zeta^{*}/(\kappa Z^{\mathbb{Q}}_{t,T}),T)],&\quad \zeta^{*}\in L^{1}_{+}\textrm{ and}\\ &\quad \quad \{\zeta^{*}>0\}\subseteq\{\kappa>0\}, \\ \infty, &\quad \textrm{otherwise}, \end{cases}\displaystyle \end{aligned}$$
(6.2)
and the function \(v^{\mathbb{Q}}_{\kappa}:L^{1}(\mathcal{F}_{t})\to(-\infty ,\infty]\) by
$$ v^{\mathbb{Q}}_{\kappa}(\eta):=\left\{ \textstyle\begin{array}{lcl}\inf_{\zeta^{*}\in\mathcal{D}^{\eta}_{t,T}}\mathbb {V}^{\mathbb{Q}}_{\kappa}(\zeta ^{*}),&&\quad \eta\in L^{1}_{+}(\mathcal{F}_{t}), \\ \infty, &&\quad \eta\in L^{1}(\mathcal{F}_{t})\setminus L^{1}_{+}(\mathcal{F}_{t}). \end{array}\displaystyle \right. $$
Finally, we introduce the auxiliary value functions \(u_{\kappa}:L^{\infty}(\mathcal{F}_{t})\to(-\infty,\infty]\) and \(v_{\kappa}:L^{1}(\mathcal{F}_{t})\to (-\infty ,\infty]\) given, respectively, by
$$ u_{\kappa}(\xi)=\sup_{g\in\mathcal{C}_{t,T}}\inf_{\mathbb{Q}\in \mathcal{Q}_{t,T}}\mathrm{E}\big[\kappa\big(Z^{\mathbb{Q}}_{t,T}U(\xi+g,T)+\gamma_{t,T}(\mathbb{Q})\big)\big] $$
and
$$ v_{\kappa}(\eta)=\inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\big(v_{\kappa}^{\mathbb{Q}}(\eta )+\mathrm{E}[\kappa \gamma_{t,T}(\mathbb{Q})]\big). $$
Results for the auxiliary value functions \(u_{\kappa}\) and \(v_{\kappa}\)
We establish in this section results for the auxiliary value functions \(u_{\kappa}\) and \(v_{\kappa}\) introduced above. First, we consider the existence of a dual optimiser.
Proposition 6.1
Suppose the assumptions of Proposition 3.4
hold and let
\(\eta\in L^{1}_{+}(\mathcal{F}_{t})\). Then there exists
\((\bar{\zeta}^{*},\bar{\mathbb{Q}})\in\mathcal {D}^{\eta }_{t,T}\times\mathcal{Q} _{t,T}\)
such that
$$ v_{\kappa}(\eta) = \mathbb{V}_{\kappa}^{\bar{\mathbb{Q}}}(\bar{\zeta}^{*})+\mathrm {E}[\kappa\gamma _{t,T}(\bar {\mathbb{Q}})]. $$
Moreover, the function
\(v_{\kappa}(\eta)\)
is convex and weakly lower semicontinuous.
Proof
First, since \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), by definition,
$$ v_{\kappa}^{\mathbb{Q}}(\eta)\;=\inf_{\zeta^{*}\in\mathcal{D}^{\eta}_{t,T}}\mathbb {V}_{\kappa}^{\mathbb{Q}}(\zeta^{*}),\qquad \mathbb{Q}\in\mathcal{Q}_{t,T}. $$
(6.3)
In turn, note that if \(\kappa\tilde{U}(x,T)\in L^{1}\) for all \(x\in \mathbb{R}\), where \(\tilde{U}(x,T):=Z^{\mathbb{Q}}_{t,T}U(x,T)\), using Assumption 3.2 allows us to apply Proposition A.3 in [69] to obtain
$$ \mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*}) \;= \sup_{\zeta\in L^{\infty}}\big(\mathrm{E}[\kappa Z^{\mathbb{Q}}_{t,T}U(\zeta ,T)]-\langle \zeta^{*},\zeta\rangle\big), \qquad \zeta^{*}\in\mathcal{D}^{\eta}_{t,T}. $$
(6.4)
On the other hand, recall that for \(\zeta^{*}\in\mathcal{D}_{t,T}\cap L^{1}_{+}\) with \(\{\zeta^{*}>0\}\subseteq\{\kappa>0\}\), by (6.2) and the fact that \(V(\, \cdot\,,T)\) is the convex conjugate of \(U(\, \cdot\,,T)\),
$$\begin{aligned} \mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*}) \ge \mathrm{E}\big[\kappa Z_{t,T}^{\mathbb{Q}}\big(U\left(\zeta,T\right )-\zeta^{*}\zeta /\kappa Z_{t,T}^{\mathbb{Q}}\big)\big] = \mathrm{E}[\kappa Z_{t,T}^{\mathbb{Q}}U(\zeta,T)] -\langle\zeta^{*},\zeta\rangle \end{aligned}$$
for all \(\zeta\in L^{\infty}\); for any other \(\zeta^{*}\in\mathcal{D}_{t,T}\), \(\mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*})=\infty\). Hence if \(\kappa\tilde{U}(x,T)\notin L^{1}\) for some \(x\in\mathbb {R}\), both sides of (6.4) must equal \(\infty\); (6.4) therefore holds for all \(\mathbb{Q}\in\mathcal{Q}_{t,T}\).
Next, note that \(\mathcal{D}_{t,T}^{\eta}\subseteq(L^{\infty})^{*}\) is included in a ball of size \(\langle\eta,1\rangle\) with respect to the operator norm, and such balls are \(\textrm{weak}^{*}\) compact according to the Banach–Alaoglu theorem. For any net \((\zeta^{*}_{\alpha})_{\alpha\in A}\) in \(\mathcal {D}_{t,T}^{\eta}\), where \(A\) is some directed set, there thus exists a subnet, which we still label by \((\zeta^{*}_{\alpha})_{\alpha\in A}\), converging in the \(\textrm {weak}^{*}\) topology to some \(\bar{\zeta}^{*}\in(L^{\infty})^{*}\). Since \(\mathcal{D}_{t,T}\) clearly is \(\textrm{weak}^{*}\) closed, \(\bar{\zeta}^{*}\in \mathcal{D}_{t,T}\). Further, since for any \(\xi\in L^{\infty}(\mathcal {F}_{t})\), \(\langle \bar{\zeta}^{*},\xi\rangle=\lim_{\alpha}\langle\zeta^{*}_{\alpha},\xi\rangle =\langle \eta,\xi\rangle\), we have that \(\bar{\zeta}^{*}\in\mathcal {D}_{t,T}^{\eta}\); in consequence, \(\mathcal{D}_{t,T}^{\eta}\) is \(\mbox{weak}^{*}\) compact. Recall that \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\}\) is weakly compact by assumption.
Fix \(\zeta\in L^{\infty}\) and recall that the set \(\{Z^{\mathbb{Q}} _{t,T}U^{-}(\zeta ,T):\mathbb{Q}\in\mathcal{Q}_{t,T}\}\) is uniformly integrable. The set \(\{Z^{\mathbb{Q}}_{t,T}: \mathbb{Q}\in\mathcal{Q}_{t,T} \textrm{ and }\mathrm{E}[\kappa Z^{\mathbb{Q}} _{t,T}U(\zeta,T)]\leq c\}\) is convex. Further, using the above uniform integrability and Fatou’s lemma, it is closed in \(L^{1}\) and hence by convexity also weakly closed. It follows that \(Z\mapsto\mathrm{E}[\kappa ZU(\zeta,T)]\) is weakly lower semicontinuous on the weakly compact set \(\{Z^{\mathbb{Q}} _{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\}\). Next, \(\zeta^{*}\mapsto \langle\zeta ^{*},\zeta \rangle\), \(\zeta\in L^{\infty}\), is trivially continuous with respect to the \(\textrm{weak}^{*}\) topology. Since the pointwise supremum preserves lower semicontinuity, we thus obtain joint lower semicontinuity of the mapping \((\zeta^{*},Z^{\mathbb{Q}}_{t,T})\mapsto \mathbb {V}^{\mathbb{Q}}_{\kappa}(\zeta^{*})\) with respect to the product topology on \(\mathcal{D} ^{\eta}_{t,T}\times\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal {Q}_{t,T}\}\). Combined with the assumed lower semicontinuity of the mapping \(Z\mapsto\mathrm {E}[\kappa\gamma _{t,T}(Z)]\) (see Definition 2.5), this implies the existence of a minimiser \((\bar{\zeta}^{*},\bar{Z})\) for which \(v_{\kappa}(\eta )\) is attained.
The convexity of \(v_{\kappa}(\eta)\) follows immediately from the joint convexity of the mapping \((\zeta^{*},Z)\mapsto\mathbb{V}^{Z}_{\kappa}(\zeta ^{*})+\mathrm{E}[\kappa\gamma_{t,T}(Z)]\) (cf. (6.4)), where we write \(\mathbb{V}^{Z}_{\kappa}=\mathbb{V}^{\mathbb{Q}}_{\kappa}\) for \(Z=Z^{\mathbb{Q}}_{t,T}\). In order to establish lower semicontinuity of \(v_{\kappa}(\eta)\), we take a directed set \(A\) and a net \((\eta_{\alpha})_{\alpha\in A}\) in \(L^{1}_{+}\) with \(\eta_{\alpha}\to\eta\) weakly. By the above, we can pick \((\zeta^{*}_{\alpha},Z^{*}_{\alpha})\in\mathcal{D}_{t,T}^{\eta _{\alpha}}\times\{ Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\}\) such that \(v_{\kappa}(\eta_{\alpha})= \mathbb{V}^{\mathbb{Q}_{\alpha}}_{\kappa}(\zeta ^{*}_{\alpha})+\mathrm{E}[\kappa\gamma_{t,T}(Z_{\alpha})]\). Thanks to the weak compactness of the set of conditional densities, passing to a subnet, \((\zeta^{*}_{\alpha},Z^{*}_{\alpha})\) converges in the product topology to some element \((\zeta^{*},Z)\) in the set \(\mathcal{D}_{t,T}\times\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal{Q} _{t,T}\}\). Since \(\langle\zeta^{*},\xi\rangle=\lim_{\alpha}\langle\eta _{\alpha},\xi \rangle=\langle\eta,\xi\rangle\), \(\xi\in L^{\infty}(\mathcal {F}_{t})\), it follows that \(\zeta^{*}\in\mathcal{D}_{t,T}^{\eta}\). The lower semicontinuity of \(v_{\kappa}(\eta)\) then follows from the joint lower semicontinuity of the mapping \((\zeta^{*},Z)\mapsto\mathbb{V}^{Z}_{\kappa}(\zeta^{*})+\mathrm{E}[\kappa \gamma _{t,T}(Z)]\) established above. □
In order to establish the conjugacy relations for \(u_{\kappa}\) and \(v_{\kappa}\), we first recall a result from [69]. To this end, take \(\kappa\in L^{\infty}_{+}(\mathcal{F}_{t})\) and \(\mathbb {Q}\in\mathcal{Q}_{t,T}\) and consider the auxiliary stochastic utility function \(\tilde{U}(x,T):=Z^{\mathbb{Q}}_{t,T}U(x,T)\), \(x\in\mathbb{R}\), with convex conjugate \(\tilde{V}(y,T)=Z^{\mathbb{Q}}_{t,T}V(y/Z^{\mathbb{Q}}_{t,T},T)\), \(y\ge0\). Suppose that \(\kappa\tilde{U}(x,T)\in L^{1}\), \(x\in\mathbb{R}\), and that the second part of Assumption 3.2 holds. Then we may apply Propositions A.1 and A.3 in [69] to obtain
$$ u^{\mathbb{Q}}_{\kappa}(\xi)=\inf_{\zeta^{*}\in\mathcal{D}_{t,T}} \big(\mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*})+\langle\zeta^{*},\xi \rangle \big), \qquad \xi\in L^{\infty}(\mathcal{F}_{t}). $$
(6.5)
According to (6.1), for each \(\zeta^{*}\in\mathcal {D}_{t,T}\cap L^{1}_{+}\), there exists \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\) such that \(\zeta^{*}\in\mathcal{D}_{t,T}^{\eta}\). Combined with the definitions of \(\mathbb{V}^{\mathbb{Q}}_{\kappa}\) and \(v^{\mathbb{Q}}_{\kappa}\), (6.5) hence implies
$$ u^{\mathbb{Q}}_{\kappa}(\xi)=\inf_{\eta\in L^{1}_{+}(\mathcal{F}_{t}) }\big(v^{\mathbb{Q}}_{\kappa}(\eta)+\langle\xi,\eta\rangle\big), \qquad \xi\in L^{\infty}(\mathcal{F}_{t}). $$
(6.6)
We now establish the conjugacy relations between \(u_{\kappa}\) and \(v_{\kappa}\). This result is the cornerstone in the proof below of the conditional versions in Theorem 3.3. As in previous works, see e.g. [60, 64, 65], we use a minimax theorem in order to reformulate the robust problem as the infimum over a class of non-robust criteria. We then apply duality to each of the inner maximisation problems. Unlike Schied [64], who used the EUM duality results of Kramkov and Schachermayer [45], we apply the relation (6.6) to suitably defined stochastic utility fields considered under the fixed reference measure. This is of technical as well as conceptual importance and makes key use of Assumption 3.2.
Proposition 6.2
Suppose that Assumption 3.2
holds and let
\(\kappa\in L^{\infty}_{+}(\mathcal{F}_{t})\). Then for all
\(\xi\in L^{\infty}(\mathcal{F}_{t})\)
and
\(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), it holds that
$$ u_{\kappa}(\xi)=\inf_{\eta\in L^{1}_{+}(\mathcal{F}_{t}) }\big(v_{\kappa}(\eta)+\langle\xi,\eta\rangle\big) \qquad \textrm{and}\qquad v_{\kappa}(\eta)=\sup_{\xi\in L^{\infty}(\mathcal{F}_{t})}\big(u_{\kappa}(\xi )-\langle \xi,\eta\rangle\big). $$
Proof
By exploiting properties (i) and (ii) of Definition 2.5 and the same arguments as used in the proof of Proposition 6.1 to establish lower semicontinuity of the function in (6.4), we obtain that for \(\xi\in L^{\infty}(\mathcal{F}_{t})\),
$$ Z\mapsto \mathrm{E}\big[\kappa\big(Z U(\xi+g,T)+ \gamma_{t,T}(Z)\big)\big], \qquad g\in\mathcal{C}_{t,T}, $$
is convex and weakly lower semicontinuous on the convex and weakly compact set \(\{Z^{\mathbb{Q}}_{t,T}:\mathbb{Q}\in\mathcal{Q}_{t,T}\}\). Moreover, \(g\mapsto\mathrm{E}[\kappa Z^{\mathbb{Q}}_{t,T}U(\xi+g)]\), \(\mathbb{Q}\in\mathcal{Q} _{t,T}\), is concave on the convex set \(\mathcal{C}_{t,T}\). Hence the assumptions of [24, Thm. 2] are satisfied, and applying that result yields
$$\begin{aligned} u_{\kappa}(\xi) =& \sup_{g\in\mathcal{C}_{t,T}}\inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}} \mathrm{E}\big[\kappa\big(Z^{\mathbb{Q}}_{t,T}U(\xi+g,T)+\gamma _{t,T}(\mathbb{Q})\big)\big] \\ =&\inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\sup_{g\in\mathcal{C}_{t,T}} \mathrm{E}\big[\kappa\big(Z^{\mathbb{Q}}_{t,T}U(\xi+g,T)+\gamma _{t,T}(\mathbb{Q})\big)\big] \\ =& \inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\big(u^{\mathbb{Q}}_{\kappa}(\xi)+\mathrm{E}[\kappa \gamma _{t,T}(\mathbb{Q})]\big), \end{aligned}$$
(6.7)
where the last equality follows directly from the definition of \(u^{\mathbb{Q}} _{\kappa}\).
Next, note that due to concavity, if \(U(x_{0},T)\in L^{1}\) for some \(x_{0}\in \mathbb{R}\), then \(U(x,T)\in L^{1}\) for all \(x\in\mathbb{R}\). Now, using the convention \(\inf{\emptyset}=\infty\), without loss of generality, we may replace the set \(\mathcal{Q}_{t,T}\) in (6.7) by
$$ \mathcal{Q}_{t,T}^{\kappa}:=\{\mathbb{Q}\in\mathcal{Q}_{t,T}: \kappa Z_{t,T}^{\mathbb{Q}}U(x,T)\in L^{1}, \; x\in\mathbb{R}\}. $$
In turn, by Assumption 3.2 and the discussion preceding this proof, for each \(\mathbb{Q}\in\mathcal{Q}_{t,T}^{\kappa}\), the conjugacy relation (6.6) applies and we obtain
$$\begin{aligned} u_{\kappa}(\xi) =& \inf_{\mathbb{Q}\in\mathcal{Q}^{\kappa}_{t,T}}\bigg(\inf_{\eta\in L^{1}_{+}(\mathcal{F} _{t})}\big(v^{\mathbb{Q}}_{\kappa}(\eta)+\langle\xi,\eta\rangle\big)+\mathrm {E}[\kappa \gamma _{t,T}(\mathbb{Q})]\bigg)\\ =& \inf_{\eta\in L^{1}_{+}(\mathcal{F}_{t})}\bigg(\inf_{\mathbb{Q}\in \mathcal{Q}^{\kappa}_{t,T}}\big(v^{\mathbb{Q}}_{\kappa}(\eta)+\mathrm{E}[\kappa\gamma_{t,T}(\mathbb {Q})]\big)+\langle \xi,\eta \rangle\bigg)\\ =& \inf_{\eta\in L^{1}_{+}(\mathcal{F}_{t})}\big(v_{\kappa}(\eta)+\langle \xi,\eta \rangle \big); \end{aligned}$$
indeed, to see the last equality, recall that \(v_{\kappa}^{\mathbb {Q}}(\eta)\), \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), is given by (6.3) with \(\mathbb{V}_{\kappa}^{\mathbb{Q}}\) admitting the representation (6.4), which implies that \(\mathcal{Q}_{t,T}^{\kappa}\) may be replaced by \(\mathcal{Q}_{t,T}\) in the second line above.
To establish that \(v_{\kappa}\) is also the convex conjugate of \(u_{\kappa}\), it now suffices to argue that \(v_{\kappa}\) is convex and weakly lower semicontinuous, which follows from Proposition 6.1. □
Proof of Theorem 3.3 and Proposition 3.4
We are now ready to prove the main results of Sect. 3.1. Our setting is dynamic, which in this generality appears novel even in the context of the classical robust EUM; compare e.g. Schied [64]. We proceed by reducing the conditional formulations to the auxiliary problem studied in Sect. 6.1.1. This is done with the help of the following lemma which uses crucially that our penalty functions satisfy condition (2.11). For \(\kappa\in L^{\infty}_{+}(\mathcal{F}_{t})\) and \(\xi\in L^{\infty}(\mathcal{F}_{t})\), we define
$$ J_{\kappa,\xi}(\mathbb{Q},g):=\kappa\mathrm{E}[Z_{t,T}^{\mathbb{Q}}U(\xi+g,T)|\mathcal{F} _{t}]+\kappa \gamma_{t,T}(\mathbb{Q}), \qquad g\in\mathcal{C}_{t,T},\mathbb{Q}\in\mathcal{Q}_{t,T}. $$
Lemma 6.3
Suppose that Assumption 3.2
holds. Given
\(\kappa\in L^{\infty}_{+}(\mathcal{F}_{t})\), \(\xi\in L^{\infty}(\mathcal{F}_{t})\)
and
\(g\in\mathcal{C}_{t,T}\), it then holds that
$$\begin{aligned} \mathrm{E}\bigg[\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,T}}J_{\kappa ,\xi}(\mathbb{Q},g)\bigg] =\inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\mathrm{E}[J_{\kappa,\xi }(\mathbb{Q},g)]. \end{aligned}$$
Proof
The inequality “≤” is trivial. To show “≥”, define \(J(\mathbb{Q} ):=J_{\kappa,\xi}(\mathbb{Q},g)\) for \(\mathbb{Q}\in\mathcal {Q}_{t,T}\). It suffices to argue that the set \(\{J(\mathbb{Q}):\mathbb{Q}\in \mathcal{Q}_{t,T}\}\) is downward directed because by Neveu [56, Proposition VI.1.1], there is then a sequence \((\mathbb{Q}_{n})\subseteq{\mathcal{Q}_{t,T}}\) such that \((J(\mathbb{Q}_{n}))\) decreases to \(\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in \mathcal{Q} _{t,T}}J(\mathbb{Q} )\). The result then follows by using monotone convergence. To argue the directedness, let \(\mathbb{Q}_{1}\), \(\mathbb {Q}_{2}\in\mathcal{Q}_{t,T}\), define the set \(A:=\{J(\mathbb{Q}_{1})\le J(\mathbb{Q}_{2})\}\in \mathcal{F}_{t}\) and let the measure \(\bar{\mathbb{Q}}\) be given by
Using property (2.11), we have 
. So \(\bar{\mathbb{Q}}\in\mathcal{Q}_{t,T}\) and \(J(\bar{\mathbb {Q}})=\min\{J(\mathbb{Q} _{1}),J(\mathbb{Q} _{2})\}\) a.s. In consequence, the set \(\{J(\mathbb{Q}):\mathbb{Q}\in \mathcal{Q}_{t,T}\}\) is closed under minimisation and thus downward directed. □
First, we establish the existence of a dual optimiser.
Proof of Proposition 3.4
Recall that \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\) is fixed and define
$$ \kappa:=\big(\max\{1,v(\eta;t,T)\}\big)^{-1}\in L^{\infty}(\mathcal{F}_{t}). $$
Note that \(\kappa\) has values in \([0,1]\) and without loss of generality, we may assume that \(\{\kappa>0\}\neq\emptyset\). Further, we have \(v_{\kappa}(\kappa\eta)<\infty\); indeed, by using Proposition 6.2, Lemma 6.3 and the weak duality between \(u(\, \cdot\,;t,T)\) and \(v(\, \cdot\,;t,T)\) (cf. (6.10) below), we obtain
$$\begin{aligned} v_{\kappa}(\kappa\eta) &= \sup_{\xi\in L^{\infty}(\mathcal{F}_{t})}\big(u_{\kappa}(\xi)-\langle \xi ,\kappa\eta \rangle\big)\\ &\le \sup_{\xi\in L^{\infty}(\mathcal{F}_{t})}\mathrm{E}[\kappa u(\xi ;t,T)-\xi\kappa\eta] \le \mathrm{E}[\kappa v(\eta;t,T)] \le1. \end{aligned}$$
According to Proposition 6.1, \(v_{\kappa}(\kappa\eta)\) is attained for some pair \((\bar{\zeta}^{*},\bar{\mathbb{Q}})\in \mathcal{D} _{t,T}^{\kappa\eta}\times\mathcal{Q}_{t,T}\). Further, since \(v_{\kappa}(\kappa\eta)<\infty\), we have \(\mathbb {V}_{\kappa}^{\bar{\mathbb{Q}}}(\bar{\zeta}^{*})<\infty\) and thus it follows from (6.2) that \(\bar{\zeta}^{*}\in L^{1}_{+}\). So \(\bar{\zeta}^{*}\in \mathcal{D} ^{\kappa \eta}_{t,T}\cap L^{1}_{+}\), and according to (6.1), there exists \(\bar{Z}\in\mathcal{Z}^{a}_{T}\) such that \(\bar{\zeta}^{*}=\kappa \eta \bar{Z}_{t,T}\). We now argue that the pair \((\bar{Z},\bar{\mathbb{Q}})\) attains the essential infimum in (3.2). By way of contradiction, suppose that there exist \(\varepsilon>0\), a pair \((Z',\mathbb{Q}')\in\mathcal{Z}^{a}_{T}\times\mathcal{Q}_{t,T}\) and a set \(B\in \mathcal{F}_{t}\) with \(\mathbb{P}[B]>0\) such that on \(B\),
$$ \mathrm{E}^{\mathbb{Q}'}\bigg[V\bigg(\eta\frac {Z_{t,T}'}{Z_{t,T}^{\mathbb{Q}'}},T\bigg)\bigg|\mathcal{F} _{t}\bigg] +\gamma_{t,T}(\mathbb{Q}')+\varepsilon < \mathrm{E}^{\bar{\mathbb{Q}}}\bigg[V\bigg(\eta\frac{\bar{Z}_{t,T}}{Z_{t,T}^{\bar {\mathbb{Q}} }},T\bigg)\bigg|\mathcal{F}_{t}\bigg] +\gamma_{t,T}(\bar{\mathbb{Q}}). $$
(6.8)
Define now \(\tilde{\zeta}^{*}\in L^{1}_{+}\) via 
; then \(\tilde{\zeta}^{*}\in\mathcal {D}^{\kappa\eta}_{t,T}\). Similarly, define \(\tilde{\mathbb{Q}}\in\mathcal{Q}_{t,T}\) via 
. Multiplying (6.8) by \(\kappa\), taking expectations on both sides – noticing that \(B\subseteq\{\kappa>0\}\) – and applying property (2.11), we then obtain
$$ \mathbb{V}_{\kappa}^{\tilde{\mathbb{Q}}}(\tilde{\zeta}^{*})+\mathrm {E}[\kappa\gamma _{t,T}(\tilde{\mathbb{Q}})] -\varepsilon\mathbb{P}[B] \le \mathbb{V}_{\kappa}^{\bar{\mathbb{Q}}}(\bar{\zeta}^{*})+\mathrm {E}[\kappa\gamma _{t,T}(\bar {\mathbb{Q}})], $$
which contradicts the choice of \((\bar{\zeta}^{*},\bar{\mathbb{Q}})\) as the minimiser. □
We now turn to Theorem 3.3. The proof proceeds by assuming that the conditional conjugacy relations do not hold; taking expectations and applying Proposition 6.2 and Lemma 6.3, we then obtain a contradiction which allows us to conclude.
Proof of Theorem 3.3
First, we consider assertion (3.4). In order to verify that the (weak) inequality “≤” holds, note that we trivially have the inequality
$$ u(\xi;t,T)\le\mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\bigg(\mathop{\mathrm{ess}\;\mathrm{sup}}_{g\in\mathcal{C}_{t,T} }\mathrm{E}^{\mathbb{Q}} [U(\xi+g,T)|\mathcal{F}_{t}]+\gamma_{t,T}(\mathbb{Q})\bigg). $$
(6.9)
Since \(\mathrm{E}^{\mathbb{Q}}[g]\le0\) for all \(\mathbb{Q}\in \mathcal{M}^{a}_{T}\), \(g\in \mathcal{C}_{t,T}\) and \(U(x,T)\le V(y,T)+xy\) a.s. for all \(x\in\mathbb{R}\), \(y\ge0\), it follows immediately from (6.9) that for all \(\eta\in L^{0}_{+}(\mathcal{F}_{t})\),
$$\begin{aligned} u(\xi;t,T) \le& \mathop{\mathrm{ess}\;\mathrm{inf}}_{\mathbb{Q}\in\mathcal{Q}_{t,T}} \bigg(\mathop{\mathrm{ess}\;\mathrm{inf}}_{Z\in\mathcal{Z}^{a}_{T}}\mathrm{E}^{\mathbb{Q}}[V(\eta Z_{t,T}/Z^{\mathbb{Q}}_{t,T},T)|\mathcal{F} _{t}]+\xi\eta +\gamma_{t,T}(\mathbb{Q})\bigg) \\ =&v(\eta;t,T)+\xi\eta. \end{aligned}$$
(6.10)
Next, we argue that the inequality “≥” holds in (3.4) with the infimum on the right-hand side taken over \(L^{1}_{+}(\mathcal{F}_{t})\); since \(L^{1}_{+}(\mathcal{F}_{t})\subseteq L^{0}_{+}(\mathcal{F}_{t})\), this trivially yields the claim. So assume to the contrary that there exist \(\xi\in L^{\infty}(\mathcal{F}_{t})\), \(\varepsilon>0\) and \(A\in\mathcal{F}_{t}\) with \(\mathbb{P}[A]>0\) such that
for all \(g\in\mathcal{K}_{t,T}\), \(Z\in\mathcal{Z}^{a}_{T}\), \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and \(\eta\in L^{1}_{+}(\mathcal {F}_{t})\). Observe that \(u(\xi ;t,T)<\infty\) a.s. on \(A\) and without loss of generality, we may assume that there is \(M<\infty\) such that \(u(\xi;t,T)\le M\) a.s. on \(A\). Multiplying the latter inequality by 
, taking expectations on both sides and applying Lemma 6.3, we then obtain
$$\begin{aligned} &\inf_{\mathbb{Q}\in\mathcal{Q}_{t,T}}\mathrm{E}\big[\kappa\big(Z^{\mathbb{Q}}_{t,T}U(\xi +g,T)+\gamma _{t,T}(\mathbb{Q})\big)\big]+\varepsilon \mathbb{P}[A] \\ &\le\mathrm{E}\bigg[\kappa Z^{\mathbb{Q}}_{t,T}V\bigg(\frac{\eta }{\kappa}\frac{Z_{t,T} }{Z_{t,T} ^{\mathbb{Q}}},T\bigg)\bigg]+\mathrm{E}[\kappa\gamma_{t,T}(\mathbb {Q})]+\mathrm{E}\left [\kappa\xi\eta \right] \end{aligned}$$
for any \(\eta\in L_{+}^{1}\) and \(Z\in\mathcal{Z}^{a}_{T}\) such that \(\{\eta Z_{t,T}>0\} \subseteq A\). According to (6.1), we have that for every \(\zeta^{*}\in \mathcal{D}_{t,T}^{\eta}\cap L^{1}_{+}\) with \(\eta\in L^{1}_{+}(\mathcal {F}_{t})\), there exists \(Z\in\mathcal{Z}^{a}_{T}\) such that \(\zeta^{*}=\eta Z_{t,T}\). Using this and taking the supremum over \(g\in\mathcal{K}_{t,T}\), we deduce that
$$ u_{\kappa}(\xi)+\varepsilon \mathbb{P}[A] \le\mathbb{V}^{\mathbb{Q}}_{\kappa}(\zeta^{*})+\mathrm{E}[\kappa\gamma _{t,T}(\mathbb{Q} )]+\langle\xi,\eta\rangle $$
for all \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and \(\zeta^{*}\in\mathcal{D}^{\eta}_{t,T}\cap L^{1}_{+}\) with \(\{\zeta^{*}>0\}\subseteq A\). Therefore, for any \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\) and \(\mathbb{Q}\in \mathcal{Q}_{t,T}\), the above inequality holds for all \(\zeta^{*}\in\mathcal{D}^{\eta}_{t,T}\). Indeed, if \(\zeta^{*}\notin L^{1}_{+}\) or \(\{\zeta^{*}>0\}\nsubseteq A\), then it holds that \(\mathbb {V}^{\mathbb{Q}} _{\kappa}(\zeta^{*})=\infty\) (cf. (6.2)). Hence,
$$ u_{\kappa}(\xi)+\varepsilon \mathbb{P}[A] \le v^{\mathbb{Q}}_{\kappa}(\eta)+\mathrm{E}[\kappa\gamma _{t,T}(\mathbb{Q})]+\langle\xi ,\eta \rangle $$
for all \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\) and \(\mathbb{Q}\in\mathcal {Q}_{t,T}\). In turn, since \(u_{\kappa}(\xi)\le M<\infty\) due to the above choice of \(\kappa\), we obtain
$$ u_{\kappa}(\xi)< u_{\kappa}(\xi)+\varepsilon \mathbb{P}[A] \le\inf_{\eta\in L^{1}(\mathcal{F}_{t})}\big(v_{\kappa}(\eta )+\langle \xi,\eta\rangle\big), $$
which according to Proposition 6.2 yields the required contradiction.
Next, we turn to relation (3.5). Note that assertion (3.4) implies that for any \(\eta\in L^{0}_{+}(\mathcal{F}_{t})\) and \(\xi\in L^{\infty}(\mathcal{F}_{t})\), we have \(v(\eta;t,T)\ge u(\xi ;t,T)-\xi\eta \). Hence the inequality “≥” follows directly. For \(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), the reverse inequality follows by similar arguments as above, specifically, by arguing by contradiction and applying Lemma 6.3 and Proposition 6.2. In turn, for \(\eta\in L^{0}_{+}(\mathcal{F}_{t})\) and \(A\in\mathcal{F}_{t}\),
for any \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and \(Z\in\mathcal{Z}^{a}_{T}\); it follows from the definition of \(v(\, \cdot\,;t,T)\) that 
a.s. For an arbitrary \(\eta\in L^{0}_{+}(\mathcal{F}_{t})\), we may then define \(A^{n}:=\{ \eta \le n\}\) and 
, \(n\in\mathbb{N}\). By using the identity 
and applying (3.5) to \(\eta^{n}\), we then obtain that (3.5) holds for \(\eta\) on \(A^{n}\) for any \(n\in\mathbb{N}\). Since \(\eta\) takes finite values a.s., we thus obtain that (3.5) holds a.s. □
Proof of Propositions 4.2, 4.4 and 4.5
In order to prove the results in Sect. 4, we first establish two lemmas. Throughout this section, we write \(\gamma_{0,t}(\mathbb{Q}) := \gamma_{0,t}(\mathbb{Q}|_{\mathcal {F}_{t}})\) for \(\mathbb{Q}\in \mathcal{Q}_{0,T}\).
Lemma 6.4
Let
\((U,\gamma)\)
be a pair of a utility random field and an admissible family of penalty functions with associated dual field
\(V\). Given
\(T>0\), let
\(v(x;t,T)\)
be the corresponding dual value field. Suppose that the infimum in (3.2) is attained for any
\(t\le T\)
and
\(\eta\in L^{1}_{+}(\mathcal{F}_{t})\), and that either Assumption 4.1
holds or (4.1) holds and
\(v^{-}(\zeta;t,T)\in L^{1}(\mathcal {F}_{t};\mathbb{Q})\)
for
\(\zeta\in L^{0}(\mathcal{F}_{t})\)
and
\(\mathbb{Q}\in\tilde{\mathcal {Q}}_{0,T}\), \(t\le T\). Then the pair
\((v,\gamma)\)
is dynamically consistent on the interval
\([0,T]\).
Proof
Let \(0\le s< t< T<\infty\) and take \(\eta\in L^{1}_{+}(\mathcal{F}_{s})\), \(Z\in\mathcal{Z}^{a}_{t}\) and \(\mathbb{Q}\in\mathcal{Q}_{s,t}\). By using similar arguments as in the proof of Lemma 6.3, we obtain that the optimisation set in (3.3) is downward directed, and so there exists a sequence \((Z^{n},\mathbb{Q}^{n})\subseteq\mathcal {Z}^{a}_{T}\times\mathcal{Q}_{t,T}\) such that the objective function evaluated at \((Z^{n},\mathbb{Q}^{n})\), \(n\in \mathbb{N}\), decreases to \(v(\eta Z_{s,t}/Z^{\mathbb{Q}}_{s,t};t,T)\). By using monotone convergence, we then obtain
$$\begin{aligned} &\mathrm{E}^{\mathbb{Q}}\bigg[v\bigg(\eta\frac{Z_{s,t}}{Z^{\mathbb{Q}}_{s,t}};t,T\bigg)\bigg|\mathcal{F}_{s}\bigg]+\gamma_{s,t}(\mathbb{Q}) \\ &=\mathrm{E}^{\mathbb{Q}}\bigg[\lim_{n\to\infty}\bigg( \mathrm{E}\Big[Z_{t,T}^{\mathbb{Q}^{n}} V\Big(\eta\frac {Z_{s,t}}{Z^{\mathbb{Q}} _{s,t}}\frac {Z^{n}_{t,T}}{Z^{\mathbb{Q}^{n}}_{t,T}},T\Big)\Big|\mathcal{F}_{t}\Big]+\gamma _{t,T}(Z_{t,T}^{\mathbb{Q}^{n}}) \bigg)\bigg|\mathcal{F}_{s}\bigg]+\gamma_{s,t}(\mathbb{Q}) \\ &=\lim_{n\to\infty}\mathrm{E}\bigg[Z_{s,t}^{\mathbb{Q}}Z_{t,T}^{\mathbb{Q}^{n}}V\bigg(\eta \frac{Z_{s,t}Z^{n}_{t,T}}{Z^{\mathbb{Q}}_{s,t}Z^{\mathbb {Q}^{n}}_{t,T}},T\bigg)\bigg|\mathcal{F} _{s}\bigg]+\gamma_{s,T}(Z_{s,t}^{\mathbb{Q}}Z_{t,T}^{\mathbb {Q}^{n}}) \\ & \ge v(\eta;s,T), \end{aligned}$$
(6.11)
where we used that \(\mathrm{E}[Z_{t}Z^{n}_{t,T}|\mathcal{F}_{u}]\), \(u\le T\), belongs to \(\mathcal{Z}^{a}_{T}\) and that \(\bar{\mathbb{Q}}\in\mathcal{Q}_{s,T}\) for \(\frac{\mathrm {d}\bar{\mathbb{Q}} }{\mathrm {d}\mathbb{P}|_{\mathcal{F}_{T}}}=Z^{\mathbb{Q}}_{t} Z^{\mathbb{Q}^{n}}_{t,T}\). Indeed, (4.2) yields immediately that \(\bar{\mathbb{Q}}\in \mathcal{Q}_{s,T}\). For the case when (4.1) holds and \(v^{-}(\zeta;s,T)\in L^{1}(\mathcal{F} _{T};\bar{\mathbb{Q}})\) for \(\zeta\in L^{0}(\mathcal{F}_{T})\), the fact that \(v(\eta;s,t)\) is finite implies without loss of generality that \(\mathrm{E}^{\mathbb{Q}}[\gamma _{t,T}(\mathbb{Q}^{n})|\mathcal{F}_{s}]<\infty\), and thus \(\bar{\mathbb {Q}}\in\mathcal{Q}_{s,T}\).
Next, let \(Z\in\mathcal{Z}^{a}_{T}\) and \(\mathbb{Q}\in\mathcal{Q}_{s,T}\) be optimal objects for which the infimum in \(v(\eta;s,T)\) is attained. From (4.1) we deduce that \(\mathbb{Q}\in\mathcal{Q}_{t,T}\) and \(\mathbb{Q}|_{\mathcal{F} _{t}}\in\mathcal{Q} _{s,t}\). It follows that
$$\begin{aligned} v(\eta;s,T) &= \mathrm{E}\bigg[Z_{s,T}^{\mathbb{Q}}V\bigg(\eta\frac {Z_{s,T}}{Z^{\mathbb{Q} }_{s,T}},T\bigg)\bigg|\mathcal{F}_{s}\bigg]+\gamma _{s,T}(Z_{s,T}^{\mathbb{Q}}) \\ &=\mathrm{E}\bigg[Z_{s,t}^{\mathbb{Q}}\bigg(\mathrm{E}\Big[Z_{t,T}^{\mathbb{Q}} V\Big(\eta \frac {Z_{s,t}}{Z^{\mathbb{Q}}_{s,t}}\frac{Z_{t,T}}{Z^{\mathbb {Q}}_{t,T}},T\Big)\Big|\mathcal{F} _{t}\Big]+\gamma_{t,T}(Z_{t,T}^{\mathbb{Q}}) \bigg)\bigg|\mathcal{F}_{s}\bigg]+\gamma_{s,t}(\mathbb{Q}) \\ &\ge\mathrm{E}\bigg[Z_{s,t}^{\mathbb{Q}}v\bigg(\eta\frac {Z_{s,t}}{Z^{\mathbb{Q}} _{s,t}};t,T\bigg)\bigg|\mathcal{F}_{s}\bigg]+\gamma_{s,t}(\mathbb {Q}) \\ &\ge v(\eta;s,T), \end{aligned}$$
(6.12)
where the last inequality is due to (6.11). Hence equality must hold throughout. Finally, the fact that property (3.2) must hold also for \(\eta \in L^{0}_{+}(\mathcal{F}_{s})\) follows by the same arguments as used at the end of the proof of Theorem 3.3. □
Lemma 6.5
Let
\((U,\gamma)\)
be a pair of a utility random field and an admissible family of penalty functions satisfying (4.1). Let
\(V\)
be the associated dual field, and suppose that the infimum in (3.2) is attained for
\(t\le T<\infty\)
and
\(\eta\in L^{1}_{+}(\mathcal{F}_{t})\). Then the following two statements are equivalent:
-
(i)
The pair
\((V,\gamma)\)
satisfies (3.2) for all
\(t\le T<\infty\)
and
\(\eta\in L^{0}_{+}(\mathcal{F}_{t})\).
-
(ii)
For any
\(s>0\)
and
\(\eta\in L^{1}_{+}(\mathcal{F}_{s})\), it holds for all
\(s\le t\le T<\infty\)
that
$$ V(\eta Z_{s,t}/Z^{\mathbb{Q}}_{s,t},t) \le\mathrm{E}^{\mathbb{Q}}[V(\eta Z_{s,T}/Z^{\mathbb{Q}}_{s,T},T)|\mathcal{F}_{t}]+\gamma_{t,T}(\mathbb{Q} ) $$
(6.13)
for all
\(\mathbb{Q}\in\mathcal{Q}_{t,T}\)
and
\(Z\in\mathcal {Z}^{a}_{T}\); moreover, for any
\(\bar{T}>s\), there are
\(\bar{\mathbb{Q}}\in\mathcal{Q}_{s,\bar{T}}\)
and
\(\bar{Z}\in\mathcal{Z} ^{a}_{\bar{T}}\)
such that (6.13) holds with equality for all
\(s\le t\le T\le \bar{T}\).
Furthermore, if either (a) \(\mathcal{Q}_{0,T}=\tilde{\mathcal {Q}}_{0,T}\), \(T>0\), orFootnote 9 (b) for any
\(T>0\)
and
\(\zeta \in L^{0}(\mathcal{F}_{T})\), we have
\(V^{-}(\zeta,T)\in L^{1}(\mathcal {F}_{T};\mathbb{Q})\)
for all
\(\mathbb{Q} \in \tilde{\mathcal{Q}}_{0,T}\), then (i) and (ii) are equivalent to the following condition:
-
(iii)
For any
\(s>0\)
and
\(\eta\in L^{1}_{+}(\mathcal{F}_{s})\), for all
\(s\le t\le T<\infty\), (6.13) holds for all
\(\mathbb{Q}\in \mathcal{Q}_{t,T}\)
and
\(Z\in\mathcal{Z}^{a}_{T}\); moreover, there exist a process
\(Z\in \mathcal{Z}^{a}\)
and a sequence of measures
\(\mathbb{Q}^{i}\in\mathcal{Q}_{s,T_{i}}\), \(i\in \mathbb{N}\), \(T_{i+1}-T_{i}>i\)
and
\(T_{1}>s\), with
\(\mathbb{Q}^{i}=\mathbb {Q}^{i+1}|_{\mathcal{F}_{T_{i}}}\)
and such that for all
\(s\le t\le T<\infty\), (6.13) holds with equality for
\((Z,\mathbb{Q}_{T})\), where
\(\mathbb{Q}_{T}:=\mathbb {Q}^{i}|_{\mathcal{F}_{T}}\in\mathcal{Q}_{s,T}\), \(T\le T_{i}\).
Proof
Footnote 10 In order to argue that (ii) implies (i), note that an application of (6.13) with \(s\equiv t\) immediately yields that the pair \((V,\gamma)\) satisfies (3.2) for all \(t\le T<\infty\) and \(\eta \in L^{1}_{+}(\mathcal{F}_{t})\); the extension to \(\eta\in L^{0}_{+}(\mathcal {F}_{t})\) then follows by the same arguments as in the proof of Theorem 3.3.
To show that (i) implies (ii), let \(s>0\) and \(\eta\in L^{1}_{+}(\mathcal{F}_{s})\). For \(s\le t\le T<\infty\), \(Z\in\mathcal{Z}^{a}_{T}\) and \(\mathbb{Q}\in \mathcal{Q}_{t,T}\), applying (3.2) with \(\eta\) replaced by \(\eta Z_{s,t}/Z^{\mathbb{Q}}_{s,t}\) then yields
$$\begin{aligned} V(\eta Z_{s,t}/Z^{\mathbb{Q}}_{s,t},t) =& \mathop{\mathrm{ess}\;\mathrm{inf}}_{\tilde{\mathbb{Q}}\in\mathcal{Q}_{t,T}}\mathop{\mathrm{ess}\;\mathrm{inf}}_{\tilde{Z}\in\mathcal{Z}^{a}_{T}}\bigg( \mathrm{E}^{\tilde{\mathbb{Q}}}\bigg[V\bigg(\eta\frac {Z_{s,t}\tilde{Z}_{t,T}}{Z^{\mathbb{Q}} _{s,t}Z_{t,T}^{\tilde{\mathbb{Q}}}},T\bigg)\bigg|\mathcal{F}_{t}\bigg]+\gamma _{t,T}(\tilde{\mathbb{Q}} )\bigg)\\ \le&\mathrm{E}^{\mathbb{Q}}[V(\eta Z_{s,T}/Z^{\mathbb{Q}}_{s,T},T)|\mathcal{F}_{t}]+\gamma _{t,T}(\mathbb{Q}), \end{aligned}$$
which implies the inequality (6.13).
Next, let \(\bar{T}>s\) and let \(\bar{Z}\in\mathcal{Z}^{a}_{\bar{T}}\) and \(\bar{\mathbb{Q}} \in \mathcal{Q}_{s,\bar{T}}\) be the optimal objects for which \(v(\eta ;s,\bar{T})\) is attained. Since \(\bar{\mathbb{Q}}\in\mathcal{Q}_{s,\bar{T}}\), we have that \(\bar {\mathbb{Q}}|_{\mathcal{F} _{T}}\in\mathcal{Q} _{s,T}\) and \(\bar{\mathbb{Q}}\in\mathcal{Q}_{T,\bar{T}}\). In turn, using that \((V,\gamma)\) is self-generating (cf. (3.2)) and performing a calculation similar to the one in (6.12) (which then holds with equalities throughout), we obtain that \(v(\eta;s,T)\) is attained for \(\bar{Z}_{s,T}\) and \(\bar{\mathbb{Q}}|_{\mathcal{F}_{T}}\), when \(T\le\bar{T}\). We now claim that for \(s\le t\le T\le\bar{T}\), (6.13) holds as equality for \(\bar{Z}\) and \(\bar{\mathbb{Q}}\). Indeed, suppose contrary to the claim that there exist \(\varepsilon>0\) and \(A\in\mathcal{F}_{t}\) with \(\mathbb{P} [A]>0\) such that
Taking expectations under \(\bar{\mathbb{Q}}\) and using (4.1) combined with the fact that \(v(\eta;s,t)\) and \(v(\eta;s,T)\) are attained by \((\bar{Z}_{s,t},\bar {\mathbb{Q}} |_{\mathcal{F}_{t}})\) and \((\bar{Z}_{s,T},\bar{\mathbb{Q}}|_{\mathcal {F}_{T}})\), we then obtain a contradiction to the identity \(v(\eta;s,t)=v(\eta;s,T)\) a.s.
Assertion (iii) trivially implies (ii). Hence, it only remains to show that (i) implies (iii). To this end, let \(s< T_{1}< T_{2}\) and let \((Z^{1},\mathbb{Q}^{1})\in\mathcal{Z} ^{a}_{T_{1}}\times\mathcal{Q} _{s,T_{1}}\) be an argument for which \(v(\eta;s,T_{1})\) is attained. In turn, let \((Z^{*},\mathbb{Q}^{*})\in\mathcal{Z}^{a}_{T_{2}}\times \mathcal{Q}_{s,T_{2}}\) be an argument for which \(v(\eta;s,T_{2})\) is attained and define \(Z^{2}\) and \(\mathbb{Q}^{2}\) by
$$ \frac{\mathrm{d}\mathbb{Q}^{2}}{\mathrm{d}\mathbb{P}|_{\mathcal {F}_{T_{2}}}}:=Z^{\mathbb{Q}^{1}}_{T_{1}} Z^{\mathbb{Q}^{*}}_{T_{1},T_{2}},\qquad Z^{2}_{u}:=\mathrm{E}[Z^{1}_{T_{1}}Z^{*}_{T_{1},T_{2}}|\mathcal{F}_{u}], \quad u\le T_{2}. $$
We next show that also \((Z^{2},\mathbb{Q}^{2})\) attains the infimum in \(v(\eta ;s,T_{2})\). To this end, recall first from the proof of “(i) ⇒ (ii)” that \(v(\eta;s,T_{1})\) is attained for \((Z^{*}_{s,T_{1}},\mathbb {Q}^{*}|_{\mathcal{F}_{T_{1}}})\). Further, note that due to the strict convexity of \(V(\, \cdot\, ,t,\omega)\), \((t,\omega)\in[0,\infty)\times{\Omega}\), we have for any \(z_{0},z_{1},y_{0},y_{1}\in(0,\infty)\) that
$$\begin{aligned} \frac{z_{0}+z_{1}}{2}V\bigg(\frac{\frac{1}{2}(y_{0}+y_{1})}{\frac {1}{2}(z_{0}+z_{1})},T_{1},\omega\bigg) \le \frac{1}{2}z_{0}V\bigg(\frac{y_{0}}{z_{0}},T_{1},\omega\bigg)+\frac {1}{2}z_{1}V\bigg(\frac{y_{1}}{z_{1}},T_{1},\omega\bigg), \end{aligned}$$
and the inequality is strict whenever \(\frac{y_{0}}{z_{0}}\neq\frac {y_{1}}{z_{1}}\); see [65, Eq. (21)]. In consequence, we must have \(Z^{1}_{s,T_{1}}/Z^{\mathbb{Q}^{1}}_{s,T_{1}} = Z^{*}_{s,T_{1}}/Z^{\mathbb{Q}^{*}}_{s,T_{1}}\) a.s. Second, using that \((V,\gamma)\) is self-generating (cf. (3.2)) and the fact that \(\mathbb{Q}^{*}\sim\mathbb{P}\), performing a similar calculation as in (6.12) (which then holds with equalities throughout), we obtain that \(v(\eta Z^{*}_{s,T_{1}}/Z^{\mathbb{Q} ^{*}}_{s,T_{1}};T_{1},T_{2})\) is attained for \((Z^{*},\mathbb{Q}^{*})\). Combining the above two facts and using once again that \((V,\gamma)\) is self-generating, we obtain
$$\begin{aligned} &\mathrm{E}^{\mathbb{Q}^{2}}\bigg[ V\bigg(\eta\frac{Z^{2}_{s,T_{2}}}{Z^{\mathbb{Q}^{2}}_{s,T_{2}}},T_{2}\bigg)\bigg|\mathcal{F} _{s}\bigg]+\gamma_{s,T_{2}}(\mathbb{Q}^{2}) \\ &= \mathrm{E}^{\mathbb{Q}^{1}}\bigg[ \mathrm{E}^{\mathbb{Q}^{*}}\Big[ V\Big(\eta\frac{Z^{1}_{s,T_{1}}}{Z^{\mathbb{Q}^{1}}_{s,T_{1}}}\frac {Z^{*}_{T_{1},T_{2}}}{Z^{\mathbb{Q}^{*}}_{T_{1},T_{2}}},T_{2}\Big)\Big|\mathcal {F}_{T_{1}}\Big] +\gamma_{T_{1},T_{2}}(Z^{\mathbb{Q}^{*}}_{T_{1},T_{2}}) \bigg|\mathcal{F}_{s}\bigg] +\gamma_{s,T_{1}}(\mathbb{Q}^{1}) \\ &= \mathrm{E}^{\mathbb{Q}^{1}}\bigg[ V\bigg(\eta\frac{Z^{1}_{s,T_{1}}}{Z^{\mathbb{Q}^{1}}_{s,T_{1}}},T_{1}\bigg) \bigg|\mathcal{F}_{s}\bigg] +\gamma_{s,T_{1}}(\mathbb{Q}^{1})\\ & \ge v(\eta;s,T_{1}) = v(\eta;s,T_{2}), \end{aligned}$$
and thus \(v(\eta;s,T_{2})\) is attained for \((Z^{2},\mathbb{Q}^{2})\); the fact that \(\mathbb{Q}^{2}\in\mathcal{Q}_{0,T_{2}}\) is immediate under the full Assumption 4.1, and follows by similar arguments as in Lemma 6.4 under the assumption (b). We note that \((Z^{2},\mathbb{Q}^{2})\) was constructed so that \(Z^{1}_{T_{1}}=Z^{2}_{T_{1}}\) and \(\mathbb{Q}^{1}=\mathbb{Q}^{2}|_{\mathcal {F}_{T_{1}}}\), and that for any \(s< T< T_{2}\), \(v(\eta;s,T)\) is attained for \(Z^{2}_{s,T}\) and \(\mathbb {Q}^{2}|_{\mathcal{F}_{T}}\).
For any sequence \(s< T_{1}< T_{2}<\cdots\), a repetition of the above pasting procedure yields a process \(Z\in\mathcal{Z}^{a}\) and a sequence of measures \(\mathbb{Q} ^{i}\in\mathcal{Q}_{s,T_{i}}\), \(i\in\mathbb{N}\), with \(\mathbb {Q}^{i}=\mathbb{Q}^{i+1}|_{\mathcal{F} _{T_{i}}}\), such that for all \(T>s\), \(v(\eta;s,T)\) is attained for \((Z_{s,T},\mathbb{Q}_{T})\) with \(\mathbb{Q}_{T}:=\mathbb{Q}^{i}|_{\mathcal {F}_{T}}\in\mathcal{Q}_{s,T}\) for \(T\le T_{i}\). In turn, applying again the same arguments as used to show that (i) implies (ii), we obtain that for any \(s\le t\le T<\infty\), (6.13) holds with equality for \((Z, \mathbb{Q}_{T})\). Hence (iii) holds and we conclude. □
We now argue that the results in Sect. 4 follow from the above lemmas. First, while Theorem 3.3, Proposition 3.4 and Lemma 6.4 readily yield Proposition 4.2, Proposition 4.5 follows from combining Theorem 3.3 and Proposition 3.4 with Lemma 6.5.
Next, we establish Proposition 4.4. To this end, without loss of generality, let \(t=0\) and \(x\in\mathbb{R}\). Recall that \(u(\, \cdot\,;0,T)\) and \(v(\, \cdot\,;0,T)\) satisfy the conjugacy relations (see Theorem 3.3) and let \(y^{*}>0\) be the value for which the infimum in (3.4) is attained; \(y^{*}\) is independent of \(T\) since \(u(x;0,T)=U(x,0)\), \(T\ge0\). By the same arguments as in the proof of Lemma 6.5 (cf. “(i) ⇒ (iii)”), it follows that there exist \(Z\in\mathcal {Z}^{a}\) and a positive martingale \(Y_{t}\), \(t\ge0\), such that for \(T\ge0\), \(\mathbb{Q} _{T}\in \mathcal{Q}_{0,T}\) with \(\frac{\mathrm{d}\mathbb{Q}_{T}}{\mathrm {d}\mathbb{P}|_{\mathcal{F}_{T}}}:=Y_{T}\), and \(v(y^{*};0,T)\) is attained for \(Z_{T}\) and \(\mathbb{Q}_{T}\). Due to the conjugacy relations and the existence of a saddle point, it follows (see e.g. the proof of Theorem 2.6 in [64]) that
$$ u(x;0,T)~=~ \sup_{\pi\in\mathcal{A}} \mathrm{E}^{\mathbb{Q}_{T}}\bigg[U\bigg(x+\int_{0}^{T}\pi_{s}dS_{s},T\bigg)\bigg] +\gamma_{0,T}(\mathbb{Q}_{T}), \qquad T>0, $$
and that the supremum on the right-hand side is attained for
$$ \bar{X}_{T}~=~ -V'(y^{*} Z_{T}/Y_{T},T),\qquad T>0. $$
The latter implies that \(\bar{X}_{T}=x-\int_{0}^{T}dF_{t}\) with \(F_{t}:=V'(y^{*}Z_{t}/Y_{t},t)\). In consequence, \(\bar{\pi}^{0,T}_{0}=\bar{\pi}^{0,\bar{T}}_{0}\), \(0\le T\le\bar{T}\). To argue that \(\pi^{t,T}_{u}(\xi)=\pi^{u,T}_{u}(\xi+\int_{t}^{u} \pi^{t,T}_{s} dS_{s})\), \(t\le u\le T\), assume contrary to the claim that there exist \(\varepsilon>0\) and \(A\in\mathcal{F}_{u}\) with \(P[A] > 0\) such that
Taking expectations under \(\mathbb{Q}_{u}\), using that \((U,\gamma)\) satisfies (2.12) and that (4.1) holds then yields
$$\begin{aligned} &\mathrm{E}^{\mathbb{Q}_{T}} \bigg[U\bigg(x+\int_{0}^{T}\bar{\pi}^{0,T}_{s}dS_{s},T\bigg)\bigg]+\gamma _{0,T}(\mathbb{Q}_{T}) \\ &< \mathrm{E}^{\mathbb{Q}_{u}} \bigg[U\bigg(x+\int_{0}^{u}\bar{\pi}^{0,T}_{s}dS_{s},u\bigg)\bigg]+\gamma _{0,u}(\mathbb{Q}_{u}), \end{aligned}$$
which gives the contradiction \(u(x;0,T)< u(x;0,u)\). Similarly, assuming the reverse strict inequality in (6.14) also gives a contradiction and we conclude. □
Proof of Propositions 2.1 and 2.2
Proof of Proposition 2.1
Let \(0\leq t\leq T<\infty\) be fixed. Throughout the proof, we write \(\hat{W}_{s}=\hat{W}^{1}_{s}\). To alleviate the notation, let \(L_{s}=\int_{0}^{s} \hat{\lambda}_{u} d\hat{W}_{u}\) and \(M_{s}=\int_{0}^{s} \frac{\hat{\lambda}_{u}}{1+\delta_{u}} d\hat{W}_{u}\). Recall that \(\hat{\mathrm{E}}[\mathrm{e}^{\kappa\langle L\rangle_{T}}]<\infty\), \(\kappa>1/2\). Take \(p,\tilde{p}>1\) such that \(p^{2}\tilde{p}^{2} \leq2 \kappa\) and, with \(\frac{1}{p}+\frac{1}{q}=1=\frac{1}{\tilde{p}}+\frac{1}{\tilde{q}}\), such that \(\tilde{q}(\frac{p^{2}\tilde{p}}{2}-\frac{p}{2})=\frac{p\tilde{p}(p\tilde{p}-1)}{2(\tilde{p} -1)}\leq\kappa\). We then have
$$\begin{aligned} \mathrm{E}^{{\bar{\eta}}}\bigg[\int_{0}^{T}\hat{\lambda}_{s}^{2}ds\bigg] & = \hat{\mathrm{E}}[D^{\bar{\eta}}_{T} \langle L\rangle_{T}] \\ &\leq\hat{\mathrm{E}}[(D_{T}^{\bar{\eta}})^{p}]^{\frac{1}{p}}\hat{\mathrm {E}}[\langle L\rangle _{T}^{q}]^{\frac{1}{q}}\\ &\leq\hat{\mathrm{E}}\big[\mathrm{e}^{-p\tilde{p}M_{T}-\frac{p^{2}\tilde{p}^{2}}{2}\langle M\rangle_{T}}\big]^{\frac{1}{p\tilde{p}}} \hat{\mathrm{E}}\big[ \mathrm{e}^{\kappa\langle M\rangle_{T}}\big]^{\frac {1}{p\tilde{q}}} \hat{\mathrm{E}}[\langle L\rangle_{T}^{q}]^{\frac{1}{q}}, \end{aligned}$$
which is finite. More precisely, out of the three factors, the first is equal to one and the other two are finite, as is easily seen using Novikov’s condition, the fact that \(\langle M\rangle_{T}\le\langle L\rangle_{T}\) and the assumed integrability of \(\langle L\rangle_{T}\). It follows that \(\gamma_{t,T}(\mathbb{Q}^{\bar{\eta}})<\infty\) and hence \(\mathbb{Q} ^{\bar{\eta}}\in\mathcal{Q}_{t,T}\). Next, let
$$ N^{\pi,\eta}_{u}:=U(X^{\pi}_{u},u)+\int_{t}^{u}\frac{\delta_{s}}{2}|\eta_{s}|^{2}ds, \qquad u\ge t. $$
It then suffices to show that we have \(\mathrm{E}^{\bar{\eta}}[N^{\pi ,\bar{\eta}}_{T}|\mathcal{F}_{t}]\le N^{\pi,\bar{\eta}}_{t}\) for all \(\pi\in\mathcal {A}^{x}_{t}\), and that \(\mathrm{E}^{\eta}[N^{\bar{\pi},\eta}_{T}|\mathcal{F}_{t}]\ge N^{\bar{\pi},\eta}_{t}\) for all \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{t,T}\). For simplicity, and without loss of generality, we establish the claim for \(t=0\). For \(\pi\in\mathcal {A}^{x}_{0}\), the wealth process then satisfies
$$ dX^{\pi}_{s}=\pi_{s}\sigma_{s}S_{s}\big((\hat{\lambda}_{s}+\eta^{1}_{s})ds+dW^{\eta}_{s}\big), \qquad s\le T, X^{\pi}_{0}=x, $$
(6.15)
where \(W^{\eta}\) is a Brownian motion under \(\mathbb{Q}^{\eta}\). Due to the form of \(U\) and \(\bar{\pi}\), a straightforward application of Itô’s lemma yields
$$\begin{aligned} dN^{\bar{\pi},\eta}_{s} =&\frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s}\big((\hat{\lambda}_{s}+\eta ^{1}_{s})ds+dW^{\eta}_{s}\big) -\frac{1}{2}\left(\frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s}\right )^{2}ds\\ & -\frac{1}{2}\frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s}^{2}ds +\frac{\delta_{s}}{2}\big((\eta^{1}_{s})^{2}+(\eta^{2}_{s})^{2}\big)ds\\ =& \frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s}\eta^{1}_{s} ds +\frac{1}{2}\frac{\delta_{s}}{(1+\delta_{s})^{2}}\hat{\lambda}_{s}^{2}ds +\frac{\delta_{s}\hat{\lambda}_{s}}{1+\delta_{s}}dW^{\eta}_{s} +\frac{\delta_{s}}{2}\big((\eta^{1}_{s})^{2}+(\eta^{2}_{s})^{2}\big)ds\\ =&\frac{\delta_{s}}{2}\bigg(\Big(\frac{\hat{\lambda}_{s} +(1+\delta_{s})\eta^{1}_{s}}{1+\delta_{s}}\Big)^{2} +(\eta^{2}_{s})^{2}\bigg)ds +\frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s} dW^{\eta}_{s}. \end{aligned}$$
Note that \(\delta_{s}/(1+\delta_{s})\) is in \((0,1)\) and thus, by the definition of \(\gamma \) in (2.3), we deduce that the process \(\int _{0}^{\cdot}\frac{\delta_{s}}{1+\delta_{s}}\hat{\lambda}_{s} dW^{\eta}_{s}\) is a martingale under \(\mathbb{Q}^{\eta}\). It follows that \(N^{\bar{\pi},\eta} \) is a submartingale for all \(\mathbb{Q}^{\eta}\in\mathcal {Q}_{0,T}\), and a martingale for \(\bar{\eta}\) as specified in (2.4). On the other hand, it holds that
$$\begin{aligned} N^{\pi,\bar{\eta}}_{T} =&U(X^{\pi}_{T},T)+\int_{0}^{T}\frac{\delta_{s}}{2}(\bar{\eta}_{s})^{2}ds = \ln X^{\pi}_{T}-\frac{1}{2}\int_{0}^{T}\frac{\delta_{s} \hat{\lambda}_{s}^{2}}{1+\delta_{s}} -\frac{\delta_{s} \hat{\lambda}_{s}^{2}}{(1+\delta_{s})^{2}}ds\\ =&\ln X^{\pi}_{T}-\frac{1}{2}\int_{0}^{T}\bigg(\frac{\delta_{s}}{1+\delta _{s}}\hat{\lambda}_{s}\bigg)^{2}ds = \ln X^{\pi}_{T}-\frac{1}{2}\int_{0}^{T}(\hat{\lambda}_{s}+\bar{\eta}^{1}_{s})^{2}ds. \end{aligned}$$
Since \(\mathrm{E}^{\mathbb{Q}^{\bar{\eta}}}[\ln X_{T}^{\pi}]\leq \mathrm{E}^{\mathbb{Q}^{\bar{\eta}}}[\ln X_{T}^{\bar{\pi}}]\) for any strategy \(\pi\in\mathcal{A}^{x}_{0}\), we conclude that
$$ \mathrm{E}^{\mathbb{Q}^{\bar{\eta}}}[N^{\pi,\bar{\eta}}_{T}] \le\mathrm{E}^{\mathbb{Q}^{\bar{\eta}}}[\ln X_{T}^{\bar{\pi}}]- \mathrm{E}^{\mathbb{Q}^{\bar{\eta}}} \bigg[\frac{1}{2}\int_{0}^{T}(\hat{\lambda}_{s}+\bar{\eta}^{1}_{s})^{2}ds\bigg] =\ln x=N_{0}, $$
where the equality follows by a direct computation (see also [39, Example 10.1]). □
Proof of Proposition 2.2
Fix \(0\le t\le T<\infty\). Note that by definition, \(\mathbb{Q}^{\bar{\eta}}\in \mathcal{Q}_{t,T}\). It suffices to show that \(\bar{\pi}\in\mathcal{A}^{x}_{t}\); that \(U(X^{\bar{\pi}}_{u},u)\), \(u\in[t,T]\), is a submartingale under any \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{t,T}\); and that for any \(\pi\in \mathcal {A}^{x}_{t}\), \(U(X^{\pi}_{u},u)\), \(u\in[t,T]\), is a supermartingale under \(\mathbb{Q}^{\bar{\eta}}\). Without loss of generality, we let \(t=0\); we also define \(\lambda^{\eta}_{s}:=\hat{\lambda}_{s}+\eta^{1}_{s}\) and write \(\hat{W}_{s}=\hat{W}^{1}_{s}\).
First note that for any strategy \(\pi\in\mathcal{A}^{x}_{0}\), recalling the form of \(U\) from (2.9) and the wealth dynamics from (6.15), we have
$$\begin{aligned} U(X^{\pi}_{s},s)=- e^{-ax} \mathcal{E}\bigg(-a\int_{0}^{s}\pi_{u}\sigma_{u}S_{u} dW^{\bar{\eta}}_{u}\bigg) \exp\bigg(\frac{1}{2}\int_{0}^{s}(\lambda^{\bar{\eta}}_{u}-a\pi_{u}\sigma _{u}S_{u})^{2}du\bigg), \end{aligned}$$
where \(W^{\bar{\eta}}\) is a Brownian motion under \(\mathbb{Q}^{\bar{\eta}}\). Using the properties imposed on the set \(\mathcal{A}^{x}_{0}\) of admissible strategies, we obtain that the Doléans-Dade exponential in the above expression is a martingale under any \(\mathbb{Q}^{\eta}\in \mathcal{Q} _{0,T}\). In particular, for any \(\pi\in\mathcal{A}^{x}_{0}\), the process \(U(X^{\pi}_{\cdot},\, \cdot\,)\) is a supermartingale on \([0,T]\) under \(\mathbb{Q} ^{\bar{\eta}}\).
Next, recall the form of the strategy \(\bar{\pi}\) from (2.8); it is clearly adapted. Using the notation \(L_{\cdot}=\int_{0}^{\cdot}\hat{\lambda}_{u}d\hat{W}_{u}\) and \(M^{\eta}_{\cdot}=\int _{0}^{\cdot}\eta^{1}_{u}d\hat{W}_{u}\), we notice that \(\langle M^{\eta}\rangle_{T}\le\langle L\rangle_{T}\), and thus it follows by the same arguments as in the proof of Proposition 2.1 that \(\mathrm{E}^{\eta}[\int_{0}^{T}(\lambda^{\bar{\eta}}_{u})^{2}du]\le\mathrm {E}^{\eta}[\langle L\rangle_{T}]<\infty\) for any \(\eta^{1}\) with \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{0,T}\); in particular, \(X^{\bar{\pi}}\) is thus well defined. Recalling again the form of \(U\) from (2.9) and the dynamics of the wealth process from (6.15), we then note that for any \(\eta\) with \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{0,T}\),
$$\begin{aligned} U(X^{\bar{\pi}}_{s},s)= -e^{-ax} \mathcal{E}\bigg(-\int_{0}^{s}\lambda^{\bar{\eta}}_{u}dW^{\eta}_{u}\bigg) \exp\bigg(-\int_{0}^{s}\lambda^{\bar{\eta}}_{u}(\lambda^{\eta}_{u}-\lambda ^{\bar{\eta}}_{u})du\bigg), \end{aligned}$$
where \(W^{\eta}\) is a Brownian motion under \(\mathbb{Q}^{\eta}\). Using that \(0\le\lambda^{\bar{\eta}}_{s}\le\lambda^{\eta}_{s}\), that the Doléans-Dade exponential of a local martingale is again a local martingale, and that any nonpositive local martingale with integrable initial value is a submartingale, we thus obtain that \(U(X^{\bar{\pi}}_{\cdot},\, \cdot\,)\) is a submartingale on \([0,T]\) under any \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{0,T}\).
Finally, to verify that \(\bar{\pi}\) is indeed in \(\mathcal{A}^{x}_{0}\), it only remains to argue that we have \(\mathrm{E}^{\eta}[e^{\frac {1}{2}\int _{0}^{T}(\lambda^{\bar{\eta}}_{u})^{2}du}]<\infty\) for any \(\mathbb{Q}^{\eta}\in\mathcal{Q}_{0,T}\). Since \(\int_{0}^{\cdot}(\lambda^{\bar{\eta}}_{u})^{2}du \le\langle L\rangle _{\cdot}\), it suffices to argue that \(\mathrm{E}^{\eta}[e^{\frac {1}{2}\langle L\rangle_{T}}]<\infty\). To this end, recall that \(\hat{\mathrm{E}}[e^{2\langle L\rangle _{T}}]<\infty\). With \(p=q=2\), we then have
$$\begin{aligned} \mathrm{E}^{\eta}\big[e^{\frac{1}{2}\langle L\rangle_{T}}\big] &= \hat{\mathrm{E}}\big[e^{M^{\eta}_{T}-\frac{1}{2}\langle M^{\eta}\rangle _{T}+\frac {1}{2}\langle L\rangle_{T}}\big] \\ &\le \hat{\mathrm{E}}\big[e^{p M^{\eta}_{T}-\frac{1}{2}p^{2}\langle M^{\eta}\rangle _{T}}\big]^{\frac{1}{p}} \hat{\mathrm{E}}\big[e^{\frac{q}{2}(p-1)\langle M^{\eta}\rangle _{T}+\frac {q}{2}\langle L\rangle_{T}}\big]^{\frac{1}{q}} \\ &\le \hat{\mathrm{E}}\big[e^{p M^{\eta}_{T}-\frac{1}{2}p^{2}\langle M^{\eta}\rangle _{T}}\big]^{\frac{1}{p}} \hat{\mathrm{E}}\big[e^{\frac{qp}{2}\langle L\rangle_{T}}\big]^{\frac{1}{q}}, \end{aligned}$$
which is finite since the first of the two factors is equal to one and the second one is finite, as can be seen by applying Novikov’s condition, the fact that \(\langle M^{\eta}\rangle_{T}\le\langle L\rangle _{T}\) and the assumed integrability of \(\langle L\rangle_{T}\). □