$Y^{\varepsilon}_{T}=y-\int_{0}^{T}H^{\rm F}_{s}(X_{\cdot\wedge
s},Z_{s},\Gamma_{s}^{\varepsilon},\alpha_{s})\mathrm{d}s+\int_{0}^{T}Z_{s}\cdot\mathrm{d}X_{s}+\frac{1}{2}\int_{0}^{T}\text{Tr}[\widehat{\sigma}^{2}_{s}\Gamma^{\varepsilon}_{s}]\mathrm{d}s=Y_{T}^{y,Z,\Gamma^{\varepsilon},\alpha},$

and, recalling that $K=K^{\varepsilon}=0$, $\mathbb{P}^{\star}\text{\rm--
a.s.}$, we see that $\Gamma^{\varepsilon}=\Gamma$,
$\mathrm{d}t\otimes\mathrm{d}\mathbb{P}^{\star}\text{\rm--a.e.}$, and deduce
that $Y=Y^{\varepsilon},$ $\mathbb{P}^{\star}\text{\rm--a.s.}$ In particular,
$Y^{\varepsilon}_{T}=g(X_{\cdot\wedge T}),\;\mathbb{P}^{\star}\text{\rm--
a.s.}$ Thus, from Proposition 4.4, we deduce that
$(\mathbb{P}^{\star},\beta^{\star})$ satisfies (4.7) and thus
$(\mathbb{P}^{\star},\beta^{\star})\in{\cal
R}^{\star}(y,Z,\Gamma^{\varepsilon},\alpha)$. Similarly to the conclusion in
part $(i)$, this implies that $\hat{V}_{L}\geq V_{L}$. ∎

##  5 Solving the problem of the leader: strong formulation

In this section, we use the techniques developed in [14, 13] based on the
geometric dynamic programming principle, see [67, 68], to study _Markovian_
stochastic target control problems. For this reason, and to take full
advantage of the standard tools from stochastic target problems, we bring
ourselves to a Markovian setting and study the strong formulation of (4.9). We
expect it to be equivalent to $\hat{V}_{L}$, see Remark 5.2.

In this setting, $(\Omega,{\cal F}_{T},\mathbb{F},\mathbb{P})$ denotes an
abstract complete probability space supporting a $\mathbb{P}$–Brownian motion,
which we still denote $W$, and $\mathbb{F}$ denotes the filtration generated
by $W$, augmented under $\mathbb{P}$ so that it satisfies the usual
conditions. In addition, the dependence of the data of the problem on
$(t,x)\in[0,T]\times{\cal C}([0,T];\mathbb{R}^{d})$ is only through
$(t,x(t))\in[0,T]\times\mathbb{R}^{d}$. With a slight abuse of notation, we
now write $c(t,x(t),a,b)$ instead of $c_{t}(x,a,b)$—and similarly for all the
other mappings introduced in the previous sections—and thus without any risk
of misunderstanding, consider now the maps as defined on $\mathbb{R}^{d}$
instead of ${\cal C}([0,T];\mathbb{R}^{d})$.

In light of Proposition 4.4, by a measurable selection argument, we introduce
${\cal B}^{\star}$ as the set of Borel-measurable maps
$b^{\star}:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times
A\longrightarrow B$ which are uniformly Lipschitz-continuous in $(x,z)$ and
such that

$\displaystyle H^{\rm F}(t,x,z,\gamma,a)=h^{\rm
F}(t,x,z,\gamma,a,b^{\star}(t,x,z,\gamma,a)),\;(t,x,z,\gamma,a)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times
A.$ inlineinlinetodo: inlineThe Lipschitz-continuity assumption requires some
comments. Also, what are we doing: assuming that all best-responses satisfy
this, or only select the ones that do? The former seems a bit nicer to me,
since this is a assumption on primitives of the model, why the latter means we
are potentially solving a sub-optimal problem for the leader.

We now topologise ${\cal B}^{\star}$. Consider the measurable space $(O,{\cal
O},\lambda)$, where
$O\coloneqq[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times
A$, and ${\cal O}$ and $\lambda$ denote the Borel $\sigma$-algebra and
Lebesgue measure on $O$, respectively. inlineinlinetodo: inlineWe use ${\cal
O}$ later, change notation. We see ${\cal B}^{\star}$ as a subspace of
$\mathbb{L}^{1}(O,\nu)$, where
$\mathrm{d}\nu=C{\mathrm{e}}^{-\|\cdot\|}\mathrm{d}\lambda$ and $C>0$ is a
normalising constant. In this way, as a subspace of a separable metric space,
${\cal B}^{\star}$ is separable. Lastly, for any $b^{\star}\in{\cal
B}^{\star}$ and
$\varphi\in\\{C,c,\lambda,\sigma,\lambda\sigma,\sigma\sigma^{\top}\\!\\}$ we
define

$\displaystyle\varphi(t,x,{\color[rgb]{0.19,0.55,0.91}\definecolor[named]{pgfstrokecolor}{rgb}{0.19,0.55,0.91}a},z,\gamma)\coloneqq\varphi(t,x,a,b^{\star}(t,x,z,\gamma,a)).$
(5.1) inlineinlinetodo: inlineI don’t think this is a meaningful definition:
first (but I guess this is a typo?) you have $\varphi$ on both sides, and more
importantly, the value has to depend on the choice of $b^{\star}$. This does
not matter for $h^{F}$ since they are all maximisers of it, but it does for
all other functions. Why not using the notation $\varphi^{b^{\text{$\star$}}}$
then?

With this, we introduce the following set of assumptions.

######  Assumption 5.1.

In addition to 3.1, we assume that $c$, $\sigma$ and $\sigma\lambda$ are
Lipschitz-continuous in $(x,b)$, uniformly in $(t,a)$.

We let $\mathfrak{C}$ be the family of tuples $(\alpha,Z,\Gamma,b^{\star})$
consisting of $\mathbb{F}$-predictable processes
$(\alpha,Z,\Gamma):[0,T]\times\Omega\longrightarrow
A\times\mathbb{R}^{d}\times\mathbb{S}^{d}$ and $b^{\star}\in{\cal B}^{\star}$
such that, for some $p>1$,

$\|Z\|_{\mathbb{H}^{\raisebox{1.0pt}{\text{$p$}}}}^{p}+\|\Gamma\|_{\mathbb{G}^{\raisebox{1.0pt}{\text{$p$}}}}^{p}+\|b^{\star}\|_{\mathbb{L}^{\raisebox{1.0pt}{\text{$p$}}}}^{p}<+\infty.$
inlineinlinetodo: inlineWhy did the norms suddenly change here?? I know
$\sigma$ is bounded, but still, why take it off the norms?

To alleviate the notation, we use $\upsilon$ to denote a generic element of
$\mathfrak{C}$ and $\hat{\upsilon}=(\alpha,Z,\Gamma)$ its first three
components. With this, given $t\in[0,T]$, $(x,y)\in\mathbb{R}^{d+1}$ and
$\upsilon\in\mathfrak{C}$, the controlled state processes are given by

$\displaystyle\begin{split}X_{u}^{t,x,\upsilon}&=x+\int_{t}^{u}\sigma\lambda(s,X_{s}^{t,x,\upsilon},\hat{\upsilon}_{s})\mathrm{d}s+\int_{t}^{u}\sigma(s,X_{s}^{t,x,\upsilon},\hat{\upsilon}_{s})\mathrm{d}W_{s},\;u\in[t,T],\\\
Y_{u}^{t,x,y,\upsilon}&=y-\int_{t}^{u}c(s,X_{s}^{t,x,\upsilon},\hat{\upsilon}_{s})\mathrm{d}s+\int_{t}^{u}Z_{s}\cdot\sigma(s,X_{s}^{t,x,\upsilon},\hat{\upsilon}_{s})\mathrm{d}W_{s},\;u\in[t,T].\end{split}$
(5.2)

With this, we define the problem

$\tilde{V}_{L}\coloneqq\sup_{y\in\mathbb{R}}V(0,x_{0},y),$ (5.3)

where

$V(t,x,y)\coloneqq\sup_{\upsilon\in{\mathfrak{C}}(t,x,y)}\mathbb{E}^{\mathbb{P}}\bigg{[}\int_{t}^{T}C(s,X_{s}^{t,x,\upsilon},\hat{\upsilon}_{s})\mathrm{d}s+G(X_{T}^{t,x,\upsilon})\bigg{|}{\cal
F}_{t}\bigg{]},$ (5.4)

and, for $(t,x,y)\in[0,T]\times\mathbb{R}^{d+1}$,

$\mathfrak{C}(t,x,y)\coloneqq\big{\\{}\upsilon\in\mathfrak{C}:\hat{\upsilon}\text{
is independent of }{\cal F}_{t}\text{, and
}Y_{T}^{t,y,x,\upsilon}=g(X_{T}^{t,x,\upsilon}),\ \mathbb{P}\text{\rm--
a.s.}\big{\\}}.$ inlineinlinetodo: inlineAs it is written, there should be an
essup in the dynamic value. Though there shouldn’t be a need to condition if
we define the state variables starting from $t$, which you do below.

######  Remark 5.2.

Let us comment on the previous formulation.

  1. $(i)$

We remind the reader that in the strong formulation, the background
probability measure $\mathbb{P}$ is fixed, and thus, the norms in the
definition of ${\mathfrak{C}}$ coincide with those in the standard literature.
In particular, contrary to the weak formulation, the family $\mathfrak{C}$
does not depend on the choice of $\alpha\in{\cal A}$. We also remark that
$\mathfrak{C}$ is a separable topological space. This guarantees the geometric
dynamic programming principle of [14], based on [67], holds.

  2. $(ii)$

The Lipschitz-continuity in $(x,b)$ of $\sigma$ and $\lambda\sigma$ in 5.1 and
in $x$ of $b^{\star}$ in the definition of ${\cal B}^{\star}$, ensure that the
process $X^{t,x,\upsilon}$ is well defined, and provide sufficient regularity
to conduct our upcoming analysis. Notice that $Y^{t,x,y,\upsilon}$ is a direct
definition. Note, we do not assume the uniqueness of maximisers of $h^{\rm F}$
in $b$. The Lipschitz-continuity in $(x,b)$ of $c$ in 5.1 together with the
Lipschitz-continuity of $b^{\star}\in{\cal B}^{\star}$ in $z$ will be used to
establish a comparison principle for the target boundaries in Section 5.1.

  3. $(iii)$

Let us also digress on the equivalence of the strong and weak formulations. A
potential roadmap to obtain this result uses [28]. Indeed, to handle the
constraint in both formulations, it is natural to embed it in the reward by
means of a Lagrange multiplier $k\geq 0$ and the continuous penalty function
$\Phi(y,x)\coloneqq|g(x)-y|^{2}$. In this way, after establishing that strong
duality holds, the results in [28] will allow us to obtain the equivalence of
the strong and weak formulations for each element of a family of penalised
problems, obtained by fixing $k$ and optimising over the corresponding
controls. The only work needed to complete this argument is the strong duality
results for the Lagrangian versions of both $\hat{V}_{L}$ and $\tilde{V}_{L}$.
We have refrained from writing such arguments as this will require, for
instance, introducing the so-called relaxed formulation of $\hat{V}_{L}$,
which will unnecessarily encumber our analysis.

As usual in stochastic target problems, we define the target reachability set
as the set set of triplets $(t,x,y)$ such that the set $\mathfrak{C}(t,x,y)$
is non-empty. That is

$V_{g}(t)\coloneqq\big{\\{}(x,y)\in\mathbb{R}^{d+1}:\exists\upsilon\in{\mathfrak{C}}(t,x,y),\;Y_{T}^{t,x,y,\upsilon}=g(X_{T}^{t,x,\upsilon}),\;\mathbb{P}\text{\rm
--a.s.}\big{\\}}.$

We are interested in characterising $V_{g}(t)$, through the auxiliary sets

$\displaystyle
V_{g}^{-}(t)\coloneqq\big{\\{}(x,y)\in\mathbb{R}^{d+1}:\exists\upsilon\in{\mathfrak{C}}(t,x,y),\;Y_{T}^{t,x,y,\upsilon}\geq
g(X_{T}^{t,x,\upsilon}),\;\mathbb{P}\text{\rm--a.s.}\big{\\}},$ $\displaystyle
V_{g}^{+}(t)\coloneqq\big{\\{}(x,y)\in\mathbb{R}^{d+1}:\exists\upsilon\in{\mathfrak{C}}(t,x,y),\;Y_{T}^{t,x,y,\upsilon}\leq
g(X_{T}^{t,x,\upsilon}),\;\mathbb{P}\text{\rm--a.s.}\big{\\}}.$

Notice that the inclusion $V_{g}(t)\subseteq V_{g}^{-}(t)\cap V_{g}^{+}(t)$ is
immediate. The set $V_{g}^{-}(t)$ has been studied by [68, 13] and its
boundary can be characterised through the auxiliary value function defined
below

$w^{-}(t,x)\coloneqq\inf\\{y\in\mathbb{R}:(x,y)\in V_{g}^{-}(t)\\}.$ (5.5)

It is known, see for instance [14, Corollary 2.1], that the closure of
$V_{g}^{-}(t)$ is given by

$\overline{V_{g}^{-}(t)}=\big{\\{}(x,y):y\geq w^{-}(t,x)\big{\\}}.$

Moreover, $w^{-}$ is a discontinuous viscosity solution of the following PDE

$-\partial_{t}w(t,x)-H^{-}\big{(}t,x,\partial_{x}w(t,x),\partial_{xx}^{2}w(t,x)\big{)}=0,\;(t,x)\in[0,T)\times\mathbb{R}^{d},\;w(T^{-},x)=g(x),\;x\in\mathbb{R}^{d},$
(5.6)

where
$H^{-}:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\longrightarrow\mathbb{R}$,
and $h^{\rm
b}:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times
A\times\mathbb{R}^{d}\times\mathbb{S}^{d}\longrightarrow\mathbb{R}$ are given
by

$\displaystyle\begin{split}H^{-}(t,x,p,Q)&\coloneqq\inf_{(a,z,\gamma,b^{\text{$\star$}})\in
N(t,x,p)}\big{\\{}h^{\rm b}(t,x,p,Q,a,z,\gamma)\big{\\}},\\\ h^{\rm
b}(t,x,p,Q,a,z,\gamma)&\coloneqq
c(t,x,a,z,\gamma)+\sigma\lambda(t,x,a,z,\gamma)\cdot
p+\frac{1}{2}\text{Tr}[\sigma{\sigma}^{\top}\\!(t,x,a,z,\gamma)Q],\end{split}$
(5.7)

and, since $\sigma\sigma^{\top}\\!$ is invertible by assumption

$\displaystyle N(t,x,p)\coloneqq\\{(a,z,\gamma,b^{\star})\in
A\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times{\cal
B}^{\star}:\sigma^{\top}\\!(t,x,a,z,\gamma)(z-p)=0\\}=A\times\\{p\\}\times\mathbb{S}^{d}\times{\cal
B}^{\star}.$ (5.8)

Similarly, by doing a change of variables and following the same ideas, the
closure of $V_{g}^{+}(t)$ can be characterised as follows

$\overline{V_{g}^{+}(t)}=\big{\\{}(x,y):y\leq w^{+}(t,x)\big{\\}},$

where the auxiliary value function $w^{+}$ is defined by

$w^{+}(t,x)\coloneqq\sup\\{y\in\mathbb{R}:(x,y)\in V_{g}^{+}(t)\\},$ (5.9)

and it is a discontinuous viscosity solution of the PDE

$-\partial_{t}w(t,x)-H^{+}\big{(}t,x,\partial_{x}w(t,x),\partial_{xx}^{2}w(t,x)\big{)}=0,\;(t,x)\in[0,T)\times\mathbb{R}^{d},\;w(T^{-},x)=g(x),\;x\in\mathbb{R}^{d},$
(5.10)

where
$H^{+}:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}\longrightarrow\mathbb{R}$
is given by

$H^{+}(t,x,p,Q)\coloneqq\sup_{(a,z,\gamma,b^{\text{$\star$}})\in
N(t,x,p)}\big{\\{}h^{\rm b}(t,x,p,Q,a,z,\gamma)\big{\\}}.$

We propose the two auxiliary value functions as the upper and lower boundaries
of $V_{g}(t)$, and thus define the set

$\hat{V}_{g}(t)\coloneqq\\{(x,y):w^{-}(t,x)\leq y\leq w^{+}(t,x)\\},$

which, provided the upper and lower boundaries are sufficiently separated
before $T$, corresponds to the closure of the reachability set $V_{G}(t)$, as
we prove next. For this, we introduce

$\delta_{\varepsilon}\coloneqq\inf_{(t,x)\in[0,T-\varepsilon]\times\mathbb{R}^{\text{$d$}}}|w^{-}(t,x)-w^{+}(t,x)|,\;\varepsilon>0.$

######  Lemma 5.3.

Let $t\in[0,T]$. The following holds

  1. $(i)$

$V_{g}(t)\subseteq\hat{V}_{g}(t).$

  2. $(ii)$

If in addition $\delta_{\varepsilon}>0$ for any $\varepsilon>0$, and $w^{-}$
and $w^{+}$ are continuous, then, ${\rm
int}\big{(}\hat{V}_{g}(t)\big{)}\subseteq V_{g}(t)$ and
$\overline{V_{g}(t)}=\hat{V}_{g}(t)$.

######  Remark 5.4.

Let us provide a sufficient structural condition for the assumption
$\delta_{\varepsilon}>0$ for any $\varepsilon>0$, before presenting the proof
of Lemma 5.3. We claim that it holds if PDE (5.6) satisfies a comparison
principle, as we will establish in Section 5.1, and there is $\eta>0$ such
that

$H^{+}(t,x,p,Q)\geq
H^{-}(t,x,p,Q)+\eta,\;\forall(t,x,p,Q)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d}.$
(5.11)

Indeed, under this condition, it is easy to see that the function
$\hat{w}^{-}(t,x)\coloneqq w^{-}(t,x)+\eta(T-t)$ is a discontinuous viscosity
sub-solution to PDE (5.6). Therefore, from the comparison principle we have
$\hat{w}^{-}\leq w^{+}$, which implies $\delta_{\varepsilon}>0$ for any
$\varepsilon>0$. A similar argument works if PDE (5.10) satisfies a comparison
principle instead.

###### Proof of Lemma 5.3.

Let us argue $(i)$. Let $(x,y)\in V_{g}(t)$, then there exists
$\upsilon\in\mathfrak{C}(t,x,y)$ such that
$Y_{T}^{t,x,y,\upsilon}=g(X_{T}^{t,x,y,\upsilon}),\;\mathbb{P}\text{\rm--
a.s.}$ Then it is clear that $(x,y)$ belongs to both auxiliary sets
$V_{g}^{-}(t)$ and $V_{g}^{+}(t)$, that is, $(x,y)\in V_{g}^{-}(t)\cap
V_{g}^{+}(t)$. Since
$\hat{V}_{g}(t)=\overline{V_{g}^{-}(t)}\cap\overline{V_{g}^{+}(t)}$, it
follows that $V_{g}(t)\subseteq\hat{V}_{g}(t)$.

As for $(ii)$, we first note that the second part of the statement, _i.e._
$\overline{V_{g}(t)}=\hat{V}_{g}(t)$, follows from the inclusions
$\text{int}(\hat{V}_{g}(t))\subseteq V_{g}(t)\subseteq\hat{V}_{g}(t)$ by
taking closure. Let us now argue $\text{int}(\hat{V}_{g}(t))\subseteq
V_{g}(t)$. To increase the readability of the proof, given
$(t,x,y)\in[0,T]\times\mathbb{R}^{d+1}$ and
$\upsilon\in{\mathfrak{C}}(t,x,y)$, we will say that $\upsilon$ satisfies
$(U)$ or $(L)$ whenever $Y_{T}^{t,x,y,\upsilon}\geq
g(X_{T}^{t,x,\upsilon}),\;\mathbb{P}\text{\rm--a.s.}$, or,
$Y_{T}^{t,x,y,\upsilon}\leq g(X_{T}^{t,x,\upsilon}),\;\mathbb{P}\text{\rm--
a.s.}$, respectively. Let $t\in[0,T]$ and $(x,y)\in{\rm int}(\hat{V}_{G}(t))$.
We argue in 2 steps.

Step 1. We fix $n\in\mathbb{N}^{\star}$ and construct an admissible control up
to $T_{n}\coloneqq T-n^{-1}$. Since $(x,y)\in{\rm int}(\hat{V}_{G}(t))$, by
continuity, we have that $w^{-}(t,x)<y<w^{+}(t,x)$. Thus, in particular, there
is $\upsilon^{0,n}\in{\mathfrak{C}}(t,x,y)$ satisfying $(U)$. Let
$X^{0,n}\coloneqq
X^{t,x,\upsilon^{\text{$0$}\text{$,$}\text{$n$}}},Y^{0,n}\coloneqq
Y^{t,x,y,\upsilon^{\text{$0$}\text{$,$}\text{$n$}}}$. By [14, Corollary 2.1],
$Y^{{0},n}_{s}\geq w^{-}(s,X^{0,n}_{s})$, $s\in[t,T]$. We have two cases. If
$Y^{0,n}_{s}=w^{-}(s,X^{0,n}_{s})$ for some $s\in[t,T]$, by the Markov
property and the continuity of $w^{-}$, we find that $(x,y)\in V_{g}(t)$ as
desired and conclude the proof. Otherwise, we have that
$Y^{0,n}_{s}>w^{-}(s,X^{0,n}_{s})$, $s\in[t,T]$. We then define the sequence
of $\mathbb{F}$–stopping times $(\tau_{k}^{n})_{k\in\\{0,\dots,k(n)\\}}$, with
$k(n)\in\mathbb{N}$ to be defined, recursively as follows

$\displaystyle\tau_{0}^{n}$ $\displaystyle\coloneqq\inf\big{\\{}s\geq
t:w^{+}\big{(}s,X^{0,n}_{s}\big{)}-Y^{0,n}_{s}\leq\delta_{n^{\text{$-$}\text{$1$}}}/3\big{\\}}\wedge
T_{n}.$

If $\tau_{0}^{n}=T_{n}$, we set $k(n)=0$ and conclude the construction.
Otherwise, by continuity, we have that
$w^{+}\big{(}\tau_{0}^{n},X_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n}\big{)}-Y_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n}=\delta_{n^{\text{$-$}\text{$1$}}}/3$.
By definition of $\delta_{\varepsilon}$, we have that

$\displaystyle\big{(}X_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n},Y_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n}\big{)}\in{\rm
int}\big{(}\hat{V}_{g}(\tau_{0}^{n})\big{)},\;\mathbb{P}\text{\rm--
a.s.},\;\text{\emph{i.e.}}\;w^{-}\big{(}\tau_{0}^{n},X_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n}\big{)}<Y_{\tau_{\text{$0$}}^{\text{$n$}}}<w^{+}\big{(}\tau_{0}^{n},X_{\tau_{\text{$0$}}^{\text{$n$}}}^{0,n}\big{)},\;\mathbb{P}\text{\rm
--a.s.}$ (5.12)

Thus, by [14, Corollary 2.1], there is
$\upsilon^{1,n}\in{\mathfrak{C}}(t,x,y)$, satisfying $(L)$ and
$\upsilon^{1,n}=\upsilon^{0,n}$ on $[t,\tau_{0}^{n})$. Let now

$\displaystyle\tau_{1}^{n}$
$\displaystyle\coloneqq\inf\big{\\{}s\geq\tau_{0}^{n}:Y_{s}^{1,n}-w^{-}(s,X_{s}^{1,n})\leq\delta_{n^{\text{$-$}\text{$1$}}}/3\big{\\}}\wedge
T_{n},\;\text{for}\;X^{1,n}\coloneqq
X^{t,x,\upsilon^{\text{$1$}\text{$,$}\text{$n$}}},\;Y^{1,n}\coloneqq
Y^{t,x,y,\upsilon^{\text{$1$}\text{$,$}\text{$n$}}}.$

Arguing as above, by definition of $\tau_{1}^{n}$, we find that
$(X_{\tau_{\text{$1$}}^{\text{$n$}}}^{1,n},Y_{\tau_{\text{$1$}}^{\text{$n$}}}^{1,n})\in{\rm
int}\big{(}\hat{V}_{g}(\tau_{1}^{n})\big{)},\;\mathbb{P}\text{\rm--a.s.}$
Thus, again by [14, Corollary 2.1], there is
$\upsilon^{2,n}\in{\mathfrak{C}}(t,x,y)$ such that $(U)$ holds and
$\upsilon^{2,n}=\upsilon^{1,n}$ on $[\tau_{0}^{n},\tau_{1}^{n})$. Recursively,
for $k\in\mathbb{N}^{\star}$ we let $X^{k,n}\coloneqq
X^{t,x,\upsilon^{\text{$k$}\text{$,$}\text{$n$}}},\;Y^{k,n}\coloneqq
Y^{t,x,y,\upsilon^{\text{$k$}\text{$,$}\text{$n$}}}$

$\displaystyle\tau_{2k}^{n}$
$\displaystyle\coloneqq\inf\big{\\{}s\geq\tau_{2k-1}^{n}:w^{+}(\tau_{k-1}^{n},X_{s}^{k,n})-Y_{s}^{k,n}\leq\delta_{n^{\text{$-$}\text{$1$}}}/3\big{\\}}\wedge
T_{n},$ $\displaystyle\tau_{2k+1}^{n}$
$\displaystyle\coloneqq\inf\big{\\{}s\geq\tau_{2k}^{n}:Y_{s}^{k,n}-w^{-}(\tau_{k-1}^{n},X_{s}^{k,n})\leq\delta_{n^{\text{$-$}\text{$1$}}}/3\big{\\}}\wedge
T_{n},$

and find $\upsilon^{k+1,n}\in\mathfrak{C}(t,x,y)$ for which
$(X_{\tau_{\text{$k$}}^{\text{$n$}}}^{k,n},Y_{\tau_{\text{$k$}}^{\text{$n$}}}^{k,n})\in{\rm
int}\big{(}\hat{V}_{g}(\tau_{k}^{n})\big{)},\;\mathbb{P}\text{\rm--a.s.}$

We now claim that there is a process $k(n)$ with values in $\mathbb{N}$ such
that $\tau^{n}_{k(n)}=T_{n}$, $\mathbb{P}\text{\rm--a.s.}$ Indeed, by
continuity of $w^{-}$ and $w^{+}$, the mappings

$[t,T_{n}]\ni s\longmapsto
w^{+}\big{(}s,X_{s}^{t,x,\upsilon}\big{)}-Y_{s}^{t,x,y,\upsilon},\;\text{and}\;[t,T_{n}]\ni
s\longmapsto
Y_{s}^{t,x,y,\upsilon}-w^{-}\big{(}s,X_{s}^{t,x,\upsilon}\big{)},$

are, $\omega$-by-$\omega$, uniformly continuous for any
$\upsilon\in{\mathfrak{C}}(t,x,y)$. Hence, there exists a constant
$\bar{\gamma}_{n}>0$ and a $[\bar{\gamma}_{n},T_{n}]$-valued random variable
$\gamma_{n}$ such that, $\|\tau_{k}^{n}-\tau_{k-1}^{n}\|_{\infty}>\gamma_{n},$
$\mathbb{P}\text{\rm--a.s.},\;k\in\mathbb{N}$. This proves the claim.

At the end of this construction, we set
$\upsilon^{n}\coloneqq\upsilon^{k(n),n}$, and notice that
$\upsilon^{n}\in{\mathfrak{C}}(t,x,y)$ and

$\displaystyle
w^{-}\big{(}T_{n},X_{T_{\text{$n$}}}^{n}\big{)}<Y_{T_{\text{$n$}}}^{n}<w^{+}\big{(}T_{n},X_{T_{\text{$n$}}}^{n}\big{)},\;\mathbb{P}\text{\rm
--a.s.},\text{ for }X^{n}\coloneqq
X^{t,x,\upsilon^{\text{$n$}}},Y^{n}\coloneqq Y^{t,x,y,\upsilon^{\text{$n$}}}.$
(5.13)

Step 2. We iterate the previous construction. From here on, we can repeat Step
1, with $(T_{n},X_{T_{\text{$n$}}}^{n})$, control $\upsilon^{n}$, and $n+1$
playing the role of $(t,x)$, $\upsilon^{0,n}$ and $n$, respectively. With
this, we obtain the existence of $\upsilon^{n+1}\in{\mathfrak{C}}(t,x,y)$,
such that, by uniform continuity, (5.13) holds at
$(T_{n+1},X^{n+1}_{T_{\text{$n$}\text{$+$}\text{$1$}}})$ and
$Y^{n+1}_{T_{\text{$n$}\text{$+$}\text{$1$}}}$. Iterating this construction,
we find $\upsilon$ which is well-defined
$\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm--a.e.}$ on
$[0,T]\times\Omega$.171717Indeed, the construction allows us to define said
process $\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm--a.e.}$ on
$[t,T)\times\Omega$, and consequently,
$\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm--a.e.}$ on
$[0,T]\times\Omega$.

To conclude $(x,y)\in V_{g}(t)$, let $n\longrightarrow\infty$ in (5.13), and
notice that by continuity of $w^{-}$ and $w^{+}$ we have that
$Y_{T}^{t,x,y,\upsilon}=g(X_{T}^{t,x,\upsilon})$ as desired.∎

###  5.1 Boundaries PDEs: comparison and verification

In this section, we prove the following verification theorem for the solutions
to PDEs (5.6) and (5.10). We remind the reader the 5.1 is in place.

######  Theorem 5.5.

Let $u$ and $v$ be continuous viscosity solutions to (5.6) and (5.10),
respectively, with linear growth. Then $u=w^{-}$ and $v=w^{+}$.

We conduct the analysis for $w^{-}$, the argument for $w^{+}$ being analogous.
We start by establishing a comparison result for viscosity solutions to (5.6).
Let us recall that $w^{-}$ is a discontinuous viscosity solution of such an
equation.

######  Lemma 5.6.

Let $u$ and $v$ be respectively an upper–semi-continuous viscosity sub-
solution and a lower–semi-continuous viscosity super-solution of (5.6), such
that for $\varphi\in\\{u,v\\}$ and some $C>0$, $\varphi(y)\leq C(1+\|y\|)$,
$y\in[0,T]\times\mathbb{R}^{d}$. If, $u(T,x)\leq v(T,x)$,
$x\in\mathbb{R}^{d}$, then $u\leq v$ on ${\cal
O}\coloneqq(0,T)\times\mathbb{R}^{d}$.

###### Proof.

Step 1. Fix postive constants $\alpha$, $\beta$, $\eta$, and $\varepsilon$,
and define $\phi(t,x,y)\coloneqq u^{\eta}(t,x)-v(t,y)$, where
$u^{\eta}(t,x)\coloneqq u(t,x)-\frac{\eta}{t},\;(t,x)\in{\cal O}$. Note that
since $\frac{\partial}{\partial t}(-\eta t^{-1})=\eta t^{-2}>0$, $u^{\eta}$ is
a viscosity sub-solution of (5.6) in ${\cal O}$. Define as well

$\psi_{\alpha,\beta,\varepsilon}(t,x,y)\coloneqq\alpha|x-y|^{2}/2+\varepsilon|x|^{2}+\varepsilon|y|^{2}-\beta(t-T).$

Now, let

$M_{\alpha,\beta,\varepsilon}\coloneqq\sup_{(t,x,y)\in(0,T]\times\mathbb{R}^{\text{$d$}}\times\mathbb{R}^{\text{$d$}}}\big{(}\phi-\psi_{\alpha,\beta,\varepsilon}\big{)}(t,x,y)=(\phi-\psi_{\alpha,\beta,\varepsilon})(t_{\alpha,\beta,\varepsilon},x_{\alpha,\beta,\varepsilon},y_{\alpha,\beta,\varepsilon}),$

for some
$(t_{\alpha,\beta,\varepsilon},x_{\alpha,\beta,\varepsilon},y_{\alpha,\beta,\varepsilon})\in(0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
thanks to the upper–semi-continuity of $u^{\eta}-v$, the growth assumptions on
$u$ and $v$ and that of $\beta(t-T)-\eta t^{-1}$. Moreover, we have that
$-\infty<\lim_{\alpha\rightarrow\infty}M_{\alpha,\beta,\varepsilon}<\infty$,
meaning that the supremum is attained on a compact subset of
$(0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$. Consequently, there is a
subsequence
$(t_{n}^{\beta,\varepsilon},x_{n}^{\beta,\varepsilon},y_{n}^{\beta,\varepsilon})\coloneqq(t_{\alpha_{\text{$n$}},\beta,\varepsilon},x_{\alpha_{\text{$n$}},\beta,\varepsilon},y_{\alpha_{\text{$n$}},\beta,\varepsilon})$
that converges to some
$(\hat{t}^{\beta,\varepsilon},\hat{x}^{\beta,\varepsilon},\hat{y}^{\beta,\varepsilon})$.
It then follows from [21, Proposition 3.7] that

$\displaystyle\hat{x}^{\beta,\varepsilon}=\hat{y}^{\beta,\varepsilon},\;\lim_{n\to\infty}\alpha_{n}|x^{\beta,\varepsilon}_{n}-y^{\beta,\varepsilon}_{n}|^{2}=0,\;M_{\beta,\varepsilon}\coloneqq\lim_{n\longrightarrow\infty}M_{\alpha_{\text{$n$}},\beta,\varepsilon}=\sup_{(t,x)\in\overline{\cal
O}}(u^{\eta}-v)(t,x)-2\varepsilon|\hat{x}^{\varepsilon}|+\beta(\hat{t}^{\beta,\varepsilon}-T).$
(5.14)

Step 2. To prove the statement, as it is standard in the literature, let us
assume by contradiction that there is $(t_{o},x_{o})\in{\cal O}$ such that
$\gamma_{o}\coloneqq(u-v)(t_{o},x_{o})>0$. We claim that there are positive
$\beta_{o}$, $\eta_{o}$, and $\varepsilon_{o}$ such that for any
$\beta_{o}\geq\beta>0$, $\eta_{o}\geq\eta>0$,
$\varepsilon_{o}\geq\varepsilon>0$,
$(t_{n}^{\beta,\varepsilon},x_{n}^{\beta,\varepsilon},y_{n}^{\beta,\varepsilon})$
is a local maximiser of
$\phi(t,x,y)-\psi_{\alpha_{\text{$n$}},\beta,\varepsilon}(t,x,y)$ on
$(0,T)\times{\cal K}^{2}$ for some ${\cal K}\subseteq\mathbb{R}^{d}$ compact.
We first note that the existence of ${\cal K}$ is clear since the supremum is
attained on a compact set. It remains to show that
$t_{n}^{\beta,\varepsilon}<T$ for all $n\in\mathbb{N}$.

Suppose by contradiction that $t_{n}^{\beta,\varepsilon}=T$ for some $n$.
Thanks to the first step, for any positive $\beta$, $\varepsilon$, and $\eta$
we have that

$\displaystyle\gamma_{o}-\frac{\eta}{t_{o}}+\beta(t_{o}-T)-2\varepsilon|x_{o}|^{2}\leq
M_{\alpha_{\text{$n$}},\beta,\varepsilon}=\sup_{(x,y)\in\mathbb{R}^{\text{$d$}}\times\mathbb{R}^{\text{$d$}}}\big{\\{}u(T,x)-v(T,y)-\alpha_{n}|x-y|^{2}/2-\varepsilon|x|^{2}-\varepsilon|y|^{2}\big{\\}}-\frac{\eta}{T}\leq-\frac{\eta}{T},$

where the rightmost inequality follows from the assumption $u(T,x)\leq
v(T,x)$, $x\in\mathbb{R}^{d}$. Consequently

$\displaystyle\gamma_{o}\leq\frac{\eta}{t_{o}}-\frac{\eta}{T}+\beta(T-t_{o})+2\varepsilon|x_{o}|^{2},$

so that for $\beta$, $\varepsilon$, and $\eta$ sufficiently small,
$\gamma_{o}$ is arbitrarily small which contradicts $\gamma_{o}>0$. This
proves the claim.

Step 3. In light of the second step, it follows from Crandall–Ishii’s lemma
for parabolic problems, [21, Theorem 8.3], applied to $u^{\eta}$ and $v$ that
we can find $(q_{n},\hat{q}_{n})$,
$q_{n}-\hat{q}_{n}=\partial_{t}\psi_{\alpha,\beta,\varepsilon}(t,x,y)=-\beta$,
and symmetric matrices $(X_{n}^{\beta,\varepsilon},Y_{n}^{\beta,\varepsilon})$
such that

$\big{(}q_{n},\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})+\varepsilon
x_{n}^{\beta,\varepsilon},X_{n}^{\beta,\varepsilon}\big{)}\in\overline{\cal
P}^{1,2,+}u^{\eta}(x_{n}^{\beta,\varepsilon}),\;\big{(}\hat{q}_{n},-\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})+\varepsilon
y_{n}^{\beta,\varepsilon},Y_{n}^{\beta,\varepsilon}\big{)}\in\overline{\cal
P}^{1,2,-}v(y_{n}^{\beta,\varepsilon}),\;$

and, for $C_{n}\coloneqq\alpha_{n}\begin{pmatrix}I_{d}&-I_{d}\\\
-I_{d}&I_{d}\end{pmatrix}+\varepsilon I_{2d}$, we have that

$-\bigg{(}\frac{1}{\lambda}+\|C_{n}\|\bigg{)}I_{2d}\leq\begin{pmatrix}X_{n}^{\beta,\varepsilon}&0\\\
0&-Y_{n}^{\beta,\varepsilon}\end{pmatrix}\leq C_{n}(I_{2d}+\lambda
C_{n}),\text{ for all }\lambda>0.$

Taking $\lambda=(\alpha_{n}+\varepsilon)^{-1}$ leads to

$\displaystyle-\Big{(}\alpha_{n}+\varepsilon+\|C_{n}\|\Big{)}I_{2d}\leq\begin{pmatrix}X_{n}^{\beta,\varepsilon}&0\\\
0&-Y_{n}^{\beta,\varepsilon}\end{pmatrix}\leq
3\alpha_{n}\begin{pmatrix}I_{d}&-I_{d}\\\
-I_{d}&I_{d}\end{pmatrix}+2\varepsilon I_{2d}.$ (5.15)

Step 4. With the notation
$(t_{n},x_{n},y_{n})\coloneqq(t_{n}^{\beta,\varepsilon},x_{n}^{\beta,\varepsilon},y_{n}^{\beta,\varepsilon})$,
$p_{n}^{x}\coloneqq\alpha_{n}(x_{n}-y_{n})-\varepsilon x_{n}$,
$p_{n}^{y}\coloneqq\alpha_{n}(x_{n}-y_{n})-\varepsilon y_{n}$, under the above
assumptions we claim that there exists a universal constant $C>0$ such that

$H^{-}(t_{n},y_{n},p_{n}^{y},Q_{2})-H^{-}(t_{n},x_{n},p_{n}^{x},Q_{1})\leq
C\big{(}1+\varepsilon^{2}\|x_{n}\|+\varepsilon^{2}\|y_{n}\|+\varepsilon\big{)}\big{(}\alpha_{n}\|x_{n}-y_{n}\|^{2}+\|x_{n}-y_{n}\|+\varepsilon\big{)}$

for matrices $Q_{1}$, $Q_{2}$ satisfying (5.15). We consider each term in
$h^{\rm b}$ separately, recall (5.7) and (5.8).181818The following estimates
hold for arbitrary, but fixed, $(a,\gamma,b^{\star})$.

Letting
$\Sigma^{x}\coloneqq\sigma(t_{n},x_{n},a,p_{n}^{x},\gamma),\Sigma^{y}\coloneqq\sigma(t_{n},y_{n},a,p_{n}^{y},\gamma)$,
note that there is $C>0$ such that

$\displaystyle\text{Tr}[\sigma\sigma^{\top}\\!(t_{n},y_{n},a,b^{\star}(t,y_{n},p_{n}^{y},\gamma,a))Q_{2}]-\text{Tr}[\sigma\sigma^{\top}\\!(t_{n},x_{n},a,b^{\star}(t,x_{n},p_{n}^{x},\gamma,a))Q_{1}]$
$\displaystyle=\text{Tr}\bigg{[}\begin{pmatrix}\Sigma^{x}{\Sigma^{x}}^{\top}&\Sigma^{x}{\Sigma^{y}}^{\top}\\\
\Sigma^{y}{\Sigma^{x}}^{\top}&\Sigma^{y}{\Sigma^{y}}^{\top}\end{pmatrix}\begin{pmatrix}Q_{2}&0\\\
0&-Q_{1}\end{pmatrix}\bigg{]}$ $\displaystyle\leq
3\alpha_{n}\text{Tr}\bigg{[}\begin{pmatrix}\Sigma^{x}{\Sigma^{x}}^{\top}&\Sigma^{x}{\Sigma^{y}}^{\top}\\\
\Sigma^{y}{\Sigma^{x}}^{\top}&\Sigma^{y}{\Sigma^{y}}^{\top}\end{pmatrix}\begin{pmatrix}I_{d}&-I_{d}\\\
-I_{d}&I_{d}\end{pmatrix}\bigg{]}+2\varepsilon\text{Tr}\bigg{[}\begin{pmatrix}\Sigma^{x}{\Sigma^{x}}^{\top}&\Sigma^{x}{\Sigma^{y}}^{\top}\\\
\Sigma^{y}{\Sigma^{x}}^{\top}&\Sigma^{y}{\Sigma^{y}}^{\top}\end{pmatrix}I_{2d}\bigg{]}$
$\displaystyle=3\alpha_{n}\text{Tr}\big{[}(\Sigma^{x}-\Sigma^{y})(\Sigma^{x}-\Sigma^{y})^{\top}\big{]}+2\varepsilon\text{Tr}\big{[}\Sigma^{x}{\Sigma^{x}}^{\top}\\!+\Sigma^{y}{\Sigma^{y}}^{\top}\big{]}$
$\displaystyle=3\alpha_{n}\|\Sigma^{x}-\Sigma^{y}\|^{2}+2\varepsilon\text{Tr}\big{[}\Sigma^{x}{\Sigma^{x}}^{\top}\\!+\Sigma^{y}{\Sigma^{y}}^{\top}\big{]}$
$\displaystyle\leq
3\alpha_{n}\|\sigma(t_{n},x_{n},p_{n}^{x},\gamma,a)-\sigma(t_{n},y_{n},p_{n}^{y},\gamma,a)\|^{2}+4\varepsilon
C_{\sigma\sigma^{\top}\\!}\leq
C\big{(}(1+\varepsilon)\alpha_{n}\|x_{n}-y_{n}\|^{2}+\varepsilon\big{)},$

where the first inequality follows from the right-hand side of (5.15),
$C_{\sigma\sigma^{\top}\\!}$ denotes the bound, assumed to exists, on
$\sigma\sigma^{\top}\\!$ and the last inequality follows from 5.1. Similarly,
note that there is a constant $C>0$ such that

$\displaystyle
c(t_{n},y_{n},p_{n}^{y},\gamma,a)-c(t_{n},x_{n},p_{n}^{x},\gamma,a)$
$\displaystyle\leq
C\big{(}\|x_{n}-y_{n}\|+\|b^{\star}(t_{n},y_{n},p_{n}^{y},\gamma,a)-b^{\star}(t_{n},x_{n},p_{n}^{x},\gamma,a)\|\big{)}$
$\displaystyle\leq C\big{(}\|x_{n}-y_{n}\|+\|p_{n}^{y}-p_{n}^{x}\|\big{)}\leq
C\big{(}\|x_{n}-y_{n}\|+\varepsilon\big{)},$

and

$\displaystyle\sigma\lambda(t_{n},x_{n},p_{n}^{x},\gamma,a)\cdot
p_{n}^{x}-\sigma\lambda(t_{n},y_{n},p_{n}^{y},\gamma,a)\cdot p_{n}^{y}$
$\displaystyle\leq\|\sigma\lambda(t_{n},x_{n},p_{n}^{x},\gamma,a)\|\|p_{n}^{x}-p_{n}^{y}\|+\|\sigma\lambda(t_{n},x_{n},p_{n}^{x},\gamma,a)-\sigma\lambda(t_{n},y_{n},p_{n}^{y},\gamma,a)\|\|p_{n}^{y}\|$
$\displaystyle\leq\varepsilon
C\|x_{n}-y_{n}\|+C\|p_{n}^{y}\|(1+\varepsilon)\|x_{n}-y_{n}\|\leq
C(1+\varepsilon+\varepsilon^{2}\|y_{n}\|)\big{(}\|x_{n}-y_{n}\|+\alpha_{n}\|x_{n}-y_{n}\|^{2}\big{)}.$

The result follows from using these estimates back in the Hamiltonian.

Step 5. We conclude. By Step 3 and the viscosity properties of $u^{\eta}$ and
$v$, we have that

$-q_{n}+H^{-}\big{(}t_{n}^{\beta,\varepsilon},x_{n}^{\beta,\varepsilon},\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})-\varepsilon
x_{n}^{\beta,\varepsilon},X_{n}^{\beta,\varepsilon}\big{)}\leq
0\leq-\hat{q}_{n}+H^{-}\big{(}t_{n}^{\beta,\varepsilon},y_{n}^{\beta,\varepsilon},\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})-\varepsilon
y_{n}^{\beta,\varepsilon},Y_{n}^{\beta,\varepsilon}\big{)}.$

Subtracting, we find from Step 4 that

$\displaystyle\beta=\hat{q}_{n}-q_{n}$ $\displaystyle\leq
H^{-}\big{(}t_{n}^{\beta,\varepsilon},y_{n}^{\beta,\varepsilon},\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})-\varepsilon
y_{n}^{\beta,\varepsilon},Y_{n}^{\beta,\varepsilon}\big{)}-H^{-}\big{(}t_{n}^{\beta,\varepsilon},x_{n}^{\beta,\varepsilon},\alpha_{n}(x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon})-\varepsilon
x_{n}^{\beta,\varepsilon},X_{n}^{\beta,\varepsilon}\big{)}$ $\displaystyle\leq
C(1+\varepsilon^{2}\|x_{n}^{\beta,\varepsilon}\|+\varepsilon^{2}\|y_{n}^{\beta,\varepsilon}\|+\varepsilon)\big{(}\alpha_{n}\|x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon}\|^{2}+\|x_{n}^{\beta,\varepsilon}-y_{n}^{\beta,\varepsilon}\|+\varepsilon\big{)}.$

Passing to the limit $n\longrightarrow\infty$ and $\varepsilon\longrightarrow
0$, thanks to (5.14), we find that $\beta\leq 0$ which is a contradiction. ∎

The next lemma proves, in particular, that the auxiliary value function
satisfies the hypotheses of Lemma 5.6.

######  Lemma 5.7.

Suppose the functions $H^{+}$ and $H^{-}$ are continuous. The functions
$w^{-}$ and $w^{+}$ from $[0,T]\times\mathbb{R}^{d}$ to $\mathbb{R}$ defined
in (5.5) are bounded and continuous.

For completing the last step in the verification result, we have assumed the
continuity of the Hamiltonian functions. We remark that this assumption holds,
for instance, if the optimisation over $\gamma$ in the definition of $H^{+}$
and $H^{-}$ can be reduced to a compact set, continuously with respect to
$(t,x,p,Q)$.

###### Proof of Lemma 5.7.

We prove the result for $w^{-}$, the other being analogous. We first argue
$w^{-}$ is bounded. Let $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and
$y>T\ell_{c}+\ell_{g}$. We claim that $(x,y)\in V_{g}(t)$. Indeed, taking the
control $Z\equiv 0$, $\Gamma\equiv 0$ and any $(\alpha,b^{\star})\in{\cal
A}\times{\cal B}^{\star}$ we have

$Y_{T}^{t,x,y,\upsilon}=y-\int_{t}^{T}c\big{(}s,X_{s}^{t,x,\upsilon},Z_{s},\hat{\upsilon}\big{)}\textrm{d}s\geq
y-T\ell_{c}>\ell_{g}\geq g(X_{T}^{t,x,\upsilon}).$

That is $w^{-}(t,x)\leq T\ell_{c}+\ell_{g}$. To obtain a lower bound take
again $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and $y<-T\ell_{c}-\ell_{g}$. Then,
it is easy to check that for any $M\in\mathbb{R}$ and any
$\upsilon\in{\mathfrak{C}}$ the following process is an
$(\mathbb{F},\mathbb{P})$–super-martingale

$A_{s}\coloneqq Y_{s}^{t,x,y,\upsilon}-s\ell_{c}+M,\;s\in[0,T].$

Thus, choosing $M=T\ell_{c}+\ell_{g}$, we have that
$\mathbb{E}^{\mathbb{P}}[Y_{T}^{t,x,y,\upsilon}-T\ell_{c}+M]\leq y+M<0$, which
implies $\mathbb{P}[Y_{T}^{t,x,y,\upsilon}+\ell_{g}<0]>0$. Therefore, for any
$\upsilon\in{\mathfrak{C}}$

$\mathbb{P}\big{[}Y_{T}^{t,x,y,\upsilon}<g(X_{T}^{t,x,\upsilon})\big{]}\geq\mathbb{P}\big{[}Y_{T}^{t,x,y,\upsilon}+\ell_{g}<0\big{]}>0,$

which means that the pair $(x,y)\not\in V_{g}^{1}(t)$. Thus,
$w^{-}(t,x)\geq-T\ell_{c}-\ell_{g}$.

Let us now prove the continuity. By [13, Theorem 2.1], $w^{-}$ is a
discontinuous viscosity solution to PDE (5.10) as long as we verify Assumption
2.1 therein. Indeed, the continuity condition on the set $N(t,x,p)$ holds in
our case given the explicit form that was obtained in (5.8). Since $H^{-}$ is
continuous, the lower– and upper–semi-continuous envelopes $w^{-}_{\star}$ and
$w^{-,\star}$ are viscosity super-solution and sub-solution, respectively, of
Equation 5.6. From [13, Theorem 2.2], which in our case is not subject to the
gradient constraints (see [13, Remark 2.1] and notice that in our setting
their set Nc is empty), we conclude that $w^{-,\star}(T,\cdot)\leq g\leq
w^{-}_{\star}(T,\cdot)$. Finally, from Lemma 5.6, we have therefore that
$w^{-,\star}\leq w^{-}_{\star}$ on $[0,T)\times\mathbb{R}^{d}$. Since the
reverse inequality holds by definition, we conclude the equality of the
semicontinuous envelopes and thus the continuity of $w^{-}$. ∎

###### Proof of Theorem 5.5.

The result is an immediate consequence of Lemmata 5.6 and 5.7. ∎

###  5.2 PDE characterisation for the problem of the leader

Having conducted the analysis of the auxiliary boundary functions $w^{-}$ and
$w^{+}$, we are in a position to provide a verification theorem for Problem
5.4. Theorem 5.8 below provides a PDE characterisation for the intermediate
problem of the leader under the CL information structure. Let us remark that
once $V(t,x,y)$ is found it only remains to optimise over $y\in\mathbb{R}$.

To ease the notation, we will use ${\rm x}\in\mathbb{R}^{d+1}$ and $u\in
A\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times{\cal B}^{\star}\eqqcolon U$,
to denote the values of the state and control processes associated with
Problem 5.4, that is, we make the convention ${\rm x}=(x,y)$ and
$u=(a,z,\gamma,b^{\star})$. In this way, recall (5.1), we let $C(t,{\rm
x},u)\coloneqq C(t,x,a,z,\gamma)$ and similarly for the other functions in the
analysis. Moreover, we denote the drift and volatility coefficients,
$(\mu,\vartheta):[0,T]\times\mathbb{R}^{d+1}\times
U\longrightarrow\mathbb{R}^{d+1}\times\mathbb{R}^{(d+1)\times n}$ associated
with the state process ${\rm X}\coloneqq(X,Y)$ by

$\displaystyle\mu(t,{\rm x},u)\coloneqq\begin{pmatrix}\sigma\lambda(t,x,u)\\\
-c(t,x,u)\end{pmatrix},\;\vartheta(t,{\rm
x},u)\coloneqq\begin{pmatrix}\sigma(t,x,u)\\\
z\cdot\sigma(t,x,u)\end{pmatrix}.$

Given $w\in C^{1,2}([0,T)\times\mathbb{R}^{d})$, we define the sets

$\displaystyle U^{-}(t,x,w)$ $\displaystyle\coloneqq\big{\\{}u\in
U:\sigma^{\top}\\!(t,x,u)(z-\partial_{x}w(t,x))=0,\;-\partial_{t}w(t,x)-h^{\rm
b}(t,x,\partial_{x}w(t,x),\partial_{xx}^{2}w(t,x),u)\geq 0\big{\\}},$
$\displaystyle U^{+}(t,x,w)$ $\displaystyle\coloneqq\big{\\{}u\in
U:\sigma^{\top}\\!(t,x,u)(z-\partial_{x}w(t,x))=0,\;-\partial_{t}w(t,x)-h^{\rm
b}(t,x,\partial_{x}w(t,x),\partial_{xx}^{2}w(t,x),u)\leq 0\big{\\}},$

and, for $i\in\\{-,+\\}$, introduce the Hamiltonians $(H^{\rm
L},H^{i,w}):[0,T]\times\mathbb{R}^{d+1}\times\mathbb{R}^{d+1}\times\mathbb{S}^{d+1}\longrightarrow\mathbb{R}$,
given by

$\displaystyle{H}^{\rm L}(t,{\rm x},{\rm p},{\rm Q})\coloneqq\sup_{u\in
U}\big{\\{}{h}^{\rm L}(t,{\rm x},{\rm p},{\rm Q},u)\big{\\}},\;{\rm
H}^{i,w}(t,{\rm x},{\rm p},{\rm Q})\coloneqq\sup_{u\in
U^{\text{$i$}}(t,x,w)}\big{\\{}{h}^{\rm L}(t,{\rm x},{\rm p},{\rm
Q},u)\big{\\}},$ (5.16)

where

${h}^{\rm L}(t,{\rm x},{\rm p},{\rm Q},u)\coloneqq C(t,x,u)+\mu(t,{\rm
x},u)\cdot{\rm p}+\frac{1}{2}{\rm Tr}[\vartheta\vartheta^{\top}\\!(t,{\rm
x},u){\rm Q}].$

Below, ${\cal T}_{T}$ denotes the family of $\mathbb{F}$–stopping times with
values on $[0,T]$. With this, we have all the elements necessary to state our
main result, which is the following verification theorem.

######  Theorem 5.8.

Let $w^{i}\in C^{1,2}([0,T)\times\mathbb{R}^{d})\cap
C^{0}([0,T]\times\mathbb{R}^{d})$, $i\in\\{-,+\\}$, be solutions to (5.6) and
(5.10), and $v\in C^{1,2}([0,T)\times\mathbb{R}^{d+1})\cap
C^{0}([0,T]\times\mathbb{R}^{d+1})$ satisfy

$\displaystyle\begin{cases}-\partial_{t}v(t,{\rm x})-{\rm H}^{\rm L}(t,{\rm
x},\partial_{\rm x}v(t,{\rm x}),\partial_{\rm xx}^{2}v(t,{\rm
x}))=0,\;(t,x,y)\in[0,T)\times\mathbb{R}^{d}\times(w^{-}(t,x),w^{+}(t,x)),\\\
-\partial_{t}v(t,{\rm x})-{\rm H}^{i,w^{\text{$i$}}}(t,{\rm x},\partial_{\rm
x}v(t,{\rm x}),\partial_{\rm xx}^{2}v(t,{\rm
x}))=0,\;(t,x,y)\in[0,T)\times\mathbb{R}^{d}\times\\{w^{i}(t,x)\\},\;i\in\\{-,+\\}.\\\
v(T^{-},{\rm x})=G(x),\;(x,y)\in\mathbb{R}^{d}\times\\{g(x)\\}.\end{cases}$
(5.17)

Moreover, suppose that

  * •

the family
$\\{v(\tau,{X}_{\tau}^{\upsilon},Y_{\tau}^{\upsilon})\\}_{\tau\in{\cal
T}_{T}}$ is uniformly integrable for all controls $\upsilon\in{\mathfrak{C}};$

  * •

there exists
$\upsilon^{\star}:[0,T]\times\mathbb{R}^{d}\times[w^{-},w^{+}]\longrightarrow
A\times\mathbb{R}^{d}\times\mathbb{S}^{d}\times{\cal B}^{\star}$ attaining the
maximisers in $H^{\rm L}$, $H^{i,w^{i}}$, $i\in\\{+,-\\};$191919Here,
$[w^{-},w^{+}]\coloneqq\\{y\in\mathbb{R}:w^{-}(t,x)\leq y\leq
w^{+}(t,x),\text{ for some }(t,x)\in[0,T]\times\mathbb{R}^{d}\\}$.

  * •

there is a strong solution to the system (5.2) with control
$(\alpha^{\star}_{\cdot},Z_{\cdot}^{\star},\Gamma_{\cdot}^{\star},b^{\star})\coloneqq\upsilon^{\star}(\cdot,X_{\cdot},Y_{\cdot});$
inlineinlinetodo: inlineUnique?

  * •

$(\alpha^{\star},Z^{\star},\Gamma^{\star},b^{\star})\in{\mathfrak{C}}$.

Then, $V(t,x,y)=v(t,x,y)$, and
$(\alpha^{\star},Z^{\star},\Gamma^{\star},b^{\star})$ is an optimal control
for the problem $V(t,x,y)$.

######  Remark 5.9.

We remark that we could build upon one of the main results of [14] to
characterise the functions $V$, $w^{+}$ and $w^{-}$ given by (5.4), (5.5), and
(5.9), respectively, as viscosity solutions to—a relaxed version of—(5.6),
(5.10) and (5.17), respectively. In particular, if one can show that $V$,
$w^{+}$, and $w^{-}$ are smooth and the associated Hamiltonians are
continuous, the relaxation reduces to the above system. We refer to [14] for
details. We have refrained from doing so as the above verification theorem
gives the result most useful in solving any example in practice. In Section
2.2, we use the above result and search for solution to the above system
directly.

###### Proof.

Let $t\in[0,T]$, $(x,y)\in V_{g}(t)$, $\upsilon\in\mathfrak{C}(t,x,y)$, and
$(X,Y)\coloneqq(X^{t,x,\upsilon},Y^{t,x,y,\upsilon})$ be given by (5.2). We
set ${\rm X}\coloneqq(X,Y)$. By Lemma 5.3, we have that $w^{-}(t,x)\leq y\leq
w^{+}(t,x),$ and by a comparison argument we find that $w^{-}(s,X_{s})\leq
Y_{s}\leq w^{+}(s,X_{s})$, $s\in[t,T]$, $\mathbb{P}\text{\rm--a.s.}$
inlineinlinetodo: inlineWhat do you mean by _comparison argument_? Let $\tau$
be given by

$\tau\coloneqq\inf\\{s>t:Y_{s}=w^{-}(s,X_{s}),\text{ or,
}Y_{s}=w^{+}(s,X_{s})\\}\wedge T.$

We now consider the process $v(t,{\rm X}_{t})\coloneqq v(t,X_{t},Y_{t})$ and
compute $v(t,{\rm X}_{t})-v(\theta,{\rm X}_{\theta})=v(t,{\rm
X}_{t})-v(\tau,{\rm X}_{\tau})+v(\tau,{\rm X}_{\tau})-v(\theta,{\rm
X}_{\theta})\eqqcolon I_{1}+I_{2}$, for $\theta\in{\cal T}_{T}$,
$\tau\leq\theta$. It follows from Itô’s formula that

$\displaystyle I_{1}$
$\displaystyle=-\int_{t}^{\tau}\Big{(}\partial_{t}v(s,{\rm
X}_{s})\mathrm{d}s+\frac{1}{2}{\rm Tr}[\partial_{\rm xx}^{2}v(s,{\rm
X}_{s})\mathrm{d}[{\rm X}]_{s}]\Big{)}-\int_{t}^{\tau}\partial_{\rm x}v(s,{\rm
X}_{s})\cdot\mathrm{d}{\rm X}_{s}$ $\displaystyle=\int_{t}^{\tau}\Big{(}{\rm
H}^{\rm L}\big{(}s,{\rm X}_{s},\partial_{\rm x}v(s,{\rm X}_{s}),\partial_{\rm
xx}^{2}v(s,{\rm X}_{s})\big{)}-{\rm h}^{\rm L}\big{(}s,{\rm
X}_{s},\partial_{\rm x}v(s,{\rm X}_{s}),\partial_{\rm xx}^{2}v(s,{\rm
X}_{s}),\upsilon_{s}\big{)}\Big{)}\mathrm{d}s$
$\displaystyle\quad+\int_{t}^{\tau}C(s,X_{s},\upsilon_{s})\mathrm{d}s-\int_{t}^{\tau}\big{(}\partial_{x}v(s,{\rm
X}_{s}),\partial_{y}v(s,{\rm
X}_{s})\big{)}^{\top}\cdot\big{(}\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s},Z_{s}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s}\big{)}^{\top}$
$\displaystyle\geq\int_{t}^{\tau}C(s,X_{s},\upsilon_{s})\mathrm{d}s-\int_{t}^{\tau}\Big{(}\partial_{x}v(s,{\rm
X}_{s})+\partial_{y}v(s,{\rm
X}_{s})Z_{s}\Big{)}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s},$ (5.18)

where we used the fact, on $[t,{\tau})$, $v$ satisfies the first equation in
(5.17), we computed the dynamics of ${\rm X}$ and added and subtracted $C$ to
complete the term $h^{\rm L}$. The inequality follows from the definition of
$H^{\rm L}$.

We now consider $I_{2}$. Without loss of generality, we assume that
$Y_{\tau}=w^{-}(\tau,X_{\tau})$, and thus, by the Markov property,
$Y_{s}=w^{-}(s,X_{s})$ for $s\in[\tau,T]$, $\mathbb{P}\text{\rm--a.s.}$
inlineinlinetodo: inlineWhy is that the Markov property? You seem to be saying
that the boundary is absorbing. Why is that true? By the uniqueness of their
Itô decomposition, we deduce that $Z_{t}=\partial_{x}w^{-}(t,X_{t})$, and
$\upsilon_{t}\in
N(t,X_{t},\partial_{x}w^{-}(t,X_{t})),\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm
--a.e.}$ With this, applying Itô’s formula to $w^{-}(\tau,X_{\tau})$ we find
that $\upsilon$ attains the infimum in (5.7); in particular, $\upsilon_{t}\in
U^{-}(t,X_{t}),\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm--a.s.}$ Let
$\bar{v}(t,x)\coloneqq v(t,x,w^{-}(t,x))$, so that

$\displaystyle I_{2}$
$\displaystyle=-\int_{\tau}^{\theta}\bigg{(}\partial_{t}\bar{v}(s,X_{s})\mathrm{d}s+\frac{1}{2}{\rm
Tr}[\partial_{xx}^{2}\bar{v}(s,X_{s})\mathrm{d}[X]_{s}]\bigg{)}-\int_{t}^{\theta}\partial_{x}\bar{v}(s,X_{s})\cdot\mathrm{d}X_{s}$
$\displaystyle=-\int_{\tau}^{\theta}\Big{(}\partial_{t}v(s,X_{s},w^{-}(s,X_{s}))+\sigma\lambda(s,X_{s},\upsilon_{s})\cdot\partial_{x}v(s,X_{s},w^{-}(s,X_{s}))\Big{)}\mathrm{d}s-\int_{\tau}^{\theta}\frac{1}{2}{\rm
Tr}[\partial_{xx}^{2}\bar{v}(s,X_{s})\mathrm{d}[X]_{s}]$
$\displaystyle\quad-\int_{\tau}^{\theta}\partial_{y}v(s,X_{s},w^{-}(s,X_{s}))\Big{(}\partial_{t}w^{-}(s,X_{s})+\sigma\lambda(s,X_{s},\upsilon_{s})\cdot\partial_{x}w^{-}(s,X_{s})\Big{)}\mathrm{d}s$
$\displaystyle\quad-\int_{\tau}^{\theta}\Big{(}\partial_{x}v(s,X_{s},w^{-}(s,X_{s}))+\partial_{y}v(s,X_{s},w^{-}(s,X_{s}))\partial_{x}w^{-}(s,X_{s})\Big{)}\cdot\sigma(s,X_{s},u_{s})\mathrm{d}W_{s}$
$\displaystyle=-\int_{\tau}^{\theta}\Big{(}\partial_{t}v(s,X_{s},w^{-}(s,X_{s}))+\sigma\lambda(s,X_{s},\upsilon_{s})\cdot\partial_{x}v(s,X_{s},w^{-}(s,X_{s}))-\partial_{y}v(s,X_{s},w^{-}(s,X_{s}))c(s,X_{s},\upsilon_{s})\Big{)}\mathrm{d}s$
$\displaystyle\quad-\int_{\tau}^{\theta}\bigg{(}\frac{1}{2}{\rm
Tr}[\partial_{xx}^{2}\bar{v}(s,X_{s})\mathrm{d}[X]_{s}]-\frac{1}{2}\partial_{y}v(s,X_{s},w^{-}(s,X_{s})){\rm
Tr}[\partial^{2}_{xx}w^{-}(s,X_{s})\mathrm{d}[X]_{s}]\bigg{)}$
$\displaystyle\quad-\int_{\tau}^{\theta}\partial_{y}v(s,X_{s},w^{-}(s,X_{s}))\bigg{(}\partial_{t}w^{-}(s,X_{s})+h^{\rm
b}(s,X_{s},\partial_{x}w^{-}(s,X_{s}),\partial^{2}_{xx}w^{-}(s,X_{s}),\upsilon_{s})\bigg{)}\mathrm{d}s$
$\displaystyle\quad-\int_{\tau}^{\theta}\Big{(}\partial_{x}v(s,X_{s},w^{-}(s,X_{s}))+\partial_{y}v(s,X_{s},w^{-}(s,X_{s}))\partial_{x}w^{-}(s,X_{s})\Big{)}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s}$

where in the first equality, we computed the time and space derivatives of
$\bar{v}$, the dynamics of $X$, and in the second equality, we added and
subtracted $\partial_{y}v\big{(}c+\frac{1}{2}{\rm
Tr}[\sigma\sigma^{\top}\\!\partial^{2}_{xx}w^{-}]\big{)}$ and use the fact
that $Z_{\cdot}=\partial_{x}w^{-}(\cdot,X_{\cdot})$ to complete the term
$h^{\rm b}$ in the third line.

Recalling that $\upsilon$ attains the infimum in (5.7), we see the term
$\partial_{t}w^{-}+h^{\rm b}$ equals $0$. Moreover, since
$Z_{\cdot}=\partial_{x}w^{-}(\cdot,X_{\cdot})$, ${\rm
Tr}\big{[}\partial^{2}_{\rm xx}v(t,{\rm X}_{t})\mathrm{d}[{\rm
X}]_{t}\big{]}={\rm
Tr}\big{[}\partial^{2}_{xx}\bar{v}(t,{X}_{t})\mathrm{d}[{X}]_{t}\big{]}-\partial_{y}v(t,X_{t},w^{-}(t,{X}_{t})){\rm
Tr}\big{[}\partial^{2}_{xx}w^{-}(t,{X}_{t})\mathrm{d}[{X}]_{t}\big{]},\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm
--a.e.}$ Consequently,

$\displaystyle I_{2}$
$\displaystyle=-\int_{\tau}^{\theta}\Big{(}\partial_{t}v(s,{\rm X}_{s})+{\rm
h}^{\rm L}\big{(}s,{\rm X}_{s},\partial_{\rm x}v(s,{\rm X}_{s}),\partial_{\rm
xx}^{2}v(s,{\rm X}_{s}),\upsilon_{s}\big{)}\Big{)}\mathrm{d}s$
$\displaystyle\quad+\int_{\tau}^{\theta}C(s,X_{s},\upsilon_{s})\mathrm{d}s-\int_{\tau}^{\theta}\Big{(}\partial_{x}v(s,{\rm
X}_{s})+\partial_{y}v(s,{\rm
X}_{s})\partial_{x}w^{-}(s,X_{s})\Big{)}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s}$
$\displaystyle=\int_{\tau}^{\theta}\Big{(}{\rm H}^{-,w^{-}}\\!\big{(}s,{\rm
X}_{s},\partial_{\rm x}v(s,{\rm X}_{s}),\partial_{\rm xx}^{2}v(s,{\rm
X}_{s})\big{)}-{\rm h}^{\rm L}\big{(}s,{\rm X}_{s},\partial_{\rm x}v(s,{\rm
X}_{s}),\partial_{\rm xx}^{2}v(s,{\rm
X}_{s}),\upsilon_{s}\big{)}\Big{)}\mathrm{d}s$
$\displaystyle\quad+\int_{\tau}^{\theta}C(s,X_{s},\upsilon_{s})\mathrm{d}s-\int_{\tau}^{\theta}\Big{(}\partial_{x}v(s,{\rm
X}_{s})+\partial_{y}v(s,{\rm
X}_{s})\partial_{x}w^{-}(s,X_{s})\Big{)}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s}$
$\displaystyle\geq\int_{\tau}^{\theta}C(s,X_{s},\upsilon_{s})\mathrm{d}s-\int_{\tau}^{\theta}\Big{(}\partial_{x}v(s,{\rm
X}_{s})+\partial_{y}v(s,{\rm
X}_{s})\partial_{x}w^{-}(s,X_{s})\Big{)}\cdot\sigma(s,X_{s},\upsilon_{s})\mathrm{d}W_{s},$
(5.19)

where in the first equality we added and subtracted $C$ to complete the
expression for $h^{\rm L}$, and in the second equality, we used the fact that
$v$ satisfies the second equation in (5.17) for ${\rm
X}_{\cdot}=(X_{\cdot},w^{-}(\cdot,X_{\cdot}))$. The inequality follows from
the definition of $H^{-,w^{-}}$ and the fact that $\upsilon_{\cdot}\in
N(\cdot,X_{\cdot},\partial_{x}w^{-}(\cdot,X_{\cdot})),\mathrm{d}t\otimes\mathrm{d}\mathbb{P}\text{\rm
--a.e.}$

Let $(\theta_{n})_{n\geq 1}\subseteq{\cal T}_{T}$,
$t\leq\theta_{n}\leq\theta_{n+1}$, $n\geq 1$, $\theta_{n}\longrightarrow T$,
$\mathbb{P}\text{\rm--a.s.}$, be such that $(X,Y)$ is bounded on
$[t,\theta_{n}]$.We now add (5.2) and (5.2), and take
$\theta=\theta_{n}$.202020Note that whenever $\tau=T$, $I_{2}=0$ and
$\theta_{n}\longrightarrow\tau=T$. By continuity, the terms $v$, $w^{+}$, and
their derivates are bounded on $[t,\theta_{n}]$. Thus, since $\sigma$ is
bounded and
$\|Z\|_{\mathbb{H}^{\raisebox{1.0pt}{\text{$p$}}}(\mathbb{F},\mathbb{P})}^{p}<\infty$,
the stochastic integrals in (5.2) and (5.2) are martingales. Consequently,

$\displaystyle
v(t,x,y)\geq\mathbb{E}\bigg{[}v(\theta_{n},X_{\theta_{n}},Y_{\theta_{n}})+\int_{t}^{\theta_{n}}C(s,X_{s},\upsilon_{s})\mathrm{d}s\bigg{|}{\cal
F}_{t}\bigg{]}.$

Thus, the uniform integrability of the family
$\\{v(\theta_{n},X_{\theta_{n}},Y_{\theta_{n}})\\}_{n\geq 1}$, the boundedness
of $C$, together with an application of dominated convergence, gives

$\displaystyle
v(t,x,y)\geq\mathbb{E}\bigg{[}G(X_{T})+\int_{t}^{T}C(s,X_{s},\upsilon_{s})\mathrm{d}s\bigg{|}{\cal
F}_{t}\bigg{]},$ (5.20)

where we used the boundary condition in time in (5.17) and that by the
feasibility of $\upsilon$, if $\tau=T$ we have that $Y_{T}=g(X_{T})$, and if
$\tau<T$ we have that $w^{-}(T^{-},x)=g(x)$, see (5.10). The arbitrariness of
$\upsilon$ gives $v\geq V$. To conclude, note that for
$(Z^{\star},\Gamma^{\star},\alpha^{\star})$ as in the statement, the
inequalities in (5.2) and (5.2) are tight. ∎

inlineinlinetodo: inlineI guess the discussion of what happens in the
pessimistic case is missing, right? We should mention there that the value
looks like the lower value of a zero-sum game where the leader would maximise
his controls, but the follower would now simply minimise the criterion of the
leader over his best responses. Then informally one would put the lower HJBI
equation for this instead of the one we put.

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##  Appendix A Illustrative example: additional proofs

###  A.1 The first-best case

The proof of Lemma 2.1 is very straightforward using standard HJB techniques
or even by pointwise optimisation, as one can compute

$\displaystyle J_{\rm
L}(\alpha,\beta)\coloneqq\mathbb{E}^{\mathbb{P}}\bigg{[}X_{T}-\dfrac{c_{\rm
L}}{2}\int_{0}^{T}\alpha_{t}^{2}\mathrm{d}t\bigg{]}=\mathbb{E}^{\mathbb{P}}\bigg{[}\int_{0}^{T}\bigg{(}\alpha_{t}+\beta_{t}-\dfrac{c_{\rm
L}}{2}\alpha_{t}^{2}\bigg{)}\mathrm{d}t\bigg{]},$

and directly verify that the optimal efforts are the one defined in Lemma 2.1.

###  A.2 ACLM formulation

######  Lemma A.1.

For $k>0$, consider the closed-loop memoryless strategy $a_{k}\in{\cal A}$
defined for all $t\in[0,T]$ by

$\displaystyle
a_{k}(t,X_{t})\coloneqq\Pi_{[0,a_{\circ}]}\bigg{(}\frac{1}{c_{\rm
L}}+k(X_{t}-X_{t}^{\star})\bigg{)},\;\text{\rm
where}\;X^{\star}_{t}=x_{0}+\dfrac{t}{c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}}{kc_{\rm F}}\big{(}1-\mathrm{e}^{-kt}\big{)}+\sigma
W_{t},\;t\in[0,T].$

Define $\bar{K}\coloneqq\frac{1}{T}\log(b_{\circ}c_{\rm F})$. Then, for a
fixed $k\in(0,\bar{K}]$ and the associated strategy $a_{k}$, the leader
obtains the following reward, which is higher than his value in the AOL
information case

$f(k)=x_{0}+\dfrac{T}{2c_{\rm L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}.$

Moreover, if $a_{\circ}>\frac{1}{c_{\rm L}}+b_{\circ}(b_{\circ}c_{\rm F}-1)$
and $a_{\circ}>\frac{1}{2c_{\rm F}}(b_{\circ}^{2}c_{\rm
F}^{2}-1)-\frac{1}{c_{\rm L}}$, then the solution to the ACLM problem is equal
to the solution to the ACLM-$\bar{K}$ problem, with value $f(\bar{K})$.

###### Proof of Lemma A.1.

$(i)$ To provide the main intuition, suppose first that the leader’s actions
are unrestricted, that is $A=\mathbb{R}$. This is the usual setting for the
ACLM problems that are solved explicitly in the literature. Then, the leader
announces her strategy $\alpha_{k}\in{\cal A}$ defined by

$a_{k}(t,X_{t})=\frac{1}{c_{\rm L}}+k(X_{t}-X_{t}^{\star}),\;t\in[0,T].$

Then, the follower’s optimisation problem originally defined in (2.2) is the
following

$\displaystyle V_{\rm F}(\alpha_{k})\coloneqq\sup_{\beta\in{\cal
B}}\mathbb{E}^{\mathbb{P}}\bigg{[}X_{T}-\dfrac{c_{\rm
F}}{2}\int_{0}^{T}\beta_{t}^{2}\mathrm{d}t\bigg{]},\;\text{subject
to}\;\mathrm{d}X_{t}=\bigg{(}\frac{1}{c_{\rm
L}}+k(X_{t}-X_{t}^{\star})+\beta_{t}\bigg{)}\mathrm{d}t+\sigma\mathrm{d}W_{t},\;t\in[0,T].$
(A.1)

As described in Section 2.1.3, one can use the stochastic maximum principle to
obtain, after solving the appropriate FBSDE system, that the optimal response
of the follower is given by

$\beta^{\star}_{t}=\Pi_{[0,b_{\circ}]}\bigg{(}\frac{\mathrm{e}^{k(T-t)}}{c_{\rm
F}}\bigg{)},\;t\in[0,T].$ (A.2)

Alternatively, one can solve this stochastic control problem in a more
straightforward way, by noticing that the follower’s problem defined above by
(A.1) can be rewritten as

$\displaystyle V_{\rm F}(\alpha_{k})=\sup_{\beta\in{\cal
B}}\mathbb{E}^{\mathbb{P}}\bigg{[}X^{\star}_{T}+\widetilde{X}_{T}-\dfrac{c_{\rm
F}}{2}\int_{0}^{T}\beta_{t}^{2}\mathrm{d}t\bigg{]}=x_{0}+\dfrac{T}{c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}+\sup_{\beta\in{\cal
B}}\mathbb{E}^{\mathbb{P}}\bigg{[}\sigma W_{T}+\widetilde{X}_{T}-\dfrac{c_{\rm
F}}{2}\int_{0}^{T}\beta_{t}^{2}\mathrm{d}t\bigg{]},$

where the process $\widetilde{X}\coloneqq X-X^{\star}$, corresponding to the
only state variable of the previous control problem, satisfies the following
controlled ODE

$\displaystyle\mathrm{d}\widetilde{X}_{t}=\bigg{(}k\widetilde{X}_{t}+\beta_{t}-\dfrac{1}{c_{\rm
F}}\mathrm{e}^{k(T-t)}\bigg{)}\mathrm{d}t,\;t\in[0,T],\;\widetilde{X}_{0}=0,$
(A.3)

whose solution is given by

$\widetilde{X}_{t}\coloneqq\mathrm{e}^{kt}\int_{0}^{t}\mathrm{e}^{-ks}\bigg{(}\beta_{s}-\frac{\mathrm{e}^{k(T-s)}}{c_{\rm
F}}\bigg{)}\mathrm{d}s=\mathrm{e}^{kt}\int_{0}^{t}\mathrm{e}^{-ks}\beta_{s}\mathrm{d}s-\frac{1}{2kc_{\rm
F}}\mathrm{e}^{kT}\big{(}\mathrm{e}^{kt}-\mathrm{e}^{-kt}\big{)},\;\forall
t\in[0,T].$

The follower’s optimisation problem thus becomes

$\displaystyle V_{\rm F}(\alpha_{k})$ $\displaystyle=x_{0}+\dfrac{T}{c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}+\sup_{\beta\in{\cal
B}}\bigg{\\{}\mathrm{e}^{kT}\int_{0}^{T}\mathrm{e}^{-kt}\beta_{t}\mathrm{d}t-\frac{1}{2kc_{\rm
F}}\mathrm{e}^{kT}\big{(}\mathrm{e}^{kT}-\mathrm{e}^{-kT}\big{)}-\dfrac{c_{\rm
F}}{2}\int_{0}^{T}\beta_{t}^{2}\mathrm{d}t\bigg{\\}}$
$\displaystyle=x_{0}+\dfrac{T}{c_{\rm L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm
F}}-\frac{1}{2kc_{\rm F}}\big{(}\mathrm{e}^{2kT}-1\big{)}+\sup_{\beta\in{\cal
B}}\int_{0}^{T}\bigg{(}\mathrm{e}^{k(T-t)}\beta_{t}-\dfrac{c_{\rm
F}}{2}\beta_{t}^{2}\bigg{)}\mathrm{d}t.$

The optimal effort $\beta^{\star}$ introduced above in (A.2) is deduced by
pointwise optimisation.

$(i.1)$ Suppose first, that $k<\bar{K}$ so then
$\beta^{\star}_{t}=\frac{1}{c_{\rm F}}\mathrm{e}^{k(T-t)}$. The associated
value for the follower is given by

$\displaystyle V_{\rm F}(\alpha_{k})$ $\displaystyle=x_{0}+\dfrac{T}{c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}-\frac{1}{2kc_{\rm
F}}\big{(}\mathrm{e}^{2kT}-1\big{)}+\frac{1}{2c_{\rm
F}}\int_{0}^{T}\mathrm{e}^{2k(T-t)}\mathrm{d}t=x_{0}+\dfrac{T}{c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}-\frac{1}{4kc_{\rm
F}}\big{(}\mathrm{e}^{2kT}-1\big{)}.$

Remark that for the optimal control $\beta^{\star}$, the controlled ODE (A.3)
simplifies, and gives the trivial solution $\widetilde{X}_{t}=0$, i.e.
$X_{t}=X_{t}^{\star}$, for all $t\in[0,T]$. In other words, the best choice
for the follower is to choose $\beta$ so that the process $X$ coincides with
the process $X^{\star}$. Given the follower’s optimal response, the objective
value of the leader for the strategy $\alpha_{k}$ simplifies to

$\displaystyle\mathbb{E}^{\mathbb{P}}\bigg{[}X_{T}-\dfrac{c_{\rm
L}}{2}\int_{0}^{T}\big{(}a_{k}(t,X_{t})\big{)}^{2}\mathrm{d}t\bigg{]}$
$\displaystyle=\mathbb{E}^{\mathbb{P}}\bigg{[}X_{T}^{\star}-\dfrac{c_{\rm
L}}{2}\int_{0}^{T}\bigg{(}\frac{1}{c_{\rm
L}}+k(X_{t}-X_{t}^{\star})\bigg{)}^{2}\mathrm{d}t\bigg{]}=x_{0}+\dfrac{T}{2c_{\rm
L}}+\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm F}}\eqqcolon f(k).$

Notice that $f$ is an increasing function of $k$, and that its limit when $k$
goes to $0$ is given by

$\displaystyle\lim_{k\rightarrow 0}f(k)=x_{0}+\dfrac{T}{2c_{\rm
L}}+\lim_{k\rightarrow 0}\dfrac{\mathrm{e}^{kT}-1}{kc_{\rm
F}}=x_{0}+\bigg{(}\dfrac{1}{2c_{\rm L}}+\dfrac{1}{c_{\rm F}}\bigg{)}T.$

As this value corresponds to the leader’s value function in the AOL case, we
immediately conclude that her value with the closed-loop memoryless strategy
$\alpha_{k}$ would provide her a better value, as soon as
$k\in(0,\frac{1}{T}\log(b_{\circ}c_{\rm F}))$. Similarly, we have

$\displaystyle\lim_{k\rightarrow 0}V_{\rm F}(\alpha_{k})$
$\displaystyle=x_{0}+\bigg{(}\dfrac{1}{c_{\rm L}}+\dfrac{1}{2c_{\rm
F}}\bigg{)}T.$

$(i.2)$ Take now $k>\bar{K}$ so we have

$\beta^{\star}_{t}=b_{\circ}{\bf 1}_{\\{t<t_{0}\\}}+\frac{e^{k(T-t)}}{c_{\rm
F}}{\bf 1}_{t\in[t_{0},T]},\;\text{where}\;t_{0}\coloneqq
T-\frac{\log(b_{\circ}c_{\rm F})}{k}.$

In this case the follower is not able to keep the process $X$ equal to
$X^{\star}$. Notice that $\beta^{\star}_{t}c_{\rm F}<\mathrm{e}^{k(T-t)}$ for
every $t\in[0,t_{0}]$ so then $\widetilde{X}_{t}$ is negative over $[0,t_{0})$
and remains so over $[t_{0},T]$. Then, we have that
$a_{k}(t,X_{t})<\frac{1}{c_{\rm L}}$ for every $t>0$. In fact, we can compute
explicitly

$\widetilde{X}_{t}=\begin{cases}\displaystyle\frac{b_{\circ}}{k}(\mathrm{e}^{kt}-1)-\frac{1}{2kc_{\rm
F}}\big{(}\mathrm{e}^{k(t+t_{\text{$0$}})}-\mathrm{e}^{k(t-t_{\text{$0$}})}\big{)},\;t\in[0,t_{0}],\\\\[5.0pt]
\displaystyle\mathrm{e}^{k(t-t_{\text{$0$}})}\widetilde{X}_{t_{\text{$0$}}^{\text{$\star$}}},\;t\in[t_{0},T].\end{cases}$

The utility for the leader from this strategy is given by

$\displaystyle J_{\rm L}(a_{k},\beta^{\star}(a_{k}))$
$\displaystyle=\mathbb{E}^{\mathbb{P}}\bigg{[}X_{T}-\dfrac{c_{\rm
L}}{2}\int_{0}^{T}\alpha_{t}^{2}\mathrm{d}t\bigg{]}$
$\displaystyle=x_{0}+\frac{T}{c_{\rm L}}+\frac{\mathrm{e}^{kT}-1}{kc_{\rm
F}}+\frac{b_{\circ}\mathrm{e}^{kT}}{k}-\frac{b_{\circ}^{2}c_{\rm
F}}{2k}-\frac{\mathrm{e}^{2kT}}{2kc_{\rm F}}-\dfrac{c_{\rm
L}}{2}\mathbb{E}^{\mathbb{P}}\bigg{[}\int_{0}^{T}\bigg{(}\frac{1}{c_{\rm
L}}+k\widetilde{X}_{t}\bigg{)}^{2}\mathrm{d}t\bigg{]}.$

It can be seen numerically that the previous expression is decreasing in $k$.
Combining this with the deductions from part $(i.1)$, the (unrestricted)
leader should choose the value $k=\bar{K}$ which provides her the highest
utility.

inlineinlinetodo: inlineThis is not an acceptable argument… We have to prove
that this is non-increasing in $k$. Also the notations are misleading since
$\widetilde{X}$ actually depends on $k$, and so does $t_{0}$. It is also
completely deterministic so the expectation is useless.

(ii) We now discuss the problem in our setting, when $A=[0,a_{\circ}]$. To
simplify the solution of this example, we can choose the parameters
$a_{\circ}$ and $b_{\circ}$ properly so the solution of the case $(i)$ is
admissible in the restricted problem. This implies that the solution found in
the case $(i)$ is also the solution to the ACLM problem with bounded effort
for the leader. Indeed, note that the process $\widetilde{X}_{t}$ defined in
(A.3) is bounded since $B=[0,b_{\circ}]$. Namely, we have

$\frac{1}{2kc_{\rm
F}}(\mathrm{e}^{k(T-t)}-\mathrm{e}^{k(T+t)})\leq\widetilde{X}_{t}\leq\frac{1}{2kc_{\rm
F}}(\mathrm{e}^{k(T-t)}-\mathrm{e}^{k(T+t)})+\frac{b_{\circ}}{k}(\mathrm{e}^{kt}-1),\;t\in[0,T].$

Therefore the strategy $a_{k}$ is guaranteed to take values in
$A=[-a_{\circ},a_{\circ}]$, for every response $\beta\in{\cal B}$, for
instance, if $a_{\circ}>\frac{1}{c_{\rm L}}+b_{\circ}(b_{\circ}c_{\rm F}-1)$
and $a_{\circ}>\frac{1}{2c_{\rm F}}(b_{\circ}^{2}c_{\rm
F}^{2}-1)-\frac{1}{c_{\rm L}}$.

∎

##  Appendix B Functional spaces

We introduce the spaces used in this paper, by following [58]. Let
$(t,x)\in[0,T]\times\Omega$, $(\mathcal{P}(t,x))_{t\in[0,T]\times x\in\Omega}$
be a family of sets of probability measures on $(\Omega,\mathcal{F}_{T})$. In
this section, we denote by $\mathbb{X}\coloneqq(\mathcal{X}_{s})_{s\in[0,T]}$
a general filtration on $(\Omega,\mathcal{F}_{T})$. Let $p\geq 1$,
$\mathbb{P}\in\mathcal{P}(t,x)$ and $\mathbb{X}_{\mathbb{P}}$ the usual
$\mathbb{P}$-augmented filtration associated with $\mathbb{X}$.

  * •

$\mathbb{H}^{p}_{t,x}(\mathbb{X},\mathbb{P})$ (resp.
$\mathbb{H}^{p}_{t,x}(\mathbb{X},\mathcal{P})$) denotes the spaces of
$\mathbb{X}$-predictable $\mathbb{R}^{d}$-valued processes $Z$ such that

$\|Z\|_{\mathbb{H}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}^{p}\coloneqq\mathbb{E}^{\mathbb{P}}\bigg{[}\bigg{(}\int_{t}^{T}\|\widehat{\sigma}_{s}^{\top}\\!Z_{s}\|^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}<+\infty,\;\;\bigg{(}\text{resp.
}\|Z\|_{\mathbb{H}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathcal{P})}^{p}\coloneqq\sup_{\mathbb{P}\in\mathcal{P}(t,x)}\|Z\|_{\mathbb{H}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}^{p}<+\infty\bigg{)}.$

  * •

$\mathbb{S}^{p}_{t,x}(\mathbb{X},\mathbb{P})$ (resp.
$\mathbb{S}^{p}_{t,x}(\mathbb{X},\mathcal{P})$) denotes the spaces of
$\mathbb{X}$–progressively measurable $\mathbb{R}$-valued processes $Y$ such
that

$\|Y\|_{\mathbb{S}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}^{p}\coloneqq\mathbb{E}^{\mathbb{P}}\bigg{[}\sup_{s\in[t,T]}|Y_{s}|^{p}\bigg{]}<+\infty,\;\;\bigg{(}\text{resp.
}\|Y\|_{\mathbb{S}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathcal{P})}^{p}\coloneqq\sup_{\mathbb{P}\in\mathcal{P}(t,x)}\|Y\|_{\mathbb{S}^{p}_{t,x}(\mathbb{X},\mathbb{P})}^{p}<+\infty\bigg{)}.$

  * •

$\mathbb{I}^{p}_{t,x}(\mathbb{X},\mathbb{P})$ (resp.
$\mathbb{I}^{p}_{t,x}(\mathbb{X},\mathcal{P})$) denotes the spaces of
$\mathbb{X}$-optional $\mathbb{R}$-valued processes $K$ with
$\mathbb{P}\text{\rm--a.s.}$ càdlàg and non-decreasing paths on $[t,T]$ with
$K_{t}=0,\;\mathbb{P}\text{\rm--a.s.}$ and

$\|K\|_{\mathbb{I}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}^{p}\coloneqq\mathbb{E}^{\mathbb{P}}[K_{T}^{p}]<+\infty,\;\;\bigg{(}\text{resp.
}\|K\|_{\mathbb{I}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathcal{P})}^{p}\coloneqq\sup_{\mathbb{P}\in\mathcal{P}(t,x)}\|K\|^{p}_{\mathbb{I}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}<+\infty\bigg{)}.$

  * •

$\mathbb{G}^{p}_{t,x}(\mathbb{X},\mathbb{P})$ denotes the spaces of
$\mathbb{X}$-predictable $\mathbb{S}^{d}$-valued processes $\Gamma$ such that

$\|\Gamma\|_{\mathbb{G}^{\text{$p$}}_{\text{$t$}\text{$,$}\text{$x$}}(\mathbb{X},\mathbb{P})}^{p}\coloneqq\mathbb{E}^{\mathbb{P}}\bigg{[}\bigg{(}\int_{t}^{T}\big{\|}\widehat{\sigma}^{2}_{s}\Gamma_{s}\big{\|}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}<+\infty.$

When $t=0$, we simplify the previous notations by omitting the dependence on
both $t$ and $x$.