Next, by Assumption 2.7 (i), for $(t,x)\in[0,T]\times\mathbb{R}$ the maps
$\mathcal{M}(\mathbb{R}^{d})\ni v\mapsto\tilde{\lambda}(t,x,v)$ and
$\mathcal{M}(\mathbb{R}^{d})\ni v\mapsto\tilde{b}_{0}(t,x,v)$ have the linear
functional derivatives $D\tilde{\lambda}$ and $D\tilde{b}_{0}$ given by

$\displaystyle D\tilde{\lambda}(t,x,v)(y)$
$\displaystyle=D\lambda\bigl{(}t,x+\sigma_{0}W_{t},(\operatorname{id}+\sigma_{0}W_{t})^{\\#}v\bigr{)}(y+\sigma_{0}W_{t}),$
$\displaystyle D\tilde{b}_{0}(t,x,v)(y)$
$\displaystyle=Db_{0}\bigl{(}t,x+\sigma_{0}W_{t},(\operatorname{id}+\sigma_{0}W_{t})^{\\#}v\bigr{)}(y+\sigma_{0}W_{t})$

for
$(t,x,v,y)\in[0,T]\times\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\times\mathbb{R}^{d}\to\mathbb{R}$.
Thus, we obtain that

$\displaystyle\tilde{\lambda}(t,x,\mu^{\epsilon}_{t})-\tilde{\lambda}(t,x,\mu_{t})$
$\displaystyle=\int_{0}^{1}\Bigl{\langle}\mu^{\epsilon}_{t}-\mu_{t},D\tilde{\lambda}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}\theta$
$\displaystyle=\int_{0}^{1}\Bigl{\langle}\epsilon\mathcal{V}^{\epsilon}_{t},D\tilde{\lambda}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}\theta$
(4.10)

and, similarly,

$\tilde{b}_{0}(t,x,\mu^{\epsilon}_{t})-\tilde{b}_{0}(t,x,\mu_{t})=\int_{0}^{1}\Bigl{\langle}\epsilon\mathcal{V}^{\epsilon}_{t},D\tilde{b}_{0}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}\theta.$
(4.11)

We set
$\Lambda^{\epsilon,\theta}_{t}(x,y)=D\tilde{\lambda}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}(y)$
and
$B^{\epsilon,\theta}_{t}(x,y)=D\tilde{b}_{0}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}(y)$
for
$(t,\omega,x,y)\in[0,T]\times\Omega\times\mathbb{R}^{d}\times\mathbb{R}^{d}$.
Then, we insert the expressions
$\tilde{b}_{1}(t,x,\gamma^{\epsilon}_{t}(x))-\tilde{b}_{1}(t,x,\gamma_{t}(x))=\epsilon(h_{t}(x))^{\top}\nabla_{g}\tilde{b}_{1}(t,x)$,
(4.3), and (4.11) into Equation (4.9), which yields

$\displaystyle\begin{split}\mathrm{d}\langle\mathcal{V}^{\epsilon}_{t},\varphi\rangle=&\bigl{\langle}\mathcal{V}^{\epsilon}_{t},\tilde{\mathcal{L}}\varphi(t,\cdot,\mu^{\epsilon}_{t},\gamma^{\epsilon}_{t})\big{\rangle}\,\mathrm{d}t+\bigl{\langle}\mu_{t},h_{t}^{\top}\nabla_{g}\tilde{b}_{1}(t,\cdot)\nabla\varphi\bigr{\rangle}\,\mathrm{d}t\\\
&+\biggl{(}\int_{0}^{1}\biggl{(}\int_{\mathbb{R}^{d}}\Bigl{\langle}\mu_{t},\Lambda^{\epsilon,\theta}_{t}(\cdot,x)\varphi+B^{\epsilon,\theta}_{t}(\cdot,x)\cdot\nabla\varphi\Bigr{\rangle}\mathcal{V}^{\epsilon}_{t}(x)\,\mathrm{d}x\biggr{)}\mathrm{d}\theta\biggr{)}\,\mathrm{d}t\end{split}$
(4.12)

for $\varphi\in C^{2}_{c}(\mathbb{R}^{d})$. We will show that the family
$(\mathcal{V}^{\epsilon})_{\epsilon}$ is Cauchy in
$C_{\mathbb{F}}^{2}([0,T];L_{\eta}^{2}(\mathbb{R}^{d}))\cap
L_{\mathbb{F}}^{2}([0,T];H_{\eta}^{1}(\mathbb{R}^{d}))$.

We begin with a uniform bound on $\sup_{0\leq t\leq
T}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}^{2}$. Clearly, the PDE (4.12)
has the same form as PDE (4.1) with $\alpha$, $\beta$, $h^{0}$, and $h^{1}$
replaced by $\mathfrak{l}^{\epsilon}$, $\mathfrak{b}^{\epsilon}$,
$\mathfrak{h}^{\epsilon,0}$, and $\mathfrak{h}^{\epsilon,1}$, respectively,
where the latter two are defined as

$\displaystyle\mathfrak{h}^{\epsilon,0}_{t}(x)$
$\displaystyle=\mu_{t}(x)\int_{0}^{1}\bigl{\langle}\Lambda^{\epsilon,\theta}_{t}(x,\cdot),\mathcal{V}^{\epsilon}_{t}\bigr{\rangle}\,\mathrm{d}\theta,$
$\displaystyle\mathfrak{h}^{\epsilon,1}_{t}(x)$
$\displaystyle=\mu_{t}(x)(h_{t}(x))^{\top}\nabla_{g}\tilde{b}_{1}(t,x)+\mu_{t}(x)\int_{0}^{1}\bigl{\langle}B^{\epsilon,\theta}_{t}(x,\cdot),\mathcal{V}^{\epsilon}_{t}\bigr{\rangle}\,\mathrm{d}\theta.$

We will now apply the estimate (4.2) to $\mathcal{V}^{\epsilon}$ and then
further bound the terms appearing on the right-hand side of this estimate.
Since
$\bigl{\lvert}\bigl{\langle}\Lambda^{\epsilon,\theta}_{t}(x,\cdot),\mathcal{V}^{\epsilon}_{t}\bigr{\rangle}\bigr{\rvert}\leq
C_{\lambda}\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}$ and
$\bigl{\lvert}\bigl{\langle}B^{\epsilon,\theta}_{t}(x,\cdot),\mathcal{V}^{\epsilon}_{t}\bigr{\rangle}\bigr{\rvert}\leq
C_{b}\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}$, we
get
$\lvert\mathcal{V}^{\epsilon}_{t}(x)\mathfrak{l}^{\epsilon}_{t}(x)\rvert+\lvert\mathfrak{h}^{\epsilon,0}_{t}(x)\rvert\leq
C_{\lambda}\lvert\mathcal{V}^{\epsilon}_{t}(x)\rvert+C_{\lambda}\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}\mu_{t}(x)$
and

$\lvert\mathcal{V}^{\epsilon}_{t}(x)\mathfrak{b}^{\epsilon}_{t}(x)\rvert+\lvert\mathfrak{h}^{\epsilon,1}_{t}(x)\rvert\leq
C_{b}\lvert\mathcal{V}^{\epsilon}_{t}(x)\rvert+C_{b}\bigl{(}1+\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}\bigr{)}\mu_{t}(x).$

Inserting this into the bound provided by (4.2) yields

$\sup_{0\leq s\leq
t}\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}^{2}+\int_{0}^{t}\lVert\nabla\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}^{2}\,\mathrm{d}s\leq
C\int_{0}^{t}\Bigl{(}1+\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}^{2}+(1+\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta})\lVert\nabla\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}\Bigr{)}\,\mathrm{d}s$
(4.13)

for a constant $C>0$ depending on $\mu$, $\eta_{0}$, and the bounds on the
coefficients $\lambda$ and $b$. Using the estimate
$(1+\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta})\lVert\nabla\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}\leq\frac{C}{2}(1+\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta})^{2}+\frac{1}{2C}\lVert\nabla\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}^{2}$,
rearranging (4.13), and applying Grönwall’s inequality implies that

$\sup_{0\leq t\leq
T}\lVert\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}^{2}+\frac{1}{2}\int_{0}^{T}\lVert\nabla\mathcal{V}^{\epsilon}_{t}\rVert_{\eta}^{2}\,\mathrm{d}t\leq
C$

for a possibly enlarged constant $C$ independent of $\epsilon$.

Using the uniform bound on the family $(\mathcal{V}^{\epsilon})_{\epsilon>0}$,
we can show that it is Cauchy in
$C_{\mathbb{F}}^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))\cap
L_{\mathbb{F}}^{2}([0,T];H^{1}_{\eta}(\mathbb{R}^{d}))$. Indeed, let us define
$\Delta^{\epsilon,\delta}=\mathcal{V}^{\epsilon}-\mathcal{V}^{\delta}$. Note
that again, the PDE satisfied by $\Delta^{\epsilon,\delta}$ is of the same
form as PDE (4.1). This time with $\alpha$, $\beta$, $h^{0}$, and $h^{1}$
replaced by $\mathfrak{l}^{\epsilon}$, $\mathfrak{b}^{\epsilon}$,
$\mathfrak{h}^{\epsilon,\delta,0}$, and $\mathfrak{h}^{\epsilon,\delta,1}$,
respectively, where

$\mathfrak{h}^{\epsilon,\delta,0}_{t}(x)=\mathcal{V}^{\delta}_{t}(x)\bigl{(}\mathfrak{l}^{\epsilon}_{t}(x)-\mathfrak{l}^{\delta}_{t}(x)\bigr{)}+\mathfrak{h}^{\epsilon,0}_{t}(x)-\mathfrak{h}^{\delta,0}_{t}(x)$

and

$\mathfrak{h}^{\epsilon,\delta,1}_{t}(x)=\mathcal{V}^{\delta}_{t}(x)\bigl{(}\mathfrak{b}^{\epsilon}_{t}(x)-\mathfrak{b}^{\delta}_{t}(x)\bigr{)}+\mathfrak{h}^{\epsilon,1}_{t}(x)-\mathfrak{h}^{\delta,1}_{t}(x).$

Once more, we make us of the estimate (4.2) to show that the expression
$\sup_{0\leq t\leq
T}\lVert\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}+\int_{0}^{T}\lVert\nabla\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}\,\mathrm{d}t$
vanishes as $\epsilon$, $\delta\to 0$. The estimate reads

$\displaystyle\begin{split}\sup_{0\leq s\leq
t}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}+\int_{0}^{t}\lVert\nabla\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}\,\mathrm{d}s&\leq
C\int_{0}^{t}\bigl{\lvert}\bigl{\langle}\Delta^{\epsilon,\delta}_{s}(\mathfrak{l}^{\epsilon}_{s}+\lvert\mathfrak{b}^{\epsilon}_{s}\rvert)+\mathfrak{h}^{\epsilon,\delta,0}_{s}+\lvert\mathfrak{h}^{\epsilon,\delta,1}_{s}\rvert,\Delta^{\epsilon,\delta}_{s}\bigr{\rangle}_{\eta}\bigr{\rvert}\,\mathrm{d}s\\\
&\ \ \
+C\int_{0}^{t}\bigl{\lvert}\bigl{\langle}\Delta^{\epsilon,\delta}_{s}\mathfrak{b}^{\epsilon}_{s}+\mathfrak{h}^{\epsilon,\delta,1}_{s},\nabla\Delta^{\epsilon,\delta}_{s}\bigr{\rangle}_{\eta}\bigr{\rvert}\,\mathrm{d}s.\end{split}$
(4.14)

Let us first derive bounds on the functions $\mathfrak{h}^{\epsilon,\delta,0}$
and $\mathfrak{h}^{\epsilon,\delta,1}$. Note that

$\displaystyle\mathfrak{h}^{\epsilon,0}_{s}(x)-\mathfrak{h}^{\delta,0}_{s}(x)=\mu_{s}(x)\int_{0}^{1}\bigl{\langle}\Lambda^{\epsilon,\theta}_{s}(x,\cdot),\Delta^{\epsilon,\delta}_{s}\bigr{\rangle}\,\mathrm{d}\theta+\mu_{s}(x)\int_{0}^{1}\bigl{\langle}\Lambda^{\epsilon,\theta}_{s}(x,\cdot)-\Lambda^{\delta,\theta}_{s}(x,\cdot),\mathcal{V}^{\delta}_{s}\bigr{\rangle}\,\mathrm{d}\theta,$

so that

$\displaystyle\lvert\mathfrak{h}^{\epsilon,\delta,0}_{s}(x)\rvert$
$\displaystyle\leq
C_{\lambda}d_{0}(\mu^{\epsilon}_{s},\mu^{\delta}_{s})\lvert\mathcal{V}^{\delta}_{s}(x)\rvert+C_{\lambda}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}\mu_{s}(x)$
$\displaystyle\ \ \
+\lVert\mathcal{V}^{\delta}_{s}\rVert_{\eta}\mu_{s}(x)\int_{0}^{1}\bigl{\lVert}e^{-\eta/2}\bigl{(}\Lambda^{\epsilon,\theta}_{s}(x,\cdot)-\Lambda^{\delta,\theta}_{s}(x,\cdot)\bigr{)}\bigr{\rVert}_{L^{2}}\,\mathrm{d}\theta$
$\displaystyle\leq C_{\lambda}\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mu^{\epsilon}_{s}-\mu^{\delta}_{s}\rVert_{\eta}\lvert\mathcal{V}^{\delta}_{s}(x)\rvert+\bigl{(}C_{\lambda}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}+\lVert\mathcal{V}^{\delta}_{s}\rVert_{\eta}I^{\epsilon,\delta,0}_{s}(x)\bigr{)}\mu_{s}(x),$

where
$I^{\epsilon,\delta,0}_{s}(x)=\int_{0}^{1}\bigl{\lVert}e^{-\eta/2}\bigl{(}\Lambda^{\epsilon,\theta}_{s}(x,\cdot)-\Lambda^{\delta,\theta}_{s}(x,\cdot)\bigr{)}\bigr{\rVert}_{L^{2}}\,\mathrm{d}\theta$.
Similarly, we obtain

$\displaystyle\lvert\mathfrak{h}^{\epsilon,\delta,1}_{s}(x)\rvert$
$\displaystyle\leq C_{b}\bigl{(}\lVert
e^{-\eta/2}\rVert_{L^{2}}\lVert\mu^{\epsilon}_{s}-\mu^{\delta}_{s}\rVert_{\eta}+\lvert\epsilon-\delta\rvert\bigr{)}\lvert\mathcal{V}^{\delta}_{s}(x)\rvert$
$\displaystyle\ \ \
+\bigl{(}C_{b}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}+\lVert\mathcal{V}^{\delta}_{s}\rVert_{\eta}I^{\epsilon,\delta,1}_{s}(x)\bigr{)}\mu_{s}(x)$

with
$I^{\epsilon,\delta,1}_{s}(x)=\int_{0}^{1}\bigl{\lVert}e^{-\eta/2}\bigl{(}B^{\epsilon,\theta}_{s}(x,\cdot)-B^{\delta,\theta}_{s}(x,\cdot)\bigr{)}\bigr{\rVert}_{L^{2}}\,\mathrm{d}\theta$.
Plugging the previous two displays into (4.14), we find

$\displaystyle\sup_{0\leq s\leq t}$
$\displaystyle\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}+\int_{0}^{t}\lVert\nabla\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}\,\mathrm{d}s$
$\displaystyle\leq
C\int_{0}^{t}\bigl{(}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}+\lVert\nabla\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}\bigr{)}\Bigl{(}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}+\lVert\mu^{\epsilon}_{s}-\mu^{\delta}_{s}\rVert_{\eta}+\lvert\epsilon-\delta\rvert\Bigr{)}\,\mathrm{d}s$
$\displaystyle\ \ \
+C\int_{0}^{t}\bigl{(}\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}+\lVert\nabla\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}\bigr{)}\bigl{\lVert}(I^{\epsilon,\delta,0}_{s}+I^{\epsilon,\delta,1}_{s})\mu_{s}\bigr{\rVert}_{\eta}\,\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\frac{C+2C^{2}}{2}\Bigl{(}3\lVert\mu^{\epsilon}_{s}-\mu^{\delta}_{s}\rVert_{\eta}^{2}+3\lvert\epsilon-\delta\rvert^{2}+\bigl{\lVert}(I^{\epsilon,\delta,0}_{s}+I^{\epsilon,\delta,1}_{s})\mu_{t}\bigr{\rVert}_{\eta}^{2}\Bigr{)}\,\mathrm{d}s$
$\displaystyle\ \ \
+\int_{0}^{t}(C+3C^{2})\lVert\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}+\frac{1}{2}\lVert\nabla\Delta^{\epsilon,\delta}_{s}\rVert_{\eta}^{2}\,\mathrm{d}s$

for a constant $C>0$ independent of $\epsilon$, $\delta>0$. Again, we
rearrange and apply Grönwall’s inequality, whereby

$\sup_{0\leq t\leq
T}\lVert\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}+\int_{0}^{T}\lVert\nabla\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}\,\mathrm{d}t\leq
C\int_{0}^{T}\Bigl{(}\lVert\mu^{\epsilon}_{t}-\mu^{\delta}_{t}\rVert_{\eta}^{2}+\lvert\epsilon-\delta\rvert^{2}+\bigl{\lVert}(I^{\epsilon,\delta,0}_{t}+I^{\epsilon,\delta,1}_{t})\mu_{t}\bigr{\rVert}_{\eta}^{2}\Bigr{)}\,\mathrm{d}t,$

where we enlarge $C$ if necessary. We claim that the three expressions on the
right-hand side vanish as $\epsilon$, $\delta\to 0$. Firstly, it holds that
$\lVert\mu^{\epsilon}_{t}-\mu^{\delta}_{t}\rVert_{\eta}^{2}\leq\lVert\epsilon\mathcal{V}^{\epsilon}_{t}-\delta\mathcal{V}^{\delta}_{t}\lVert_{\eta}^{2}$,
which tends to zero due to the uniform in $\epsilon$ bound on $\sup_{0\leq
s\leq T}\lVert\mathcal{V}^{\epsilon}_{s}\rVert_{\eta}^{2}$. The convergence of
the second term is clear. For the third term, we note that owing to the
continuity of $D\tilde{\lambda}$ and $D\tilde{b}_{0}$ in the measure argument,
we have $I^{\epsilon,\delta,0}_{t}(x)+I^{\epsilon,\delta,0}_{t}(x)\to 0$ for
a.e. $(t,x)\in[0,T]\times\mathbb{R}^{d}$. Since $I^{\epsilon,\delta,0}_{t}(x)$
and $I^{\epsilon,\delta,1}_{t}(x)$ are bounded uniformly over
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, the dominated convergence theorem implies
that
$\bigl{\lVert}(I^{\epsilon,\delta,0}_{t}+I^{\epsilon,\delta,1}_{t})\mu_{t}\bigr{\rVert}_{\eta}\to
0$. Hence, $\sup_{0\leq t\leq
T}\lVert\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}+\int_{0}^{T}\lVert\nabla\Delta^{\epsilon,\delta}_{t}\rVert_{\eta}^{2}\,\mathrm{d}t\to
0$ as $\epsilon$, $\delta\to 0$, so that
$(\mathcal{V}^{\epsilon})_{\epsilon>0}$ is Cauchy in
$C_{\mathbb{F}}^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))\cap
L_{\mathbb{F}}^{2}([0,T];H^{1}_{\eta}(\mathbb{R}^{d}))$.

Let $\mathcal{V}$ denote the limit of $(\mathcal{V}^{\epsilon})_{\epsilon>0}$
in $C_{\mathbb{F}}^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))\cap
L_{\mathbb{F}}^{2}([0,T];H^{1}_{\eta}(\mathbb{R}^{d}))$. Using the convergence
of $\mathcal{V}^{\epsilon}$ to $\mathcal{V}$, we can pass to the limit in PDE
(4.12) to see that $\mathcal{V}$ solves PDE (4.8). Uniqueness for PDE (4.8)
can be easily deduced from (4.2). This concludes the proof. ∎

Next, we use the adjoint process $(u,m)$, characterised by BSPDE (2.10), and
the variation $\mathcal{V}$ to derive an expression for the derivative of the
cost function $\tilde{J}$ defined in (2.5). Inserting the convexity of
$g\mapsto H_{1}(t,x,p,g)=\tilde{b}_{1}(t,x,g)\cdot p+\tilde{f}_{1}(t,x,g)$
implied by Assumption 2.7 (iii) into this expression then yields the necessary
SMP.

######  Proposition 4.4.

Let Assumptions 2.1 and 2.7 be satisfied. Then for any control
$\gamma\in\mathbb{G}$ and any $\mathbb{F}$-progressively measurable random
function
$h\mathpunct{:}[0,T]\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d_{G}}$,
such that $\gamma+\epsilon h$ maps into $G$, it holds that

$\frac{\mathrm{d}}{\mathrm{d}\epsilon}\tilde{J}(\gamma+\epsilon
h)\Big{|}_{\epsilon=0}=\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\nabla_{g}H_{1}(t,\cdot,\nabla
u_{t},\gamma_{t})\cdot h_{t}\bigr{\rangle}\,\mathrm{d}t.$ (4.15)

###### Proof.

Set $\gamma^{\epsilon}=\gamma+\epsilon h$ and denote the solutions to PDE
(2.4) corresponding to the inputs $\gamma^{\epsilon}$ and $\gamma$ by
$\mu^{\epsilon}$ and $\mu$, respectively. Further, let
$\mathcal{V}^{\epsilon}=\frac{1}{\epsilon}(\mu^{\epsilon}-\mu)$ and let
$\mathcal{V}$ denote the limit of $(\mathcal{V}^{\epsilon})_{\epsilon>0}$ in
$C_{\mathbb{F}}^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))\cap
L_{\mathbb{F}}^{2}([0,T];H^{1}_{\eta}(\mathbb{R}^{d}))$, which exists by
Proposition 4.3. Then, we decompose

$\displaystyle\begin{split}\tilde{J}(g^{\epsilon}&)-\tilde{J}(g)\\\
&=\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu^{\epsilon}_{t}-\mu_{t},\tilde{f}(t,\cdot,\mu^{\epsilon}_{t},\gamma^{\epsilon}_{t})\bigr{\rangle}\,\mathrm{d}t+\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\tilde{f}_{1}(t,\cdot,\gamma^{\epsilon}_{t})-\tilde{f}_{1}(t,\cdot,\gamma_{t})\bigr{\rangle}\,\mathrm{d}t\\\
&\ \ \
+\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\tilde{f}_{0}(t,\cdot,\mu^{\epsilon}_{t})-\tilde{f}_{0}(t,\cdot,\mu_{t})\bigr{\rangle}\,\mathrm{d}t+\mathbb{E}[\tilde{\psi}(\mu^{\epsilon}_{T})-\tilde{\psi}(\mu_{T})].\end{split}$
(4.16)

We divide the equation by $\epsilon$ and then take the limit as $\epsilon$
tends to zero. Owing to Assumption 2.1 (i), we can pass to the limit in the
first expression on the right-hand side above, which is given by
$\bigl{\langle}\mathcal{V}_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}$.
For the second term, we use Assumption 2.7 (ii), whereby

$\displaystyle\frac{1}{\epsilon}\Bigl{(}\tilde{f}_{1}\bigl{(}t,x,\gamma^{\epsilon}_{t}(x)\bigr{)}-\tilde{f}_{1}\bigl{(}t,x,\gamma_{t}(x)\bigr{)}\Bigr{)}=\int_{0}^{1}\nabla_{g}f_{1}\bigl{(}t,x,\gamma_{t}(x)+\epsilon\theta
h_{t}(x)\bigr{)}\cdot h_{t}(x)\,\mathrm{d}\theta,$

where $\nabla_{g}\tilde{f}_{1}$ denotes the shift of $\nabla_{g}f_{1}$ along
$W$. Since $g\mapsto\nabla_{g}f_{1}(t,x,g)$ is continuous and bounded by
Assumption 2.7 (ii), we conclude by the dominated convergence theorem that the
second expression tends to
$\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\nabla_{g}\tilde{f}_{1}(t,\cdot,\gamma_{t})\cdot
h_{t}\bigr{\rangle}\,\mathrm{d}t$. Next, by Assumption 2.7 (i), for any
$(t,x)\in[0,T]\times\mathbb{R}^{d}$ the map $v\mapsto f_{0}(t,x,v)$ has a
linear functional derivative, which is bounded and continuous in the measure
argument by Assumption 2.7 (i). Hence, we can write

$\displaystyle\frac{1}{\epsilon}\Bigl{(}\tilde{f}_{0}(t,x,\mu^{\epsilon}_{t})-\tilde{f}_{0}(t,x,\mu_{t})\Bigr{)}$
$\displaystyle=\int_{0}^{1}\Bigl{\langle}\mathcal{V}^{\epsilon}_{t},D\tilde{f}_{0}\bigl{(}t,x,\theta\mu^{\epsilon}_{t}+(1-\theta)\mu_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}\theta.$

Again, the dominated convergence theorem allows us to pass to the limit, so
the third term on the right-hand side of (4.16) tends to
$\mathbb{E}\int_{0}^{T}\bigl{(}\int_{\mathbb{R}^{d}}\bigl{\langle}\mu_{t},D\tilde{f}_{0}(t,\cdot,\mu_{t})(x)\bigr{\rangle}\mathcal{V}_{t}(x)\,\mathrm{d}x\bigr{)}\,\mathrm{d}t$.
The same reasoning allows us to take the limit as $\epsilon\to 0$ in the last
term on the right-hand side of Equation (4.16). Putting everything together,
we obtain

$\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}\epsilon}&\tilde{J}(\gamma+\epsilon
h)\Big{|}_{\epsilon=0}\\\
&=\mathbb{E}\int_{0}^{T}\bigl{\langle}\mathcal{V}_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t+\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\nabla_{g}\tilde{f}_{1}(t,\cdot,\gamma_{t})\cdot
h_{t}\bigr{\rangle}\,\mathrm{d}t\\\ &\ \ \
+\mathbb{E}\int_{0}^{T}\biggl{(}\int_{\mathbb{R}^{d}}\bigl{\langle}\mu_{t},D\tilde{f}_{0}(t,\cdot,\mu_{t})(x)\bigr{\rangle}\mathcal{V}_{t}(x)\,\mathrm{d}x\biggr{)}\,\mathrm{d}t+\mathbb{E}\bigl{\langle}\mathcal{V}_{t},D\tilde{\psi}(\mu_{T})\bigr{\rangle}.\end{split}$
(4.17)

Next, we apply the generalisation of Itô’s formula from Corollary 3.3 to
$\langle\mathcal{V}_{t},u_{t}\rangle$, where $(u,m)$ is the adjoint process,
solving BSPDE (2.10). This implies that

$\displaystyle\bigl{\langle}\mathcal{V}_{T},D\tilde{\psi}(\mu_{T})\bigr{\rangle}$
$\displaystyle=\langle\mathcal{V}_{0},u_{0}\rangle+\int_{0}^{T}\biggl{(}\Bigl{\langle}\mathcal{V}_{t},\tilde{\lambda}(t,\cdot,\mu_{t})u_{t}+\tilde{b}(t,\cdot,\mu_{t},\gamma_{t})\cdot\nabla
u_{t}\Bigr{\rangle}-\bigl{\langle}\nabla\cdot(\tilde{a}(t,\cdot)\mathcal{V}_{t}),\nabla
u_{t}\big{\rangle}\biggr{)}\,\mathrm{d}t$ $\displaystyle\ \ \
+\int_{0}^{T}\biggl{(}\int_{\mathbb{R}^{d}}\Bigl{\langle}\mu_{t},D\tilde{\lambda}(t,\cdot,\mu_{t})(x)u_{t}+D\tilde{b}_{0}(t,\cdot,\mu_{t})(x)\cdot\nabla
u_{t}\Bigr{\rangle}\mathcal{V}_{t}(x)\,\mathrm{d}x\biggr{)}\,\mathrm{d}t$
$\displaystyle\ \ \
+\int_{0}^{T}\bigl{\langle}\mu_{t},h_{t}^{\top}\nabla_{g}\tilde{b}_{1}(t,\cdot)\nabla
u_{t}\bigr{\rangle}\,\mathrm{d}t$ $\displaystyle\ \ \
-\int_{0}^{T}\biggl{(}\Bigl{\langle}\tilde{\lambda}(t,\cdot,\mu_{t})u_{t}+\tilde{b}(t,\cdot,\mu_{t},\gamma_{t})\cdot\nabla
u_{t}+\tilde{f}(t,\cdot,\mu_{t},\gamma_{t}),\mathcal{V}_{t}\Bigr{\rangle}-\bigl{\langle}\nabla
u_{t},\nabla\cdot(\tilde{a}(t,\cdot)\mathcal{V}_{t})\bigr{\rangle}\biggr{)}\,\mathrm{d}t$
$\displaystyle\ \ \
-\int_{0}^{T}\biggl{(}\int_{\mathbb{R}^{d}}\Bigl{\langle}\mu_{t},D\tilde{\lambda}(t,\cdot,\mu_{t})(x)u_{t}+D\tilde{b}_{0}(t,\cdot,\mu_{t})(x)\cdot\nabla
u_{t}+D\tilde{f}_{0}(t,\cdot,\mu_{t})(x)\Bigr{\rangle}\mathcal{V}_{t}(x)\,\mathrm{d}x\biggr{)}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\biggl{(}\bigl{\langle}\mu_{t},h_{t}^{\top}\nabla_{g}\tilde{b}_{1}(t,\cdot)\nabla
u_{t}\bigr{\rangle}-\bigl{\langle}\mathcal{V}_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\int_{\mathbb{R}^{d}}\bigl{\langle}\mu_{t},D\tilde{f}_{0}(t,\cdot,\mu_{t})(x)\bigr{\rangle}\mathcal{V}_{t}(x)\,\mathrm{d}x\biggr{)}\,\mathrm{d}t,$
(4.18)

where we used that $\mathcal{V}_{0}=0$. Now, we take expectations on both
sides of (4.3) and then plug the resulting expression for
$\mathbb{E}\bigl{\langle}\mathcal{V}_{T},D\tilde{\psi}(\mu_{T})\bigr{\rangle}$
into Equation (4.17). This yields the desired identity (4.15). ∎

From (4.15) and the convexity of $H_{1}$ in the control argument, guaranteed
by Assumption 2.7 (iii), we can deduce the necessary SMP.

###### Proof of Theorem 2.9.

Let $\gamma$, $\gamma^{\prime}\in\mathbb{G}$. Since $G$ is convex, for any
$\epsilon>0$ the random function
$\gamma^{\epsilon}=(1-\epsilon)\gamma+\epsilon\gamma^{\prime}=\gamma+\epsilon(\gamma^{\prime}-\gamma)$
maps into $G$. Hence, we can apply Proposition 4.4 with $h$ replaced by
$\gamma^{\prime}-\gamma$, which implies that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\epsilon}\tilde{J}(\gamma^{\epsilon})\Big{|}_{\epsilon=0}$
$\displaystyle=\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},\nabla_{g}H_{1}(t,\cdot,\nabla
u_{t},\gamma_{t})\cdot(\gamma^{\prime}_{t}-\gamma_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},H_{1}(t,\cdot,\nabla
u_{t},\gamma^{\prime}_{t})-H_{1}(t,\cdot,\nabla
u_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t,$

where we used that the function $g\mapsto H_{1}(t,x,p,g)$ is convex by
Assumption 2.7 (iii). On the other hand, we know that $\gamma$ is a minimiser
of the cost functional $\tilde{J}$, so that

$\frac{\mathrm{d}}{\mathrm{d}\epsilon}\tilde{J}(\gamma^{\epsilon})\Big{|}_{\epsilon=0}=\lim_{\epsilon\to
0}\frac{1}{\epsilon}\bigl{(}\tilde{J}(\gamma+\epsilon(\gamma^{\prime}-\gamma))-\tilde{J}(\gamma)\bigr{)}\geq
0.$

Combining the previous two inequalities yields

$\mathbb{E}\int_{0}^{T}\bigl{\langle}\mu_{t},H_{1}(t,\cdot,\nabla
u_{t},\gamma^{\prime}_{t})-H_{1}(t,\cdot,\nabla
u_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t\geq 0.$ (4.19)

Now, for any $\mathbb{F}$-progressively measurable set
$A\subset[0,T]\times\Omega\times\mathbb{R}^{d}$ and any $g\in G$ we can define
a control $\gamma^{\prime}\in\mathbb{G}$ by $\gamma^{\prime}_{t}(x)=g$ if
$(t,\omega,x)\in A$ and $\gamma^{\prime}_{t}(x)=\gamma_{t}(x)$ otherwise.
Applying (4.19) with this choice of $\gamma^{\prime}$ implies that

$\mathbb{E}\int_{0}^{T}\biggl{(}\int_{\mathbb{R}^{d}}\mathbf{1}_{A}(t,\cdot,x)\Bigl{(}H_{1}\bigl{(}t,x,\nabla
u_{t}(x),g\bigr{)}-H_{1}\bigl{(}t,x,\nabla
u_{t}(x),\gamma_{t}(x)\bigr{)}\Bigr{)}\,\mathrm{d}\mu_{t}(x)\biggr{)}\,\mathrm{d}t$

is nonnegative. Since $A$ and $g$ were arbitrary, we conclude that for all
$g\in G$ it holds that

$H_{1}\bigl{(}t,x,\nabla u_{t}(x),g\bigr{)}\geq H_{1}\bigl{(}t,x,\nabla
u_{t}(x),\gamma_{t}(x)\bigr{)}$

for $\mu_{t}$-a.e. $x\in\mathbb{R}^{d}$ and
$\mathrm{Leb}\otimes\mathbb{P}$-a.e. $(t,\omega)\in[0,T]\times\Omega$. ∎

###  4.4 The Sufficient Stochastic Maximum Principle

Lastly, we shall establish the sufficient SMP under Assumptions 2.1 and 2.7
(i). The idea of its proof is to compare the cost of the control
$\gamma\in\mathbb{G}$ from the statement of the sufficient SMP, Theorem 2.12,
with that of any other control, and then to insert the convexity assumption on
the functions $\psi$ and $\mathcal{H}_{0}$.

By Lemma A.5, for any convex function
$F\mathpunct{:}\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$, which has a linear
functional derivative $DF$ such that for all $v$,
$v^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$ it holds that $DF(\theta
v+(1-\theta)v^{\prime})(x)\to DF(v^{\prime})(x)$ as $\theta\in(0,1]$ tends to
zero, we have $F(v)-F(v^{\prime})\geq\langle
v-v^{\prime},DF(v^{\prime})\rangle$. Under Assumptions 2.7 (i), the function
$v\mapsto\mathcal{H}_{0}(t,v,r,p)$ satisfies the above continuity condition
for any
$(t,\omega,r,p)\in[0,T]\times\Omega\times\mathcal{B}_{b}(\mathbb{R}^{d})\times\mathcal{B}_{b}(\mathbb{R}^{d};\mathbb{R}^{d})$,
so if $v\mapsto\mathcal{H}_{0}(t,v,r,p)$ is convex in the measure argument,
then

$\mathcal{H}_{0}(t,v,r,p)-\mathcal{H}_{0}(t,v^{\prime},r,p)\geq\bigl{\langle}v-v^{\prime},D\mathcal{H}_{0}(t,v^{\prime},r,p)\bigr{\rangle}.$

This inequality readily extends to $v$, $v^{\prime}\in
L^{2}_{\eta}(\mathbb{R}^{d};[0,\infty))$, $r\in
L^{2}_{\bar{\eta}}(\mathbb{R}^{d})$, and $p\in
L^{2}_{\bar{\eta}}(\mathbb{R}^{d};\mathbb{R}^{d})$. Consequently, writing
$\mathcal{H}(t,v,r,p,g)=\mathcal{H}_{0}(t,v,r,p)+\bigl{\langle}v,H_{1}(t,\cdot,p,g)\bigr{\rangle}$,
if $g^{\prime}$ is any other element of $\mathcal{G}$, we obtain that

$\displaystyle\mathcal{H}(t,v,r,p,g)-\mathcal{H}(t,v^{\prime},r,p,g^{\prime})-\bigl{\langle}v-v^{\prime},D\mathcal{H}(t,v,r,p,g)\bigr{\rangle}$
$\displaystyle=\mathcal{H}_{0}(t,v,r,p)-\mathcal{H}_{0}(t,v^{\prime},r,p)-\bigl{\langle}v-v^{\prime},D\mathcal{H}_{0}(t,v,r,p)\bigr{\rangle}$
$\displaystyle\ \ \
+\bigl{\langle}v,H_{1}(t,\cdot,p,g)\bigr{\rangle}-\bigl{\langle}v^{\prime},H_{1}(t,\cdot,p,g^{\prime})\bigr{\rangle}-\bigl{\langle}v-v^{\prime},H_{1}(t,\cdot,p,g)\bigr{\rangle}$
$\displaystyle\leq\bigl{\langle}v^{\prime},H_{1}(t,\cdot,p,g)-H_{1}(t,\cdot,p,g^{\prime})\bigr{\rangle}.$
(4.20)

Now, if $g\in\mathcal{G}$ is a pointwise minimiser of the map $h\mapsto
H_{1}(t,x,p,h)$, then clearly the expression on the right-hand side is
nonpositive. This is the key insight in the proof of Theorem 2.12.

###### Proof of Theorem 2.12.

Let $\gamma\in\mathbb{G}$, $\mu$, and $(u,m)$ be as in the statement of the
theorem. We may assume without loss of generality that Equation (2.14) holds
for $\mathrm{Leb}$-a.e. $x\in\mathbb{R}^{d}$ and
$\mathrm{Leb}\otimes\mathbb{P}$-a.e. $(t,\omega)\in[0,T]\times\Omega$. Indeed,
otherwise we follow the steps outlined below Theorem 2.9 to obtain a new
control $\gamma^{\ast}$, which coincides with $\gamma$ for $\mu_{t}$-a.e.
$x\in\mathbb{R}^{d}$ and $\mathrm{Leb}\otimes\mathbb{P}$-a.e.
$(t,\omega)\in[0,T]\times\Omega$, and a corresponding adjoint process
$(u^{\ast},m^{\ast})$, such that Equation (2.14) holds with $\gamma$ replaced
by $\gamma^{\ast}$ and $(u,m)$ replaced by $(u^{\ast},m^{\ast})$ for
$\mathrm{Leb}$-a.e. $x\in\mathbb{R}^{d}$ and
$\mathrm{Leb}\otimes\mathbb{P}$-a.e. $(t,\omega)\in[0,T]\times\Omega$.

Next, fix $\gamma^{\prime}\in\mathbb{G}$ and denote the corresponding solution
of PDE (2.4) by $\mu^{\prime}$. We compute

$\displaystyle\tilde{J}(\gamma)-\tilde{J}(\gamma^{\prime})$
$\displaystyle=\mathbb{E}\biggl{[}\int_{0}^{T}\bigl{\langle}\mu_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\bigl{\langle}\mu^{\prime}_{t},\tilde{f}(t,\cdot,\mu^{\prime}_{t},\gamma^{\prime}_{t})\bigr{\rangle}\,\mathrm{d}t\biggr{]}+\mathbb{E}\bigl{[}\tilde{\psi}(\mu_{T})-\tilde{\psi}(\mu^{\prime}_{T})\bigr{]}$
$\displaystyle\leq\mathbb{E}\biggl{[}\int_{0}^{T}\bigl{\langle}\mu_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\bigl{\langle}\mu^{\prime}_{t},\tilde{f}(t,\cdot,\mu^{\prime}_{t},\gamma^{\prime}_{t})\bigr{\rangle}\,\mathrm{d}t\biggr{]}+\mathbb{E}\langle\mu_{T}-\mu^{\prime}_{T},u_{T}\rangle,$
(4.21)

where we used that the map $v\mapsto\psi(v)$ is convex and has a linear
functional derivative that is continuous in the measure argument, so that
Lemma A.5 applies. Note further that $D\tilde{\psi}(\mu_{T})=u_{T}$. Then, as
in the proof of Proposition 4.4, we apply the generalisation of Itô’s formula
from Corollary 3.3 to $\langle\mu_{t}-\mu^{\prime}_{t},u_{T}\rangle$, whereby

$\displaystyle\langle\mu_{T}-\mu^{\prime}_{T},u_{T}\rangle$
$\displaystyle=\langle\mu_{0}-\mu^{\prime}_{0},u_{0}\rangle+\int_{0}^{T}\bigl{\langle}\mu_{t},\tilde{\lambda}(t,\cdot,\mu_{t})u_{t}+\tilde{b}_{0}(t,\cdot,\mu_{t},\gamma_{t})\cdot\nabla
u_{t}\bigr{\rangle}\,\mathrm{d}t$ $\displaystyle\ \ \
-\int_{0}^{T}\bigl{\langle}\mu^{\prime}_{t},\tilde{\lambda}(t,\cdot,\mu^{\prime}_{t})u_{t}+\tilde{b}_{0}(t,\cdot,\mu^{\prime}_{t},\gamma_{t})\cdot\nabla
u_{t}\bigr{\rangle}\,\mathrm{d}t$ $\displaystyle\ \ \
-\int_{0}^{T}\bigl{\langle}\mu_{t}-\mu^{\prime}_{t},D\mathcal{H}(t,\mu_{t},u_{t},\nabla
u_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle=-\int_{0}^{T}\Bigl{(}\bigl{\langle}\mu_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\bigl{\langle}\mu^{\prime}_{t},\tilde{f}(t,\cdot,\mu^{\prime}_{t},\gamma^{\prime}_{t})\bigr{\rangle}\Bigr{)}\,\mathrm{d}t$
$\displaystyle\ \ \ +\int_{0}^{T}\Bigl{(}\mathcal{H}(t,\mu_{t},u_{t},\nabla
u_{t},\gamma_{t})-\mathcal{H}(t,\mu^{\prime}_{t},u_{t},\nabla
u_{t},\gamma^{\prime}_{t})\Bigr{)}\,\mathrm{d}t$ $\displaystyle\ \ \
-\int_{0}^{T}\bigl{\langle}\mu_{t}-\mu^{\prime}_{t},D\mathcal{H}(t,\mu_{t},u_{t},\nabla
u_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle\leq-\int_{0}^{T}\Bigl{(}\bigl{\langle}\mu_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\bigl{\langle}\mu^{\prime}_{t},\tilde{f}(t,\cdot,\mu^{\prime}_{t},\gamma^{\prime}_{t})\bigr{\rangle}\Bigr{)}\,\mathrm{d}t$
$\displaystyle\ \ \
+\int_{0}^{T}\bigl{\langle}\mu^{\prime}_{t},H_{1}(t,\cdot,\nabla
u_{t},\gamma_{t})-H_{1}(t,\cdot,\nabla
u_{t},\gamma^{\prime}_{t})\bigr{\rangle}\,\mathrm{d}t,$

where we made use of inequality (4.4), which followed from the convexity
assumption on $\mathcal{H}_{0}$. By Equation (2.14), the last term on the
right-hand side is nonpositive, so that

$\langle\mu_{T}-\mu^{\prime}_{T},u_{T}\rangle\leq-\int_{0}^{T}\Bigl{(}\bigl{\langle}\mu_{t},\tilde{f}(t,\cdot,\mu_{t},\gamma_{t})\bigr{\rangle}-\bigl{\langle}\mu^{\prime}_{t},\tilde{f}(t,\cdot,\mu^{\prime}_{t},\gamma^{\prime}_{t})\bigr{\rangle}\Bigr{)}\,\mathrm{d}t.$
(4.22)

Taking expectation on both sides above and inserting the resulting expression
into (4.4) yields $\tilde{J}(\gamma)-\tilde{J}(\gamma^{\prime})\leq 0$. Since
$\gamma^{\prime}$ was arbitrary, we obtain
$\tilde{J}(\gamma)=\inf_{\gamma^{\prime}\in\mathbb{G}}\tilde{J}(\gamma^{\prime})$,
so $\gamma$ is optimal.

Next, assume that $G\ni g\mapsto H_{1}\bigl{(}t,x,\nabla u_{t}(x),g\bigr{)}$
has a unique minimum for $\mathrm{Leb}$-a.e. $x\in\mathbb{R}^{d}$ and
$\mathrm{Leb}\otimes\mathbb{P}$-a.e. $(t,\omega)\in[0,T]\times\Omega$. Our
goal is to show that $\gamma$ is the unique optimal control up to modification
on null sets of $\mu_{t}$. If $\gamma^{\prime}\in\mathbb{G}$ is not a
modification of $\gamma$ on null sets of $\mu_{t}$, either (4.22) holds with
strict inequality with positive probability, so that
$\tilde{J}(\gamma)-\tilde{J}(\gamma^{\prime})<0$ or
$\gamma_{t}(x)=\gamma^{\prime}_{t}(x)$ for $\mu^{\prime}_{t}$-a.e.
$x\in\mathbb{R}^{d}$ and $\mathrm{Leb}\otimes\mathbb{P}$-a.e.
$(t,\omega)\in[0,T]\times\Omega$. In the former case, we are done, so let us
lead the latter case to a contradiction. If $\gamma$ and $\gamma^{\prime}$ are
modifications of each other on null sets of $\mu^{\prime}_{t}$, then the
solution to the random Fokker–Planck equation (2.4) with input $\gamma$ is
$\mu^{\prime}_{t}$. Since PDE (2.4) has a unique solution by Proposition 2.3
and $\mu$ is another solution with input $\gamma$, it follows that
$\mu^{\prime}=\mu$. Thus, $\gamma^{\prime}$ is a modification of $\gamma$ on
null sets of $\mu_{t}$, in contradiction to our choice of $\gamma^{\prime}$.
This concludes the proof. ∎

##  5 Application to a Model of Government Interventions in Financial Systems

In this final section we discuss the application of the theory developed in
this article to the mean-field model for government interventions in financial
markets from [28]. The model consists of an infinite system of banks, each
represented by the capital it holds in excess of the regulatory capital
requirements. The institutions have common exposures, modelled by the noise
$W$, and mutual obligations. Whenever the excess capital of a bank is
negative, so that it is in breach of the capital requirements, it has a
positive default intensity, expressed by the coefficient $\lambda$. Through
the mutual obligations, the default of an institution leads to losses at other
banks, making them more likely to default as well. This contagion may
precipitate systemic events, where a large portion of the system is wiped out
in a short period of time. The controller, which should be thought of as a
government or another central authority, can inject capital into the system to
prevent defaults.

The excess capital $X=(X_{t})_{0\leq t\leq T}$ of a representative bank in the
mean-field system follows the McKean–Vlasov SDE

$X_{t}=(\xi-c)+\int_{0}^{t}\alpha_{s}\,\mathrm{d}s+\sigma
B_{t}+\sigma_{0}W_{t}-\kappa L_{t},$ (5.1)

where $\xi$ is the initial capital and $c>0$ denotes the regulatory capital
requirements, $\alpha_{s}\in[0,g_{\text{max}}]$ is the amount of capital
injected by the government at time $t\in[0,T]$, $B$ and $W$ are independent
Brownian motions that represent the fluctuations of banks’ assets values
correlated through the common noise $W$, $\kappa>0$ measures the size of the
mutual obligations, and $L_{t}$ denotes the fraction of banks that have
defaulted up to time $t\in[0,T]$ conditional on the common noise $W$. In
particular, we assume that the losses from default are proportional to the
fraction of defaulted entities $L_{t}$. Let us try to give a more precise
account of the latter quantity. As mentioned above, the representative
institution’s instantaneous default intensity is $\lambda(X_{t})$, where we
assume that $\lambda$ is bounded and $\lambda(x)=0$ for $x\geq 0$ while
$\lambda(x)>0$ for $x<0$. Default occurs, when the cumulative intensity
$\Lambda_{t}=\int_{0}^{t}\lambda(X_{s})\,\mathrm{d}s$ exceeds a standard
exponentially distributed random variable $\theta$, which is independent of
$X_{0}$, $B$, $W$, and $\alpha$. Hence,
$L_{t}=\mathbb{P}(\theta\geq\Lambda_{t}|W)=1-\mathbb{E}[e^{-\Lambda_{t}}]=\int_{0}^{t}\langle\nu_{s},\lambda\rangle\,\mathrm{d}s$,
where $\nu_{t}=\mathbb{P}(X_{t}\in\cdot,\,\theta>\Lambda_{t}|W)$ is the
conditional subprobability distribution of the solvent banks. The government’s
objective is to minimise the cost functional

$\alpha\mapsto\mathbb{E}\biggl{[}\int_{0}^{T}\mathbf{1}_{\theta>\Lambda_{t}}w\alpha_{t}+L_{T}\biggr{]}=\mathbb{E}\biggl{[}\int_{0}^{T}e^{-\Lambda_{t}}w\alpha_{t}+L_{T}\biggr{]}$

over $\mathbb{F}^{\xi,B,W}$-progressively measurable
$[0,g_{\text{max}}]$-valued processes $\alpha=(\alpha_{t})_{0\leq t\leq T}$.
The weight $w>0$ trades off the negative externalities caused by the bailout,
such as moral hazard, measured by the size of the capital injections with the
cost of a potential systemic crisis, captured by the fraction of defaulted
entities $L_{T}$ at time $T$.

If $\alpha$ takes the form $\alpha_{t}=\gamma_{t}(X_{t})$ for an
$\mathbb{F}^{W}$-progressively measurable random function
$\gamma\mathpunct{:}[0,T]\times\Omega\times\mathbb{R}\to[0,g_{\text{max}}]$,
then one can show that $\nu=(\nu_{t})_{0\leq t\leq T}$ satisfies the nonlinear
stochastic Fokker–Planck equation

$\mathrm{d}\langle\nu_{t},\varphi\rangle=\Bigl{\langle}\nu_{t},-\lambda\varphi+\gamma_{t}\partial_{x}\varphi-\kappa\langle\nu_{t},\lambda\rangle\partial_{x}\varphi+\tfrac{1}{2}(\sigma^{2}+\sigma_{0}^{2}\bigr{)}\partial_{x}^{2}\varphi\Bigr{\rangle}\,\mathrm{d}t+\langle\nu_{t},\sigma_{0}\partial_{x}\varphi\rangle\,\mathrm{d}W_{t}$
(5.2)

for $\varphi\in C_{c}^{2}(\mathbb{R})$ with initial condition
$\nu_{0}=\mathcal{L}(X_{0})$. This SPDE is a special case of SPDE (2.1) (note,
however, the sign change of $\lambda$) and we can rewrite the cost functional
in terms of $\nu$ as

$J(\gamma)=\mathbb{E}\biggl{[}\int_{0}^{T}w\langle\nu_{t},\gamma_{t}\rangle\,\mathrm{d}t+1-\nu_{T}(\mathbb{R})\biggr{]}.$

This setup fits the framework introduced in Section 2. Moreover, by Theorem
A.2, the control of McKean–Vlasov SDE (5.1) over
$\mathbb{F}^{X_{0},B,W}$-progressively measurable $[0,g_{\text{max}}]$-valued
controls yields the same optimal value as the control of the nonlinear
stochastic Fokker–Planck equation (5.2) over $\mathbb{F}^{W}$-progressively
measurable random functions
$[0,T]\times\Omega\times\mathbb{R}\to[0,g_{\text{max}}]$. (Note that here we
use that the weak formulation for the McKean–Vlasov control problem as
introduced in Appendix A.1 is equivalent to the strong formulation with
$\mathbb{F}^{X_{0},B,W}$-progressively measurable controls considered here by
[28, Theorem 2.8], and that the value for the control problem of the nonlinear
stochastic Fokker–Planck equation is independent of the weak setup, cf.
Theorem 2.4.) The nonlinear random Fokker–Planck equation associated to SPDE
(5.2) is

$\mathrm{d}\langle\mu_{t},\varphi\rangle=\Bigl{\langle}\mu_{t},-\tilde{\lambda}\varphi+\gamma_{t}\partial_{x}\varphi-\kappa\langle\mu_{t},\tilde{\lambda}\rangle\partial_{x}\varphi+\tfrac{\sigma^{2}}{2}\partial_{x}^{2}\varphi\Bigr{\rangle}\,\mathrm{d}t$
(5.3)

for $\varphi\in C_{c}^{2}(\mathbb{R})$ with initial condition
$\mu_{0}=\mathcal{L}(X_{0})$, where $\gamma$ is still an
$\mathbb{F}^{W}$-progressively measurable random function
$[0,T]\times\Omega\times\mathbb{R}\to[0,g_{\text{max}}]$.

After replacing $\mathbb{F}^{W}$ with a larger filtration $\mathbb{F}$ if
necessary, Theorem 2.4 implies that there exists an optimal
$\mathbb{F}$-progressively measurable control
$\gamma\mathpunct{:}[0,T]\times\Omega\times\mathbb{R}\to[0,g_{\text{max}}]$
for $g_{\text{max}}>0$. Now, for $p\in\mathbb{R}$, the reduced Hamiltonian
$g\mapsto H_{1}(p,g)=gp+wg=(p+w)g$ is linear and, therefore, convex. Hence,
the necessary SMP, Theorem 2.9, implies that the optimal control takes the
form $\gamma_{t}(x)=g_{\text{max}}\mathbf{1}_{\partial_{x}u_{t}(x)\leq-w}$,
where we use that the minimiser of the map $g\mapsto H_{1}(p,g)$ over
$g\in[0,g_{\text{max}}]$ is given by $p\mapsto
g_{\text{max}}\mathbf{1}_{p\leq-w}$. The adjoint process $(u,m)$ solves the
semilinear BSPDE

$\displaystyle\begin{split}\mathrm{d}u_{t}(x)&=\Bigl{(}\tilde{\lambda}(x)u_{t}(x)+g_{\text{max}}(\partial_{x}u_{t}(x)+w)_{-}+\kappa\langle\mu_{t},\tilde{\lambda}\rangle\partial_{x}u_{t}(x)+\kappa\langle\mu_{t},\partial_{x}u_{t}\rangle\tilde{\lambda}(x)\Bigr{)}\,\mathrm{d}t\\\
&\ \ \
-\tfrac{\sigma^{2}}{2}\partial_{x}^{2}u_{t}(x)\,\mathrm{d}t+\mathrm{d}m_{t}(x)\end{split}$
(5.4)

with terminal condition $u_{T}(x)=D(1-\nu_{T}(\mathbb{R}))=-1$. The
nonlinearity $p\mapsto-g_{\text{max}}(p+w)_{-}$ arises from the minimisation
of $(p+w)g$ over $g\in[0,g_{\text{max}}]$. As one would expect, the optimal
control is of bang-bang type, i.e. either the controller injects the maximal
amount of capital allowed (this happens when $\partial_{x}u_{t}(x)\leq-w$) or
none at all. Naturally, a smaller weight lowers the threshold at which the
controller intervenes.

Let us assume for the moment, that $\kappa$ and $\sigma_{0}$ vanish, so that
BSPDE (5.4) becomes a local and deterministic PDE:

$-\partial_{t}u_{t}(x)+\lambda(x)u_{t}(x)+g_{\text{max}}(\partial_{x}u_{t}(x)+w)_{-}-\frac{\sigma^{2}}{2}\partial_{x}^{2}u_{t}(x)=0.$
(5.5)

The functions which are constantly equal to $-1$ and $0$ are sub- and
supersolution to this PDE, respectively, so by the comparison principle we
have $-1\leq u_{t}(x)\leq 0$. Next, set $u^{h}_{t}(x)=u_{t}(x+h)$ for $h>0$.
Then, since $\lambda$ is nonincreasing and $u_{t}$ is nonpositive, it is easy
to see that $u^{h}$ is a subsolution, whence $u_{t}(x+h)=u^{h}_{t}(x)\leq
u_{t}(x)$. Since $h>0$ is arbitrary, we see that $x\mapsto u_{t}(x)$ is
nonincreasing, in other words $\partial_{x}u_{t}(x)\leq 0$. One may then
conjecture that for each time $t\in[0,T)$ the set
$\\{x\in\mathbb{R}\mathpunct{:}\partial_{x}u_{t}(x)\leq-w\\}$, where the
controller is active, is a possibly empty and otherwise compact interval
$[a_{t},b_{t}]$ in $\mathbb{R}$. That is, banks whose excess capital is above
$b_{t}$ are safe and do not obtain any capital injections while banks with an
excess capital below $a_{t}$ are not worth saving. This conjecture is
confirmed by a finite-difference approximation of the solution to PDE (5.5),
depicted on the left-hand plot in Figure 1, but at this stage we are unable to
prove it.

Figure 1: Plot of numerical approximations of the adjoint process $u_{t}(x)$
for $\kappa=0$ (left-hand side) and $\kappa=1$ (right-hand side). The dotted
lines indicate the interval for which $\partial_{x}u_{t}(x)\leq-w$.

Things become more complicated if $\kappa>0$, in which case the state equation
(5.2) becomes nonlinear. Then, owing to the nonlocal term
$\kappa\langle\mu_{t},\partial_{x}u_{t}\rangle\tilde{\lambda}(x)$ in BSPDE
(5.4), it is no longer necessarily the case that $u_{t}$ is nonpositive.
However, as for the local PDE we conjecture that the set
$\\{x\in\mathbb{R}\mathpunct{:}\partial_{x}u_{t}(x)\leq-w\\}$ is either empty
or a compact interval in $\mathbb{R}$. This is once again confirmed by a
numerical approximation of the solution to PDE (5.4) in the setting without
noise, i.e. where $\sigma_{0}=0$ so that $m$ vanishes. The result is plotted
on the right-hand side of Figure 1. To compute this approximation, we
discretised the noiseless versions of both the Fokker–Planck equation (5.3)
and PDE (5.4) using a finite-difference scheme and then employed a Picard
iteration to simulate the forward-backward system. The code used to produce
both plots in Figure 1 can be found on
GitHub111https://github.com/philkant/fokker-planck-control.

In conclusion, let us remark that the sufficient SMP, Theorem 2.12, cannot be
applied to the nonlinear model, because if $\kappa>0$, then the map
$\mathcal{M}(\mathbb{R})\ni
v\mapsto\mathcal{H}_{0}(t,v,u_{t},\partial_{x}u_{t})=\bigl{\langle}v,-\tilde{\lambda}u_{t}-\kappa\langle
v,\tilde{\lambda}\rangle\partial_{x}u_{t}\bigr{\rangle}$ will not be convex.
This is due to the quadratic term $v\mapsto-\kappa\langle
v,\tilde{\lambda}\rangle\langle v,\partial_{x}u_{t}\rangle$, which is convex
if and only if $\tilde{\lambda}=-\partial_{x}u_{t}$ and the latter will not be
satisfied. On the other hand, if $\kappa$ vanishes, this issue disappears and
the sufficient SMP holds. In particular, since the map $g\mapsto
g\partial_{x}u_{t}(x)+wg$ has a unique minimiser, namely
$g_{\text{max}}\mathbf{1}_{\partial_{x}u_{t}(x)\leq-w}$, the optimal control
$\gamma_{t}(x)=g_{\text{max}}\mathbf{1}_{\partial_{x}u_{t}(x)\leq-w}$ is
unique (up to modifications on null sets of $\mu_{t}$).

##  Appendix A Appendix

###  A.1 Equivalence between the Mean-Field Control Problem and the Control of
the Nonlinear Stochastic Fokker–Planck Equation

In this section we establish the equivalence between the optimal control of
the nonlinear stochastic Fokker–Planck equation (2.1) and the associated
McKean–Vlasov SDE.

We will first introduce a weak formulation of the control problem for the
McKean–Vlasov SDE. We adopt the concept from [36, Definition 8.1].

######  Definition A.1.

A weak control is a tuple
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{F}^{0},\mathbb{P},B,W,X,\nu,\alpha)$
such that

  1. (i)

$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a filtered probability space
and $\mathbb{F}^{0}$ is a subfiltration of $\mathbb{F}$;

  2. (ii)

$B$ is a $d$-dimensional $\mathbb{F}$-Brownian motion, $W$ is a
$d_{W}$-dimensional $\mathbb{F}^{0}$-adapted $\mathbb{F}$-Brownian motion, $X$
is a continuous $\mathbb{R}^{d}$-valued $\mathbb{F}$-adapted process, $\nu$ is
a continuous $\mathcal{M}(\mathbb{R}^{d})$-valued $\mathbb{F}^{0}$-adapted
process, and $\alpha$ is a $G$-valued $\mathbb{F}$-progressively measurable
process;

  3. (iii)

$X_{0}$, $B$, and $\mathcal{F}^{0}_{T}$ are independent;

  4. (iv)

for all $t\in[0,T]$ we have
$\mathcal{F}^{X,\alpha}_{t}\perp\mathcal{F}^{B}_{T}\lor\mathcal{F}^{0}_{T}|\mathcal{F}^{B}_{t}\lor\mathcal{F}^{0}_{t}$;

  5. (v)

for all $t\in[0,T]$ we have

$\mathrm{d}X_{t}=b(t,X_{t},\nu_{t},\alpha_{t})\,\mathrm{d}t+\sigma(t,X_{t})\,\mathrm{d}B_{t}+\sigma_{0}\,\mathrm{d}W_{t},\quad
X_{0}\sim\nu_{0},$ (A.1)

and $\nu_{t}=\mathbb{E}[e^{\Lambda_{t}}\delta_{X_{t}}|\mathcal{F}^{0}_{T}]$,
where $\Lambda_{t}=\int_{0}^{t}\lambda(s,X_{s},\nu_{s})\,\mathrm{d}s$.

The coefficient
$\sigma\mathpunct{:}[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ in
(A.1) is a square root of the diffusion coefficient $a$ appearing in the
generator $\mathcal{L}$ of the stochastic Fokker–Planck equation (2.1).

We have the following result that establishes the equivalence between the two
control formulations. Note that its proof still holds if both $\sigma$ and
$\sigma_{0}$ depend on space and the measure argument as long as they are
uniformly Lipschitz continuous in the space variable.

######  Theorem A.2.

Let Assumption 2.1 be satisfied. If, in addition, the map $G\ni g\mapsto
b_{1}(t,x,g)$ is linear and $G\ni g\mapsto f_{1}(t,x,g)$ is convex for all
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, then for any weak control
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{F}^{0},\mathbb{P},B,W,X,\nu,\alpha)$,
there exists a probabilistic setup
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{F}},\tilde{\mathbb{P}},\tilde{W})$
and a control $\gamma\in\mathbb{G}_{\tilde{\mathbb{F}}}$ with a corresponding
solution $\tilde{\nu}$ to SPDE (2.1) such that
$\mathcal{L}^{\tilde{\mathbb{P}}}(\tilde{\nu}_{t})=\mathcal{L}^{\mathbb{P}}(\nu_{t})$
for all $t\in[0,T]$ and

$\tilde{\mathbb{E}}\biggl{[}\int_{0}^{T}\langle\tilde{\nu}_{t},f(t,\cdot,\tilde{\nu}_{t},\gamma_{t})\rangle\,\mathrm{d}t+\psi(\tilde{\nu}_{T})\biggr{]}\leq\mathbb{E}\biggl{[}\int_{0}^{T}e^{\Lambda_{t}}f(t,X_{t},\nu_{t},\alpha_{t})\,\mathrm{d}t+\psi(\nu_{T})\biggr{]}.$

Conversely, if $\nu$ is a solution to SPDE (2.1) for a control
$\gamma\in\mathbb{G}_{\mathbb{F}}$ on some probabilistic setup
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},W)$, then the probabilistic setup
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},W)$ together with $\nu$ and
$\gamma$ can be extended to a weak control
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{F}},\tilde{\mathbb{F}}^{0},\tilde{\mathbb{P}},\tilde{B},\tilde{W},\tilde{X},\tilde{\nu},\tilde{\alpha})$
such that $\tilde{\mathbb{F}}^{0}$, $\tilde{W}$, and $\tilde{\nu}$ are the
extensions of $\mathbb{F}$, $W$, and $\nu$, respectively, and
$\tilde{\alpha}_{t}=\tilde{\gamma}_{t}(\tilde{X}_{t})$ for
$\mathrm{Leb}\otimes\tilde{\mathbb{P}}$-a.e.
$(t,\omega)\in[0,T]\times\tilde{\Omega}$, where $\tilde{\gamma}$ denotes the
extension of $\gamma$.

###### Proof.

Fix a weak control
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{F}^{0},\mathbb{P},B,W,X,\nu,\alpha)$
and define measurable functions $r$,
$g\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$
by $r(t,x,v)=\log\mathbb{E}[e^{\Lambda_{t}}|X_{t}=x,\,\nu_{t}=v]$ and

$g(t,x,v)=\mathbb{E}\bigl{[}e^{\Lambda_{t}-r(t,X_{t},\nu_{t})}\alpha_{t}\bigr{|}X_{t}=x,\,\nu_{t}=v\bigr{]},$

so that
$\langle\nu_{t},\varphi\rangle=\mathbb{E}[e^{r(t,X_{t},\nu_{t})}\varphi(X_{t})|\nu_{t}]$
for any $\varphi\mathpunct{:}\mathbb{R}^{d}\to\mathbb{R}$ measurable and
bounded. Next, fix $\varphi\in C^{2}_{c}(\mathbb{R}^{d})$ and apply Itô’s
formula to $e^{\Lambda_{t}}\varphi(X_{t})$, whereby

$\mathrm{d}(e^{\Lambda_{t}}\varphi(X_{t}))=\mathcal{L}\varphi(t,X_{t},\nu_{t},\alpha_{t})\,\mathrm{d}t+e^{\Lambda_{t}}(\nabla\varphi(X_{t}))^{\top}\sigma(t,X_{t})\,\mathrm{d}B_{t}+e^{\Lambda_{t}}(\nabla\varphi(X_{t}))^{\top}\sigma_{0}\,\mathrm{d}W_{t}.$

We take conditional expectation with respect to $\mathcal{F}^{0}_{T}$ on both
sides to obtain

$\mathrm{d}\langle\nu_{t},\varphi\rangle=\mathbb{E}[\mathcal{L}\varphi(t,X_{t},\nu_{t},\alpha_{t})|\mathcal{F}^{0}_{T}]\,\mathrm{d}t+\langle\nu_{t},(\nabla\varphi)^{\top}\sigma_{0}\rangle\,\mathrm{d}W_{t},$

where we use the stochastic Fubini theorem to exchange the conditional
expectation with respect to $\mathcal{F}^{0}_{T}$ and the stochastic
integration against $W$. The stochastic integral with respect to $B$ vanishes,
since $B$ is independent of $\mathcal{F}^{0}_{T}$. Now, we may proceed as in
the proof of Corollary 1.6 in [36], to show that the law
$P_{t}=\mathcal{L}(\nu_{t})$ satisfies the Fokker–Planck equation

$\displaystyle\int_{\mathcal{M}(\mathbb{R}^{d})}\Phi$ $\displaystyle(\langle
v,\varphi\rangle)\,\mathrm{d}(P_{t}-P_{0})(v)$
$\displaystyle=\sum_{i=1}^{k}\int_{0}^{t}\biggl{(}\int_{\mathcal{M}(\mathbb{R}^{d})}\partial_{x_{i}}\Phi(\langle
v,\varphi\rangle)\bigl{\langle}v,\mathcal{L}\varphi_{i}\bigl{(}s,\cdot,v,g(s,\cdot,v)\bigr{)}\bigr{\rangle}\,\mathrm{d}P_{s}(v)\biggr{)}\,\mathrm{d}s$
$\displaystyle\ \ \
+\frac{1}{2}\sum_{i,j=1}^{k}\int_{0}^{t}\biggl{(}\int_{\mathcal{M}(\mathbb{R}^{d})}\partial^{2}_{x_{i}x_{j}}\Phi(\langle
v,\varphi\rangle)\langle
v,(\nabla\varphi_{i})^{\top}\sigma_{0}\rangle\cdot\langle
v,(\nabla\varphi_{j})^{\top}\sigma_{0}\rangle\,\mathrm{d}P_{s}(v)\biggr{)}\,\mathrm{d}s$

for $\varphi=(\varphi_{1},\dots,\varphi_{k})\in
C^{2}_{c}(\mathbb{R}^{d};\mathbb{R}^{k})$ and $\Phi\in
C^{2}_{c}(\mathbb{R}^{k})$ on $\mathcal{M}(\mathbb{R}^{d})$. Here we used that

$\displaystyle\mathbb{E}\bigl{[}\partial_{x_{i}}\Phi$
$\displaystyle(\langle\nu_{t},\varphi\rangle)e^{\Lambda_{t}}\nabla\varphi_{i}(X_{t})\cdot
b_{1}(t,X_{t})\alpha_{t}\bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\partial_{x_{i}}\Phi(\langle\nu_{t},\varphi\rangle)e^{r(t,X_{t},\nu_{t})}\nabla\varphi_{i}(X_{t})\cdot
b_{1}(t,X_{t})\mathbb{E}\bigl{[}e^{\Lambda_{t}-r(t,X_{t},\nu_{t})}\alpha_{t}\big{|}X_{t},\,\nu_{t}\bigr{]}\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\partial_{x_{i}}\Phi(\langle\nu_{t},\varphi\rangle)e^{r(t,X_{t},\nu_{t})}\nabla\varphi_{i}(X_{t})\cdot
b_{1}(t,X_{t})g(t,X_{t},\nu_{t})\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\partial_{x_{i}}\Phi(\langle\nu_{t},\varphi\rangle)\bigl{\langle}\nu_{t},\nabla\varphi_{i}\cdot
b_{1}(t,\cdot)g(t,\cdot,\nu_{t})\bigr{\rangle}\Bigr{]},$

where by a slight abuse of notation we write $b_{1}(t,x)g$ instead of
$b_{1}(t,x,g)$ to emphasise the linearity of $b_{1}$ in $g$. Hence, we can
apply the superposition principle for stochastic Fokker–Planck equations from
[36, Theorem 1.5], which can easily be extended to equations with zeroth-order
terms, to show there exists a probabilistic setup
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{F}},\tilde{\mathbb{P}},\tilde{W})$
with $\tilde{\mathcal{F}}$ countable generated, supporting a continuous
$\tilde{\mathbb{F}}$-adapted $\mathcal{M}(\mathbb{R}^{d})$-valued process
$\tilde{\nu}$ that solves SPDE (2.1) with input
$\gamma\in\mathbb{G}_{\tilde{\mathbb{F}}}$ given by
$\gamma_{t}(x)=g(t,x,\tilde{\nu}_{t})$ for
$(t,\omega,x)\in[0,T]\times\tilde{\Omega}\times\mathbb{R}^{d}$. Moreover, by
convexity of $f_{1}$ in the control argument, it holds that

$\displaystyle\mathbb{E}\biggl{[}\int_{0}^{T}e^{\Lambda_{t}}$ $\displaystyle
f(t,X_{t},\nu_{t},\alpha_{t})\,\mathrm{d}t+\psi(\nu_{T})\biggr{]}$
$\displaystyle=\mathbb{E}\biggl{[}\int_{0}^{T}e^{r(t,X_{t},\nu_{t})}\mathbb{E}\bigl{[}e^{\Lambda_{t}-r(t,X_{t},\nu_{t})}f(t,X_{t},\nu_{t},\alpha_{t})\big{|}X_{t},\,\nu_{t}\bigr{]}\,\mathrm{d}t+\psi(\nu_{T})\biggr{]}$
$\displaystyle\geq\mathbb{E}\biggl{[}\int_{0}^{T}e^{r(t,X_{t},\nu_{t})}f\bigl{(}t,X_{t},\nu_{t},\mathbb{E}\bigl{[}e^{\Lambda_{t}-r(t,X_{t},\nu_{t})}\alpha_{t}\big{|}X_{t},\,\nu_{t}\bigr{]}\bigr{)}\,\mathrm{d}t+\psi(\nu_{T})\biggr{]}$
$\displaystyle=\mathbb{E}\biggl{[}\int_{0}^{T}\bigl{\langle}\nu_{t},f\bigl{(}t,\cdot,\nu_{t},g(t,\cdot,\nu_{t})\bigr{)}\bigr{\rangle}\,\mathrm{d}t+\psi(\nu_{T})\biggr{]}$
$\displaystyle=\tilde{\mathbb{E}}\biggl{[}\int_{0}^{T}\bigl{\langle}\tilde{\nu}_{t},f\bigl{(}t,\cdot,\tilde{\nu}_{t},g(t,\cdot,\tilde{\nu}_{t})\bigr{)}\bigr{\rangle}\,\mathrm{d}t+\psi(\tilde{\nu}_{T})\biggr{]},$

where we applied Jensen’s inequality using that $\mathcal{F}\ni
A\mapsto\mathbb{E}[e^{\Lambda_{t}-r(t,X_{t},\nu_{t})}\mathbf{1}_{A}|X_{t},\nu_{t}]$
is a (random) probability measure. This concludes the proof of the first part
of the proposition.

Next, let us fix a probabilistic setup
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},W)$ and let $\nu$ be a solution to
SPDE (2.1) for a control $\gamma\in\mathbb{G}_{\mathbb{F}}$. Then we let
$\mu=(\mu_{t})_{0\leq t\leq T}$ be the unique solution to the linear SPDE

$\displaystyle\mathrm{d}\langle\mu_{t},\varphi\rangle=\bigl{\langle}\mu_{t},\mathcal{L}\varphi(t,\cdot,\nu_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t-\langle\mu_{t},\lambda(t,\cdot,\nu_{t})\rangle\,\mathrm{d}t+\bigl{\langle}\mu_{t},(\nabla\varphi)^{\top}\sigma_{0}\bigr{\rangle}\,\mathrm{d}W_{t}$

for $\varphi\in C_{c}^{2}(\mathbb{R}^{d})$ with initial condition
$\mu_{0}=\nu_{0}$. Since this stochastic Fokker–Planck equation does not have
a zeroth-order term and $\nu_{0}\in\mathcal{P}(\mathbb{R}^{d})$, it follows
that $\mu_{t}$ takes values in $\mathcal{P}(\mathbb{R}^{d})$. Consequently, by
the superposition principle for SDEs with random coefficients from [36,
Theorem 1.3], there exists an extension
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{F}},\tilde{\mathbb{P}})$
of $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ supporting a $d$-dimensional
$\tilde{\mathbb{F}}$-Brownian motion $B$ independent of $\mathcal{F}_{T}$ and
a continuous $\mathbb{R}^{d}$-valued $\tilde{\mathbb{F}}$-adapted process $X$
such that

  1. (i)

for all $t\in[0,T]$ we have
$\mathcal{F}^{X}_{t}\perp\mathcal{F}^{B}_{T}\lor\mathcal{F}_{T}|\mathcal{F}^{B}_{t}\lor\mathcal{F}_{t}$;

  2. (ii)

$W$ is an $\tilde{\mathbb{F}}$-Brownian motion;

  3. (iii)

for all $t\in[0,T]$ we have

$\mathrm{d}X_{t}=b\bigl{(}t,X_{t},\nu_{t},\gamma_{t}(X_{t})\bigr{)}\,\mathrm{d}t+\sigma(t,X_{t})\,\mathrm{d}B_{t}+\sigma_{0}\,\mathrm{d}W_{t},\quad
X_{0}\sim\nu_{0},$

and $\mu_{t}=\mathcal{L}^{\tilde{\mathbb{P}}}(X_{t}|\mathcal{F}_{T})$.

Let us set $\Lambda_{t}=\int_{0}^{t}\lambda(s,X_{s},\nu_{s})\,\mathrm{d}s$ and
then define the $\mathcal{M}(\mathbb{R}^{d})$-valued process
$\tilde{\nu}=(\tilde{\nu}_{t})_{0\leq t\leq T}$ by
$\tilde{\nu}_{t}=\tilde{\mathbb{E}}[e^{\Lambda_{t}}\delta_{X_{t}}|\mathcal{F}_{T}]$
for $t\in[0,T]$. Proceeding as at the beginning of the proof, we can show that
$\tilde{\nu}$ solves the linear SPDE

$\mathrm{d}\langle\tilde{\nu}_{t},\varphi\rangle=\bigl{\langle}\tilde{\nu}_{t},\mathcal{L}\varphi(t,\cdot,\nu_{t},\gamma_{t})\bigr{\rangle}\,\mathrm{d}t+\bigl{\langle}\tilde{\nu}_{t},(\nabla\varphi)^{\top}\sigma_{0}\bigr{\rangle}\,\mathrm{d}W_{t}$

for $\varphi\in C_{c}^{2}(\mathbb{R}^{d})$ with initial condition
$\tilde{\nu}_{0}=\mu_{0}=\nu_{0}$. But the unique solution to this SPDE is
$\nu$, so we conclude that $\tilde{\nu}_{t}=\nu_{t}$, meaning that $(X,\nu)$
is a solution to McKean–Vlasov SDE (A.1). Consequently,
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{F}},\mathbb{F},B,W,X,\nu,\alpha)$
with $\alpha=(\alpha_{t})_{0\leq t\leq T}$ defined by
$\alpha_{t}=\gamma_{t}(X_{t})$ is the desired weak control. ∎

###  A.2 Some Properties of the Linear Functional Derivative

######  Lemma A.3.

Let $F\mathpunct{:}\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$ be a measurable
function. If $F$ has a linear functional derivative
$DF\mathpunct{:}\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$
such that $v\mapsto DF(v)(x)$ is continuous for every $x\in\mathbb{R}^{d}$,
then

$DF(v)(x)=\lim_{\epsilon\to
0}\frac{1}{\epsilon}\bigl{(}F(v+\epsilon\delta_{x})-F(v)\bigr{)}$

for all $(x,v)\in\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})$. Conversely,
if there exists a measurable and bounded function
$\mathcal{F}\mathpunct{:}\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$
such that

$\lim_{\epsilon\to
0}\frac{1}{\epsilon}\bigl{(}F(v+\epsilon(v^{\prime}-v))-F(v)\bigr{)}=\langle
v^{\prime}-v,\mathcal{F}(v)\rangle$ (A.2)

for all $v$, $v^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$, then $\mathcal{F}$ is
a linear functional derivative of $F$.

###### Proof.

For the first statement, it follows from the definition of the linear
functional derivative that

$\lim_{\epsilon\to
0}\frac{1}{\epsilon}\bigl{(}F(v+\epsilon\delta_{x})-F(v)\bigr{)}=\lim_{\epsilon\to
0}\int_{0}^{1}\langle\delta_{x},DF(v+\epsilon\theta\delta_{x})\rangle\,\mathrm{d}\theta=DF(v)(x),$

where we used the continuity of $DF$ in the measure argument in the last step.
Note that in fact much less than continuity with respect to $d_{0}$ is
required here.

To establish the second statement, let us fix $v$,
$v^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$ and define the function
$f\mathpunct{:}[0,1]\to\mathbb{R}$ by $f(\theta)=F(v+\theta(v^{\prime}-v))$.
From (A.2), it follows that $f$ is differentiable with
$f^{\prime}(\theta)=\bigl{\langle}v^{\prime}-v,\mathcal{F}(v+\theta(v^{\prime}-v))\bigr{\rangle}$.
Thus, by the fundamental theorem of calculus it holds that

$F(v^{\prime})-F(v)=f(1)-f(0)=\int_{0}^{1}f^{\prime}(\theta)\,\mathrm{d}\theta=\int_{0}^{1}\bigl{\langle}v^{\prime}-v,\mathcal{F}(\theta
v^{\prime}+(1-\theta)v)\bigr{\rangle}\,\mathrm{d}\theta.$

This shows that $\mathcal{F}$ is indeed a linear functional derivative of $F$.
∎

######  Lemma A.4.

Let
$F\mathpunct{:}\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$
be a measurable and bounded function such that $\mathcal{M}(\mathbb{R}^{d})\ni
v\mapsto F(x,v)$ has a linear functional derivative $DF(x,\cdot)$ for all
$x\in\mathbb{R}^{d}$, such that
$DF\mathpunct{:}\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\times\mathbb{R}^{d}\to\mathbb{R}$
is measurable, bounded, and continuous in the measure argument. Then the map
$\mathcal{M}(\mathbb{R}^{d})\ni v\mapsto\langle v,F(\cdot,v)\rangle$ possesses
the linear functional derivative $F(x,v)+\langle v,DF(\cdot,v)(x)\rangle$.

###### Proof.

Set $\mathbb{F}(v)=\langle v,F(\cdot,v)\rangle$ for
$v\in\mathcal{M}(\mathbb{R}^{d})$. By the second part of Lemma A.3 it is
enough to show that for any $v$, $v^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$
the limit $\lim_{\epsilon\to
0}\frac{1}{\epsilon}\bigl{(}\mathbb{F}(v+\epsilon(v^{\prime}-v))-\mathbb{F}(v)\bigr{)}$
exists and is equal to

$\int_{\mathbb{R}^{d}}\bigl{(}F(x,v)+\langle
v,DF(\cdot,v)(x)\rangle\bigr{)}\,\mathrm{d}(v^{\prime}-v)(x).$ (A.3)

We compute

$\displaystyle\begin{split}\frac{1}{\epsilon}\bigl{(}\mathbb{F}(v+\epsilon(v^{\prime}-v))-\mathbb{F}(v)\bigr{)}&=\bigl{\langle}v^{\prime}-v,F(\cdot,v+\epsilon(v^{\prime}-v))\bigr{\rangle}\\\
&\ \ \
+\frac{1}{\epsilon}\bigl{\langle}v,F(\cdot,v+\epsilon(v^{\prime}-v))-F(\cdot,v)\bigr{\rangle}.\end{split}$
(A.4)

Now, since for any $x\in\mathbb{R}^{d}$, the map $m\mapsto F(x,m)$ has a
linear functional derivative, its continuous. Consequently, the first term on
the right-hand side above converges to $\langle
v^{\prime}-v,F(\cdot,v)\rangle$ by the dominated convergence theorem. Next,
for fixed $x\in\mathbb{R}^{d}$, it follows from the continuity of $m\mapsto
DF(x,m)(y)$ for all $y\in\mathbb{R}^{d}$ and the first part of Lemma A.3 that

$\frac{1}{\epsilon}\bigl{(}F(x,v+\epsilon(v^{\prime}-v))-F(x,v)\bigr{)}\to\langle
v^{\prime}-v,DF(x,v)\rangle.$

Applying the dominated convergence theorem once more we conclude that the
second expression on the right-hand side of (A.4) tends to

$\int_{\mathbb{R}^{d}}\langle
v,DF(\cdot,v)(x)\rangle\,\mathrm{d}(v^{\prime}-v)(x).$

Combining the two expression shows that
$\frac{1}{\epsilon}\bigl{(}\mathbb{F}(v+\epsilon(v^{\prime}-v))-\mathbb{F}(v)\bigr{)}$
indeed tends to the expression in (A.3). ∎

######  Lemma A.5.

Let $F\mathpunct{:}\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$ be convex and
assume $F$ has a linear functional derivative $DF$ such that for all $v$,
$v^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$ it holds
that $DF(\theta v+(1-\theta)v^{\prime})(x)\to DF(v^{\prime})(x)$ as
$\theta\in(0,1]$ tends to zero. Then we have $F(v)-F(v^{\prime})\geq\langle
v-v^{\prime},DF(v^{\prime})\rangle$.

###### Proof.

By convexity of $F$, we have for any $\epsilon\in(0,1]$ that

$\displaystyle F(v)-F(v^{\prime})$
$\displaystyle\geq\frac{1}{\epsilon}\bigl{(}F(v^{\prime}+\epsilon(v-v^{\prime}))-F(v^{\prime})\bigr{)}$
$\displaystyle=\int_{0}^{1}\bigl{\langle}v-v^{\prime},DF\bigl{(}v^{\prime}+\epsilon\theta(v-v^{\prime})\bigr{)}\bigr{\rangle}\,\mathrm{d}\theta.$

The expression on the right-hand side converges to $\langle
v-v^{\prime},DF(v^{\prime})\rangle$ as $\epsilon\to 0$, as required. ∎

###  A.3 Auxiliary Results for Section 3

For the rest of this section we assume that $\mathbb{H}$ is a separable
Hilbert space. We begin with an infinite-dimensional analogue of the
Kunita–Watanabe inequality.

######  Proposition A.6.

Let $m$, $\tilde{m}$ be elements of
$\mathcal{M}_{\mathbb{F}}^{2}([0,T];\mathbb{H})$ and let $F$, $\tilde{F}$ be
measurable functions $[0,T]\times\Omega\to B(\mathbb{H})$. Then for all
$t\in[0,T]$ it holds that

$\int_{0}^{t}\operatorname{tr}\bigl{(}\lvert\tilde{F}_{s}F_{s}\rvert\,\mathrm{d}A_{s}\bigr{)}\leq\biggl{(}\int_{0}^{t}\lVert
F^{2}_{s}\rVert\,\mathrm{d}[m]_{s}\biggr{)}^{1/2}\biggl{(}\int_{0}^{t}\lVert\tilde{F}^{2}_{s}\rVert\,\mathrm{d}[\tilde{m}]_{s}\biggr{)}^{1/2},$

where $A$ is the total variation of $[[m,\tilde{m}]]$ and $\lvert\cdot\rvert$
denotes the absolute value of a bounded linear operator. Here
$\lVert\cdot\rVert$ is the operator norm for elements of $B(\mathbb{H})$.

###### Proof.

Using the approximation of the tensor quadratic variation by sums of square
increments, see [46, Theorem 26.11 ], we can show that for $0\leq s\leq t\leq
T$ and $L$, $\tilde{L}\in B(\mathbb{H})$ it holds that

$\displaystyle\operatorname{tr}\bigl{(}\lvert\tilde{L}L\rvert(A_{t}-A_{s})\bigr{)}\leq\lVert
L\rVert\lVert\tilde{L}\rVert\operatorname{tr}(A_{t}-A_{s})\leq\lVert
L\rVert\lVert\tilde{L}\rVert\sqrt{[m]_{t}-[m]_{s}}\sqrt{[\tilde{m}]_{t}-[\tilde{m}]_{s}}.$

With this inequality established one can follow the proof of the classical
one-dimensional Kunita–Watanabe inequality. See e.g. [41, Proposition 4.18]. ∎

######  Lemma A.7.

Suppose that $(u^{n})_{n\geq 1}$ is a sequence in
$D_{\mathbb{F}}^{2}([0,T];\mathbb{H})$ that converges to some process $u\in
D_{\mathbb{F}}^{2}([0,T];\mathbb{H})$, and that $(m^{n})_{n\geq 1}$,
$(\tilde{m}^{n})_{n\geq 1}$ are sequences in
$\mathcal{M}_{\mathbb{F}}^{2}([0,T];\mathbb{H})$ that converge to processes
$m$, $\tilde{m}\in\mathcal{M}_{\mathbb{F}}^{2}([0,T];\mathbb{H})$,
respectively. Next, let $F_{1}\mathpunct{:}\mathbb{H}\to\mathbb{H}$ and
$F_{2}\mathpunct{:}\mathbb{H}\to B(\mathbb{H})$ be continuous and bounded on
bounded sets of $\mathbb{H}$. Then

$\displaystyle\int_{0}^{t}\langle
F_{1}(u^{n}_{s-}),\mathrm{d}m^{n}_{s}\rangle_{\mathbb{H}}$
$\displaystyle\to\int_{0}^{t}\langle
F_{1}(u_{s-}),\mathrm{d}m_{s}\rangle_{\mathbb{H}},$ (A.5)
$\displaystyle\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}(u^{n}_{s-})\,\mathrm{d}[[m^{n},\tilde{m}^{n}]]^{c}_{s}\bigr{)}$
$\displaystyle\to\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}(u_{s-})\,\mathrm{d}[[m,\tilde{m}]]^{c}_{s}\bigr{)}$
(A.6)

uniformly in $t\in[0,T]$ in probability.

###### Proof.

We prove the convergences in (A.5) and (A.6) separately. For the first one we
write

$\displaystyle\begin{split}\int_{0}^{t}\langle
F_{1}(u^{n}_{s-})&,\mathrm{d}m^{n}_{s}\rangle_{\mathbb{H}}-\int_{0}^{t}\langle
F_{1}(u_{s-}),\mathrm{d}m_{s}\rangle_{\mathbb{H}}\\\
&=\int_{0}^{t}\bigl{\langle}F_{1}(u^{n}_{s-})-F_{1}(u_{s-}),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}+\int_{0}^{t}\bigl{\langle}F_{1}(u_{s-}),\mathrm{d}(m^{n}_{s}-m_{s})\bigr{\rangle}_{\mathbb{H}}.\end{split}$
(A.7)

We will deal with the terms on the right-hand side in order. Our goal is to
prove that both converge to zero in probability. For $k\geq 1$ let
$\pi_{k}\mathpunct{:}\mathbb{H}\to\mathbb{H}$ denote the projection onto the
centred ball $B_{k}$ of radius $k$ in $\mathbb{H}$. Now, since
$\mathbb{E}\sup_{0\leq s\leq t}\lVert
u^{n}_{s}-u_{s}\rVert_{\mathbb{H}}^{2}\to 0$, for any $\delta>0$ we can find
$K\geq 1$ such that for all $k\geq K$ it holds that

$\displaystyle\mathbb{P}\Bigl{(}\pi_{k}(u^{n}_{s-})=u^{n}_{s-},\,\pi_{k}(u_{s-})=u_{s-}\
\text{for }s\in(0,t]\Bigr{)}$ $\displaystyle=\mathbb{P}\biggl{(}\sup_{0\leq
s<t}\lVert u^{n}_{s}\rVert_{\mathbb{H}}^{2}\lor\sup_{0\leq s<t}\lVert
u_{s}\rVert_{\mathbb{H}}^{2}\leq k\biggr{)}$ $\displaystyle\geq 1-\delta.$

Hence, for any $\epsilon>0$ we obtain

$\displaystyle\begin{split}\mathbb{P}\biggl{(}\sup_{0\leq t\leq
T}\biggl{\lvert}&\int_{0}^{t}\bigl{\langle}F_{1}(u^{n}_{s-})-F_{1}(u_{s-}),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}>\epsilon\biggr{)}\\\
&\leq\mathbb{P}\biggl{(}\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}(\pi_{k}(u^{n}_{s-}))-F_{1}(\pi_{k}(u_{s-})),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}>\epsilon\biggr{)}+\delta.\end{split}$
(A.8)

Next, using Proposition 3.1 we find that

$\displaystyle\mathbb{P}\biggl{(}\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}$
$\displaystyle(\pi_{k}(u^{n}_{s-}))-F_{1}(\pi_{k}(u_{s-})),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}>\epsilon\biggr{)}$
$\displaystyle\leq\frac{1}{\epsilon}\mathbb{E}\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}(\pi_{k}(u^{n}_{t-}))-F_{1}(\pi_{k}(u_{t-})),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}$
$\displaystyle\leq\frac{C_{1}}{\epsilon}\mathbb{E}\biggl{(}\int_{0}^{T}\bigl{\lVert}F_{1}(\pi_{k}(u^{n}_{t-}))-F_{1}(\pi_{k}(u_{t-}))\bigr{\rVert}_{\mathbb{H}}^{2}\,\mathrm{d}[m^{n}]_{t}\biggr{)}^{1/2}$
$\displaystyle\leq\frac{C_{1}}{\epsilon}\mathbb{E}\biggl{[}\sup_{0\leq t\leq
T}\bigl{\lVert}F_{1}(\pi_{k}(u^{n}_{t}))-F_{1}(\pi_{k}(u_{t}))\bigr{\rVert}_{\mathbb{H}}\bigl{(}[m^{n}]_{T}-[m^{n}]_{0}\bigr{)}^{1/2}\biggr{]}$
$\displaystyle\leq\frac{C_{1}^{2}}{\delta\epsilon^{2}}\mathbb{E}\sup_{0\leq
t\leq
T}\bigl{\lVert}F_{1}(\pi_{k}(u^{n}_{t}))-F_{1}(\pi_{k}(u_{t}))\bigr{\rVert}_{\mathbb{H}}^{2}+\delta\mathbb{E}\bigl{(}[m^{n}]_{T}-[m^{n}]_{0}\bigr{)}.$
(A.9)

Now, since $m^{n}$ converges to $m$ in
$\mathcal{M}_{\mathbb{F}}^{2}([0,T];\mathbb{H})$, the quantity
$\mathbb{E}\bigl{(}[m^{n}]_{T}-[m^{n}]_{0}\bigr{)}$ is bounded uniformly in
$n\geq 1$ by some constant $C>0$. Moreover, the map $F_{1}\circ\pi_{k}$ is
continuous and bounded, so since $\mathbb{E}\sup_{0\leq t\leq T}\lVert
u^{n}_{t}-u_{t}\rVert_{\mathbb{H}}^{2}\to 0$ by the dominated convergence
theorem, we may choose $N\geq 1$ large enough so that for all $n\geq N$ we
have $\mathbb{E}\sup_{0\leq t\leq
T}\bigl{\lVert}F_{1}(\pi_{k}(u^{n}_{t}))-F_{1}(\pi_{k}(u_{t}))\bigr{\rVert}_{\mathbb{H}}^{2}\leq\frac{\delta^{2}\epsilon^{2}}{C_{1}^{2}}$.
Plugging this into (A.3) and combining it with Equation (A.8) implies that

$\mathbb{P}\biggl{(}\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}(u^{n}_{s-})-F_{1}(u_{s-}),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}>\epsilon\biggr{)}\leq(2+C)\delta.$

Since $\delta$ and $\epsilon$ were arbitrary this implies that
$\int_{0}^{t}\bigl{\langle}F_{1}(u^{n}_{s-})-F_{1}(u_{s-}),\mathrm{d}m^{n}_{s}\bigr{\rangle}_{\mathbb{H}}$
converges uniformly in $t\in[0,T]$ to zero in probability.

Next, let us consider the second term on the right-hand side of (A.7). Arguing
as above for any $\delta>0$ we can choose $K\geq 1$ and $k\geq K$ such that
for any $\epsilon>0$,

$\displaystyle\mathbb{P}\biggl{(}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}$
$\displaystyle(u_{s-}),\mathrm{d}(m^{n}_{s}-m_{s})\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}>\epsilon\biggr{)}$
$\displaystyle\leq\frac{1}{\epsilon}\mathbb{E}\biggl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}(\pi_{k}(u_{s-})),\mathrm{d}(m^{n}_{s}-m_{s})\bigr{\rangle}_{\mathbb{H}}\biggr{\rvert}+\delta$
$\displaystyle\leq\frac{\delta}{c_{k}^{2}}\mathbb{E}\sup_{0\leq t\leq T}\lVert
F_{1}(\pi_{k}(u_{t}))\rVert_{\mathbb{H}}^{2}+\frac{C_{1}^{2}c_{k}^{2}}{\delta\epsilon^{2}}\mathbb{E}\bigl{[}[m^{n}]_{t}-[m^{\epsilon}]_{0}-([m]_{t}-[m]_{0})\bigr{]}+\delta$
$\displaystyle\leq
2\delta+\frac{C_{1}^{2}c_{k}^{2}}{\delta\epsilon^{2}}\mathbb{E}\bigl{[}[m^{n}]_{t}-[m^{n}]_{0}-([m]_{t}-[m]_{0})\bigr{]},$

where $c_{k}=\sup_{h\in B_{k}}\lVert F_{1}(h)\rVert_{\mathbb{H}}$. Then we
choose $N\geq 1$ large enough such that for all $n\geq 1$ we have
$\mathbb{E}\bigl{[}[m^{n}]_{t}-[m^{n}]_{0}-([m]_{t}-[m]_{0})\bigr{]}\leq\frac{\delta^{2}\epsilon^{2}}{C_{1}^{2}c_{n}^{2}}$
to obtain $\mathbb{P}\bigl{(}\sup_{0\leq t\leq
T}\bigl{\lvert}\int_{0}^{t}\bigl{\langle}F_{1}(u_{s-}),\mathrm{d}(m^{n}_{s}-m_{s})\bigr{\rangle}_{\mathbb{H}}\bigr{\rvert}>\epsilon\bigr{)}\leq
3\delta$, which means that
$\int_{0}^{t}\bigl{\langle}F_{1}(u_{s-}),\mathrm{d}(m^{n}_{s}-m_{s})\bigr{\rangle}_{\mathbb{H}}$
converges uniformly in $t\in[0,T]$ to zero in probability as well. This
concludes our treatment of the convergence in (A.5).

We continue with (A.6). By appealing to the polarisation identity
$[[m,\tilde{m}]]=\frac{1}{4}([[m+\tilde{m}]]-[[m-\tilde{m}]])$, we can reduce
to the case $\tilde{m}^{n}=m^{n}$ and $\tilde{m}=m$. Then similarly to
Equation (A.7), we decompose

$\displaystyle\int_{0}^{t}\operatorname{tr}\bigl{(}$ $\displaystyle
F_{2}(u^{n}_{s-})\,\mathrm{d}[[m^{n}]]^{c}_{s}\bigr{)}-\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}(u_{s-})\,\mathrm{d}[[m]]^{c}_{s}\bigr{)}$
$\displaystyle=\int_{0}^{t}\operatorname{tr}\bigl{(}(F_{2}(u^{n}_{s-})-F_{2}(u_{s-}))\,\mathrm{d}[[m^{n}]]^{c}_{s}\bigr{)}+\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}(u_{s-})\,\mathrm{d}([[m^{n}]]^{c}_{s}-[[m]]^{c}_{s})\bigr{)}.$

For the first term we apply the inequality

$\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\operatorname{tr}\bigl{(}(F_{2}(u^{n}_{s-})-F_{2}(u_{s-}))\,\mathrm{d}[[m^{n}]]^{c}_{s}\bigr{)}\biggr{\rvert}\leq\int_{0}^{T}\bigl{\lVert}F_{2}(u^{n}_{t-})-F_{2}(u_{t-})\bigr{\rVert}\,\mathrm{d}[m^{n}]^{c}_{t},$
(A.10)

where $\bigl{\lVert}F_{2}(u^{n}_{t-})-F_{2}(u_{t-})\bigr{\rVert}$ denotes the
operator norm of
$F_{2}(u^{n}_{t-})-F_{2}(u_{t-})\mathpunct{:}\mathbb{H}\to\mathbb{H}$ and then
proceed as above to show that the right-hand side converges to zero in
probability as $n\to\infty$. To deal with the second term, we write
$[[m^{n}]]^{c}_{s}-[[m]]^{c}_{s}=[[m^{n}-m,m^{n}]]^{c}_{s}+[[m,m^{n}-m]]^{c}_{s}$
and then apply the Kunita–Watanabe inequality, Proposition A.6, to obtain

$\displaystyle\sup_{0\leq t\leq
T}\biggl{\lvert}\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}$
$\displaystyle(u_{s-})\,\mathrm{d}([[m^{n}]]^{c}_{s}-[[m]]^{c}_{s})\bigr{)}\biggr{\rvert}$
$\displaystyle\leq\sqrt{2}\biggl{(}\int_{0}^{T}\lVert
F_{2}(u_{t-})\rVert\,\mathrm{d}([m^{n}]_{t}+[m]_{t})\biggr{)}^{1/2}\biggl{(}\int_{0}^{T}\lVert
F_{2}(u_{t-})\rVert\,\mathrm{d}[m^{n}-m]_{t}\biggr{)}^{1/2}$
$\displaystyle\leq\delta\int_{0}^{T}\lVert
F_{2}(u_{t-})\rVert\,\mathrm{d}([m^{n}]_{t}+[m]_{t})+\frac{1}{2\delta}\int_{0}^{T}\lVert
F_{2}(u_{t-})\rVert\,\mathrm{d}[m^{n}-m]_{t}.$

The first expression on the right-hand can be made arbitrarily small by
choosing $\delta>0$ appropriately. The second term tends to zero in
probability as $n\to\infty$. Hence, $\sup_{0\leq t\leq
T}\bigl{\lvert}\int_{0}^{t}\operatorname{tr}\bigl{(}F_{2}(u_{s-})\,\mathrm{d}([[m^{n}]]^{c}_{s}-[[m]]^{c}_{s})\bigr{)}\bigr{\rvert}\to
0$ in probability, which concludes the proof. ∎

###  A.4 Auxiliary Results for Section 4

In what follows, $\eta\mathpunct{:}\mathbb{R}^{d}\to\mathbb{R}$ is defined by
$\eta(x)=\eta_{0}\sqrt{1+\lvert x\rvert^{2}}$ for $x\in\mathbb{R}^{d}$ and
some $\eta_{0}>0$. Recall the distance $d_{0}$ on the space
$\mathcal{M}(\mathbb{R}^{d})$ of finite measures on $\mathbb{R}^{d}$ defined
in Equation (1.2).

######  Lemma A.8.

Let $(\mu^{n})_{n}$ be a bounded sequence in
$L([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))$ that converges to some $\mu\in
L([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))$ in
$L([0,T];L^{2}_{\eta,\textup{loc}}(\mathbb{R}^{d}))$. Then it holds that

$\int_{0}^{T}\sup_{\lVert\varphi\rVert_{\infty}\leq
1}\langle\mu^{n}_{t}-\mu_{t},\varphi\rangle\,\mathrm{d}t\to 0$

as $n\to\infty$, where the supremum is taken over all bounded and measurable
functions $\varphi\mathpunct{:}\mathbb{R}^{d}\to\mathbb{R}$ with
$\lVert\varphi\rVert_{\infty}\leq 1$.

###### Proof.

For $R>0$, we can write

$\displaystyle\int_{0}^{T}\sup_{\lVert\varphi\rVert_{\infty}\leq 1}\langle$
$\displaystyle\mu^{n}_{t}-\mu_{t},\varphi\rangle\,\mathrm{d}t$
$\displaystyle\leq
2\int_{0}^{T}\langle\lvert\mu^{n}_{t}-\mu_{t}\rvert,\mathbf{1}_{B_{R}}\rangle\,\mathrm{d}t+2\int_{0}^{T}\bigl{\langle}\lvert\mu^{n}_{t}-\mu_{t}\rvert,\mathbf{1}_{B_{R}^{C}}\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle\leq 2\lVert
e^{-\eta/2}\rVert_{L^{2}}\int_{0}^{T}\lVert\mathbf{1}_{B_{R}}(\mu^{n}_{t}-\mu_{t})\rVert_{\eta}\,\mathrm{d}t+2\bigl{\lVert}\mathbf{1}_{B_{R}^{C}}\bigr{\rVert}_{\bar{\eta}}\int_{0}^{T}\lVert\mu^{n}_{t}\rVert_{\eta}+\lVert\mu_{t}\rVert_{\eta}\,\mathrm{d}t.$

The second integral in the last line is bounded uniformly in $n\geq 1$ while
$\bigl{\lVert}\mathbf{1}_{B_{R}^{C}}\bigr{\rVert}_{\bar{\eta}}$ vanishes as
$R\to\infty$. Thus, for any $\epsilon>0$, choosing $R$ large enough, we can
make the supremum in $n\geq 1$ of the second summand in the last line smaller
than $\epsilon/2$. Then, we pick $N\geq 1$ large enough, such that for any
$n\geq N$, the first term in the last line is bounded by $\epsilon/2$.
Together, we get $\int_{0}^{T}\sup_{\lVert\varphi\rVert_{\infty}\leq
1}\langle\mu^{n}_{t}-\mu_{t},\varphi\rangle\,\mathrm{d}t\leq\epsilon$ for all
$n\geq N$. Since $\epsilon>0$ was arbitrary, the desired convergence follows.
∎

######  Lemma A.9.

Let $(\mu^{n})_{n}$ and $\mu$ be as in the statement of Lemma A.8. Next, let
$(g^{n})_{n}$ be a sequence in
$L^{2}([0,T];L^{2}_{\textup{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$, such
that $g^{n}_{t}(x)\in G$ for all $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and which
converges weakly to some $g\in
L^{2}([0,T];L^{2}_{\textup{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$.
Finally, let $(w^{n})_{n}$ be a convergent sequence in
$C([0,T];\mathbb{R}^{d})$ with limit $w$. Then,

  1. (i)

if
$F_{1}\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})\to\mathbb{R}$
is a bounded and measurable function, such that $v\mapsto F_{1}(t,x,y,v)$ is
continuous for all $(t,x,y)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
and for which there exists bounded modulus of continuity
$\omega\mathpunct{:}[0,\infty)\to[0,\infty)$ with $\lim_{\delta\to
0}\omega(\delta)=\omega(0)=0$ and

$\lvert F_{1}(t,x,y,v)-F_{1}(t,x,y^{\prime},v)\rvert\leq\omega(\lvert
y-y^{\prime}\rvert)$

for all $y$, $y^{\prime}\in\mathbb{R}^{d}$ and every
$(t,x,v)\in[0,T]\times\mathbb{R}^{d}\times\mathcal{M}(\mathbb{R}^{d})$, it
follows that

$\displaystyle\lim_{n\to\infty}\int_{0}^{T}\bigl{\langle}\mu^{n}_{t},F_{1}\bigl{(}t,\cdot+w^{n}_{t}$
$\displaystyle,\cdot,(\operatorname{id}+w^{n}_{t})^{\\#}\mu^{n}_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\bigl{\langle}\mu_{t},F_{1}\bigl{(}t,\cdot+w_{t},\cdot,(\operatorname{id}+w_{t})^{\\#}\mu_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t;$

  2. (ii)

if
$F_{2}\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$
is bounded and measurable and such that there exists a bounded set
$K\subset\mathbb{R}^{d}$ such that $F_{2}(t,x,y)=0$ for $t\in[0,T]$ and $x$,
$y$ outside of $K$, it follows that

$\lim_{n\to\infty}\int_{0}^{T}\bigl{\langle}\mu^{n}_{t},F_{2}(t,\cdot+w^{n}_{t},\cdot)g^{n}_{t}\bigr{\rangle}\,\mathrm{d}t=\int_{0}^{T}\bigl{\langle}\mu_{t},F_{2}(t,\cdot+w_{t},\cdot)g_{t}\bigr{\rangle}\,\mathrm{d}t;$

  3. (iii)

if $F_{3}\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times G\to\mathbb{R}$ is
bounded and measurable, such that $h\mapsto F_{3}(t,x,h)$ is convex for every
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, it follows that

$\liminf_{n\to\infty}\int_{0}^{T}\bigl{\langle}\mu^{n}_{t},F_{3}\bigl{(}t,\cdot+w^{n}_{t},g^{n}_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t\geq\int_{0}^{T}\bigl{\langle}\mu_{t},F_{3}\bigl{(}t,\cdot+w_{t},g_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t.$

  4. (iv)

if $(g^{n})_{n}$ converges in the strong topology on
$L^{2}([0,T];L^{2}_{\textup{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$ and
$F_{4}\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times
G\to\mathbb{R}$ is bounded and measurable, such that $h\mapsto F_{4}(t,x,y,h)$
is continuous for all
$(t,x,y)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ and for which there
exists bounded modulus of continuity
$\omega\mathpunct{:}[0,\infty)\to[0,\infty)$ with $\lim_{\delta\to
0}\omega(\delta)=\omega(0)=0$ and

$\lvert F_{1}(t,x,y,h)-F_{1}(t,x,y^{\prime},h)\rvert\leq\omega(\lvert
y-y^{\prime}\rvert)$

for all $y$, $y^{\prime}\in\mathbb{R}^{d}$ and every
$(t,x,h)\in[0,T]\times\mathbb{R}^{d}\times G$, it follows that

$\lim_{n\to\infty}\int_{0}^{T}\bigl{\langle}\mu^{n}_{t},F_{4}\bigl{(}t,\cdot+w^{n}_{t},\cdot,g^{n}_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t=\int_{0}^{T}\bigl{\langle}\mu_{t},F_{4}\bigl{(}t,\cdot+w_{t},\cdot,g_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t;$

  5. (v)

if $(g^{n})_{n}$ converges in the strong topology on
$L^{2}([0,T];L^{2}_{\textup{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$ and
$F_{5}\mathpunct{:}[0,T]\times\mathbb{R}^{d}\times G\to\mathbb{R}$ is bounded
and measurable, such that $h\mapsto F_{5}(t,x,h)$ is upper semicontinuous for
every $(t,x)\in[0,T]\times\mathbb{R}^{d}$, it follows that

$\limsup_{n\to\infty}\int_{0}^{T}\bigl{\langle}\mu^{n}_{t},F_{5}\bigl{(}t,\cdot+w^{n}_{t},g^{n}_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t\leq\int_{0}^{T}\bigl{\langle}\mu_{t},F_{5}\bigl{(}t,\cdot+w_{t},g_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t.$

###### Proof.

Let us define $\tilde{\mu}^{n}$, $\tilde{\mu}\in
L^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))$ by
$\tilde{\mu}^{n}_{t}=(\operatorname{id}+w^{n}_{t})^{\\#}\mu^{n}_{t}$ and
$\tilde{\mu}_{t}=(\operatorname{id}+w_{t})^{\\#}\mu_{t}$ and, similarly,
$\tilde{g}^{n}$, $\tilde{g}\in
L^{2}([0,T];L^{2}_{\text{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$ by
$\tilde{g}^{n}_{t}(x)=g^{n}_{t}(x-w^{n}_{t})$ and
$\tilde{g}_{t}(x)=g_{t}(x-w_{t})$. Then $(\tilde{\mu}^{n})_{n}$ is still a
bounded sequence in $L^{2}([0,T];L^{2}_{\eta}(\mathbb{R}^{d}))$ which
converges to $\tilde{\mu}$ in
$L^{2}([0,T];L^{2}_{\eta,\text{loc}}(\mathbb{R}^{d}))$, and
$(\tilde{g}^{n})_{n}$ converges weakly to $\tilde{g}$ in the space
$L^{2}([0,T];L^{2}_{\text{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$. Thus, by
Lemma A.8 it holds that

$\int_{0}^{T}\sup_{\lVert\varphi\rVert_{\infty}\leq
1}\langle\tilde{\mu}^{n}_{t}-\tilde{\mu}_{t},\varphi\rangle\,\mathrm{d}t\to 0$
(A.11)

and, in particular,
$\int_{0}^{T}d_{0}^{2}(\tilde{\mu}^{n}_{t},\tilde{\mu}_{t})\,\mathrm{d}t\to
0$. Next, since $G$ is convex and $(g^{n})_{n}$ converges weakly to $g$ in
$L^{2}([0,T];L^{2}_{\text{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$,
$g^{n}_{t}(x)\in G$ for all $(t,x)\in[0,T]\times\mathbb{R}^{d}$ implies the
same is true for $g$, i.e. $g_{t}(x)\in G$ for all
$(t,x)\in[0,T]\times\mathbb{R}^{d}$. We will prove the five statements (i) to
(v) in sequence.

Item (i): First, we decompose

$\displaystyle\begin{split}\int_{0}^{T}&\Bigl{(}\bigl{\langle}\mu^{n}_{t},F_{1}\bigl{(}t,\cdot+w^{n}_{t},\cdot,(\operatorname{id}+w^{n}_{t})^{\\#}\mu^{n}_{t}\bigr{)}\bigr{\rangle}-\bigl{\langle}\mu_{t},F_{1}\bigl{(}t,\cdot+w_{t},\cdot,(\operatorname{id}+w_{t})^{\\#}\mu_{t}\bigr{)}\bigr{\rangle}\Bigr{)}\,\mathrm{d}t\\\
&=\int_{0}^{T}\bigl{\langle}\tilde{\mu}^{n}_{t}-\tilde{\mu}_{t},F_{1}\bigl{(}t,\cdot,\cdot-w^{n}_{t},\tilde{\mu}^{n}_{t}\bigr{)}\bigr{\rangle}\,\mathrm{d}t\\\
&\ \ \
+\int_{0}^{T}\Bigl{\langle}\tilde{\mu}_{t},F_{1}\bigl{(}t,\cdot,\cdot-w^{n}_{t},\tilde{\mu}^{n}_{t}\bigr{)}-F_{1}\bigl{(}t,\cdot,\cdot-
w_{t},\tilde{\mu}^{n}_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}t\\\ &\ \ \
+\int_{0}^{T}\Bigl{\langle}\tilde{\mu}_{t},F_{1}\bigl{(}t,\cdot,\cdot-
w_{t},\tilde{\mu}^{n}_{t}\bigr{)}-F_{1}\bigl{(}t,\cdot,\cdot-
w_{t},\tilde{\mu}_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}t.\end{split}$ (A.12)

The first expression on the right-hand sides vanishes as $n\to\infty$ by
(A.11). For the second term in (A.12), we have by assumption that

$\bigl{\lvert}F_{1}\bigl{(}t,x,x-w^{n}_{t},\tilde{\mu}^{n}_{t}\bigr{)}-F_{1}\bigl{(}t,x,x-w_{t},\tilde{\mu}^{n}_{t}\bigr{)}\bigr{\rvert}\leq\omega(\lvert
w^{n}_{t}-w_{t}\rvert),$

so that

$\biggl{\lvert}\int_{0}^{T}\Bigl{\langle}\tilde{\mu}_{t},F_{1}\bigl{(}t,\cdot,\cdot-w^{n}_{t},\tilde{\mu}^{n}_{t}\bigr{)}-F_{1}\bigl{(}t,\cdot,\cdot-
w_{t},\tilde{\mu}^{n}_{t}\bigr{)}\Bigr{\rangle}\,\mathrm{d}t\biggr{\rvert}\leq\lVert
e^{-\eta/2}\rVert_{L^{2}}\int_{0}^{T}\lVert\tilde{\mu}_{t}\rVert_{\eta}\omega(\lvert
w^{n}_{t}-w_{t}\rvert)\,\mathrm{d}t.$

The bound on the right-hand side clearly vanishes as $n\to\infty$. The
convergence of the last summand in (A.12) is an immediate consequence of the
continuity of $F_{1}$ in the measure argument, the fact that
$\int_{0}^{T}d_{0}^{2}(\tilde{\mu}^{n}_{t},\tilde{\mu}_{t})\,\mathrm{d}t\to
0$, and the dominated convergence theorem.

Item (ii): Again, we split the difference into three terms:

$\displaystyle\begin{split}\int_{0}^{T}&\Bigl{(}\bigl{\langle}\mu^{n}_{t},F_{2}(t,\cdot+w^{n}_{t},\cdot)g^{n}_{t}\bigr{\rangle}-\bigl{\langle}\mu_{t},F_{2}(t,\cdot+w_{t},\cdot)g_{t}\bigr{\rangle}\Bigr{)}\,\mathrm{d}t\\\
&=\int_{0}^{T}\bigl{\langle}\tilde{\mu}^{n}_{t}-\tilde{\mu}_{t},F_{2}(t,\cdot,\cdot-w^{n}_{t})\tilde{g}^{n}_{t}\bigr{\rangle}\,\mathrm{d}t\\\
&\ \ \
+\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{2}(t,\cdot,\cdot-w^{n}_{t})\tilde{g}^{n}_{t}-F_{2}(t,\cdot,\cdot-
w_{t})\tilde{g}^{n}_{t}\bigr{\rangle}\,\mathrm{d}t\\\ &\ \ \
+\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{2}(t,\cdot,\cdot-
w_{t})(\tilde{g}^{n}_{t}-\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t.\end{split}$
(A.13)

The convergence to zero of the first term on the right-hand side follows along
the same lines as the convergence of the first integral on the right-hand side
of (A.12). For the second, we estimate

$\displaystyle\biggl{\lvert}\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{2}$
$\displaystyle(t,\cdot,\cdot-w^{n}_{t})\tilde{g}^{n}_{t}-F_{2}(t,\cdot,\cdot-
w_{t})\tilde{g}^{n}_{t}\bigr{\rangle}\,\mathrm{d}t\biggr{\rvert}$
$\displaystyle\leq\sup_{g\in G}\lvert
g\rvert\int_{0}^{T}\lVert\tilde{\mu}_{t}\rVert_{\eta}\bigl{\lVert}F_{2}(t,\cdot,\cdot-w^{n}_{t})-F_{2}(t,\cdot,\cdot-
w_{t})\bigr{\rVert}_{\bar{\eta}}\,\mathrm{d}t,$

where we exploited that $g^{n}$ maps into the bounded set $G$. Since $w^{n}$
tends to $w$ uniformly on $[0,T]$, convergence of the integral on the right-
hand side above is standard. Finally, the last summand in (A.13) vanishes as
$n\to\infty$ because $\tilde{g}^{n}$ converges weakly to $\tilde{g}$ in
$L([0,T];L^{2}_{\textup{loc}}(\mathbb{R}^{d}))$, $\tilde{\mu}\in
L([0,T];L^{2}(\mathbb{R}^{d}))$, and $(x,y)\mapsto F_{2}(t,x,y)$ has compact
support uniformly in $t\in[0,T]$.

Item (iii): We write

$\displaystyle\begin{split}\int_{0}^{T}&\Bigl{(}\bigl{\langle}\mu^{n}_{t},F_{3}\bigl{(}t,\cdot+w^{n}_{t},g^{n}_{t}\bigr{)}\bigr{\rangle}-\bigl{\langle}\mu_{t},F_{3}\bigl{(}t,\cdot+w_{t},g_{t}\bigr{)}\bigr{\rangle}\Bigr{)}\,\mathrm{d}t\\\
&=\int_{0}^{T}\bigl{\langle}\tilde{\mu}^{n}_{t}-\tilde{\mu}_{t},F_{3}(t,\cdot,\tilde{g}^{n}_{t})\bigr{\rangle}\,\mathrm{d}t\\\
&\ \ \
+\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{3}(t,\cdot,\tilde{g}^{n}_{t})-F_{3}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t.\end{split}$
(A.14)

The convergence of the first term to zero follows as before. For the second
term, we notice that the convexity of $F_{3}$ in its last component implies
convexity of the function
$L^{2}([0,T];L^{2}_{\text{loc}}(\mathbb{R}^{d};G))\ni
h\mapsto\int_{0}^{T}\langle\mu_{t},F_{3}(t,\cdot,h_{t})\rangle\,\mathrm{d}t$
and, therefore, its weak lower semicontinuity. Since $(\tilde{g}^{n})_{n}$
converges weakly to $\tilde{g}$ in
$L^{2}([0,T];L^{2}_{\text{loc}}(\mathbb{R}^{d};\mathbb{R}^{d_{G}}))$, we get
that

$\liminf_{n\to\infty}\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{3}(t,\cdot,\tilde{g}^{n}_{t})-F_{3}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t\geq
0.$

This gives (iii).

Item (iv): The convergence can be established analogously to Item (i).

Item (v): We can decompose the difference into two summands the same way as
(A.14). The convergence of the first summand is clear. To deal with the second
term
$\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{5}(t,\cdot,\tilde{g}^{n}_{t})-F_{5}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t$,
let us pick a subsequence $(n_{k})_{k}$ that achieves the limit superior of
this expression as $n\to\infty$. By choosing a further subsequence (which we
still denote by $(n_{k})_{k}$) if necessary, we may assume that
$\tilde{g}^{n_{k}}_{t}(x)$ converges to $\tilde{g}_{t}(x)$ for a.e.
$(t,x)\in[0,T]\times\mathbb{R}^{d}$. Then upper semicontinuity of $F_{5}$ in
its last argument implies that

$\limsup_{k\to\infty}F_{5}\bigl{(}t,x,\tilde{g}^{n_{k}}_{t}(x)\bigr{)}\leq
F_{t}\bigl{(}t,x,\tilde{g}_{t}(x)\bigr{)}.$

Multiplying both sides by $\tilde{\mu}_{t}(x)$, integrating over
$[0,T]\times\mathbb{R}^{d}$, and then applying the reverse version of Fatou’s
lemma gives

$\displaystyle\limsup_{n\to\infty}\int_{0}^{T}$
$\displaystyle\bigl{\langle}\tilde{\mu}_{t},F_{5}(t,\cdot,\tilde{g}^{n}_{t})-F_{5}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle=\lim_{k\to\infty}\int_{0}^{T}\bigl{\langle}\tilde{\mu}_{t},F_{5}(t,\cdot,\tilde{g}^{n_{k}}_{t})-F_{5}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle\leq\int_{0}^{T}\limsup_{k\to\infty}\bigl{\langle}\tilde{\mu}_{t},F_{5}(t,\cdot,\tilde{g}^{n_{k}}_{t})-F_{5}(t,\cdot,\tilde{g}_{t})\bigr{\rangle}\,\mathrm{d}t$
$\displaystyle\leq 0.$

This establishes Item (v) and concludes the proof. ∎

## Acknowledgement

This research has been supported by the EPSRC Centre for Doctoral Training in
Mathematics of Random Systems: Analysis, Modelling and Simulation
(EP/S023925/1).

## References

  * [1] Yves Achdou and Mathieu Laurière  “On the system of partial differential equations arising in mean field type control”  In _Discrete Contin. Dyn. Syst._ 35.9, 2015, pp. 3879–3900  DOI: 10.3934/dcds.2015.35.3879
  * [2] Daniel Andersson and Boualem Djechiche  “A Maximum Principle for SDEs of Mean-Field Type”  In _Appl. Math. Optim._ 63.3, 2011, pp. 341–356  DOI: 10.1007/s00245-010-9123-8
  * [3] Ştefana-Lucia Aniţa  “Optimal Control of Stochastic Differential Equations via Fokker–Planck Equations”  In _Appl. Math. Opt._ 84, 2021, pp. 1555–1583  DOI: 10.1007/s00245-021-09804-5
  * [4] Ştefana-Lucia Aniţa  “Controlling a generalized Fokker–Planck equation via inputs with nonlocal action”  In _Nonlinear Anal._ 241, 2024, pp. 113476  DOI: 10.1016/j.na.2023.113476
  * [5] Viorel Barbu  “Analysis and Control of Nonlinear Infinite Dimensional Systems” 190, Mathematics in Science and Engineering  Academic Press, Inc., 1993
  * [6] Viorel Barbu  “Existence of Optimal Control for Nonlinear Fokker–Planck Equations in $\boldsymbol{L^{1}(\mathbb{R}^{d})}$.”  In _SIAM J. Control_ 61.3, 2023, pp. 1213–1230  DOI: 10.1137/22M1485243
  * [7] Alain Bensoussan  “Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions”  In _Stoch._ 9.3 Taylor & Francis, 1983, pp. 169–222  DOI: 10.1080/17442508308833253
  * [8] Alain Bensoussan  “Stochastic maximum principle for distributed parameter systems”  In _J. Franklin Inst._ 315.5, 1983, pp. 387–406  DOI: 10.1016/0016-0032(83)90059-5
  * [9] Alain Bensoussan, Jens Frehse and Phillip Yam  “Mean Field Games and Mean Field Type Control Theory”, SpringerBriefs in Mathematics  Springer, New York, NY, 2013  DOI: 10.1007/978-1-4614-8508-7
  * [10] Tim Breitenbach and Alessio Borzì  “The Pontryagin maximum principle for solving Fokker–Planck optimal control problems”  In _Comput. Optim. Appl._ 76, 2020, pp. 499–533  DOI: 10.1007/s10589-020-00187-x
  * [11] Matteo Burzoni and Luciano Campi  “Mean field games with absorption and common noise with a model of bank run”  In _Stoch. Process. Their Appl._ 164, 2023, pp. 206–241  DOI: 10.1016/j.spa.2023.07.007
  * [12] Luciano Campi and Markus Fischer  “$N$-player games and mean-field games with absorption”  In _Ann. Appl. Probab._ 28.4 Institute of Mathematical Statistics, 2018, pp. 2188–2242  DOI: 10.1214/17-AAP1354
  * [13] Luciano Campi, Maddalena Ghio and Giulia Livieri  “N-Player games and mean-field games with smooth dependence on past absorptions”  In _Ann. Inst. H. Poincaré Probab. Statist._ 57.4 Institut Henri Poincaré, 2021, pp. 1901–1939  DOI: 10.1214/20-AIHP1138
  * [14] Pierre Cardaliaguet, Benjamin Seeger and Panagiotis Souganidis  “Mean field games with common noise and degenerate idiosyncratic noise”, 2022  arXiv:2207.10209
  * [15] Pierre Cardaliaguet and Panagiotis E. Souganidis  “On first order mean field game systems with a common noise”  In _Ann. Appl. Probab._ 32.3 Institute of Mathematical Statistics, 2022, pp. 2289–2326  DOI: 10.1214/21-AAP1734
  * [16] Pierre Cardaliaguet, P.\@ Jameson Graber, Alessio Porretta and Daniela Tonon  “Second order mean field games with degenerate diffusion and local coupling”  In _Nonlinear Differ. Equ. Appl._ 22, 2014, pp. 1287–1317  DOI: 10.1007/s00030-015-0323-4
  * [17] Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry and Pierre-Louis Lions  “The Master Equation and the Convergence Problem in Mean Field Games”, Annals of Mathematics Studies  Princeton University Press, 2019
  * [18] Pierre Cardaliaguet, Samuel Daudin, Joe Jackson and Panagiotis Souganidis  “An algebraic convergence rate for the optimal control of McKean–Vlasov dynamics”, 2023  arXiv:2203.14554
  * [19] René Carmona and François Delarue  “Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics”  In _Ann. Probab._ 43.5 Institute of Mathematical Statistics, 2015, pp. 2647–2700  DOI: 10.1214/14-AOP946
  * [20] René Carmona and François Delarue  “Probabilistic Theory of Mean Field Games with Applications I” 83, Probability Theory and Stochastic Modelling  Springer International Publishing, 2018  DOI: 10.1007/978-3-319-58920-6
  * [21] René Carmona and François Delarue  “Probabilistic Theory of Mean Field Games with Applications II” 84, Probability Theory and Stochastic Modelling  Springer International Publishing, 2018  DOI: 10.1007/978-3-319-56436-4
  * [22] René Carmona, Mathieu Laurière and Pierre-Louis Lions  “Non-standard Stochastic Control with Nonlinear Feynman-Kac Costs”, 2023  arXiv:2312.00908
  * [23] José A.\@ Carrillo, Edgard A.\@ Pimentel and Vardan K.\@ Voskanyan  “On a mean field optimal control problem”  In _Nonlinear Anal._ 199, 2020, pp. 112039  DOI: 10.1016/j.na.2020.112039
  * [24] Samuel Daudin  “Optimal control of the Fokker–Planck equation under state constraints in the Wasserstein space”  In _J. Math. Pures Appl._ 175, 2023, pp. 37–75  DOI: 10.1016/j.matpur.2023.05.002
  * [25] Mao Fabrice Djete, Dylan Possamaï and Xiaolu Tan  “McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations”  In _Math. Oper. Res._ 47.4, 2022, pp. 2547–3399  DOI: 10.1287/moor.2021.1232
  * [26] Arthur Fleig and Roberto Guglielmi  “Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls”  In _J. Optim. Theory Appl._ 174, 2017, pp. 408–427  DOI: 10.1007/s10957-017-1120-5
  * [27] Diogo A.\@ Gomes and Edgard Pimentel  “Local Regularity for Mean-Field Games in the Whole Space”  In _Minimax Theory Appl._ 1.1, 2016, pp. 65–82
  * [28] Ben Hambly and Philipp Jettkant  “Control of McKean–Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity”, 2023  arXiv:2310.15854
  * [29] Ben Hambly and Sean Ledger  “A stochastic McKean–Vlasov equation for absorbing diffusions on the half-line”  In _Ann. Appl. Probab._ 27.5, 2017, pp. 2698–2752  DOI: 10.1214/16-AAP1256
  * [30] Ben Hambly, Sean Ledger and Andreas Søjmark  “A McKean-Vlasov Equation with Positive Feedback and Blow-Ups”  In _Ann. Appl. Probab._ 29.4, 2019, pp. 2338–2373  DOI: 10.1214/18-AAP1455
  * [31] Ben Hambly and Andreas Søjmark  “An SPDE model for systemic risk with endogenous contagion”  In _Financ. Stoch._ 23.3, 2019, pp. 535–594  DOI: 10.1007/s00780-019-00396-1
  * [32] Ying Hu and Shige Peng  “Adapted solution of a backward semilinear stochastic evolution equation”  In _Stoch. Anal. Appl._ 9.4 Taylor & Francis, 1991, pp. 445–459  DOI: 10.1080/07362999108809250
  * [33] Abdul Rahman Al-Hussein  “Backward stochastic partial differential equations in infinite dimensions”  In _Random Oper. and Stoch. Equ._ 14.1, 2006, pp. 1–22  DOI: 10.1515/156939706776138020
  * [34] AbdulRahman Al-Hussein  “Backward stochastic partial differential equations driven by infinite-dimensional martingales and applications”  In _Stoch._ 81.6 Taylor & Francis, 2009, pp. 601–626  DOI: 10.1080/17442500903370202
  * [35] Thomas Kurtz  “Weak and strong solutions of general stochastic models”  In _Electron. Commun. Probab._ 19 Institute of Mathematical StatisticsBernoulli Society, 2014, pp. 1–16  DOI: 10.1214/ECP.v19-2833
  * [36] Daniel Lacker, Mykhaylo Shkolnikov and Jiacheng Zhang  “Superposition and mimicking theorems for conditional McKean–Vlasov equations”  In _J. Eur. Math. Soc._ 25.8, 2023, pp. 3229–3288  DOI: 10.4171/JEMS/1266
  * [37] Jean-Michel Lasry and Pierre-Louis Lions  “Jeux à champ moyen. I – Le cas stationnaire”  In _C. R. Math._ 343.9, 2006, pp. 619–625  DOI: 10.1016/j.crma.2006.09.019
  * [38] Jean-Michel Lasry and Pierre-Louis Lions  “Jeux à champ moyen. II – Horizon fini et contrôle optimal”  In _C. R. Math._ 343.10, 2006, pp. 679–684  DOI: 10.1016/j.crma.2006.09.018
  * [39] Jean-Michel Lasry and Pierre-Louis Lions  “Mean field games”  In _Jpn. J. Math._ 2.1, 2007, pp. 229–260  DOI: 10.1007/s11537-007-0657-8
  * [40] Mathieu Laurière and Olivier Pironneau  “Dynamic programming for mean-field type control”  In _C. R. Math._ 352.9, 2014, pp. 707–713  DOI: 10.1016/j.crma.2014.07.008
  * [41] Jean-François Le Gall  “Brownian Motion, Martingales, and Stochastic Calculus”, Graduate Texts in Mathematics  Springer International Publishing, 2016  DOI: doi.org/10.1007/978-3-319-31089-3
  * [42] Jacques-Louis Lions and Enrico Magenes  “Non-Homogeneous Boundary Value Problems and Applications”, Grundlehren der mathematischen Wissenschaften  Springer-Verlag, Berlin Heidelberg, 1972  DOI: 10.1007/978-3-642-65161-8
  * [43] Pierre-Louis Lions  “Équations de HJB et extensions de la théorie classique du contrôle stochastique”, Collège de France, 2016
  * [44] Robert S.\@ Liptser and Albert N.\@ Shiryaev  “Statistics of Random Processes I: General Theory”, Stochastic Modelling and Applied Probability  Springer New York, 1977  DOI: 10.1007/978-1-4757-1665-8
  * [45] Jin Ma and Jiongmin Yong  “On linear, degenerate backward stochastic partial differential equations”  In _Probab. Theory Relat. Fields_ 113.2, 1999, pp. 135–170  DOI: 10.1007/s004400050205
  * [46] Michel Métivier  “Semimartingales” 2, De Gruyter Studies in Mathematics  Berlin, New York: De Gruyter, 1982  DOI: doi:10.1515/9783110845563
  * [47] Sergey Nadtochiy and Mykhaylo Shkolnikov  “Particle systems with singular interaction through hitting times: Application in systemic risk modeling”  In _Ann. Appl. Probab._ 29.1 Institute of Mathematical Statistics, 2019, pp. 89–129  DOI: 10.1214/18-AAP1403
  * [48] Sergey Nadtochiy and Mykhaylo Shkolnikov  “Mean field systems on networks, with singular interaction through hitting times”  In _Ann. Probab._ 48.3 Institute of Mathematical Statistics, 2020, pp. 1520–1556  DOI: 10.1214/19-AOP1403
  * [49] Étienne Pardoux  “Stochastic partial differential equations and filtering of diffusion processes”  In _Stoch._ 3.1–4 Taylor & Francis, 1980, pp. 127–167  DOI: 10.1080/17442507908833142
  * [50] Shige Peng  “Stochastic Hamilton–Jacobi–Bellman Equations”  In _SIAM J. Control Optim._ 30.2, 1992, pp. 284–304  DOI: 10.1137/0330018
  * [51] Huyên Pham and Xiaoli Wei  “Dynamic Programming for Optimal Control of Stochastic McKean–Vlasov Dynamics”  In _SIAM J. Control Optim._ 55.2, 2017, pp. 1069–1101  DOI: 10.1137/16M1071390
  * [52] Songmu Zheng  “Nonlinear Evolution Equations”, Monographs and Surveys in Pure and Applied Math  Chapman & Hall/CRC, 2004  DOI: 10.1201/9780203492222
  * [53] Xun Yu Zhou  “A duality analysis on stochastic partial differential equations”  In _J. Funct. Anal._ 103.2, 1992, pp. 275–293  DOI: 10.1016/0022-1236(92)90122-Y
  * [54] Xun Yu Zhou  “On the Necessary Conditions of Optimal Controls for Stochastic Partial Differential Equations”  In _SIAM J. Control Optim._ 31.6, 1993, pp. 1462–1478  DOI: 10.1137/0331068