# Optimal Transport-based Adaptation in Dysarthric Speech Tasks

###### Abstract

In many real-world applications, the mismatch between distributions of
training data (source) and test data (target) significantly degrades the
performance of machine learning algorithms. In speech data, causes of this
mismatch include different acoustic environments or speaker characteristics.
In this paper, we address this issue in the challenging context of dysarthric
speech, by multi-source domain/speaker adaptation (MSDA/MSSA). Specifically,
we propose the use of an optimal-transport based approach, called MSDA via
Weighted Joint Optimal Transport (MSDA-WDJOT). We confront the mismatch
problem in dysarthria detection for which the proposed approach outperforms
both the Baseline and the state-of-the-art MSDA models, improving the
detection accuracy of 0.9 $\%$ over the best competitor method. We then employ
MSDA-WJDOT for dysarthric speaker adaptation in command speech recognition.
This provides a Command Error Rate relative reduction of 16$\%$ and 7$\%$ over
the baseline and the best competitor model, respectively. Interestingly, MSDA-
WJDOT provides a similarity score between the source and the target, i.e.
between speakers in this case. We leverage this similarity measure to define a
Dysarthric and Healthy score of the target speaker and diagnose the dysarthria
with an accuracy of 95$\%$.

Index Terms: dysarthric speech, speaker adaptation, multi-source domain
adaptation, dysarthria detection

##  1 Introduction

Many machine learning algorithms assume that the test and training datasets
are sampled from the same distribution. However, in many real-world
applications, new data can exhibit a distribution change (domain shift) that
degrades the algorithm performance. In speech tasks, such as speech
recognition, this shift can be observed when the recording conditions are
varying (e.g., different noisy/clean environments). This problem can be solved
through Domain Adaptation (DA) [1, 2], that is a particular case of transfer
learning [3]. The population of interest (e.g., the test dataset) is called
target domain, for which the labels are usually assumed not available. DA
methods leverage a similar labelled dataset, called source domain, in order to
learn a classifier for the target domain. Beside the recording conditions
mismatching, speech tasks suffer of differences between speakers (e.g.,
different accents, different speaking style). This mismatch is even more
evident in speech-impaired people as the speech characteristics also depend on
the type and the severity of the disorder. We refer to this problem as speaker
adaptation.

In this work, we focus on unsupervised domain adaptation and unsupervised
speaker adaptation in pathological speech, when multiple sources are
available. Therefore, in this framework, we assume to have access to multiple
training data sets (e.g., multiple speakers data) that are different, but
related, to the dataset of interest (e.g., the speaker of interest). We
consider speech recorded from people affected by dysarthria, that is a motor
disorder consisting in the disruption of the normal control of the vocal tract
musculature. Dysarthria can express different speech characteristics based on
the location of the neurological damage. However, it is possible to
individuate common tendencies in dysarthric patients, such as mumbled speech,
an acceleration or deceleration in the speaking rate, abnormal pitch and
rhythm. Further, dysarthric individuals are more subject to a fatigue factor
that affects the voice. We here investigate two case studies, that are domain
adaptation in dysarthria detection and dysarthric speaker adaptation for
command speech recognition.

Dysarthria detection is currently evaluated by neurology experts through the
use of clinical assessment tools that attribute a score to the capacity of the
subject to perform perceptual and/or acoustic tasks. This procedure can be
laborious and time consuming. Therefore, a rapid and objective dysarthria
detection procedure could help the therapist in the diagnosis.
In the last years, the research community started to look at dysarthria
detection by learning a mapping from the acoustic features to the text label
[4, 5]. In [6], the authors proposed an interpretable DNN model in which they
added an intermediate layer that acts as a bottle-neck feature extractor
providing nasality, vocal quality, articulatory precision and prosody
features. They showed that this interpretable features are highly correlated
with the evaluation of speech-language pathologists. In [7] the authors
propose a deep learning approach to compute phonological posteriors from the
speech signal and model the voice quality spectrum in patient affected by
Parkinson’s Disease (PD). In [8], articulation impairments of PD patients are
analyzed by time–frequency representations (TFRs), used to feed a
convolutional neural network (CNN) and discriminate between patients and
healthy speakers.
However, none of the previous studies considered the mismatch between training
and testing data. To the best of our knowledge, this is the first work in
which domain adaptation is applied to dysarthria detection. We take into
account the case in which we have access to multiple dysarthric datatsets with
different types of noise. While the domain adaptation problem for dysarthric
speech has not been addressed yet, the adaptation of a model to a new unseen
speech dataset with a different noise type is an old problem that always
attracted the attention of the research community [9, 10, 11]. Due to the fact
that both dysarthria detection and noise robustness are challenging tasks,
they tend to be treated separately. However, it is worthwhile to consider them
as a unique problem. Indeed, dysarthric speech corpora are usually collected
at the hospitals that are not equipped to record speech in absence of noise.

Speaker Adaptation (SA) techniques have been widely investigated by the
automatic speech recognition (ASR) community [12, 13, 14, 15, 16, 17].
However, conventional SA algorithms perform poorly in dysarthric speech when
they present a low intelligibility. As reported in [18, 19, 20], even though
commercial ASR systems can achieve up to 90$\%$ for some dysarthric speakers
with high intelligibility, the recognition performance still remains
inadequate with the decreasing of the speech intelligibility. These commercial
systems usually incorporate SA techniques to adapt the model to the voice of
that speaker that require some audio samples from the speaker of interest. It
follows that traditional adaptation techniques are not sufficient to deal with
gross abnormalities [21]. Further, they results to be inefficient even for
speakers with mild to moderate dysarthria when the vocabulary size is larger
than 30 words [22].
In the last decades, some attempts to move forward and solve the training-
testing mismatch for pathological speech have been done. [23] adapts the ASR
system trained on large datasets to a dysarthric dataset. In [24], the authors
leverage articulatory features, as well as the acoustic ones, achieving a
4-8$\%$ of World Error Rate (WER) relative reduction. However, these
techniques require a large amount of data for fine-tuning, that is usually not
available. To overcome this problem, [25] proposes to fine-tune only a subset
of the network layers to better adapt an ASR model to the dysarthric speech.
[26], rather than adapting the acoustic models, models the errors at phonetic
level made by the speaker and attempts to correct them by two possible
strategies, that incorporate the language model and find the best estimate of
the correct word sequence.
However, all the aforementioned studies are limited to the supervised
framework, whereas the unsupervised speaker adaptation is the most common real
scenario.

We propose to solve both adaptation problems by adopting an Optimal Transport
(OT) [27, 28] based approach, named Multi-Source Domain Adaptation Weighted
Joint Distribution Optimal Transport (MSDA-WJDOT), firstly introduced in [29].
This method looks for a convex combination of the joint source distributions
with minimal Wasserstein distance to the proxy joint target distribution, in
which the labels are replaced by the prediction of a classifier. The source
weights and the target classifier are simultaneously learned. The advantage of
MSDA-WJDOT over the other MSDA approaches is that the source weights provide a
similarity measure between each source and the target domain. This can be
leveraged to select only the relevant sources and it also offers an
interpretable model. For instance, in the case of speaker adaptation, these
weights reflect the similarity between speakers.

##  2 MSDA-WJDOT

In this section, we recall MSDA-WJDOT firstly proposed in [29]. This method
approaches to the MSDA problem by estimating the similarity between the source
and target domains, in the Wasserstein sense, and learning a target classifier
using only the most similar source datasets.

Let us suppose to have $J$ sources $(X_{j},Y_{j})$ with $N_{j}$ samples. MSDA-
WJDOT assumes to have access to a differentiable embedding function $g$ from
the input space to an embedding space $\mathcal{G}$. If the embedding is not
available MSDA-WJDOT becomes a two-step procedure where $g$ is learned in the
first step. This can be done by

$\operatorname*{min}_{g,f_{S}}\quad\sum_{j=1}^{J}\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\mathcal{L}(f_{S}\circ
g(x_{j}^{i}),y_{j}^{i}),$ (1)

where $f_{S}$ is a global classifier common to all sources. Alternatively, the
authors in [29] suggests to estimate $g$ with the Multi-Task Learning (MTL)
framework [30], i.e. by

$\operatorname*{min}_{g,\\{f_{j}\\}_{j=1}^{J}}\quad\sum_{j=1}^{J}\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\mathcal{L}(f_{j}\circ
g(x_{j}^{i}),y_{j}^{i}),$ (2)

where $f_{j}$ represents the classification function of the $j$-th source.
Note that Eq. 1 and 2 provide two different ways to estimate $g$, while the
source classifiers $f_{S}$ and $f_{j}$ are not further used in MSDA-WJDOT.

Once $g$ is given, we can define the source joint distributions $p_{S,j}$, for
$1\leq j\leq J$, with support on the product space
$\mathcal{G}\times\mathcal{Y}$ , where $\mathcal{Y}$ is the label space. We
define a convex combination of the source distributions

$p_{S}^{\alpha}=\sum_{j=1}^{J}\alpha_{j}p_{S,j}$

with $\boldsymbol{\alpha}\in\Delta^{J}$. MSDA-WJDOT aims at finding a
classification function $f$ that aligns the target distribution $p_{T}^{f}$
with a convex combination $\sum_{j=1}^{J}\alpha_{j}p_{S,j}$ of the source
distributions with convex weights $\boldsymbol{\alpha}$ on the simplex
$\Delta^{J}$. This can be expressed as

$\operatorname*{min}_{\boldsymbol{\alpha},f}\quad
W_{D}\left(\hat{p}_{T}^{f},\sum_{j=1}^{J}\alpha_{j}\hat{p}_{S,j}\right),$ (3)

where $W_{D}$ is the Wasserstein distance. A remarkable advantage of this
approach is that the weights $\boldsymbol{\alpha}$ are learned simultaneously
with the classifier $f$, which allows to distribute the mass based on the
similarity between the source and the target domains, both in the feature and
in the output spaces. Consequently, misleading and unrelated source domains
are in practice not used and this allows to learn a more accurate classifier.

In the following we refer to MSDA-WJDOT as MSDA-WJDOT or MSDA-WJDOTMTL to
specify if the embedding is learned by Eq. 1 or 2, respectively.

Table 1: Dysarthria detection accuracy on four target datasets: F16, Buccaneer2 (B2), Factory2 (F2), Destroyerengine (D). The mean and standard deviation of the accuracy are reported for the Baseline, IWERM, two extensions of JDOT and the proposed MSDA-WJDOT approach. Target domain | F16 | B2 | F2 | D | Average
---|---|---|---|---|---
Baseline |  93.59 $\pm$ 0.38 |  93.76 $\pm$ 0.22 |  93.23 $\pm$ 0.66 |  92.46 $\pm$ 0.82 | 93.26
IWERM [31] |  66.22 $\pm$ 0.01 |  66.38 $\pm$ 0.01 |  66.25 $\pm$ 0.05 |  66.30 $\pm$ 0.09 | 66.29
CJDOTMTL [32] |  95.35 $\pm$ 0.55 |  97.39 $\pm$ 0.09 |  96.71 $\pm$ 0.07 |  92.98 $\pm$ 0.75 | 95.61
MJDOTMTL [32] |  95.81 $\pm$ 0.42 |  97.22 $\pm$ 0.09 |  96.53 $\pm$ 0.12 |  93.12 $\pm$ 0.67 | 95.67
MSDA-WJDOTMTL |  97.32 $\pm$ 0.36 | 97.82 $\pm$ 0.13 |  97.76$\pm$ 0.10 |  95.42 $\pm$ 0.24 | 97.08

##  3 Experimental setup

###  3.1 Dysarthria detection

####  3.1.1 Dataset

TORGO is one of the most popular dysarthric speech corpora [33]. It consists
of aligned acoustic and articulatory recordings from 15 speakers. Seven of
these speakers are control speakers without any speech disorders, while the
remaining eight speakers present different levels of dysarthria. We used the
TORGO dataset to generate multiple-sources and target domain. In particular,
we generated 15 noisy datasets by combining the raw signals with different
types of noises from a noise dataset (available here.). Each noisy dataset has
been synthesized by PyDub python library [34] summing the same type of noise
signal to the raw data. We then used the libROSA Python library [35] to
extract 13 MFCCs plus deltas and delta-deltas, computed every 10ms from 25ms
Hamming windows followed by a z-normalization per track. We fixed the number
of sources equal to 14 and we tested 4 noisy domains as target: F16,
Buccaneer2 (B2), Factory2 (F2), Destroyerengine (D).

####  3.1.2 MSDA-WJDOT model details

The feature extraction $g$ is performed by a Bidirectional Long Short-Term
Memory (BLSTM) recurrent network with two hidden layers, each containing 50
memory blocks. We train the BLSTM as sequence-to-vector model, by taking the
final hidden state at the last time step as a fixed-length utterance
representation to generate the command probability given the whole sequence.
The source and target classifier functions $\\{f_{j}\\}$,$f_{S}$, $f$ are
learned as one feed-forward layer taking the last hidden state as input. The
weights were initialized with Xavier initialization. Training is performed
with Adam optimizer with 0.9 momentum and $\epsilon=e^{-8}$. The learning rate
exponentially decays at every epoch. We grid-research the initial learning
rate value and the decay rate.

###  3.2 Command speech recognition

####  3.2.1 Dataset

To investigate the Dysarthric Speaker Adaptation we employ the AllSpeak
dataset [36], that consists of speech recordings from 29 Italian native
speakers. Seventeen of these are affected by Amyotrophic Lateral Sclerosis,
while the remaining thirteen are speakers of control. The dataset contains 25
commands in Italian, relative to basic needs such as “I am thirsty”. This
dataset is very challenging due to the small amount of recordings. Indeed,
only 2387 and 1857 examples have been recorded from control and dysarthric
speakers, respectively.
We perform speaker adaptation of each dysarthric speaker by using all the
remaining speakers as training dataset. The unlabelled target speaker data is
split into adaptation set (80$\%$) and testing (20$\%$) set, which contains
one example of each command. To train MSDA-WJDOT, we simultaneously employ the
source training dataset and the adaptation target speaker dataset.

####  3.2.2 MSDA-WJDOT model details

The embedding function $g$ is represented by a BLSTM with five hidden layers,
each containing 250 memory blocks. Finally, a softmax layer performs the
classification task. Here, we do not consider the MTL variant as the dataset
size is limited and learning a source-classifier $f_{j}$ with a very small
amount of data may result hard. All weights are initialized with Xavier
initialization. Training is performed with Adam optimizer with 0.9 momentum
and $\epsilon=e^{-8}$. Learning rate is fixed to 0.001.

##  4 Results

###  4.1 Competitor models

We compare the proposed approach with well-established MSDA approaches:
Importance Weighted Empirical Risk Minimization (IWERM) [31] and two
extensions of JDOT [32] to the MSDA case (see [29] for more details). To
better measure the adaptation contribution, we also report the performance of
a Baseline model in which the global classifier $f_{S}$, learned on the source
domains, is used as target classifier. In addition, in Sec. 4.3, we provide
the performance of a supervised speaker adaptation (SSA) model, in which we
add a feed-forward linear layer atop the input to the Baseline model and train
it on the target dataset [37]. We consider this approach as the lower bound of
the SA performance error.

###  4.2 Dysarthria detection

In all experiments, we observed that learning $g$ by the Multi-Task Learning
approach always provides a better performance. Hence, for an easier reading of
Table 1, we only report the performances in which the extractor $g$ is given
by the MTL.

For all the target domains, IWERM performs poorly, even underperforming the
Baseline. This is probably due to the difficulties in computing the
probability density function of the acoustic input, that presents a high
complexity. Indeed, to make the computation feasible, we firstly extracted a
lower-dimensional vector from the audio signal by PCA. All the remaining MSDA
methods outperform the Baseline. Among them our MSDA-WJDOT provides the best
accuracy for all target domains. More precisely, MSDA-WJDOT provides a
relative error reduction of 56.7$\%$ and 32.69$\%$ over the error of the
Baseline and the best competitor model, respectively.

###  4.3 Command speech recognition

Table 2: Command Error Rate (CER) for each dysarthric target speaker provided
by the Baseline, SSA, MSDA-WJDOT and the competitor models.

Speaker | Baseline |  CJDOT [32] |  MJDOT [32] | MSDA-WJDOT | SSA
---|---|---|---|---|---
M01 | 35.79 | 32.77 | 31.93 | 31.93 | 25.00
M02 | 34.26 | 36.75 | 36.75 | 37.71 | 54.17
F01 | 63.16 | 52.63 | 49.12 | 49.12 | 60.00
F02 | 48.50 | 40.00 | 40.00 | 40.00 | 36.00
M03 | 64.44 | 68.89 | 71.11 | 57.78 | 55.56
M04 | 30.00 | 31.20 | 32.00 | 30.40 | 24.00
M05 | 18.62 | 17.46 | 15.08 | 14.29 | 17.39
F03 | 68.33 | 61.74 | 70.43 | 62.61 | 56.52
M06 | 48.67 | 34.78 | 33.91 | 35.65 | 26.09
M07 | 11.00 | 7.20 | 8.00 | 8.80 | 20.00
M08 | 39.50 | 36.00 | 41.60 | 33.60 | 32.00
M09 | 24.79 | 16.81 | 18.49 | 19.33 | 13.04
F04 | 48.07 | 38.60 | 38.60 | 38.60 | 40.91
M10 | 18.00 | 12.80 | 12.00 | 12.80 | 16.00
M11 | 56.50 | 47.20 | 48.80 | 45.60 | 40.00
M12 | 7.50 | 5.60 | 7.20 | 4.80 | 4.00
M13 | 30.91 | 45.45 | 43.64 | 21.82 | 9.09
A. CER | 38.11 | 34.46 | 34.37 | 32.05 | 31.16

Table 2 reports the results in terms of Command Error Rate (CER). A first
remark is that, although the Baseline always achieved a CER between $15\%$ and
$20\%$ on the validation set, it often had low accuracy on the target speaker.
Once again, this emphasizes the difficulty of an ASR system to generalize to a
new dysarthric speaker and the importance of the speaker adaptation in this
context.
The unsupervised speaker adaptation carried out by MSDA-WJDOT outperforms all
the methods by providing the best Average CER. Indeed, it reduces the CER of
$16\%$ and $7\%$ over the Baseline and the MSDA competitors, respectively.
Surprisingly, MSDA-WDJOT achieves an Average CER similar to the SSA approach,
in which the labels are used.

It is crucial to recall that MSDA-WJDOT provides a measure of similarity
between the target and the sources and, hence, between speakers. We found that
the recovered $\boldsymbol{\alpha}$ always attributes highest similarity
scores to dysarthric speakers rather than healthy ones, when the target
speaker is dysarthric, and vice versa for healthy target speakers. This
suggests that this approach can realistically estimate speaker closeness. Fig.
1 shows the recovered weights for a dysarthric target speaker. The mass is
spread along dysarthric speakers while the $\alpha_{j}$ values are close to
zero for healthy speakers.

Figure 1: $\boldsymbol{\alpha}$ coefficients recovered during MSDA-WJDOT
training for dysarthric target speaker M13. The $\alpha_{j}$ coefficients are
close to zero for healthy speakers (in green), while the highest weights are
attributed to dysarthric speakers (in red)

####  4.3.1 The $\boldsymbol{\alpha}$ weight and the dysarthria detection

As we showed in Fig. 1, MSDA-WJDOT associates speakers with similar voice
characteristics. As additional analysis, we investigated the possibility of
leveraging the $\boldsymbol{\alpha}$ weights of the command classifier to
detect dysarthria. Specifically, we attempt to classify a speaker as healthy
or dysarthric based on his/her similarity with the other subjects.

Let define $I_{c}$ as the set indexing the control speakers and $I_{d}$ as the
set of indices related to dysarthric speakers. We then define the Healthy
Score (HS) and the Dysarthric Score (DS) as follow

$HS=\sum_{j\in I_{c}}\alpha_{j},\quad DS=\sum_{j\in I_{d}}\alpha_{j}.$ (4)

We can use these scores to perform dysarthria detection by stating that

$A\;speaker\;is\;affected\;by\;dysarthria\;if\;DS>HS.$

Fig. 2 reports the computed scores for all dysarthric speakers and for 5
control speakers. As we can see the controls subjects are always classified as
healthy while for the patients, except for F02, we have $DS>HS$. This results
in a final accuracy of 95$\%$.

Figure 2: HS and DS computed for healthy (in green) and dysarthric (in red)
speakers.

##  5 Conclusions

In this work, we addressed usupervised domain and speaker adaptation in the
challenging context of pathological speech by employing MSDA-WJDOT [38]. We
have shown the effectiveness of our proposed method on dysarthria detection
and spoken command recognition, by comparing it with with well-established
MSDA approches and a Baseline model, in which a global classifier is learned
on the source datasets and directly tested on the target data, without
adaptation. Our method provides the best performance on both applications.
Interestingly, MSDA-WJDOT also provides source-target similarity coefficients
$\boldsymbol{\alpha}$ that, in speaker adaptation, result in a measure of
speaker relatedness. From this, we derived the Healthy (HI) and Dysarthric
(DI) Index of a target speaker and diagnosed the dysarthria, achieving an
accuracy of 95$\%$.
Future directions could delve into the dysarthria assessment by individuating
intervals of values in which the DI corresponds to dysarthria severity levels
(e.g., mild, moderate, severe).This may bring to very efficient ASR systems
that simultaneously improve their performance via SA, and compute the DI
warning the subject when the index is close to the right endpoint of its
interval. Such a device could predict the disease degeneration and allow the
patient to act in time in order to prevent it.

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# Lifting of Volterra processes:
optimal control in UMD Banach spaces

Giulia di Nunno†,∗ and Michele Giordano‡
June 22nd 2023

###### Abstract.

We study a stochastic control problem for a Volterra-type controlled forward
equation with past dependence obtained via convolution with a deterministic
kernel. To be able to apply dynamic programming to solve the problem, we lift
it to infinite dimensions and we formulate a UMD Banach-valued Markovian
problem, which is shown to be equivalent to the original finite-dimensional
non-Markovian one. We characterize the optimal control for the infinite
dimensional problem and show that this also characterizes the optimal control
for the finite dimensional problem.

†††Department of Mathematics, University of Oslo, P:O: Box 1053 Blindern,
N-0316 Oslo, Email<EMAIL_ADDRESS>of Business and
Management Science, NHH Norwegian School of Economics, Helleveien 30, N-5045
Bergen.††‡Department of Mathematics, University of Oslo, P:O: Box 1053
Blindern, N-0316 Oslo, Email<EMAIL_ADDRESS>
Keywords: Backward stochastic integral equation; Dynamic programming
principle;
Hamilton Jacobi Bellman; Optimal control; UMD Banach space; Markovian Lift;
MSC 2020: 60H10; 60H20; 93E20; 35R15; 49L20; 91B70;

##  1\. Introduction

We intend to minimize a performance functional of the form

$J(t,x,u)=\mathbb{E}\left[\int_{t}^{T}F(\tau,X^{u}_{\tau},u_{\tau})d\tau+G(X^{u}_{T})\right],$
(1.1)

where $t\in[0,T]$, and $x$ is given in the controlled Volterra-type dynamics
of the process $X_{\tau}^{u}$:

$\displaystyle X^{u}(\tau)$
$\displaystyle=x(\tau)+\int_{t}^{\tau}K(\tau-s)\Big{[}\beta(s,X^{u}(s))+\sigma(s,X^{u}(s))R(s,X^{u}(s),u(s))\Big{]}ds$
$\displaystyle\quad+\int_{t}^{\tau}K(\tau-s)\sigma(s,X^{u}(s))dW(s).$ (1.2)

Here $x:[0,T]\longrightarrow\mathbb{R}$,
$\beta:[0,T]\times\mathbb{R}\longrightarrow\mathbb{R}$,
$\sigma:[0,T]\times\mathbb{R}\longrightarrow\mathbb{R}$,
$R:[0,T]\times\mathbb{R}\times\mathcal{U}\longrightarrow\mathbb{R}$, and the
convolution kernel $K:[0,T]\longrightarrow\mathbb{R}^{+}$ are all measurable
mappings on which additional hypothesis will be stated later on, and
$u:[0,T]\times\Omega\longrightarrow\mathcal{U}\subset\mathbb{R}$ is an
admissible control. Also, $W$ is a real-valued Brownian motion on a complete
filtered probability space $(\Omega,\mathcal{F},\mathbb{P})$.

Stochastic control problems as (1.1)-(1.2) appear, e.g. when studying optimal
advertising strategies (see e.g. [15] for the case of Volterra dynamics in
(1.2) and [17, 16, 20] for the case of delay). Other cases of applications are
found in electrodynamics [24] and in epidemiology [25]. When dealing with such
problems, one cannot directly apply a dynamic programming principle (DPP) in
view of the non Markovianity of the framework. While in some particular cases
is still possible to derive the DPP also for Volterra forward dynamics (see
[1, 18, 5]), most authors approached the general problem by means of a maximum
principle (see, e.g., [12, 29, 30, 3, 2] and references therein). Even though
the maximum principle approach might seem practical, one usually has to impose
regularity conditions on both the drift and volatility which are not always
easy to satisfy. In this paper, thanks to the developments on the _lift_
theory for Volterra processes (see [1, 9, 8, 7, 15, 5]) we aim to move the
stochastic control problem (1.1)-(1.2) to an infinite dimensional UMD-Banach
setting and solve the newly formulated problem by means of DPP.

The main purpose of this lifting approach is to recover the Markov property
for the forward process (1.2) which, in turns, allows us to derive a DPP in
terms of the _Hamilton-Jacobi-Bellman_ (HJB) equations. In fact, one can show
that solving the lifted problem is equivalent to solving the original one,
with the fundamental difference that, by moving to an infinite dimensional
setting, we work in a Markovian framework. Focusing on Markovian lifts, we
assume that the kernel $K$ can be represented as $K(t)=\langle
g,\mathcal{S}_{t}^{*}\nu\rangle_{Y\times Y^{*}}$, for $\mathcal{S}_{t}^{*}$ a
uniformly continuous semigroup acting on a Banach space $Y^{*}$, $\nu\in
Y^{*}$, $g\in Y$ with $Y$ the pre dual of $Y^{*}$ and pairing
$\langle\cdot,\cdot\rangle_{Y\times Y^{*}}$. Examples of kernels that satisfy
this condition can be found both in [9, 8, 15] and in the last section of this
paper.

Our goal is to find $\hat{u}$ such that, for all $t\in[0,T]$,

$J(t,x,\hat{u})=\inf_{u\in\mathds{A}}J(t,x,u),$ (1.3)

with $J(t,x,u)$ as in (1.1) and for $u$ belonging to some admissible control
set $\mathds{A}$ defined as

$\mathds{A}=\left\\{u:[0,T]\times\Omega\longrightarrow\mathcal{U},\ \text{s.t.
$u$ is predictable}\right\\},$

where $\mathcal{U}$ is a closed convex subset of $\mathbb{R}$ and the
information flow is associated to the Brownian motion in (1.2). Our approach
consists in formulating a new infinite-dimensional Banach-valued optimization
problem that can be shown to be equivalent to (1.1)-(1.2) (in the sense that
the optimal control $\hat{u}$ and optimal value $J(t,x,\hat{u})$, $t\in[0,T]$
are the same of the original one) and then solve such infinite dimensional
optimization problem, which is Markovian. The solution is achieved exploiting
Malliavin calculus for _unconditional martingale differences_ (UMD) Banach
spaces.

The Markovian lift to the infinite dimensional setting that we present here
was originally introduced in [9], and then developed in [8] for the multi-
dimensional case, and in [7] for a Lévy drivers. Our work can be seen as a
generalization of the case presented in [1], in which a kernel $K$ that can be
expressed as the Laplace transform of a measure is considered, and where the
performance functional (1.1) is of linear-quadratic type. Our work differs
from [1] as we are able to consider a broader class of kernels and performance
functionals thanks to the nature of the lift we apply.

The present work introduces an element of novelty also with respect to
infinite dimensional stochastic control. Indeed, we consider a setting which
is different from both the ones presented in [14] and [22]. In [14] the
authors consider a Hilbert valued forward controlled process, whereas in [22]
the forward process has values in a general Banach space, but with a
volatility term $\sigma$ not depending on $X$. Here we are able to take
general volatility dynamics for the forward process and this will require to
work in Banach spaces of the UMD type so to be able to apply Malliavin
calculus techniques.

In the context of optimal control for lifted process, we also mention [5].
There, the authors follow an approach close to the one presented here.
However, we remark that, in our framework, we are able to consider a wider
class of kernel $K$ thanks to the nature of our lift, which allows us to work
in UMD Banach spaces instead of Hilbert spaces. On the other side, in [5] the
authors consider a forward equation driven by a Lévy process instead of a
Brownian motion (like (1.2)). While the lift theory for Lévy-driven forward
processes is available (see [7]), the optimal control of an infinite-
dimensional Lévy-driven forward equation in the present setting is a topic for
future research.

This paper is structured as follows: in Section 2 we present some preliminary
results both on the Gâteaux differentiability in general Banach spaces and on
the lift for Volterra processes. We recall the essentials on UMD Banach spaces
and some results of Malliavin calculus in this framework. In Section 3 we give
an existence and continuity result for the forward equation, and in Section 4
we introduce the backward equation and the Hamiltonian function associated
with the lifted optimization problem. Here we present a solution method via
HJB equations. To conclude, in Section 5 we present a problem of optimal
consumption where we obtain a characterization of the optimal control $u$ via
DPP.

##  2\. Some preliminary results

We recall some useful notions and results that are used throughout the paper.
Then we show how the Markovian lift is performed in the present context of
stochastic control (1.1)-(1.2). Lastly, we introduce UMD Banach spaces and
state some crucial results for Malliavin calculus in this setting. We refer to
[14] for the results on Gâteaux derivatives and Banach spaces, to [9, 8, 7,
15] for the ones concerning Markovian lifts, and to [19, 23, 26, 21] for the
results concerning UMD Banach spaces.

###  2.1. The class of Gâteaux differentiable functions

For a mapping $F:U\longrightarrow V$, with $U,V$ two Banach spaces, we say
that the _directional derivative at $u\in U$ in the direction $h\in U$_ is
defined as

$\nabla F(u;h):=\lim_{s\rightarrow 0}\frac{F(u+sh)-F(u)}{s},$

whenever the limit exists in the topology of $V$. The mapping $F$ is said to
be _Gâteaux differentiable at the point $u$_ if it has directional derivative
at $u$ in every direction and there exists an element $\nabla F(u)$ in
$L(U,V)$ such that $\nabla F(u;h)=\nabla F(u)h$ for every $h\in U$. We call
$\nabla F(u)$ the _Gâteaux derivative at $u$_.

######  Definition 2.1.

A mapping $F:U\longrightarrow V$ belongs to $\mathcal{G}^{1}(U;V)$ if it is
continuous, Gâteaux differentiable for all $u\in\mathcal{U}$ and $\nabla
F:U\longrightarrow L(U;V)$ is strongly continuous, i.e. the map $\nabla
F(\cdot)h:U\longrightarrow V$ is continuous for every $h\in U$.

######  Remark 2.2.

Let $U,V,Z$ be three Banach spaces and $F\in\mathcal{G}^{1}(U,V)$. If
$G\in\mathcal{G}^{1}(V,Z)$, then $G(F)\in\mathcal{G}^{1}(U,Z)$ and
$\nabla(G(F))(u)=\nabla G(F(u))\nabla F(u)$.

We also introduce the _partial directional derivative_ for a mapping
$F:U\times V\longrightarrow Z$, with $U$, $V$, $Z$ Banach spaces as

$\nabla_{u}F(u,v;h):=\lim_{s\rightarrow 0}\frac{F(u+sh,v)-F(u,v)}{h},$

with $u,h\in U$, $v\in V$ and the limit taken in the topology of $Z$. We say
that $F$ is _partially Gâteaux differentiable with respect to $u$ at $(u,v)\in
U\times V$_ if there exists $\nabla_{u}F:U\times V\longrightarrow L(U,Z)$ such
that $\nabla_{u}F(u,v;h)=\nabla_{u}F(u,v)h$ for all $h\in U$.

######  Definition 2.3.

We say that $F:U\times V\longrightarrow Z$ belongs to the class
$\mathcal{G}^{1,0}(U\times V;Z)$ if it is continuous, Gâteaux differentiable
with respect to $u$, for all $(u,v)\in U\times V$ and $\nabla_{u}F:U\times
V\longrightarrow L(U,Z)$ is strongly continuous.

For $F$ depending on additional arguments, the definition above can be easily
generalized.

######  Lemma 2.4.

Given $U,V,Z$ three Banach spaces, a continuous map $F:U\times
V\longrightarrow Z$ belongs to $\mathcal{G}^{1,0}(U\times V,Z)$ provided the
following conditions hold:

  1. (1)

The partial directional derivatives $\nabla_{u}F(u,v;h)$ exist at every point
$(u,v)\in U\times V$ and in every directoin $h\in U$.

  2. (2)

For every $(u,v)$ the mapping $h\longmapsto\nabla_{u}F(u,v;h)$ is continuous
from $U$ to $Z$.

  3. (3)

For every $h$, the mapping $\nabla_{u}F(\cdot,\cdot;h):U\times
V\longrightarrow Z$ is continuous.

We are going to use the following parameter depending contraction principle to
study the regular dependence of the solution of stochastic differential
equations on their initial data.

######  Proposition 2.5.

Let $U,V,Z$ be Banach spaces and let $F:U\times V\times Z\longrightarrow U$ a
continuous mapping satisfying

$|F(u_{1},v,z)-F(u_{2},v,z)|\leq\alpha|u_{1}-u_{2}|,$

for some $\alpha\in[0,1)$ and every $u_{1},u_{2}\in U$, $v\in V$, $z\in Z$.
Let $\phi(v,z)$ denote the unique fixed point of the mapping
$F(\cdot,v,z):U\longrightarrow U$. Then $\phi:V\times Z\longrightarrow U$ is
continuous. If, in addition $F\in\mathcal{G}^{1,1,0}(U\times V\times Z,U)$,
then $\phi\in\mathcal{G}^{1,0}(V\times Z,U)$ and

$\nabla_{v}\phi(v,z)=\nabla_{u}F\big{(}\phi(v,z),v,z\big{)}\nabla_{v}\phi(v,z)+\nabla_{v}F(\phi(v,z),v,z).$

###  2.2. Lift approach to optimal control

In the sequel, we exploit an infinite dimensional lift to reformulate the
optimization problem (1.1)-(1.2) in an infinite dimensional setting. Our first
step is to rewrite $X^{u}$ in (1.2) in terms of a process $\mathcal{Z}^{u}$
with values in a Banach space, using the lift procedure presented in [9].
Notice that we do not actually work in the affine framework of [9], but the
approach presented here is actually a particular case of the one introduced in
[7].

######  Definition 2.6.

Let $Y$ be a Banach space with dual $Y^{*}$ and denote with
$\langle\cdot,\cdot\rangle_{Y\times Y^{*}}$ the pairing between $Y$ and
$Y^{*}$. We say that a kernel $K\in L^{2}_{loc}(\mathbb{R}_{+},\mathbb{R})$ is
_liftable_ if there exist $g\in Y$, $\nu\in Y^{*}$ and a uniformly continuous
semigroup $\mathcal{S}_{t}^{*}$, $t\in[0,T]$ with generator $\mathcal{A}^{*}$,
acting on $Y^{*}$, such that

  * •

$K(t)=\langle g,\mathcal{S}_{t}^{*}\nu\rangle_{Y\times Y^{*}}$

  * •

$\mathcal{S}_{t}^{*}\nu\in Y^{*}$ for all $t>0$

  * •

$\int_{0}^{t}\|\mathcal{S}_{s}^{*}\nu\|^{2}_{Y^{*}}ds<\infty$ for all $t>0$.

For notational simplicity we write $\langle\cdot,\cdot\rangle$ for
$\langle\cdot,\cdot\rangle_{Y\times Y^{*}}$ when no confusion arises.

From now on, we make the following assumption:

######  Hypothesis 2.7.

The kernel $K$ in (1.2) is liftable.

We rewrite $X^{u}$ as

$\displaystyle X^{u}(\tau)$
$\displaystyle=x(\tau)+\int_{t}^{\tau}K(\tau-s)\Big{[}\beta(s,X^{u}(s))+\sigma(s,X^{u}(s))R(s,X^{u}(s),u(s))\Big{]}ds$
$\displaystyle\quad+\int_{t}^{\tau}K(\tau-s)\sigma(s,X^{u}(s))dW(s)$
$\displaystyle:=x(\tau)+\int_{t}^{\tau}K(\tau-s)dV^{u}(s),$

where

$dV^{u}(s):=\Big{[}\beta(s,X^{u}(s))+\sigma(s,X^{u}(s))R(s,X^{u}(s),u(s))\Big{]}ds+\sigma(s,X^{u}(s))dW(s).$
(2.1)

Defining $\zeta$ as an element in $Y^{*}$ such that $x(\tau)=:\langle
g,\mathcal{S}_{\tau}^{*}\zeta\rangle$ we can now rewrite (1.2) as follows:

$\displaystyle X^{u}(\tau)$
$\displaystyle=x(\tau)+\int_{t}^{\tau}K(\tau-s)dV^{u}(s)$
$\displaystyle=\langle
g,\mathcal{S}_{\tau}^{*}\zeta\rangle+\int_{t}^{\tau}\langle
g,\mathcal{S}_{\tau-s}^{*}\nu\rangle dV^{u}(s)$
$\displaystyle=\Bigg{\langle}g,\mathcal{S}_{\tau}^{*}\zeta+\int_{t}^{\tau}\mathcal{S}_{\tau-s}^{*}\nu
dV^{u}(s)\Bigg{\rangle}$ $\displaystyle=:\langle
g,\mathcal{Z}_{\tau}^{u}\rangle,$ (2.2)

where
$\mathcal{Z}_{\tau}^{u}:=\mathcal{S}_{\tau}^{*}\zeta+\int_{t}^{\tau}\mathcal{S}_{\tau-s}^{*}\nu
dV^{u}(s)$. One can then check that $\mathcal{Z}_{\tau}^{u}$ follows the
dynamics:

$\mathcal{Z}_{\tau}^{u}=\mathcal{S}_{t}^{*}\zeta+\int_{t}^{\tau}\mathcal{A}^{*}\mathcal{Z}_{s}^{u}ds+\int_{t}^{\tau}\nu
dV^{u}(s),$ (2.3)

In fact, we have that

$\displaystyle\int_{t}^{\tau}\mathcal{A}^{*}\mathcal{Z}^{u}_{s}ds$
$\displaystyle=\int_{t}^{\tau}\mathcal{A}^{*}\left[\mathcal{S}_{s}^{*}\zeta+\int_{t}^{s}\mathcal{S}^{*}_{s-v}\nu
dV^{u}(v)\right]ds$
$\displaystyle=e^{\mathcal{A}^{*}\tau}\zeta-e^{\mathcal{A}^{*}t}\zeta+\int_{t}^{\tau}\\!\\!\\!\int_{v}^{\tau}\mathcal{A}^{*}e^{\mathcal{A}^{*}(s-v)}\nu
dsdV^{u}(v)$
$\displaystyle=e^{\mathcal{A}^{*}\tau}\zeta-e^{\mathcal{A}^{*}t}\zeta+\int_{t}^{\tau}e^{\mathcal{A}^{*}(\tau-v)}\nu
dV^{u}(v)-\int_{t}^{\tau}\nu dV^{u}(v)$
$\displaystyle=\mathcal{Z}_{\tau}^{u}-e^{\mathcal{A}^{*}t}\zeta-\int_{t}^{\tau}\nu
dV^{u}(v),$

and, rearranging the terms we obtain (2.3).

######  Remark 2.8.

By defining
$B(t,X^{u}(t),u(t)):=\beta(s,X^{u}(s))\sigma(s,X^{u}(s))R(s,X^{u}(s),u(s))$,
and exploiting (2.2), we actually get that the function $x(\tau)=\langle
g,\mathcal{S}_{t}^{*}\zeta\rangle$ is given by the expression

$\displaystyle x(\tau)$
$\displaystyle=\mathbb{E}\left[X^{u}(\tau)-\int_{t}^{\tau}K(\tau-s)B(s,X^{u}(s),u(s))ds\right]$
$\displaystyle=\left\langle
g,\mathbb{E}\left[\mathcal{Z}^{u}_{\tau}-\int_{t}^{\tau}\mathcal{S}_{\tau-s}^{*}\nu
B(s,X^{u}(s),u(s))ds\right]\right\rangle$

Set $\mathcal{Z}_{\tau}^{u,g}:=\langle
g,\mathcal{Z}_{\tau}^{u}\rangle=X^{u}(\tau)$, and plug (2.1) into (2.3), then
we can rewrite (2.3) in differential notation as

$d\mathcal{Z}_{\tau}^{u}=\mathcal{A}^{*}\mathcal{Z}_{\tau}^{u}d{\tau}+\nu\Big{(}\beta({\tau},\mathcal{Z}_{\tau}^{u,g})d\tau+\sigma({\tau},\mathcal{Z}_{\tau}^{u,g})\big{[}R({\tau},\mathcal{Z}_{\tau}^{u,g},u_{\tau})d{\tau}+dW_{\tau}\big{]}\Big{)},$
(2.4)

with $\mathcal{Z}^{u}_{t}=e^{\mathcal{A}^{*}t}\zeta:=\zeta_{t}$. We also
rewrite (2.3) as

$d\mathcal{Z}_{\tau}^{u}=\mathcal{A}^{*}\mathcal{Z}_{\tau}^{u}d{\tau}+\nu\beta^{g}({\tau},\mathcal{Z}_{\tau}^{u})d\tau+\nu\sigma^{g}({\tau},\mathcal{Z}_{\tau}^{u})[R^{g}({\tau},\mathcal{Z}_{\tau}^{u},u_{\tau})d\tau+dW_{\tau}],$
(2.5)

where

$\displaystyle\beta^{g}(s,\mathcal{Z}_{s}^{u})$
$\displaystyle:=\beta(s,\mathcal{Z}_{s}^{u,g})=\beta(s,\langle
g,Z_{s}^{u}\rangle)=\beta(s,X(s)),$
$\displaystyle\sigma^{g}(s,\mathcal{Z}_{s}^{u})$
$\displaystyle:=\sigma(s,\mathcal{Z}_{s}^{u,g})=\sigma(s,\langle
g,Z_{s}^{u}\rangle)=\sigma(s,X(s)),$ $\displaystyle
R^{g}(s,\mathcal{Z}_{s}^{u},u_{s})$
$\displaystyle:=R(s,\mathcal{Z}_{s}^{u,g},u_{s})=R(s,\langle
g,Z_{s}^{u}\rangle,u_{s})=R(s,X(s),u_{s}).$

We are going to discuss existence and uniqueness results for equation (2.4) in
Section 3.1.

######  Remark 2.9.

We point out that the term

$\int_{0}^{\tau}\nu\sigma^{g}(s,\mathcal{Z}_{s})dW_{s},$ (2.6)

in (2.5) can be regarded in two different ways. On the one hand, it can be
seen as the element of $Y^{*}$:

$\nu\left(\int_{0}^{\tau}\sigma^{g}(s,\mathcal{Z}_{s})dW_{s}\right),$

where the integration of
$\int_{0}^{\tau}\sigma^{g}(s,\mathcal{Z}_{s})dW_{s}=\int_{0}^{\tau}\sigma(s,X(s))dW_{s}$
is done on $\mathbb{R}$ and then lifted to $Y^{*}$ by multiplying it by $\nu$.
On the other hand, by writing (2.6) as

$\int_{0}^{\tau}\sigma^{g}(s,\mathcal{Z}_{s})d(\nu W_{s}),$

we have that $\nu W_{s}$ can be considered as a cylindrical Wiener process on
$\mathbb{H}^{\nu}:=\left\\{\nu x,\ x\in\mathbb{R}\right\\}$, which is a
Hilbert space with the scalar product
$\langle\cdot,\cdot\rangle_{\mathbb{H}^{\nu}}:=\|\nu\|_{Y^{*}}\langle\cdot,\cdot\rangle_{\mathbb{R}}$.
In this case we also see that $\mathbb{H}^{\nu}\subsetneq Y^{*}$.

In section 3.1 we are going to provide sufficient conditions that guarantee
the existence of a solution of (2.3)-(2.4). Due to the nature of the lift and
identification (2.2), this will, in turn, provide sufficient conditions also
for the existence of a solution to (1.2).

######  Remark 2.10.

From [9, 8] we see that we could perform the lift also under weaker
hypothesis, by taking a subspace $Z\subset Y$ with their relative duals
$Y^{*}\subset Z^{*}$ such that:

  * •

$Z$ and $Y$ are Banach spaces $Z\subset Y$ and $Z$ embeds continuously into
$Y$.

  * •

The semigroup $\mathcal{S}^{*}$ with generator $\mathcal{A}^{*}$ acts in a
strongly continuous way on $Y^{*}$ and $Z^{*}$ with respect to the respective
norm topologies.

  * •

The map $\mathcal{Z}\longmapsto\mathcal{S}^{*}_{t}\mathcal{Z}$ is weak-*
continuous on $Y^{*}$ and on $Z^{*}$ for every $t\geq 0$.

  * •

The pre-adjoint operator of $\mathcal{A}^{*}$, generates a strongly continuous
semigroup on $Z$ with respect to the respective norm topology (but not
necessarily on $Y$).

In this case every kernel of the form $K(t)=\langle
g,\mathcal{S}_{t}^{*}\nu\rangle$ with $\nu\in Z^{*}$ and $S_{t}^{*}\nu\in
Y^{*}$ is liftable. While this setting would allow to work with a wider class
of kernels, we would not be able to formulate the HJB equations. This is due
to the fact that, when considering a kernel $K(t)=\langle
g,\mathcal{S}_{t}^{*}\nu\rangle$ with $\nu\in Z^{*}$, some of the inner
products in the definition of the Hamilton-Jacobi-Bellman equation (3.16),
would not be well defined. Being the goal of this work a control problem, we
restrict ourselves to the case $\nu\in Y^{*}$, as originally stated.

In a similar fashion to what we did for (1.2), recalling that
$X^{u}(\tau):=\mathcal{Z}_{\tau}^{u,g}$, we rewrite the performance functional
(1.1) so to make its dependence from the lifted process
$\mathcal{Z}^{u}_{\tau}$ explicit:

$\displaystyle J(t,x,u)$
$\displaystyle=\mathbb{E}\left[\int_{t}^{T}F(\tau,\mathcal{Z}^{u,g}_{\tau},u_{\tau})d\tau+G(\mathcal{Z}^{u,g}_{T})\right]$
$\displaystyle:=\mathbb{E}\left[\int_{t}^{T}F^{g}(\tau,\mathcal{Z}_{\tau}^{u},u_{\tau})d\tau+G^{g}(\mathcal{Z}_{T}^{u})\right]:=J^{g}(t,\zeta,u),$
(2.7)

where the functions
$F:[0,T]\times\mathbb{R}\times\mathcal{U}\longrightarrow\mathbb{R}$,
$G:\mathbb{R}\longrightarrow\mathbb{R}$ are _lifted_ to the functions

$\displaystyle F^{g}$ $\displaystyle:[0,T]\times
Y^{*}\times\mathcal{U}\longrightarrow\mathbb{R},$ $\displaystyle G^{g}$
$\displaystyle:Y^{*}\longrightarrow\mathbb{R}$

where $Y^{*}$ is the Banach space associated to the liftable kernel, see
Definition 2.6. The lifted maps $F^{g}$ and $G^{g}$ are

$\displaystyle F^{g}(\cdot,\mathcal{Z}^{u}_{\tau},\cdot)$
$\displaystyle:=F(\cdot,\mathcal{Z}^{u,g}_{\tau},\cdot)=F(\cdot,\langle
g,\mathcal{Z}^{u}_{\tau}\rangle,\cdot)=F(\cdot,X^{u}(\tau),\cdot),$
$\displaystyle G^{g}(\mathcal{Z}^{u}_{\tau})$
$\displaystyle:=G(\mathcal{Z}^{u,g}_{\tau})=G(\langle
g,\mathcal{Z}^{u}_{\tau}\rangle)=G(X^{u}(\tau)).$

The stochastic optimal control problem (1.1)-(1.3) is then lifted to

$J^{g}(t,\zeta,\hat{u})=\inf_{u\in\mathds{A}}J^{g}(t,\zeta,u)=\inf_{u\in\mathds{A}}\mathbb{E}\left[\int_{t}^{T}F^{g}(\tau,\mathcal{Z}_{\tau}^{u},u)d\tau+G^{g}(\mathcal{Z}_{T}^{u})\right],$
(2.8)

where the process $\mathcal{Z}^{u}$ takes values in the Banach space $Y^{*}$,
and where the dynamics for the controlled process are given by (2.3)-(2.4).
Notice that, while the performance functional has not changed, we write
$J^{g}$ instead of $J$ in order to highlight the dependence on
$\mathcal{Z}^{u}_{t}$ instead of $X^{u}(t)$, as underneath there is a passage
from finite to infinite dimensions. Indeed, this change of notation embodies a
crucial change of framework from a finite to an infinite dimensional setting,
allowing us to move from functions $\beta$, $\sigma$, $R$, $F$ and $G$ taking
values from $\mathbb{R}$ to new functions $\beta^{g}$, $\sigma^{g}$, $R^{g}$,
$F^{g}$ and $G^{g}$ that now take values from $Y^{*}$. This lift allows us to
consider a new optimization problem, written on a space which is not the
original one. Nonetheless, we have that $J(t,x,u)=J^{g}(t,\zeta,u)$ for
$t\in[0,T]$, $u\in\mathcal{U}$. Also, being $g$ fixed and only depending on
the kernel representation, finding the pair $(\hat{u},\mathcal{Z}^{\hat{u}})$
that minimizes (2.5)-(2.8) is equivalent to finding the pair
$(\hat{u},X^{\hat{u}})$ that solves (1.1)-(1.2).

###  2.3. UMD Banach spaces

In the sequel we use techniques of Malliavin calculus on the space $Y^{*}$.
For this, we assume:

######  Hypothesis 2.11.

The space $Y^{*}$ is a unconditional martingale differences (UMD) Banach
space.

For convenience we report here below the essentials on UMD Banach spaces.

######  Definition 2.12.

Let $(M_{n})_{n=1}^{N}$ be a Banach-space valued martingale, the sequence
$d_{n}=M_{n+1}-M_{n}$ is called the martingale difference sequence associated
with $(M_{n})_{n=1}^{N}$. A Banach space $E$ is said to be a $UMD_{p}$
$(1<p<\infty)$, space if there exists a constant $\beta$ such that for all
$E$-valued $L^{p}$-martingale difference sequences $(d_{n})_{n=1}^{N}$ we have

$\mathbb{E}\left\|\sum_{n=1}^{N}\epsilon_{n}d_{n}\right\|^{p}\leq\beta^{p}\mathbb{E}\left\|\sum_{n=1}^{N}d_{n}\right\|^{p},$

where $\epsilon_{n}\in\mathbb{R}$ for all $n$ and $|\epsilon_{n}|=1.$ Thanks
to [26] we also know that, if a Banach space $E$ is $UMD_{p}$ for some
$1<p<\infty$, then $E$ is a $UMD_{p}$ Banach space for all $p\in(1,\infty)$,
and we simply call it a _UMD Banach space_.

In the context of stochastic analysis in Banach spaces, martingale difference
sequences provide a substitute for orthogonal sequences. In the following
parts, we will see that this hypothesis is not very restrictive, as the UMD
Banach spaces include all Hilbert spaces, $L^{q}$ spaces for $q\in(1,\infty)$,
reflective Sobolev spaces and many others, thus allowing us to consider a wide
class of liftable kernels. In our framework, the process $\mathcal{Z}^{u}$
takes values in a UMD Banach space whenever we consider, for example, a shift
operator or a quasi-exponential kernel or a kernel that can be expressed as
the Laplace transform of a measure with density in $L^{q}([0,\infty))$,
$q\in(1,\infty)$.

Assuming that $Y^{*}$ is UMD, allows us to define the Malliavin derivative
operator $D$ on $L^{p}(\Omega,Y^{*})$. From [23, Proposition 2.5], we know
that $D$ is a closed operator and we denote with $\mathbb{D}^{1,p}(Y^{*})$ the
closure of the domain.

For the results on UMD Banach spaces exploited in the following parts, we
refer to [28] for the BDG inequality, [27] for the Fubini Theorem and to [23]
for general Malliavin calculus results. In this framework we will also use a
Clark-Okone formula for UMD Banach spaces (see [21]) and the following chain
rule linking the Malliavin derivative and the Gâteaux derivative (see [23])

######  Proposition 2.13.

Let $E$ be a UMD Banach space and let $p\in(1,\infty)$. Suppose that
$\varphi\in\mathcal{G}^{1}(E,E)$. If $F\in\mathbb{D}^{1,p}(E)$, then
$\varphi(F)\in\mathbb{D}^{1,p}(E)$ with

$D(\varphi(F))=\nabla\varphi(F)DF.$

##  3\. The optimal control problem

We are now interested in solving the lifted optimal control problem (2.8),
where the process $\mathcal{Z}$ follows the controlled dynamics given by

$\begin{cases}d\mathcal{Z}^{u}_{\tau}=&\mathcal{A}^{*}\mathcal{Z}_{\tau}^{u}d\tau+\nu\beta^{g}(\tau,\mathcal{Z}^{u}_{\tau})d\tau+\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau})R^{g}(\tau,\mathcal{Z}^{u}_{\tau},u_{\tau})d\tau\\\
&+\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau})dW_{\tau},\\\
\mathcal{Z}^{u}_{t}=&\zeta_{t}.\end{cases}$ (3.1)

For our results to hold, we add some Hypothesis on $R$, which directly
translates into hypothesis on $R^{g}$.

######  Hypothesis 3.1.

$R:[0,T]\times\mathbb{R}\times\mathcal{U}\longrightarrow\mathbb{R}$ is
measurable and $\|R(\tau,x,u)\|_{\mathbb{R}}\leq K_{R}$ for a suitable
positive constant $K_{R}>0$ and every $\tau\in[0,T]$, $x\in\mathbb{R}$,
$u\in\mathcal{U}$.

In order to find the optimal value $J(\hat{u})$, we associate the following
partially coupled system of forward-backward equations

$\begin{cases}d\mathcal{Z}_{\tau}&=\mathcal{A}^{*}\mathcal{Z}_{\tau}d\tau+\nu\beta^{g}(\tau,\mathcal{Z}_{\tau})d\tau+\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau})dW_{\tau},\
\ \tau\in[t,T],\\\ \mathcal{Z}_{t}&=\zeta_{t},\\\
dp_{\tau}&=-\mathcal{H}(\tau,\mathcal{Z}_{\tau},q_{\tau})d\tau+q_{\tau}\nu
dW_{\tau},\quad\tau\in[t,T],\\\ p_{T}&=G(\mathcal{Z}_{T}),\end{cases}$ (3.2)

to (3.1). Here above $\mathcal{H}:[0,T]\times Y^{*}\times
Y^{**}\longrightarrow\mathbb{R}$ is the Hamiltonian function defined as

$\mathcal{H}(t,z,\xi)=\inf_{u\in\mathcal{U}}\left[F^{g}(t,z,u)+\xi\nu
R^{g}(t,z,u)\right].$ (3.3)

Notice that the control $u$ only appears in the Hamiltonian functional. The
solution of the backward equation is denoted by $(p,q)$. We often write
$(p_{\tau},q_{\tau})=(p(\tau,t,\zeta_{t}),q(\tau,t,\zeta_{t}))$,
$\tau\in[t,T]$ when we want to emphasize the dependence of $p$ and $q$ on the
parameter $\zeta_{t}$ at time $t$. Analogously, when we want to emphasize the
dependence of $\mathcal{Z}$ on the initial value $\zeta_{t}$ at time $t$, we
write $\mathcal{Z}_{\tau}=\mathcal{Z}(\tau,t,\zeta_{t})$.

Define now

$v(t,z):=p(t,t,z),$ (3.4)

with $p$ the solution to the backward SDE in (3.2). In the sequel we show that
$J^{g}(t,\zeta_{t},\hat{u})$ in (2.8) is such that

$J^{g}(t,\zeta_{t},\hat{u})=v(t,\zeta_{t})$ (3.5)

and that the optimal control $\hat{u}$ can be retrieved explicitly via a
verification theorem once $v(t,\zeta_{t})$ is known. In order to achieve (3.5)
we proceed as follow. First we study the forward equation
$\mathcal{Z}(\tau,t,\zeta_{t})$ in Section 3.1, then we study the backward
equation $(p(\tau,t,\zeta_{t}),q(\tau,t,\zeta_{t}))$ in Section 3.2 and there
we prove the crucial identification:

$q_{t}=\nabla_{z}v(t,\mathcal{Z}_{t})\nu\sigma^{g}(t,\mathcal{Z}_{t}),$ (3.6)

(see Proposition 3.11). In Section 3.3 we provide an approach to find $v(t,z)$
through HJB equations and at last, in Section 3.4, we prove (3.5) and we
provide a characterization of the optimal control $\hat{u}$.

Notice that, for (3.6) to hold, the backward process $p(\cdot,t,z)$ has to be
differentiable with respect to $z$. This can be obtained by showing that
$\mathcal{Z}_{t}^{u}$ is differentiable with respect to the initial condition
$\zeta_{t}$, and by assuming the following:

######  Hypothesis 3.2.

Let us assume that

  * 1)

There exists $L_{1}>0$ such that

$|\mathcal{H}(t,z,\xi_{1})-\mathcal{H}(t,z,\xi_{2})|\leq
L_{1}\|\xi_{1}-\xi_{2}\|_{Y^{**}}$

for every $t\in[0,T]$, $z\in Y^{*}$ and $\xi_{1},\xi_{2}\in Y^{**}$.

  * 2)

For all $t\in[0,T]$,
$\mathbb{E}\left[\int_{t}^{T}|\mathcal{H}(s,0,0)|^{2}ds\right]<\infty$.

  * 3)

For every $t\in[0,T]$ we have
$\mathcal{H}(t,\cdot,\cdot)\in\mathcal{G}^{1,1}(Y^{*}\times Y^{**})$.

  * 4)

There exist $L_{2}>0$ and $m\geq 0$ such that

$|\nabla_{z}\mathcal{H}(t,z,\xi)h|\leq
L_{2}\|h\|_{Y^{*}}(1+\|z\|_{Y^{*}})^{m}(1+\|\xi\|_{Y^{**}})$

for every $t\in[0,T]$, $z,h\in Y^{*}$ and $\xi\in Y^{**}$.

  * 5)

$G^{g}\in\mathcal{G}^{1}(Y^{*})$ and there exists $L_{3}>0$ such that, for
every $z_{1},z_{2}\in Y^{*}$

$|G^{g}(z_{1})-G^{g}(z_{2})|\leq L_{3}\|z_{1}-z_{2}\|_{Y^{*}}.$

Further details on the continuous dependence on $\zeta_{t}$ of the forward
equation can be found in Section 3.1, while we refer to Section 3.2 for
details on the differentiability of $p(\cdot,t,z)$ with respect to $z$.

######  Remark 3.3.

The identification (3.6), in the case where $\sigma^{g}$ does not depend on
$\mathcal{Z}$ in (3.2), can be proven following [22] and dropping the UMD
hypothesis on $Y^{*}$. In our case though, being $\sigma^{g}$ dependent on
$\mathcal{Z}_{t}$, we need to exploit some Malliavin calculus techniques on
Banach spaces, and thus assume that $Y^{*}$ is UMD.

###  3.1. On the lifted forward equation

In this section we study the lifted forward equation $\mathcal{Z}_{t}$ in
(3.2). In particular, we prove that it admits a unique Markovian solution and
we study its continuous dependence from the initial parameter $\zeta_{t}$. We
thus take

$\begin{cases}d\mathcal{Z}_{\tau}&=\mathcal{A}^{*}\mathcal{Z}_{\tau}d\tau+\nu\beta^{g}(\tau,\mathcal{Z}_{\tau})d\tau+\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau})dW_{\tau},\quad\tau\in[t,T],\\\
\mathcal{Z}_{t}&=\zeta_{t},\end{cases}$ (3.7)

we recall that $\mathcal{A}^{*}$ is the generator of a uniformly continuous
semigroup on the Banach space $Y^{*}.$ We assume the following:

######  Hypothesis 3.4.

Suppose that

  * i)

$\beta^{g}:[0,T]\times Y^{*}\longrightarrow\mathbb{R}$ is continuous and, for
all $t\in[0,T]$ $z_{1},z_{2}\in Y^{*}$, there exists a constant $L_{1}>0$ such
that

$|\nu\beta^{g}(t,z_{1})-\nu\beta^{g}(t,z_{2})|\leq
L_{1}\|z_{1}-z_{2}\|_{Y^{*}},$

the map $\nu\beta^{g}:[0,T]\times Y^{*}\longrightarrow Y^{*}$ is measurable.
Moreover, for all $t\in[0,T]$ and $z\in Y^{*}$,

$|\nu\beta^{g}(t,z)|^{2}\leq L_{2}(1+\|z\|_{Y^{*}}^{2}),$

for some constant $L_{2}>0$.

  * ii)

$\sigma^{g}:[0,T]\times Y^{*}\longrightarrow\mathbb{R}$ is such that, for
every $v\in Y^{**}$ the map $\nu\sigma^{g}v:[0,T]\times
Y^{*}\longrightarrow\mathbb{R}$ is measurable,
$e^{s\mathcal{A}^{*}}\nu\sigma^{g}(t,z)\in L^{2}(Y^{*})$ for every $s>0$,
$t\in[0,T]$ and $z\in Y^{*}$, and

$\|\nu\sigma^{g}(t,z)\|^{2}_{L(Y^{*})}\leq L_{3}(1+\|z\|_{Y^{*}}^{2}),$

for some constant $L_{3}>0$.

Moreover, for $s>0$, $t\in[0,T]$, $z_{1},z_{2}\in Y^{*}$ there exists a
constant $L_{4}>0$ such that

$\|\nu\sigma^{g}(t,\zeta_{1})-\nu\sigma^{g}(t,z_{2})\|_{L^{2}(Y^{*})}\leq
L_{4}\|z_{1}-z_{2}\|_{Y^{*}},$

  * iii)

For every $s>0$, $t\in[0,T]$,
$\nu\beta^{g}(t,\cdot)\in\mathcal{G}^{1}(Y^{*},Y^{*})$.

Our first result is the following:

######  Proposition 3.5.

Assume Hypothesis 3.4 holds. For every $p\in[2,\infty)$, we have that:

  * i)

The map $(t,z)\longmapsto\mathcal{Z}(\cdot,t,z)$ is in
$\mathcal{G}^{0,1}([0,T]\times Y^{*};L^{p}(\Omega;C([0,T];Y^{*})))$.

  * ii)

For every $h\in Y^{*}$ the partial directional derivative process
$\nabla_{z}\mathcal{Z}(\tau,t,z)h$, $\tau\in[0,T]$ solves $\mathbb{P}$-a.s.
the equation

$\displaystyle\nabla_{z}\mathcal{Z}(\tau,t,z)h$
$\displaystyle=e^{(\tau-t)\mathcal{A}^{*}}h+\int_{t}^{\tau}e^{(\tau-s)\mathcal{A}^{*}}\nabla_{z}\nu\beta^{g}(s,\mathcal{Z}(s,t,z))\nabla_{z}\mathcal{Z}(s,t,z)h\
ds$
$\displaystyle+\int_{t}^{\tau}\nabla_{z}\left(e^{(\tau-s)\mathcal{A}^{*}}\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))\right)\nabla_{z}\mathcal{Z}(s,t,z)h\
dW_{s},\quad\tau\in[t,T],$ $\displaystyle\nabla_{z}\mathcal{Z}(\tau,t,z,)h$
$\displaystyle=h,\quad\tau\in[0,t).$

  * iii)

$\|\nabla_{z}\mathcal{Z}(\tau,t,z)h\|_{L^{p}(\Omega;C([0,T];Y^{*}))}\leq
c\|h\|_{Y^{*}}$ for some constant $c$.

We also find that

  * iv)

(3.7) admits a unique adapted solution $\mathcal{Z}\in
L^{p}(\Omega,C([t,T]);Y^{*})$.

Moreover, we have the following estimate

$\|\mathcal{Z}\|_{p}^{p}:=\mathbb{E}\left[\sup_{\tau\in[t,T]}\|\mathcal{Z}_{\tau}\|_{Y^{*}}^{p}\right]\leq
C(1+\|\zeta_{t}\|_{Y^{*}}^{p}),$ (3.8)

where $C$ is a constant depending only on $p,T,L$, where
$L:=\max\\{L_{1},L_{2},L_{3},L_{4}\\}$ and
$M:=\sup_{\tau\in[t,T]}\|e^{\tau\mathcal{A}^{*}}\|_{L^{2}(Y^{*})}$.

###### Proof.

The proof is inspired by [10, Theorem 7.4] and [14, Proposition 3.2] The main
difference with our work are the spaces at play. Consider the map

$\Phi(\mathcal{Z},t,z)_{\tau}:L^{p}(\Omega;C([0,T];Y^{*}))\times[0,T]\times
Y^{*}\longrightarrow L^{p}(\Omega;C([0,T];Y^{*}))$

defined as

$\displaystyle\Phi(\mathcal{Z},t,z)_{\tau}$
$\displaystyle:=e^{\mathcal{A}^{*}(\tau-t)}z+\int_{0}^{\tau}\mathds{1}_{[t,T]}(s)e^{\mathcal{A}^{*}(\tau-s)}\nu\beta^{g}(s,\mathcal{Z}_{s})ds$
$\displaystyle+\int_{0}^{\tau}\mathds{1}_{[t,T]}(s)e^{\mathcal{A}^{*}(\tau-s)}\nu\sigma^{g}(s,\mathcal{Z}_{s})dW_{s}$
$\displaystyle:=S_{0}(\mathcal{Z},t,z)_{\tau}+S_{1}(\mathcal{Z},t,z)_{\tau}+S_{2}(\mathcal{Z},t,z)_{\tau}\quad\tau\in[0,T].$

We want to show that $\Phi$ is a contraction with respect to the first
variable. We notice that

$\displaystyle\|S_{1}(\mathcal{Z},t,z)\|^{p}$ $\displaystyle\leq
M^{p}\mathbb{E}\left[\left(\int_{0}^{T}\|\nu\beta^{g}(s,\mathcal{Z}_{s})\|_{Y^{*}}ds\right)^{p}\right]$
$\displaystyle\leq
T^{p-1}M^{p}\mathbb{E}\int_{0}^{T}\mathbb{E}\left[\int_{0}^{T}\|\nu\beta^{g}(s,\mathcal{Z}_{s})\|_{Y^{*}}^{p}ds\right]$
$\displaystyle\leq
2^{p/2-1}T^{p-1}M^{p}L^{p}\mathbb{E}\left[\int_{0}^{T}(1+\|\mathcal{Z}_{s}\|_{Y^{*}}^{p})ds\right]$
$\displaystyle\leq 2^{p/2-1}(TLM)^{p}(1+\|\mathcal{Z}\|_{p}),$

and

$\displaystyle\|S_{2}(\mathcal{Z},t,z)\|^{p}$
$\displaystyle\leq\sup_{\tau\in[0,T]}\mathbb{E}\left[\left\|\int_{0}^{\tau}e^{(\tau-s)\mathcal{A}^{*}}\nu\sigma^{g}(s,\mathcal{Z}_{s})dW(s)\right\|_{Y^{*}}^{p}\right]$
$\displaystyle\leq
M^{p}C_{p/2}LT^{p/2-1}2^{p/2-1}\mathbb{E}\left[\int_{0}^{T}(1+\|\mathcal{Z}_{s}\|^{p}_{Y^{*}})ds\right]$
$\displaystyle\leq M^{p}C_{p/2}L(2T)^{p/2-1}(T+\|\mathcal{Z}\|_{p}),$

where we used the linear growth conditions on $\beta^{g}$ and $\sigma^{g}$ and
the Burkholder-Davis-Gundy inequality for $UMD$ Banach spaces (see [28]). We
thus have showed that $\Phi(\cdot,t,z)$ is a well defined mapping. Now, taking
$\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ arbitrary processes in $Y^{*}$, then

$\displaystyle\|\Phi(\mathcal{Z}_{1},t,z)-\Phi(\mathcal{Z}_{2},t,z)\|_{p}$
$\displaystyle\leq\|S_{1}(\mathcal{Z}_{1},t,z)-S_{1}(\mathcal{Z}_{2},t,z)\|_{p}+\|S_{2}(\mathcal{Z}_{1},t,z)-S_{2}(\mathcal{Z}_{2},t,z)\|_{p}$
$\displaystyle:=I_{1}+I_{2}.$

With computations similar to the ones above, exploiting the Lipschitz
condition on $\beta^{g}$ and $\sigma^{g}$ (see Hypothesis 3.4, _i)_ and
_ii)_), one finds that

$I_{1}^{p}\leq(TML)^{p}\|\mathcal{Z}_{1}-\mathcal{Z}_{2}\|_{p}^{p},$

and

$I_{2}^{p}\leq
C_{p/2}(ML)^{p}T^{p/2}\|\mathcal{Z}_{1}-\mathcal{Z}_{2}\|_{p}^{p}.$

Summing up, we have that

$\|\Phi(\mathcal{Z}_{1})-\Phi(\mathcal{Z}_{2})\|_{p}\leq
LM(T^{p}+C_{p/2}T^{p/2})^{1/p}\|\mathcal{Z}_{1}-\mathcal{Z}_{2}\|_{p}.$

This means that $\Phi(\mathcal{Z},t,z)$ is a contraction only for $t\in[0,T]$
when $T$ satisfies

$LM(T^{p}+C_{p/2}T^{p/2})^{1/p}<1.$ (3.9)

Condition (3.9) on $T$ can be easily removed by considering the equation on
intervals $[0,\tilde{T}]$, $[\tilde{T},2\tilde{T}]$,…, where $\tilde{T}$
satisfies (3.9). Thanks to the fixed point theorem we find that (2.5) admits a
unique solution. We conclude that (3.8) holds by applying Gronwall’s Lemma
with arguments in line with [10, Theorem 7.4 (iii)]. Notice now that, being
$\Phi(\cdot,t,z)$ a contraction uniformly with respect to $t\in[0,T]$, $z\in
Y^{*}$, by Proposition 2.5 we obtain $ii)$ if

$\Phi\in\mathcal{G}^{1,0,1}\left(L^{p}(\Omega;C([0,T];Y^{*}))\times[0,T]\times
Y^{*},L^{p}(\Omega;C([0,T];Y^{*}))\right).$

This is verified by a (slight modification) of Lemma 2.4. Indeed we notice
that $\Phi$ is differentiable in $z$. For more details we refer to [10, 14]. ∎

######  Remark 3.6.

We notice that $\mathcal{Z}$ is Markovian (see e.g. [13, Theorem 1.157]).

######  Corollary 3.7.

Assume Hypothesis 3.1 and 3.4 hold. Then (3.1) admits a unique solution.

###### Proof.

Thanks to the boundedness of $R$, one can apply the Girsanov Theorem, see e.g.
[10, Theorem 10.14], and proceed like in the proof of Proposition (3.5). ∎

###  3.2. On the backward equation

In this section we study the backward equation

$p_{\tau}=G^{g}(\mathcal{Z}_{T})+\int_{\tau}^{T}\mathcal{H}(s,\mathcal{Z}_{s},q_{s})ds-\int_{\tau}^{T}q_{s}\nu
dW_{s},$ (3.10)

introduced in (3.2). We study existence and uniqueness of a solution as well
as its continuous dependence from the parameter $\zeta_{t}$. Later on we will
exploit (3.10) to prove (3.6), as well as show that the optimal value
$J^{g}(t,z,\hat{u})$ for the optimization problem (2.5) - (2.8) is achieved
for $J^{g}(t,\zeta_{t},\hat{u})=v(t,\zeta_{t})=p(t,t,\zeta_{t})$.

We observe that the following a priori estimate for the pair process $(p,q)$
holds (see [22] and [14, Proposition 4.3] ):

$\mathbb{E}\Bigg{[}\sup_{\tau\in[t,T]}|p_{\tau}|^{2}\Bigg{]}+\mathbb{E}\Bigg{[}\int_{t}^{T}\|q_{\tau}\|^{2}_{Y^{**}}d\tau\Bigg{]}\leq
c\mathbb{E}\Bigg{[}\int_{t}^{T}|\mathcal{H}(\tau,0,0)|^{2}d\tau\Bigg{]}+c\mathbb{E}\Bigg{[}|G^{g}(\mathcal{Z}_{T})|^{2}\Bigg{]},$

where $c$ is a constant depending on $T$ and $L:=\max\\{L_{1},L_{2},L_{3}\\}$,
where $L_{i}$, $i=1,..,3$ are the coefficients in Hypothesis 3.2.

######  Proposition 3.8.

Assume that Hypotheses 3.2 and 3.4 hold true. Then (3.10) admits a unique
solution $(p,q)\in L^{2}(\Omega,C[0,T];Y^{*})\times
L^{2}(\Omega,L^{2}[0,T];L^{2}(Y^{*}))$ such that the map

$z\longmapsto(p(\cdot,\cdot,z),q(\cdot,\cdot,z))\text{ \emph{ belongs to }
}\mathcal{G}^{1}(L^{\eta}(\Omega;C([0,T];Y^{*})),\mathcal{K}_{cont}([0,T]))$

for $\eta=\ell(m+1)(m+2)$, where $\mathcal{K}_{cont}([0,T])$ is the space of
adapted processes $(p,q)$ taking values in $\mathbb{R}\times Y^{**}$ such that
$p$ has continuous paths and

$\mathbb{E}\left[\sup_{\tau\in[0,T]}|p_{\tau}|^{2}\right]+\mathbb{E}\left[\int_{\tau}^{T}\|q_{s}\|_{Y^{**}}^{2}ds\right]<\infty.$

Moreover, for every $\ell\geq 2$.

$\left(\mathbb{E}\left[\sup_{t\in[0,T]}|\nabla_{z}p(t,z)h|^{\ell}\right]\right)^{1/\ell}\leq
C\|h\|_{Y^{*}}\left(1+\|z\|_{Y^{*}}^{(m+1)^{2}}\right)$

###### Proof.

See [22] Proposition 4.2. ∎

Still aiming to prove (3.6), we provide yet another crucial result that links
the directional derivative of $\mathcal{Z}$ to its Malliavin derivative.

######  Proposition 3.9.

Assume that Hypothesis 3.4 holds. Then for almost all $s,\tau$ such that
$t\leq s\leq\tau<T$ we have that

$D_{s}\mathcal{Z}(\tau,t,z)=\nabla_{z}\mathcal{Z}(\tau,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z)),\
\mathbb{P}-a.s.$ (3.11)

moreover

$D_{s}\mathcal{Z}(T,t,z)=\nabla_{z}\mathcal{Z}(T,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z)),\
\mathbb{P}-a.s.\text{ for almost all }s.$ (3.12)

###### Proof.

Thanks to Proposition 3.5, for every $s\in[0,T]$ and every direction $h\in
Y^{*}$, the directional derivative process $\nabla_{z}\mathcal{Z}(\tau,s,z)h$,
$\tau\in[s,T)$ solves $\mathbb{P}$-a.s. the equation

$\displaystyle\nabla_{z}\mathcal{Z}(\tau,t,z)h$
$\displaystyle=e^{(\tau-t)\mathcal{A}^{*}}h+\int_{t}^{\tau}e^{(\tau-s)\mathcal{A}^{*}}\nabla_{z}\nu\beta^{g}(s,\mathcal{Z}(s,t,z))\nabla_{z}\mathcal{Z}(s,t,z)hds$
$\displaystyle+\int_{t}^{\tau}\nabla_{z}\left(e^{(\tau-s)\mathcal{A}^{*}}\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))\right)\nabla_{z}\mathcal{Z}(s,t,z)hdW_{s},\quad\tau\in[t,T],$
$\displaystyle\nabla_{z}\mathcal{Z}(\tau,t,z)h$
$\displaystyle=h,\quad\tau\in[0,t),$

Given $v\in Y^{*}$ and $t\in[0,s]$, we can replace $z$ by $\mathcal{Z}(s,t,z)$
and $h$ by $\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))v$ in the previous equation,
since $\mathcal{Z}(s,t,z)$ is $\mathcal{F}_{s}$ measurable. Note now that

$\mathcal{Z}(\eta,s,\mathcal{Z}(s,t,z))=\mathcal{Z}(\eta,t,z)\quad\mathbb{P}-a.s.,$

for $\eta\in[s,T)$, as a consequence of the uniqueness of the solution of
(3.7). This yields

$\displaystyle\nabla_{z}$
$\displaystyle\mathcal{Z}(\tau,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))v=e^{(\tau-s)\mathcal{A}^{*}}\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))v$
$\displaystyle+\int_{s}^{\tau}e^{(\tau-\eta)\mathcal{A}^{*}}\nabla_{z}\nu\beta^{g}(\eta,\mathcal{Z}(\eta,t,z))\nabla_{z}\mathcal{Z}(\eta,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))vd\eta$
$\displaystyle+\int_{s}^{\tau}\nabla_{z}(e^{(\tau-\eta)\mathcal{A}^{*}}\nu\sigma^{g}(\eta,\mathcal{Z}(\eta,t,z)))\nabla_{z}\mathcal{Z}(\eta,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))vdW_{\eta},$

for $\tau\in[s,T)$, $\mathbb{P}$-a.s. This shows that the process

$\left\\{\nabla_{z}\mathcal{Z}(\tau,t,\mathcal{Z}(s,t,z,))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))v\right\\}_{t\leq
s\leq\tau<T},$

is a solution of the equation

$\displaystyle Q_{s,\tau}$
$\displaystyle=e^{(\tau-s)\mathcal{A}^{*}}\nu\sigma^{g}(s,\mathcal{Z}_{s})v+\int_{s}^{\tau}e^{(\tau-\eta)\mathcal{A}^{*}}\nabla_{z}\nu\beta^{g}(\eta,\mathcal{Z}_{\eta})Q_{s,\eta}d\eta$
$\displaystyle+\int_{s}^{\tau}\nabla_{z}(e^{(\tau-\eta)\mathcal{A}^{*}}\nu\sigma^{g}(\eta,\mathcal{Z}_{\eta}))Q_{s,\eta}dW_{\eta},$

where $Q_{s,\tau}:=D_{s}\mathcal{Z}_{\tau}v$. The thesis now follows from the
uniqueness property, as proved e.g. in [14] Proposition 3.5. To complete the
proof of (3.12), we take a sequence $\tau_{n}\uparrow T$ such that (3.11)
holds for every $\tau_{n}$, and we let $n\rightarrow\infty$ (see [14]). The
result follows from the regularity properties of $D\mathcal{Z}$ and
$\nabla_{z}\mathcal{Z}$, as well as the closedness of the operator $D$ on UMD
Banach spaces. ∎

In this framework, using the results presented in [21] and [23] we find that:

######  Proposition 3.10.

Assume Hypotheses 3.2 \- 3.4 Then for a.a. $s,\tau$ such that $t\leq
s\leq\tau\leq T$ we have that

$\displaystyle D_{s}p(\tau,t,z)$
$\displaystyle=\nabla_{z}p(\tau,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))\quad\mathbb{P}-a.s.,$
(3.13) $\displaystyle D_{s}q(\tau,t,z)$
$\displaystyle=\nabla_{z}q(\tau,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))\quad\mathbb{P}-a.s..$
(3.14)

Moreover, for a.a. $s\in[t,T]$,

$q(s,t,z)=\nabla_{z}p(s,s,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z))\
\mathbb{P}-a.s.$ (3.15)

###### Proof.

The proof follows the same arguments as [14, Proposition 5.6] though the
spaces at play are different. Indeed, the main tools are provided in
Proposition 3.9. So, thanks to the extension of Malliavin calculus to UMD
Banach spaces, and the chain rule linking Malliavin derivative and Gateaux
derivative (see Proposition 2.13), the result is secured. ∎

Finally, the next result provides the proof of (3.6).

######  Proposition 3.11.

Assume that Hypothesis 3.2 and 3.4 hold true. Then the function
$v(t,z):=p(t,t,z)$ in (3.4) is continuous and for every $t\in[0,T]$,
$v(t,\cdot)$ belongs to $\mathcal{G}^{1}(Y^{*},\mathbb{R})$ and there exists
$C>0$ such that

$|\nabla_{z}v(t,z)h|\leq C\|h\|_{Y^{*}}(1+\|z\|_{Y^{*}}^{(m+1)^{2}}).$

Moreover we have that

$q(s,t,z)=\nabla_{z}v(t,\mathcal{Z}(s,t,z))\nu\sigma^{g}(s,\mathcal{Z}(s,t,z)).$

###### Proof.

The first part is a corollary of Proposition 3.8. The second is derived from
(3.15). ∎

###  3.3. The HJB equation

Formally define

$\displaystyle\mathcal{L}_{t}[f](x)$
$\displaystyle:=\frac{1}{2}\text{Trace}(G^{g}(t,x)G^{g}(t,x)^{*}\
\nabla^{2}f(x))+\langle\mathcal{A}^{*}x+\nu\beta^{g}(t,x),\nabla
f(x)\rangle_{Y^{*}\times Y^{**}}.$

We can consider the Hamilton-Jacobi-Bellman equation associated with the
control problem (2.5) - (2.8), which is given by

$\begin{cases}\frac{\partial w}{\partial
t}(t,z)&=-\mathcal{L}_{t}w(t,z)-\mathcal{H}(t,z,\nabla
w(t,z)\nu\sigma^{g}(t,z)),\\\ w(T,z)&=G^{g}(z).\end{cases}$ (3.16)

A solution of this equation provides a way to compute $v(t,z)$ in (3.4) by PDE
methods (see e.g. [6]). The connection between (3.16) and (3.4) is actually
detailed in the forthcoming Theorem 3.13 by means of the forward backward
system (3.2). Later on, in Theorem 3.18, we shall see how $w(t,z)$ is
connected with the optimal performance, see (3.5). Thus, we are interested in
finding mild solutions to the previous equation, which we are going to defined
soon. This problem was tackled in [14] (for a Hilbert space) in the case of a
general $\nu\sigma^{g}$, and in [22] (for a Banach space) in the case of a
constant $\nu\sigma^{g}$. Our result is then extending the on of [22].

Let $\mathcal{Z}(\tau,t,\zeta_{t})$ be a solution to (3.7), with
$\mathcal{A}^{*}$, $\beta^{g}$ and $\sigma^{g}$ satisfying Hypotheses 3.2 \-
3.4. We recall that this solution is a $Y^{*}$-valued Markov process (see
Remark 3.6). We can then define the transition semigroup on continuous and
bounded functions $\varphi:Y^{*}\longrightarrow\mathbb{R}$ as

$P_{t,\tau}[\varphi](z)=\mathbb{E}\left[\varphi(\mathcal{Z}(\tau,t,z))\right].$

Moreover, we have that this semigroup is also well defined on continuous
functions $\varphi:Y^{*}\longrightarrow\mathbb{R}$ with polynomial growth with
respect to $z$.

######  Definition 3.12.

A function $w:[0,T]\times Y^{*}\longrightarrow\mathbb{R}$ is a mild solution
of the Hamilton-Jacobi-Bellman equation (3.16) if:

  * •

For every $t\in[0,T]$ $w(t,\cdot)\in\mathcal{G}^{1}(Y^{*})$, $w$ is continuous
and $(t,z)\longmapsto w(t,z)$ is measurable from $[0,T]\times Y^{*}$ with
values in $Y^{**}$

  * •

For every $t\in[0,T]$, there exists $C>0$ such that $|w(t,z)|\leq
C(1+\|z\|^{j}_{Y^{*}})$ and $|\nabla_{z}w(t,z)h|\leq
C\|h\|_{Y^{*}}(1+\|z\|^{k}_{Y^{*}})$, with $z,h\in Y^{*}$ and $j$ and $k$
positive integers.

  * •

The following equality holds.

$w(t,z)=P_{t,T}[G^{g}](z)+\int_{t}^{T}P_{t,\tau}[\mathcal{H}(\tau,\cdot,\nabla
w(\tau,\cdot)\nu\sigma^{g}(t,\cdot))](z)d\tau,\quad t\in[0,T],z\in Y^{*}.$

In order to prove that there exists a unique solution of (3.16) we need once
again the forward-backward system (3.2):

$\begin{cases}d\mathcal{Z}_{\tau}&=\mathcal{A}^{*}\mathcal{Z}_{\tau}d\tau+\nu\beta^{g}(\tau,\mathcal{Z}_{\tau})+\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau})dW_{\tau},\qquad\tau\in[t,T]\\\
\mathcal{Z}_{t}&=\zeta_{t}\\\
dp_{t}&=-\mathcal{H}(\tau,\mathcal{Z}_{\tau},q_{\tau})d\tau+q_{\tau}\nu
dW_{\tau},\qquad\tau\in[t,T]\\\ p_{T}&=G^{g}(\mathcal{Z}_{T})\end{cases}$

######  Theorem 3.13.

Assume that $G^{g}$ and $\mathcal{H}$ satisfy Hypothesis 3.2 and that
Hypothesis 3.4 hold true. Then there exists a unique mild solution of the
Hamilton-Jacobi-Bellman equation (3.16) given by

$w(t,\zeta_{t})=v(t,\zeta_{t})$ (3.17)

where $(\mathcal{Z},p,q)$ is the solution of (3.2) and $v(t,z)=p(t,t,z)$, see
(3.4).

###### Proof.

The proof is based on arguments similar to [22, Theorem 6.2] and [14, Theorem
6.2], though adapted to the current framework. Note that the main difference
with [14] is the nature of the spaces considered. ∎

###  3.4. Solving the optimal control problem

As we have proven the identification (3.6) to be true (see Proposition 3.11),
we can finally move to the study of the optimal control problem (2.5) - (2.8).
As anticipated, we want to show that the optimal value

$\inf_{u\in\mathds{A}}J^{g}(t,\zeta_{t},u)=J^{g}(t,\zeta_{t},\hat{u})=v(t,\zeta_{t}),$

where we have defined $v(t,\zeta_{t})=p(t,t,\zeta_{t})$ in (3.4), $(p,q)$
solve the backward stochastic differential equation (3.2) and a solution of
$v(t,\zeta_{t})$ can be obtained through the HJB equation (3.16) (see Theorem
3.13). We define the, possibly empty, set

$\Gamma(\tau,z,\xi)=\left\\{u\in\mathcal{U}:F^{g}(\tau,z,u)+\xi\nu
R^{g}(\tau,z,u)=\mathcal{H}(\tau,z,\xi)\right\\},$ (3.18)

where $\tau\in[t,T]$, $z\in Y^{*}$, $\xi\in Y^{**}$.

######  Hypothesis 3.14.

We notice that, intuitively, $\Gamma(t,z,\xi)$ represents the set of cotrols
that allow us to obtain the minimum in the Hamiltonian (3.3). We will thus
assume that for all $\tau\in[t,T]$, $z\in Y^{*}$, $\xi\in Y^{**}$,
$\Gamma(\tau,z,\xi)\neq\emptyset$

######  Remark 3.15.

Thanks to the Filippov theorem (see [4]), being $\Gamma(\tau,z,\xi)$ non empty
for all $\tau\in[t,T]$, $z\in Y$, $\xi\in Y^{**}$, there exists a Borel
measurable map $\Gamma_{0}:[0,T]\times Y^{*}\times
Y^{**}\longrightarrow\mathcal{U}$ such that, for $t\in[0,T]$, $z\in Y^{*}$ and
$\xi\in Y^{**}$, $\Gamma_{0}(t,z,\xi)\in\Gamma(t,z,\xi)$.

######  Proposition 3.16.

Assume that Hypothesis 3.1 \- 3.2 and 3.4 and hold true, and let $v$ be
defined in (3.4) and $u$ in $\mathds{A}$. Then for all $t\in[0,T]$ and $z\in
Y^{*}$, we have that

$J^{g}(t,z,u)\geq v(t,z).$

###### Proof.

Let $u$ in $\mathds{A}$ and take $\mathcal{Z}_{\tau}^{u}$ to be a solution of
(3.1) corresponding to the control $u$ (for the existence of such a solution
see Corollary 3.7). Define

$W_{\tau}^{u}=W_{\tau}+\int_{t\vee\tau}^{\tau}R^{g}(s,\mathcal{Z}_{s}^{u},u_{s})ds,\quad\tau\in[0,T].$

We notice that $\mathcal{Z}_{\tau}^{u}$ solves the equation

$\begin{cases}d\mathcal{Z}^{u}_{\tau}=&\mathcal{A}^{*}\mathcal{Z}_{\tau}^{u}d\tau+\nu\beta^{g}(\tau,\mathcal{Z}^{u}_{\tau})d\tau+\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau})dW^{u}_{\tau},\\\
\mathcal{Z}_{t}=&\zeta_{t},\end{cases}$

and, being $R$ and thus $R^{g}$ bounded, we can find a probability
$\mathbb{P}^{u}$ equivalent to $\mathbb{P}$ such that $W^{u}$ is a Wiener
process on $\mathbb{R}$ and thus $\nu W^{u}$ is a cylindrical Wiener process
with values in $Y^{*}$ (see [10, Theorem 7.2 (iii)]). We consider the backward
equation w.r.t. $\mathbb{P}^{u}$ for the unknowns
$(p_{\tau}^{u},q_{\tau}^{u})$, $\tau\in[t,T]$ given by

$p_{\tau}^{u}+\int_{\tau}^{T}q_{s}^{u}\nu
dW_{s}^{u}=G^{g}(\mathcal{Z}_{T}^{u})+\int_{\tau}^{T}\mathcal{H}(s,\mathcal{Z}_{s}^{u},q_{s}^{u})ds.$
(3.19)

By taking $(p^{u},q^{u})$ at $\tau=t$ in (3.19), we get that
$p(t,t,\zeta_{t})$ depends only on $t,\zeta_{t},\beta^{g},\sigma^{g},G^{g}$
and $\mathcal{H}$. With the same approach as in [22, Proposition 5.5], one
obtains immediately that $\int_{\tau}^{T}q_{s}^{u}\nu dW^{u}_{s}$ is actually
a $\mathbb{P}^{u}$-martingale.

By recalling that $v(t,z):=p(t,t,z)$ (see (3.4)) and taking expectation with
respect to the original probability $\mathbb{P}$, we obtain that

$v(t,\zeta_{t})=\mathbb{E}[G^{g}(\mathcal{Z}_{T}^{u})]+\mathbb{E}\left[\int_{t}^{T}\mathcal{H}(s,\mathcal{Z}_{s}^{u},q_{s}^{u})-q_{s}^{u}\nu
R(s,\mathcal{Z}_{s}^{u},u_{s})ds\right].$

Adding and subtracting
$\mathbb{E}\left[\int_{t}^{T}F^{g}(s,\mathcal{Z}_{s}^{u},u_{s})ds\right]$, we
arrive at

$v(t,\zeta_{t})=J^{g}(t,\zeta_{t},u)+\mathbb{E}\left[\int_{t}^{T}\mathcal{H}(s,\mathcal{Z}_{s}^{u},q_{s}^{u})-q_{s}^{u}\nu
R^{g}(s,\mathcal{Z}_{s}^{u},u_{s})-F^{g}(s,\mathcal{Z}_{s}^{u},u_{s})ds\right].$
(3.20)

Noticing that

$\mathcal{H}(s,\mathcal{Z}_{s}^{u},q_{s}^{u})-q_{s}^{u}\nu
R^{g}(s,\mathcal{Z}_{s}^{u},u_{s})-F^{g}(s,\mathcal{Z}_{s}^{u},u_{s})\leq 0,$

by definition of $\mathcal{H}$ in (3.3), we conclude. ∎

######  Corollary 3.17.

Let $t\in[0,T]$ and $\zeta_{t}\in Y^{*}$. If
$J^{g}(t,\zeta_{t},u^{*})=v(t,\zeta_{t})$ then $u^{*}$ is optimal for the
control problem starting from $\zeta_{t}$ at time $t$. Assume Hypothesis 3.14
holds and take $\Gamma_{0}(\tau,z,\xi)$ be the Borel measurable map defined in
Remark 3.15. Then, an admissible control satisfying

$\hat{u}_{\tau}=\Gamma_{0}(\tau,\mathcal{Z}_{\tau}^{\hat{u}},q_{\tau}^{\hat{u}}),\quad\mathbb{P}-a.s.\text{
for a.e. }\tau\in[t,T]$

is optimal and $J^{g}(t,z,\hat{u})=v(t,z)$.

###### Proof.

The proof is in line with the one in [22, Corollary 5.6]. ∎

######  Theorem 3.18.

Assume that Hypothesis 3.1, 3.2, 3.4 and 3.14 hold true. For all admissible
controls $u$ in $\mathds{A}$, we have that

${J^{g}}(t,\zeta_{t},u)\geq v(t,\zeta_{t}),$

and the equality holds true if and only if

$\hat{u}_{\tau}\in\Gamma(\tau,\mathcal{Z}^{u}_{\tau},\nabla
v(\tau,\mathcal{Z}_{\tau}^{u})\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau}))\quad\mathbb{P}-a.s.\text{
for a.a. }\tau\in[t,T].$ (3.21)

Moreover, let us denote by $\Gamma_{0}(\tau,z,\xi)$ be the measurable
selection of $\Gamma(t,z,\xi)$ defined in Remark 3.15. A control satisfying
the _feedback law_ , defined as:

$u_{\tau}=\Gamma_{0}(\tau,\mathcal{Z}^{u}_{\tau},\nabla
v(\tau,\mathcal{Z}^{u}_{\tau})\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau}^{u}))\quad\mathbb{P}-a.s.\text{
for a.a. }\ \tau\in[t,T]$ (3.22)

is optimal. Define the _closed loop equation_ :

$\begin{cases}\widetilde{\mathcal{Z}}_{\tau}=&\left[\mathcal{A}^{*}\widetilde{\mathcal{Z}}_{\tau}+\nu\beta^{g}(\tau,\widetilde{\mathcal{Z}}_{\tau})+\nu\sigma^{g}(\tau,\widetilde{\mathcal{Z}}_{\tau})R^{g}\Bigg{(}\tau,\widetilde{\mathcal{Z}}_{\tau},\Gamma_{0}\Big{(}\nabla
v(\tau,\widetilde{\mathcal{Z}}_{\tau})\nu\sigma^{g}(\tau,\widetilde{\mathcal{Z}}_{\tau})\Big{)}\Bigg{)}\right]d\tau\\\
&+\nu\sigma^{g}(\tau,\widetilde{\mathcal{Z}}_{\tau})dW_{\tau},\ \
\tau\in[t,T]\\\ \widetilde{\mathcal{Z}}_{t}=&\zeta_{t}.\end{cases}$ (3.23)

Then (3.23) admits a weak solution which is unique in law, and the
corresponding pair $(u,\widetilde{\mathcal{Z}}^{u})$ is optimal.

For more details about the definition of _feedback law_ and _closed loop
equation_ in the case of optimal control for Hilbert spaces we refer to [13,
Section 2.5].

###### Proof.

Using (3.20) and Proposition 3.10 we can rewrite $v(t,\zeta_{t})$ as

$\displaystyle v(t,\zeta_{t})=$ $\displaystyle
J^{g}(t,\zeta_{t},u)+\int_{t}^{T}\left[\mathcal{H}(\tau,\mathcal{Z}^{u}_{\tau},\nabla
v(\tau,\mathcal{Z}^{u}_{\tau})\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau}))\right.$
$\displaystyle\left.-\nabla
v(\tau,\mathcal{Z}^{u}_{\tau})\nu\sigma^{g}(\tau,\mathcal{Z}^{u}_{\tau}))R^{g}(\tau,\mathcal{Z}_{t}^{u},u_{\tau})-F^{g}(\tau,\mathcal{Z}^{u}_{\tau},u_{\tau})\right]d\tau.$

The proof of the first statement now follows from Corollary 3.17. The closed
loop equation can be solved in the weak sense via a Girsanov change of
measure. Recall that $(\Omega,\mathcal{F},\mathbb{P})$ is the probability
space on which the Wiener process $(W_{\tau})_{\tau\geq 0}$ in (3.2) is
defined. Define $(\hat{W}_{\tau})_{\tau\geq 0}$ as

$\hat{W}_{t}:=W_{t}-\int_{0}^{t}R^{g}\Bigg{(}\tau,\tilde{\mathcal{Z}}_{\tau}^{u},\Gamma_{0}\Big{(}\tau,\tilde{\mathcal{Z}}_{\tau},\nabla
v(s,\tilde{\mathcal{Z}}_{\tau})\nu\sigma^{g}(\tau,\tilde{\mathcal{Z}}_{\tau})\Big{)}\Bigg{)}d\tau.$

Due to the Girsanov theorem there exists a probability $\hat{\mathbb{P}}$ on
$\Omega$ such taht $\hat{W}_{\tau}$ is a Wiener process. We then notice that
$\nu W$ and $\nu\hat{W}$ are cylindrical Wiener processes with values in
$Y^{*}$, and that the closed loop equation (3.23) can be rewritten under
$\hat{\mathbb{P}}$ as

$\begin{cases}d\tilde{\mathcal{Z}}^{u}_{\tau}=&\mathcal{A}^{*}\tilde{\mathcal{Z}}_{\tau}^{u}d\tau+\nu\beta^{g}(\tau,\tilde{\mathcal{Z}}^{u}_{\tau})d\tau+\nu\sigma^{g}(\tau,\tilde{\mathcal{Z}}^{u}_{\tau})d\hat{W}_{\tau},\\\
\mathcal{Z}_{t}=&\zeta_{t}.\end{cases}$

Then, thanks to Proposition 3.5, we have a unique solution to this new process
related to the probability $\hat{\mathbb{P}}$ and the Wiener process
$\hat{W}_{\tau}$, which implies that also the closed loop equation (3.23)
always admits a solution in the weak sense. Thanks to Hypothesis 3.14, we know
that $\Gamma(\tau,\mathcal{Z}_{\tau}^{u},\nabla
v(\tau,\mathcal{Z}_{\tau}^{u})\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau}^{u}))$ is
non empty and thus, by the Filippov theorem, a measurable selection
$\Gamma_{0}(\tau,\mathcal{Z}_{\tau}^{u},\nabla
v(\tau,\mathcal{Z}_{\tau}^{u})\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau}^{u}))$ of
$\Gamma$ exists and the optimal control can be obtained. This proof is in line
with [22, Theorem 5.7] and [14, Theorem 7.2]. ∎

######  Remark 3.19.

Notice that, having solved the lifted optimization problem, thanks to
(2.2)-(2.7), we have also solved the original problem (1.1)-(1.2). Indeed, we
have that a control $\hat{u}$ which is optimal for (2.8) where the dynamics
for the forward process are given by $\mathcal{Z}^{u}_{\tau}$ in (3.1) is also
optimal for the original problem (1.1), as $J^{g}(t,\zeta_{t},u)=J(t,x,u)$ by
definition and $X^{u}(t)=\langle g,\mathcal{Z}_{t}^{u}\rangle$.

##  4\. A problem of optimal consumption

A cash flow admits consumption with rate $c$ according to the forward dynamics

$X^{c}_{t}=x(t)+\int_{0}^{t}K(t-\tau)\
\mu(\tau,X^{c}_{\tau})d\tau+\int_{0}^{t}K(t-\tau)\
\sigma(\tau,X_{\tau}^{c})(-R(c_{\tau})d\tau+dW_{\tau}),$ (4.1)

where $x:[0,T]\longrightarrow\mathbb{R}$, $K(t-\tau)=\sqrt{t-\tau}$, $\mu$ and
$\sigma$ satisfy Hypothesis 3.4, and $R$ satisfy Hypothesis 3.1. In this case
we lift $K(t):=\sqrt{t}$ on $L^{q}(\mathbb{R})\times L^{p}(\mathbb{R})$ for
$q\in(1,2)$ and $p$ such that $\frac{1}{p}+\frac{1}{q}=1$ by considering

$e^{\mathcal{A}^{*}t}:L^{p}(\mathbb{R})\longrightarrow L^{p}(\mathbb{R})$

the left shift semigroup defined as $(e^{\mathcal{A}^{*}t}f)(s)=f(s-t)$, for
all $f\in L^{p}(\mathbb{R})$, $g(x)=\frac{1}{2\sqrt{x}}\mathds{1}_{[0,1]}(x)$,
$\nu(x)=\mathds{1}_{[-1,0]}(x)$. Then we have that

$K(t)=\langle g,e^{\mathcal{A}^{*}t}\nu\rangle_{L^{p}(\mathbb{R})\times
L^{q}(\mathbb{R})}=\int_{\mathbb{R}}\frac{1}{2\sqrt{x}}\mathds{1}_{[0,1]}(x)\mathds{1}_{[-1,0]}(x-t)dx=\int_{0}^{t}\frac{dx}{2\sqrt{x}}=\sqrt{t}.$

We consider a classical optimal control problem given by the maximization of
the performance functional

$J(t,x,c)=\mathbb{E}\left[\int_{t}^{T}F(\tau,X_{\tau}^{c},c_{\tau})d\tau+G(X_{T}^{c})\right],$
(4.2)

for some functions
$F(\tau,X_{\tau}^{c},c_{\tau})=F(c_{\tau}):=-a_{1}c_{\tau}^{2}$ and
$G(X_{t}^{c}):=a_{2}X_{T}^{c}$ ($a_{1},a_{2}\in\mathbb{R}_{>0}$) satisfying
Hypothesis 3.2. Linear-quadratic performance functionals such as (4.2) appear,
for example, when considering optimal advertising problems (see e.g. [16, 17]
and [15]). In this case we have that the stochastic control problem can be
reformulated in $Y^{*}$ with forward dynamics given by

$\mathcal{Z}_{\tau}^{c}=\zeta_{0}+\int_{0}^{\tau}\mathcal{A}^{*}\mathcal{Z}_{s}^{c}+\nu\mu^{g}(s,\mathcal{Z}_{s}^{c})ds+\nu\sigma^{g}(s,\mathcal{Z}_{s}^{c})(-R(c_{s})ds+dW_{s}),$
(4.3)

with $\zeta_{t}=e^{\mathcal{A}^{*}t}\zeta$ such that $x(t)=\langle
g,\zeta_{t}\rangle$ for all $t\in[0,T]$. The goal is to minimize

$J^{g}(0,\zeta_{0},c)=\mathbb{E}\left[\int_{0}^{T}-a_{1}c_{\tau}^{2}d\tau+a_{2}\langle
g,\mathcal{Z}^{c}_{T}\rangle\right].$ (4.4)

In this case the Hamiltonian functional (3.3) is given by

$\mathcal{H}(t,\mathcal{Z},\xi)=\inf_{c\in\mathcal{U}}[-a_{1}c^{2}-\xi\nu
R(c)]$ (4.5)

and the forward-backward system is

$\begin{cases}d\mathcal{Z}_{\tau}&=\mathcal{A}^{*}\mathcal{Z}_{\tau}d\tau+\nu\mu^{g}(s,\mathcal{Z}_{\tau})d\tau+\nu\sigma^{g}(s,\mathcal{Z}_{s})dW_{\tau},\quad\tau\in[0,T],\\\
\mathcal{Z}_{0}&=\zeta_{0},\\\
dp_{\tau}&=-\mathcal{H}(\tau,\mathcal{Z}_{\tau}^{c},q_{\tau})d\tau+q_{\tau}\nu
dW_{\tau},\quad\tau\in[t,T],\\\ p_{T}&=a_{2}\langle
g,\mathcal{Z}_{T}\rangle.\end{cases}$ (4.6)

In particular, using (4.5), we have that

$p_{t}=a_{2}\langle
g,\mathcal{Z}_{T}\rangle-\int_{t}^{T}\inf_{c\in\mathcal{U}}\left(-a_{1}c^{2}-q_{s}\nu
R(c)\right)ds+q_{s}\nu dW_{s}\ s\in[t,T].$

We thus get that the set $\Gamma$ defined in (3.18) is

$\Gamma(t,\mathcal{Z},\xi)=\Big{\\{}c\in\mathcal{U}:-a_{1}c^{2}-\xi\nu
R(c)=\mathcal{H}(t,\mathcal{Z},c)\Big{\\}},$ (4.7)

and thus the optimal $u_{\tau}$ can be characterized by Theorem 3.18 as

$c_{\tau}=\Gamma_{0}(\tau,\mathcal{Z}^{c}_{\tau},\nabla
v(\tau,\mathcal{Z}_{\tau}^{c})\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau}^{c}))\quad\
\mathbb{P}-a.s.\text{ for a.a. }\tau\in[0,T],$ (4.8)

for a certain function $\Gamma_{0}$ such that
$\Gamma_{0}(t,\mathcal{Z},\xi)\in\Gamma(t,\mathcal{Z},\xi)$. In this case the
HJB equations (3.16) become

$\begin{cases}\frac{\partial v}{\partial
t}(t,\zeta_{0})&=-\mathcal{L}_{t}v(t,\mathcal{Z}(t,0,\zeta_{0}))\\\
&\quad-\inf_{c\in\mathcal{U}}[F^{g}(t,\mathcal{Z}(t,0,\zeta_{0}),c_{t})-\nabla
v(t,\mathcal{Z}(t,0,\zeta_{0}))\nu\sigma(t,\mathcal{Z}(t,0,\zeta_{0}))c_{t}],\\\
v(T,\zeta_{0})&=a_{2}X(0).\end{cases}$ (4.9)

Where, we remind, $\mathcal{Z}(\tau,0,\zeta_{0})=\mathcal{Z}_{\tau}$,
$\tau\in[t,T]$, $\mathcal{Z}(0)=\zeta_{0}$ and

$\displaystyle\mathcal{L}_{t}[v](t,t,z)$
$\displaystyle:=\frac{1}{2}\text{Trace}(G^{g}(v(t,z)G^{g}(v(t,z))^{*}\
\nabla^{2}v(t,t,z))$
$\displaystyle\quad+\langle\mathcal{A}^{*}v(t,t,z)+\nu\beta^{g}(t,v(t,t,z)),\nabla
v(t,t,z)\rangle_{L^{q}([0,\infty))\times L^{p}([0,\infty))}.$

For more details about solving the HJB equation (4.9) we refer to [15]. Now,
thanks to Theorem 3.13 we have that

######  Theorem 4.1.

Equation (4.9) has a unique mild solution $v$. If the cost is given by (4.4),
then for all admissible couples $(c,z)$ we have that $J(t,z,c)\geq v(t,z)$,
and the equality holds if and only if $\hat{c}\in\Gamma(t,\mathcal{Z},\xi)$ in
(4.7), characterized as (4.8). Vice versa, if (4.8) holds, then

$\displaystyle\tilde{\mathcal{Z}}_{\tau}$
$\displaystyle=\mathcal{A}^{*}\tilde{\mathcal{Z}}_{\tau}+\nu\mu^{g}(\tau,\mathcal{Z}_{\tau}^{c})d\tau$
$\displaystyle\quad+\nu\sigma^{g}(\tau,\mathcal{Z}_{\tau}^{c})\Big{(}-R(\Gamma_{0}(\tau,\tilde{\mathcal{Z}}^{c}_{\tau},\nabla
v(\tau,\tilde{\mathcal{Z}}_{\tau}^{c})\nu\sigma^{g}(\tau,\tilde{\mathcal{Z}}_{\tau}^{c})))d\tau+dW_{\tau}\Big{)},\quad\tau\in[0,T],$

with initial condition $\tilde{\mathcal{Z}}_{0}=\zeta_{0}$, admits a weak
solution, which is unique in law, and the corresponding pair
$(c,\tilde{\mathcal{Z}}^{c})$ is optimal.

######  Remark 4.2.

This gives us a characterization of the optimal control for the lifted problem
(4.4) and thus also for (4.2): in fact in our case we have that $X_{t}=\langle
1,\mathcal{Z}_{t}\rangle$. Thanks to this we are able to find the optimal
process $X^{\hat{u}}$. Moreover, we note that, the optimal control for the
lifted problem and the optimal control for the original problem coincide.,
This allows us to retrieve the optimal pair $(\hat{u},X^{\hat{u}})$ for the
optimal control problem (4.2). Lastly we notice that the HJB equations (4.9),
gives us the optimal value $J^{g}(0,\zeta_{0},\hat{u})$, which in turn gives
us the optimal value of $J(t,x,\hat{u})$, thanks to (2.7).

######  Remark 4.3.

Inspired by [11], we could also have considered the kernel
$K(t)=\frac{1}{t+\varepsilon}$, $\varepsilon>0$ in (4.1). In this case we can
take the space of $L^{p}([0,\infty))$ of $L^{p}$ functions on $[0,\infty)$ and
its dual $L^{q}([0,\infty))$ of measures with density in $L^{q}$, where
$\frac{1}{p}+\frac{1}{q}=1$ and $p>1$. We have that

$K(t)=\frac{1}{t+\varepsilon}=\langle
g,\mathcal{S}_{t}^{*}\nu\rangle_{L^{p}([0,\infty))\times L^{q}([0,\infty))},$

where $g=\nu=e^{-x\varepsilon/2}$, and $\mathcal{S}_{t}^{*}=e^{-tx}$ i.e.
$K(t)$ is the Laplace trasnform of $e^{-x\varepsilon}$. We notice that kernel
is liftable (see Definition 2.6) and that we are in a UMD Banach space. It is
clear that $e^{-\varepsilon x/2}$ is actually in $L^{p}([0,\infty))$ for all
$p\geq 1$. In particular we can also take $p=q=2$ and work on the Hilbert
space $L^{2}([0,\infty))$.

### Acknowledgment

We would like to thank Anton Yurchenko-Tytarenko and Dennis Schroers for the
nice input on kernel decompositions. The research leading to these results is
within the project STORM: Stochastics for Time-Space Risk Models, receiving
founding from the Research Council of Norway (RCN). Project number: 274410.

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