# Speaker dependent acoustic-to-articulatory inversion
using real-time MRI of the vocal tract

###### Abstract

Acoustic-to-articulatory inversion (AAI) methods estimate articulatory
movements from the acoustic speech signal, which can be useful in several
tasks such as speech recognition, synthesis, talking heads and language
tutoring. Most earlier inversion studies are based on point-tracking
articulatory techniques (e.g. EMA or XRMB). The advantage of rtMRI is that it
provides dynamic information about the full midsagittal plane of the upper
airway, with a high ’relative’ spatial resolution. In this work, we estimated
midsagittal rtMRI images of the vocal tract for speaker dependent AAI, using
MGC-LSP spectral features as input. We applied FC-DNNs, CNNs and recurrent
neural networks, and have shown that LSTMs are the most suitable for this
task. As objective evaluation we measured normalized MSE, Structural
Similarity Index (SSIM) and its complex wavelet version (CW-SSIM). The results
indicate that the combination of FC-DNNs and LSTMs can achieve smooth
generated MR images of the vocal tract, which are similar to the original MRI
recordings (average CW-SSIM: 0.94).

Index Terms: magnetic resonance imaging, acoustic-to-articulatory inversion,
deep learning

##  1 Introduction

Articulation is directly linked with the acoustic speech signal in the speech
production process. The acoustic-to-articulatory inversion (AAI) methods
estimate articulatory movements from the acoustic speech signal [1]. Recently,
there has been a significant interest in AAI, because learning the correlation
between articulation and acoustics could improve the performance of several
tasks such as speech recognition [2], synthesis [3] and talking heads [4]. It
can help the visualization of speech production as 3D articulatory animations
for pronunciation training and language tutoring [5].

Statistical mapping techniques are suitable for the conversion of articulatory
movements into speech and vice versa [6, 7, 8, 9, 10]. Both speaker dependent
(SD-AAI) and independent (SI-AAI) approaches are available [11]. Several
methods have been proposed to tackle the SD-AAI problem including codebooks
[12], Gaussian Mixture Models (GMM) [13, 14], Hidden Markov Models (HMM) [15],
and Mixture Density Networks [16]. Furthermore, during the past few years,
researchers started to use Deep Neural Networks (DNN) [17, 18, 19],
convolutional [20] and recurrent networks [21]. The lowest error in predicting
articulatory position was achieved with the combination of acoustic and
textual input, using a bottleneck long-term recurrent convolutional neural
network (BLTRCNN) [22].

Definitely, all these approaches need parallel acoustic-articulatory data for
training the AAI model. Hence, most above inversion studies are based on
Electromagnetic Articulography (EMA) or X-ray Microbeam (XRMB) data, which can
track only several points of the articulatory organs, and therefore provide
limited input information. Compared to EMA and XRMB, imaging methods (e.g.
UTI: Ultrasound Tongue Imaging and MRI: Magnetic Resonance Imaging) have the
advantage that the tongue surface is fully visible [9, 10, 23, 24]. The
typical result of 2D ultrasound and 2D MRI recordings is a series of gray-
scale mid-sagittal images in which the tongue surface contour has a greater
brightness than the surrounding tissue and air. We started to use raw scanline
tongue ultrasound data for SD-AAI using feedforward DNNs, with 25-dimensional
MFCC as input [25]. The results indicated that already with a simple DNN
consisting of two hidden layers (1000 neurons each) relatively good accuracy
could be achieved. These initial results showed that SD-AAI is feasible using
ultrasound tongue imaging – but until now, only simple DNNs were used for this
task.

Recently, significant advances in MR research (software, hardware, and
reconstruction strategies) have allowed real-time MRI (rtMRI) to be a powerful
modality for speech production research and for investigating the movement of
the articulators [23, 24, 26]. The advantage of rtMRI is that it provides
dynamic information about the full midsagittal plane of the upper airway, even
during continuous spoken utterances. It can capture not only lingual, labial,
and jaw motion but also the articulatory motion of the velum and the
pharyngeal region, which is typically not possible with other articulatory
acquisition techniques. Besides, such imaging data helps to comprehend the
generation of coronal, pharyngeal, and nasal segments. The sampling rates of
rtMRI are relatively low (around 20–25 fps), but are acceptable for running
speech. A disadvantage is the large background noise in speech recordings, but
noise cancellation can yield an acceptable speech signal, which is
synchronized to the articulatory signal. Also, the presence of a substantial
number of artifacts and noise make automatic extraction and interpretation of
features a difficult problem. Overall, rtMRI provides high relative spatial
information in the midsagittal view with relatively low temporal resolution
[24]; therefore it is a potentially suitable technique for the target of
acoustic-to-articulatory inversion.

Several studies have applied MRI for articulatory-related speech technologies:
e.g. phoneme classification from articulation [27, 28] and acoustic-to-
articulatory inversion [29, 30]. Saha and his colleagues experimented with
identifying different vowel-consonant-vowel (VCV) sequences from vocal tract
rtMRI [27]. Long-term Recurrent Convolutional Networks (including a pretrained
ResNet50) models were used, which make the network spatiotemporally deep
enough to capture the sequential nature of the articulatory data, for the
classification task. Van Leeuwen et al. trained a convolutional neural network
(CNN) for the classification of 27 different sustained phonemes from MRI, and
reveal that the network has learned to focus on those parts of the images that
represent the crucial articulatory positions needed to distinguish the
different phonemes [28].

Kaburagi presented a method to estimate the cross-sectional area and length of
the vocal tract from a speech spectrum [29]. For data, a set of vocal-tract
area functions were used for 10 English vowels obtained from static MRI
measurements (the subject was a male native speaker of English). However, the
estimation result is the parameterized vocal tract, not the MR images itself.
As the first and single inversion study having MR images as the target, Li et
al. present a system for AAI using midsagittal rtMRI, where restricted
Boltzmann machine, GMM and linear regression are applied for the mapping [30].
In this task, the inputs of the machine learning models are acoustic feature
vectors (24-order line spectral pairs, with a context window of 10 acoustic
frames), whereas the targets are the gray value vectors of the 68$\times$68
pixel MR images. According to the results, deep architectures are able to
obtain better inversion accuracy than the GMM-based method, in terms of RMSE
[30]. However, only a single speaker (f1) was used from the USC-TIMIT database
[26].

Based on this overview, rtMRI data has sporadically been used previously for
direct acoustic-to-articulatory inversion. In the current paper, we train
various DNNs (fully connected, convolutional, and recurrent neural networks)
for acoustic-to-articulatory inverse mapping, using real-time magnetic
resonance images of the vocal tract, applying the data in a speaker-specific
way.

##  2 Methods

###  2.1 Data

We used two male (m1 and m2) and two female (f1 and f2) speakers from the
freely available USC-TIMIT MRI database [26]. This contains large-scale data
of synchronized audio and rtMRI for speech research, from American English
subjects. The vocal tracts were imaged in the mid-sagittal plane while lying
supine and reading 460 MOCHA-TIMIT sentences. The MRI data were acquired using
a Signa Excite HD 1.5T scanner with an image resolution in the mid-sagittal
plane of 68$\times$68 pixels (2.9$\times$2.9mm). The image data were
reconstructed as 23.18 frames/second. The audio was simultaneously recorded at
a sampling frequency of 20 kHz inside the MRI scanner while subjects were
imaged. Noise cancellation was also performed on the acoustic data.

###  2.2 Vocoder

To create the acoustic inputs, we encoded the audio recordings using an MGLSA
vocoder [31] at a frame shift of 1 / (23.18 fps) = 863 samples, which resulted
in F0 and 24-order spectral (Mel-Generalized Cepstrum, Line Spectral Pair,
MGC-LSP) features. The spectral features served as the training inputs of the
DNN.

###  2.3 Deep neural network architectures

In our earlier studies on ultrasound-based inversion [25], we were using
fully-connected feed-forward neural networks (FC-DNN). For articulatory
mapping, we showed that CNN and CNN-LSTM were also beneficial [10]. Here, we
test similar DNN types. In all cases and for all speakers, we trained speaker-
specific models, and we split the data into 430 sentences for training, 20
sentences for validation, and 10 sentences for testing. We used Adam
optimizer, trained the networks for 100 epochs, and applied early stopping
with a patience of 5 on the validation loss. The target MRI pixels were scaled
to [0-1], while the input spectral features were normalized to zero mean, unit
variance. The data is passed to the networks in batches of 128 frames. The
cost function applied for the MRI pixel regression task was the mean-squared
error (MSE). In all hidden layers, ReLU activation was used.

####  2.3.1 FC-DNN (baseline)

In the simplest case, we trained fully connected (FC) DNNs with 5 hidden
layers, each hidden layer consisting of 1000 neurons. The input layer
consisted of 25 neurons (taking the MGC-LSP dimensions). The output layer was
a linear one (4 624 neurons), with one neuron for each MRI pixel.

####  2.3.2 CNN

Next, we tested convolutional neural networks, as typically, they are more
suitable for the processing of images than simple FC-DNNs. The CNN input was
25-dimensional MGC-LSP. First, two dense layers were used with 500 and
17$\times$17$\times$8 = 2312 neurons, respectively. There was one
convolutional layer (kernel size: 3$\times$3, number of filters: 8), preceded
and followed by max-pooling. The last layer was a convolutional one with
linear activation and 68$\times$68 pixel MR image as the target.

####  2.3.3 LSTM

Figure 1: Block diagram of the LSTM network.

Also, we hypothesized that using multiple consecutive spectral features as
input can increase the accuracy of the regression. The most ambitious network
in this work is a recurrent one consisting of a combination of FC layers and
Long Short-Term Memory units (LSTMs). The motivation for designing this
network comes from the fact that in [10], we achieved better results when
using consecutive ultrasound frames. Figure 1 shows the architecture of the
LSTM network. It consists of two distinct parts: a fully connected beginning
(three fully connected layers with 575 neurons each), and a recurrent end (two
LSTM layers with 575 neurons each). We use a sequence size of 10 (accounting
for roughly 430 ms of the input spectral data) in order to incorporate time
information. To keep the FC-DNN (baseline) and the LSTM comparable with
respect to parameter count, both models have approximately 8.6 million tunable
parameters – this was the reason for having 575 neurons in the layers of the
recurrent neural network.

Figure 2: Original and predicted MRI image sequence by the three DNNs, from
speaker f1. The sentence starts with ‘Her…’ (f1_146)

##  3 Results

After training the neural networks, the prediction accuracy was evaluated on
the test set (10 sentences for each speaker). We generated MR image sequences
using the trained DNN models by having the MGC-LSP features of the test
sentences as input.

###  3.1 Demonstration sample

A sample test sentence from speaker ’f1’ (not appearing in the training data)
was chosen for demonstrating how the three systems deal with the prediction of
the MRI pixels. Fig. 2 shows the same image sequence from the original
recording and the predictions of the DNNs. In general, the DNN-predicted MR
images are close to the original, but the FC-DNN and CNN cannot follow the
fast changes of the reference, while the LSTM seems to be smoother. More video
samples can be found at
http://smartlab.tmit.bme.hu/interspeech2020_speech2mri.

###  3.2 Objective evaluation

On the validation set and on the synthesized sentences (being the test set),
we first measured the Mean Square Error (MSE) between the original and
predicted MRI pixels. The calculations were done on the [0-1] normalized
features. The normalized MSE values calculated on the validation and test are
shown in Table 1, separately for each speaker. Overall, the tendencies are the
same for all speakers: the weakest network seems to be the basic fully-
connected one (test NMSE: 0.0043), followed by the CNN (test NMSE: 0.0049),
and finally, the LSTM having the smallest error (test NMSE: 0.0036). The
difference between the FC-DNN and CNN is small, whereas the LSTM is
significantly better than these two systems. Interestingly, the results show
some speaker dependency: the test NMSE is smallest for f1 (0.0023), and
highest for f2 (0.0046), thus being roughly twice as bad. Another observation
is that for speaker m1, the LSTM network is not better than the FC-DNN and
CNN.

Table 1: NMSE scores on the validation and test set. | Normalized MSE (validation)
---|---
speaker |  FC-DNN |  CNN |  LSTM
f1 | 0.0024 | 0.0026 | 0.0021
f2 | 0.0041 | 0.0042 | 0.0038
m1 | 0.0028 | 0.0030 | 0.0026
m2 | 0.0031 | 0.0034 | 0.0028
average | 0.0031 | 0.0033 | 0.0029
speaker | Normalized MSE (test)
f1 | 0.0028 | 0.0032 | 0.0023
f2 | 0.0060 | 0.0065 | 0.0046
m1 | 0.0035 | 0.0039 | 0.0035
m2 | 0.0050 | 0.0058 | 0.0038
average | 0.0043 | 0.0049 | 0.0036

To measure the naturalness of the reconstructed MR images, two other metrics
were chosen, which have a higher correlation with subjective quality. Both of
these are calculated over each frame, where $y$ is the original image and
$\hat{y}$ is the estimated image from the DNN architecture. Structural
Similarity Index (SSIM) [32] (which we used earlier to compare ultrasound
tongue images [33]) measures three kinds of visual impact of changes in
luminance $l$, contrast $c$ and structure $s$ between two images:

$SSIM(y,\hat{y})=[l(y,\hat{y})]^{\alpha}[c(y,\hat{y})]^{\beta}[s(y,\hat{y})]^{\gamma}$

In our experiment, the SSIM index is calculated by 11$\times$11 circular-
symmetric Gaussian weighting function, with a standard deviation of 1.5
pixels.

Complex Wavelet Structural Similarity (CW-SSIM) [34] is an extension of the
SSIM method to the complex wavelet domain, which is a novel image similarity
measurement robust to small distortions:

$CW-
SSIM(y,\hat{y})=\frac{2\arrowvert\sum_{l=1}^{L}w_{y,l}w_{\hat{y},l}^{*}\arrowvert+K}{\sum_{l=1}^{L}|w_{y,l}|^{2}+\sum_{l=1}^{L}|w_{\hat{y},l}|^{2}+K}$

where $w$ represents the complex wavelet coefficients of the two images. The ∗
indicates the complex conjugate of $w$, and $K$ is a small positive
stabilizing constant [33]. In case of both SSIM and CW-SSIM, the resulting
range is between [0-1], and the higher value means more similar images
(whereas zero is for the most diverse images).

Table 2 shows the SSIM and CW-SSIM scores measured on the synthesized
sentences, separately for each speaker and DNN type. The scores are not as
speaker dependent as it was the case with NMSE. The tendencies across the
neural network type show that the CNN achieved the lowest scores, followed by
the FC-DNN, and finally, the LSTM having the most similar predicted images to
the original MR frames; but all of the differences are small and not
significant. Fig. 3 presents SSIM and CW-SSIM over time, on the f1_146 sample
sentence. The utterance starts around frame 28 and ends around frame 92.
Frames 1–27 and 93–125 are silence. The figure shows that the FC-DNN and CNN
can create videos that have quick variations across consecutive frames (e.g.
between frames 28–40, where the sentence starts), whereas the result of LSTM
is smoother. In general, the MR frames for the silence period are most similar
to the original MR images, while the LSTM can achieve high scores even in case
of the parts where the articulators are moving, which is clearly beneficial.

According to these objective experiments, all measures have shown the
advantage of using recurrent networks (namely, LSTM), instead of the networks
which are taking single images as input (the FC-DNN and CNN types).

Table 2: SSIM and CW-SSIM scores on the test set. | SSIM / CW-SSIM
---|---
speaker |  FC-DNN |  CNN |  LSTM
f1 | 0.77 / 0.94 | 0.75 / 0.94 | 0.80 / 0.95
f2 | 0.70 / 0.92 | 0.68 / 0.92 | 0.73 / 0.94
m1 | 0.80 / 0.92 | 0.77 / 0.91 | 0.81 / 0.92
m2 | 0.75 / 0.92 | 0.73 / 0.92 | 0.77 / 0.94
average | 0.76 / 0.93 | 0.73 / 0.92 | 0.78 / 0.94

Figure 3: SSIM and CW-SSIM over time, from speaker f1. The utterance, between
frames 28–92 is ‘Her auburn hair reminded him of autumn leaves.’ (f1_146)

##  4 Discussion

In general, recurrent neural networks are more suitable to process sequential
data than convolutional or simple fully-connected networks, and they can
generate more smooth consecutive images. As Saha et al. compare for MRI-based
phone recognition [27], it is important to incorporate spatial and temporal
feature extraction steps to capture complementary information from the
individual consecutive still frames as well as between the frames. In the
current study, we showed that LSTMs are more suitable to process spectral
features and MR images than FC-DNNs and (2D) CNNs.

Li and his colleagues presented the first study in SD-AAI with rtMRI of the
vocal tract as the target [30]. With stacked RBMs and Gaussian-Bernoulli RBMs
they achieved an average RMSE of 17.74 (measured on the unnormalized pixels).
However, when checking the generated images, we can see that the shape of the
vocal tract is over-smoothed. They indicate that in the resulting MR images,
there is a relatively higher error in the outline of nose which is believed to
be caused by the instability of head position in the USC-TIMIT database.

We can compare the results of this study to acoustic-to-articulatory inversion
experiments that were using other imaging techniques. Ultrasound can only
capture the movement of the tongue, but with higher frame rates (around 100
fps). In our earlier experiment for speech-to-ultrasound conversion, the
typical values of SSIM were around 0.7 and CW-SSIM around 0.8, with a 2-layer
FC-DNN for a single female speaker [25], whereas for rtMRI with the LSTM here
we achieved SSIM between 0.73–0.81 and CW-SSIM between 0.92–0.95, depending on
the speaker. Wei et al. used GMMs and DNNs for ultrasound-based AAI and
achieved (unnormalized) MSE around 32–35, which is not directly comparable to
our results [35]. However, their study was quite limited as they only focused
on Chinese vowels, i.e. they did not test longer speech.

Although the resolution of the target MR images was only 68$\times$68 pixels,
this accounts for a larger ’relative’ spatial resolution compared to
ultrasound, as MRI can show the whole vocal tract. Our experiments have
indicated that this is an advantage of rtMRI, and the high relative spatial
resolution is more important than the relatively low (around 20–25 fps) time
resolution.

##  5 Conclusions

In this work, we used midsagittal rtMRI images of the vocal tract for speaker
dependent acoustic-to-articulatory inversion. We applied FC-DNNs, CNNs, and
recurrent neural networks and have shown that LSTMs are the most suitable for
this task.

The target realtime MR images have a relatively low spatial and temporal
resolution (but high ’relative’ spatial resolution) and are infested with
noises and reconstruction artifacts [27]. In our work, we were using raw MR
images and did not apply any preprocessing. However, noise and artifact
reduction on the target images, or other spectral feature extraction methods
[20] might enhance the accuracy of the mapping. Stabilizing the head position
on the MR images can also be beneficial [30].

As pointed out in Section 1, the results in AAI might be useful for speech
recognition [2], synthesis [3], talking heads [4], and for pronunciation
training and language tutoring [5].

The keras implementations are accessible at https://github.com/BME-
SmartLab/speech2mri/.

##  6 Acknowledgements

The author was funded by the National Research, Development and Innovation
Office of Hungary (FK 124584 and PD 127915 grants). The Titan X GPU for the
deep learning experiments was donated by the NVIDIA Corporation. We would like
to thank USC for providing the USC-TIMIT articulatory database.

## References

  * [1] K. Richmond, “Estimating articulatory parameters from the acoustic speech signal,” PhD thesis, University of Edinburgh, 2002.
  * [2] S. King, J. Frankel, K. Livescu, E. McDermott, K. Richmond, and M. Wester, “Speech production knowledge in automatic speech recognition,” _The Journal of the Acoustical Society of America_ , vol. 121, no. 2, pp. 723–742, feb 2007.
  * [3] Z.-H. Ling, K. Richmond, J. Yamagishi, and R.-H. Wang, “Integrating Articulatory Features Into HMM-Based Parametric Speech Synthesis,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 17, no. 6, pp. 1171–1185, aug 2009.
  * [4] L. Wang, X. Qian, W. Han, and F. K. Soong, “Synthesizing photo-real talking head via trajectory-guided sample selection,” in _Proc. Interspeech_ , Makuhari, Japan, 2010, pp. 446–449.
  * [5] W. Katz, T. Campbell, J. Wang, E. Farrar, J. Eubanks, A. Balasubramanian, B. Prabhakaran, and R. Rennaker, “Opti-speech: A real-time, 3D visual feedback system for speech training,” in _Proc. Interspeech_ , Singapore, Singapore, 2014, pp. 1174–1178.
  * [6] T. Hueber, G. Bailly, P. Badin, and F. Elisei, “Speaker adaptation of an acoustic-articulatory inversion model using cascaded Gaussian mixture regressions,” in _Proc. Interspeech_ , Lyon, France, 2013, pp. 2753–2757.
  * [7] T. Hueber, G. Bailly, and B. Denby, “Continuous Articulatory-to-Acoustic Mapping using Phone-based Trajectory HMM for a Silent Speech Interface,” in _Proc. Interspeech_ , Portland, OR, USA, 2012, pp. 723–726.
  * [8] T. Hueber, L. Girin, X. Alameda-Pineda, and G. Bailly, “Speaker-Adaptive Acoustic-Articulatory Inversion Using Cascaded Gaussian Mixture Regression,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 23, no. 12, pp. 2246–2259, dec 2015.
  * [9] T. G. Csapó, T. Grósz, G. Gosztolya, L. Tóth, and A. Markó, “DNN-Based Ultrasound-to-Speech Conversion for a Silent Speech Interface,” in _Proc. Interspeech_ , Stockholm, Sweden, 2017, pp. 3672–3676.
  * [10] E. Moliner and T. G. Csapó, “Ultrasound-based silent speech interface using convolutional and recurrent neural networks,” _Acta Acustica united with Acustica_ , vol. 105, no. 4, pp. 587–590, 2019.
  * [11] A. Illa and P. K. Ghosh, “Low Resource Acoustic-to-articulatory Inversion Using Bi-directional Long Short Term Memory,” in _Proc. Interspeech_ , Hyderabad, India, 2018, pp. 3122–3126.
  * [12] S. Ouni and Y. Laprie, “Modeling the articulatory space using a hypercube codebook for acoustic-to-articulatory inversion,” _The Journal of the Acoustical Society of America_ , vol. 118, no. 1, pp. 444–460, jul 2005.
  * [13] T. Toda, A. W. Black, and K. Tokuda, “Statistical mapping between articulatory movements and acoustic spectrum using a Gaussian mixture model,” _Speech Communication_ , vol. 50, no. 3, pp. 215–227, mar 2008\.
  * [14] A. Sepúlveda, R. C. Guido, and G. Castellanos-Dominguez, “Estimation of relevant time–frequency features using Kendall coefficient for articulator position inference,” _Speech Communication_ , vol. 55, no. 1, pp. 99–110, jan 2013.
  * [15] L. Zhang and S. Renals, “Acoustic-Articulatory Modeling With the Trajectory HMM,” _IEEE Signal Processing Letters_ , vol. 15, pp. 245–248, 2008.
  * [16] K. Richmond, “A trajectory mixture density network for the acoustic-articulatory inversion mapping,” in _Proc. Interspeech_ , Pittsburgh, PA, USA, 2006, pp. 577–580.
  * [17] B. Uria, I. Murray, S. Renals, and K. Richmond, “Deep architectures for articulatory inversion,” in _Proc. Interspeech_ , Portland, OR, USA, 2012, pp. 867–870.
  * [18] Z. Wu, K. Zhao, X. Wu, X. Lan, and H. Meng, “Acoustic to articulatory mapping with deep neural network,” _Multimedia Tools and Applications_ , vol. 74, no. 22, pp. 9889–9907, 2015.
  * [19] P. Liu, Q. Yu, Z. Wu, S. Kang, H. Meng, and L. Cai, “A deep recurrent approach for acoustic-to-articulatory inversion,” in _Proc. ICASSP_ , Brisbane, Australia, 2015, pp. 4450–4454.
  * [20] A. Illa and P. K. Ghosh, “Representation Learning Using Convolution Neural Network for Acoustic-to-articulatory Inversion,” in _Proc. ICASSP_ , Brighton, UK, 2019, pp. 5931–5935.
  * [21] ——, “An Investigation on Speaker Specific Articulatory Synthesis with Speaker Independent Articulatory Inversion,” in _Proc. Interspeech_ , Graz, Austria, 2019, pp. 121–125.
  * [22] L. Yu, J. Yu, and Q. Ling, “BLTRCNN Based 3D Articulatory Movement Prediction: Learning Articulatory Synchronicity From Both Text and Audio Inputs,” _IEEE Transactions on Multimedia_ , vol. PP, no. c, p. 1, 2018\.
  * [23] V. Ramanarayanan, S. Tilsen, M. Proctor, J. Töger, L. Goldstein, K. S. Nayak, and S. Narayanan, “Analysis of speech production real-time MRI,” _Computer Speech and Language_ , vol. 52, pp. 1–22, 2018.
  * [24] A. Toutios, D. Byrd, L. Goldstein, and S. Narayanan, “Advances in vocal tract imaging and analysis,” in _The Routledge Handbook of Phonetics_. Taylor and Francis, jan 2019, pp. 34–50.
  * [25] D. Porras, A. Sepúlveda-Sepúlveda, and T. G. Csapó, “DNN-based Acoustic-to-Articulatory Inversion using Ultrasound Tongue Imaging,” in _International Joint Conference on Neural Networks_ , Budapest, Hungary, 2019, pp. N–19 221.
  * [26] S. Narayanan, A. Toutios, V. Ramanarayanan, A. Lammert, J. Kim, S. Lee, K. Nayak, Y.-C. Kim, Y. Zhu, L. Goldstein, D. Byrd, E. Bresch, P. Ghosh, A. Katsamanis, and M. Proctor, “Real-time magnetic resonance imaging and electromagnetic articulography database for speech production research (TC),” _The Journal of the Acoustical Society of America_ , vol. 136, no. 3, pp. 1307–1311, sep 2014.
  * [27] P. Saha, P. Srungarapu, and S. Fels, “Towards automatic speech identification from vocal tract shape dynamics in real-time MRI,” in _Proc. Interspeech_ , Hyderabad, India, 2018, pp. 1249–1253.
  * [28] K. G. Van Leeuwen, P. Bos, S. Trebeschi, M. J. Van Alphen, L. Voskuilen, L. E. Smeele, F. Van Der Heijden, and R. J. Van Son, “CNN-based phoneme classifier from vocal tract MRI learns embedding consistent with articulatory topology,” in _Proc. Interspeech_ , Graz, Austria, 2019, pp. 909–913.
  * [29] T. Kaburagi, “A method for estimating vocal-tract shape from a target speech spectrum,” _Acoustical Science and Technology_ , vol. 36, no. 5, pp. 428–437, 2015.
  * [30] H. Li, J. Tao, M. Yang, and B. Liu, “Estimate articulatory MRI series from acoustic signal using deep architecture,” in _Proc. ICASSP_ , Brisbane, Australia, 2015, pp. 4854–4858.
  * [31] S. Imai, K. Sumita, and C. Furuichi, “Mel Log Spectrum Approximation (MLSA) filter for speech synthesis,” _Electronics and Communications in Japan (Part I: Communications)_ , vol. 66, no. 2, pp. 10–18, 1983.
  * [32] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” _IEEE Transactions on Image Processing_ , vol. 13, no. 4, pp. 600–612, apr 2004.
  * [33] K. Xu, T. Gábor Csapó, P. Roussel, and B. Denby, “A comparative study on the contour tracking algorithms in ultrasound tongue images with automatic re-initialization,” _The Journal of the Acoustical Society of America_ , vol. 139, no. 5, pp. EL154–EL160, may 2016.
  * [34] M. P. Sampat, Z. Wang, S. Gupta, A. C. Bovik, and M. K. Markey, “Complex Wavelet Structural Similarity: A New Image Similarity Index,” _IEEE Transactions on Image Processing_ , vol. 18, no. 11, pp. 2385–2401, nov 2009.
  * [35] J. Wei, Q. Fang, X. Zheng, W. Lu, Y. He, and J. Dang, “Mapping ultrasound-based articulatory images and vowel sounds with a deep neural network framework,” _Multimedia Tools and Applications_ , vol. 75, no. 9, pp. 5223–5245, may 2016.


$\displaystyle={\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}\big{[}\alpha_{{t_{k+1}}}^{1+}\mu_{c^{2}}^{+}\mu_{c}^{-}+\alpha_{t_{k+1}}^{1-}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mu_{c^{2}}^{-}\mu_{c}^{+}}-\mu_{c}^{+}\mu_{c}^{-}\big{]}<0,$

where for the second- and last inequality above we used that
$\alpha_{{t_{k+1}}}^{1\pm}=\mathbb{E}[\alpha_{t_{k+1}}\,|\,\mathcal{F}_{t_{k}},\mathbbm{1}^{\pm}_{t_{k+1}}=1]\leq{}0$
since, by assumption, $\alpha_{t_{k+1}}<0$.

  * (ii)

Let us first rearrange the terms in (A-8) as follows:

$\displaystyle N_{k}$
$\displaystyle=(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha_{t_{k+1}}^{1-}\mu_{c^{2}}^{-}-\mu_{c}^{-})+(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}(\alpha_{t_{k+1}}^{1+}\mu_{c^{2}}^{+}-\mu_{c}^{+})$
(A-14)
$\displaystyle\quad\qquad-2\alpha_{t_{k+1}}^{1+}\alpha_{t_{k+1}}^{1-}{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}\pi_{t_{k+1}}(1,1)\alpha^{1,1}_{{t_{k+1}}}(\mu_{c}^{+}\mu_{c}^{-})^{2}$
$\displaystyle=\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\mu_{c}^{+})^{2}\big{[}\alpha_{{t_{k+1}}}^{1+}\pi_{t_{k+1}}^{+}\mu_{c^{2}}^{-}-\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)(\mu_{c}^{-})^{2}\big{]}$
$\displaystyle\quad\qquad+\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\mu_{c}^{-})^{2}\big{[}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{-}\mu_{c^{2}}^{+}-\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)(\mu_{c}^{+})^{2}\big{]}$
$\displaystyle\quad\qquad-(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}{\pi_{t_{k+1}}^{-}}\mu_{c}^{-}-(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}{\pi_{t_{k+1}}^{+}}\mu_{c}^{+}.$

Then, using once more the inequalities (A-12),

$\displaystyle N_{k}$
$\displaystyle\leq\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\mu_{c}^{+})^{2}\pi_{t_{k+1}}(1,1)\alpha^{1,1}_{{t_{k+1}}}[\mu_{c^{2}}^{-}-(\mu_{c}^{-})^{2}]$
(A-15)
$\displaystyle\quad\qquad+\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\mu_{c}^{+})^{2}\pi_{t_{k+1}}(1,1)\alpha^{1,1}_{{t_{k+1}}}[\mu_{c^{2}}^{+}-(\mu_{c}^{+})^{2}]$
$\displaystyle\quad\qquad-(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}{\pi_{t_{k+1}}^{-}}\mu_{c}^{-}-(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}{\pi_{t_{k+1}}^{+}}\mu_{c}^{+}$
$\displaystyle\leq-(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}{\pi_{t_{k+1}}^{-}}\mu_{c}^{-}-(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}{\pi_{t_{k+1}}^{+}}\mu_{c}^{+}<0,$

where in the second inequality we used once more that
$\max\\{\alpha_{t_{k+1}}^{1,1},\alpha_{t_{k+1}}^{+1},\alpha_{t_{k+1}}^{-1}\\}\leq{}0$.

  * (iii)

Note that, since we already proved that $\gamma_{t_{k}}<0$, proving that
$N_{k}/\gamma_{t_{k}}<-\alpha_{t_{k+1}}^{0}$ a.s. whenever
$\alpha_{t_{k+1}}<0$ is equivalent to showing that
$\alpha_{t_{k+1}}^{0}\gamma_{t_{k}}+N_{k}>0$ a.s. whenever
$\alpha_{t_{k+1}}<0$. Define

$\displaystyle\varphi(\mathbf{\hat{x}},\mathbf{\hat{y}}):=$
$\displaystyle\;\alpha_{t_{k+1}}^{0}\Big{(}\big{[}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{{t_{k+1}}}\mu_{c}^{+}\mu_{c}^{-}\big{]}^{2}-{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mathbf{\hat{x}}-\mu_{c}^{+})(\alpha^{1-}_{{t_{k+1}}}\mathbf{\hat{y}}-\mu_{c}^{-})\Big{)}$
$\displaystyle\qquad+(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha_{t_{k+1}}^{1-}\mathbf{\hat{y}}-\mu_{c}^{-})+(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}(\alpha_{t_{k+1}}^{1+}\mathbf{\hat{x}}-\mu_{c}^{+})$
$\displaystyle\qquad-2\alpha_{t_{k+1}}^{1+}\alpha_{t_{k+1}}^{1-}{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}\pi_{t_{k+1}}(1,1)\alpha^{1,1}_{{t_{k+1}}}(\mu_{c}^{+}\mu_{c}^{-})^{2}$

Notice that
$\varphi(\mu_{c^{2}}^{+},\mu_{c^{2}}^{-})=\alpha_{t_{k+1}}^{0}\gamma_{t_{k}}+N_{k}$.
Taking partial derivatives,

$\begin{split}\frac{\partial\varphi}{\partial\mathbf{\hat{x}}}&=-\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\Big{[}\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\mathbf{\hat{y}}-\alpha_{{t_{k+1}}}^{1+}\mu_{c}^{-}\Big{]}+\Big{(}\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-}\Big{)}^{2}\pi_{t_{k+1}}^{+}\alpha_{t_{k+1}}^{1+}\\\
&=\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\Big{[}\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}(\mu_{c}^{-})^{2}-\alpha_{t_{k+1}}^{0}\mathbf{\hat{y}}\Big{]}+\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1+}\mu_{c}^{-}.\end{split}$
(A-16)
$\begin{split}\frac{\partial\varphi}{\partial\mathbf{\hat{y}}}&=-\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\Big{[}\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\mathbf{\hat{x}}-\alpha_{{t_{k+1}}}^{1-}\mu_{c}^{+}\Big{]}+\Big{(}\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+}\Big{)}^{2}\pi_{t_{k+1}}^{-}\alpha_{t_{k+1}}^{1-}\\\
&=\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\Big{[}\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}(\mu_{c}^{+})^{2}-\alpha_{t_{k+1}}^{0}\mathbf{\hat{x}}\Big{]}+\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1-}\mu_{c}^{+}.\end{split}$
(A-17)

Recall that by assumption $\alpha_{t_{k+1}}<0$ a.s. and thus,

$\displaystyle\alpha_{t_{k+1}}^{0}$
$\displaystyle=\mathbb{E}\big{[}\alpha_{t_{k+1}}\big{|}\mathcal{F}_{t_{k}}\big{]}=\mathbb{E}\bigg{[}\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{\pm}\big{]}\bigg{|}\mathcal{F}_{t_{k}}\bigg{]}$
$\displaystyle=\alpha_{t_{k+1}}^{1\pm}\pi_{t_{k+1}}^{\pm}+\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{\pm}=0\big{]}(1-\pi_{t_{k+1}}^{\pm})\leq\alpha_{t_{k+1}}^{1\pm}\pi_{t_{k+1}}^{\pm}.$
(A-18)

Using the last inequality in (A-16), we obtain that

$\frac{\partial\varphi}{\partial\mathbf{\hat{x}}}\geq\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{t_{k+1}}^{0}[(\mu_{c}^{-})^{2}-\mathbf{\hat{y}}]+\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1+}\mu_{c}^{-}.\\\
$

Therefore, using that $(\mu_{c}^{-})^{2}\leq\mu^{-}_{c^{2}}$ and
$\max\\{\alpha_{t_{k+1}}^{0},\alpha_{t_{k+1}}^{1+},\alpha_{t_{k+1}}^{1-}\\}\leq{}0$,
we have

$\begin{split}\frac{\partial\varphi}{\partial\mathbf{\hat{x}}}\big{(}\;\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
\;,\mu_{c^{2}}^{-}\big{)}&\geq\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{t_{k+1}}^{0}[(\mu_{c}^{-})^{2}-\mu_{c^{2}}^{-}]+\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1+}\mu_{c}^{-}\\\
&\geq\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1+}\mu_{c}^{-}\geq
0,\end{split}$ (A-19)

which implies that
$\varphi(\;\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
\;,\mu_{c^{2}}^{-})$ is a non-decreasing function of the first parameter.
Thus,

$\varphi\big{(}(\mu_{c}^{+})^{2},\mu_{c^{2}}^{-}\big{)}\leq\varphi(\mu_{c^{2}}^{+},\mu_{c^{2}}^{-})=\alpha_{t_{k+1}}^{0}\gamma_{t_{k}}+N_{k}.$
(A-20)

Using (A-17), (A-18), and
$\max\\{\alpha_{t_{k+1}}^{0},\alpha_{t_{k+1}}^{1+},\alpha_{t_{k+1}}^{1-}\\}\leq{}0$,

$\displaystyle\frac{\partial\varphi}{\partial\mathbf{\hat{y}}}\big{(}(\mu_{c}^{+})^{2},\;\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
\;\big{)}$
$\displaystyle=\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}[\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}(\mu_{c}^{+})^{2}-\alpha_{t_{k+1}}^{0}(\mu_{c}^{+})^{2}]+\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1-}\mu_{c}^{+}$
$\displaystyle\geq\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{{t_{k+1}}}^{1-}\mu_{c}^{+}\geq
0,$

which shows that
$\varphi\big{(}(\mu_{c}^{+})^{2},\;\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
\;\big{)}$ is a nondecreasing function in the second argument. Using that
$(\mu_{c}^{-})^{2}\leq\mu_{c^{2}}^{-}$ and (A-20), we finally get that

$\varphi\big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\big{)}\leq\varphi\big{(}(\mu_{c}^{+})^{2},\mu_{c^{2}}^{-}\big{)}\leq\alpha_{t_{k+1}}^{0}\gamma_{t_{k}}+N_{k}.$
(A-21)

At this point, we only need to prove that
$0\leq\varphi\big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\big{)}$. Evaluating,
we have that

$\displaystyle\varphi\big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\big{)}$
$\displaystyle=\mu_{c}^{+}\mu_{c}^{-}\bigg{\\{}\alpha_{t_{k+1}}^{0}\Big{[}(\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1})^{2}\mu_{c}^{+}\mu_{c}^{-}-{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mu_{c}^{+}-1)(\alpha^{1-}_{{t_{k+1}}}\mu_{c}^{-}-1)\Big{]}$
$\displaystyle\qquad\qquad+\mu_{c}^{+}(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha^{1-}_{{t_{k+1}}}\mu_{c}^{-}-1)+\mu_{c}^{-}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}(\alpha^{1+}_{{t_{k+1}}}\mu_{c}^{+}-1)$
$\displaystyle\qquad\qquad-2\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1}\mu_{c}^{+}\mu_{c}^{-}\bigg{\\}}$
$\displaystyle=\mu_{c}^{+}\mu_{c}^{-}\ell(\mu_{c}^{+}),$ (A-22)

where

$\displaystyle\ell(\mathbf{\hat{z}})$
$\displaystyle:=\alpha_{t_{k+1}}^{0}\Big{[}\mathbf{\hat{z}}\mu_{c}^{-}(\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1})^{2}-{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mathbf{\hat{z}}-1)(\alpha^{1-}_{{t_{k+1}}}\mu_{c}^{-}-1)\Big{]}$
$\displaystyle\qquad\qquad+\mathbf{\hat{z}}(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha^{1-}_{{t_{k+1}}}\mu_{c}^{-}-1)+\mu_{c}^{-}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}(\alpha^{1+}_{{t_{k+1}}}\mathbf{\hat{z}}-1)$
$\displaystyle\qquad\qquad-2\,\mathbf{\hat{z}}\,\mu_{c}^{-}\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1}.$

Due to (A-18), we have

$\displaystyle\ell(0)$
$\displaystyle=\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}-1)-\mu_{c}^{-}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}$
$\displaystyle=\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})-\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}$
$\displaystyle\geq-\alpha_{t_{k+1}}^{0}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}>0.$
(A-23)

We shall prove that $d\ell/d\mathbf{\hat{z}}\geq 0$, which will imply that
$\ell(\mu_{c}^{+})\geq\ell(0)>0$ and, finally, that
$\varphi\big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\big{)}>0$ due to Equation
((iii)). Finally, by using the previous equation together with (A-21) we get
that

$0<\varphi\big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\big{)}\leq\alpha_{t_{k+1}}^{0}\gamma_{t_{k}}+N_{k},$

and will conclude the proof.

It remains to show that $d\ell/d\mathbf{\hat{z}}\geq 0$. Taking the derivative
with respect to $\mathbf{\hat{z}}$,

$\displaystyle\frac{d\ell}{d\mathbf{\hat{z}}}$
$\displaystyle=\alpha_{t_{k+1}}^{0}\big{[}(\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1})^{2}\mu_{c}^{-}-\pi^{+}_{{t_{k+1}}}{\pi^{-}_{{t_{k+1}}}}\alpha^{1+}_{{t_{k+1}}}(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}-1)\big{]}$
$\displaystyle\quad+(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha^{1-}_{{t_{k+1}}}\mu_{c}^{-}-1)+\mu_{c}^{-}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}\alpha^{1+}_{{t_{k+1}}}$
$\displaystyle\quad-2\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}\pi_{t_{k+1}}(1,1)\alpha_{t_{k+1}}^{1,1}\mu_{c}^{-}$
$\displaystyle=\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}\alpha^{1+}_{{t_{k+1}}}(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})+\mu_{c}^{-}\Psi({\alpha_{t_{k+1}}^{1,1}}\pi_{t_{k+1}}(1,1)),$
(A-24)

where

$\displaystyle\Psi(\mathbf{\hat{z}})$
$\displaystyle:=\alpha_{t_{k+1}}^{0}\mathbf{\hat{z}}^{2}-2\mathbf{\hat{z}}\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}+(\alpha_{{t_{k+1}}}^{1+}\pi_{t_{k+1}}^{+})^{2}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{-}$
(A-25)
$\displaystyle\qquad+(\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{-})^{2}\alpha_{{t_{k+1}}}^{1+}\pi_{t_{k+1}}^{+}-\alpha_{t_{k+1}}^{0}\alpha_{{t_{k+1}}}^{1+}\alpha_{{t_{k+1}}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}.$

Due to (A-18), the first term of ((iii)) is nonegative. It is also easy to see
that the quadratic function $\Psi$ attaining its minimum value over any
interval $[a,b]$ at the endpoints since $\alpha_{t_{k+1}}^{0}\leq 0$. On the
other hand, observe that
$\alpha_{t_{k+1}}(\mathbbm{1}_{t_{k+1}}^{+}+\mathbbm{1}_{t_{k+1}}^{-}-1)\geq\alpha_{t_{k+1}}\mathbbm{1}_{t_{k+1}}^{+}\mathbbm{1}_{t_{k+1}}^{-}$.
Then, by taking conditional expectation with respect to $\mathcal{F}_{t_{k}}$
and using (A-12), we obtain the bounds

$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\vee\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}\leq\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)\leq(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\wedge
0.$ (A-26)

Therefore, to show that ((iii)) is nonnegative, it suffices to check that

${\rm(1)}\;\Psi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\vee\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\geq{}0,\quad{\rm(2)}\;\Psi((\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\wedge
0)\geq{}0.$

To show (1), we consider the cases (a)
$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\geq\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}$
or (b)
$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\leq\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}$.
To show (2) above, we consider the cases (c)
$(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\geq{}0$;
or (d)
$(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\leq{}0$.

Suppose that (a)
$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\geq\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}$.
Then, in light of (A-18),

$\displaystyle\Psi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})$
$\displaystyle=\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\Big{[}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}-(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})+\alpha_{t_{k+1}}^{0}(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\Big{]}$
$\displaystyle=\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\geq
0.$

Similarly, (b) if
$\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}\geq\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}$,
then $\Psi(\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\geq 0$.

Now, suppose that (c)
$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0}\geq
0$. Then,

$\Psi(0)=\alpha_{t_{k+1}}^{1+}\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{+}\pi_{t_{k+1}}^{-}(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})\geq
0.$

The only case left is when (d)
$\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0}\leq
0$. Together with (A-18), we will have

$\alpha_{t_{k+1}}^{0}\leq\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}\leq\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-},$
(A-27)

and, by rearranging terms,

$\displaystyle\Psi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})$
$\displaystyle=(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})^{2}+\alpha_{t_{k+1}}^{0}(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}$
$\displaystyle\quad-(2\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})$
$\displaystyle=:\Xi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}),$

where

$\Xi(\mathbf{\hat{z}})=(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\mathbf{\hat{z}}^{2}-(2\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\mathbf{\hat{z}}+\alpha_{t_{k+1}}^{0}(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})^{2}$

is a quadratic function, opening downward due to (A-18). Furthermore, at the
lower and upper bounds of (A-27), we have
$\Xi(\alpha_{t_{k+1}}^{0})=\Xi(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})=0$.
Therefore, $\Xi(\mathbf{\hat{z}})\geq 0$, for all
$\mathbf{\hat{z}}\in\big{[}\alpha_{t_{k+1}}^{0},(\alpha_{t_{k+1}}^{0}-\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-})\big{]}$.
Finally, we conclude that
$\Psi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0})=\Xi(\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+})\geq{}0$
in the case (iv) as desired.

∎

Proof of Corollary 3.

Refer to the proof of Theorem 1. Denote the right hand side of Equation (A-5)
as $\sup_{L_{t_{k}}^{\pm}}{f}(L_{t_{k}}^{+},L_{t_{k}}^{-})$. It is easy to
observe that ${f}(L_{t_{k}}^{+},L_{t_{k}}^{-})$ is a quadratic function of
$L_{t_{k}}^{+}$ and $L_{t_{k}}^{-}$. Setting the partial derivatives with
respect to $L_{t_{k}}^{+}$ and $L_{t_{k}}^{-}$ equal to $0$, we have that

$\displaystyle\begin{split}0=\frac{\partial f}{\partial
L_{t_{k}}^{+}}&=2{\pi^{+}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mu_{c^{2}}^{+}-\mu_{c}^{+})L_{t_{k}}^{+}+{\pi^{+}_{{t_{k+1}}}}\big{[}\mu_{cp}^{+}+{h}_{{t_{k+1}}}^{1+}\mu_{c}^{+}+\alpha^{1+}_{{t_{k+1}}}(2\mu_{c}^{+}I_{t_{k}}-2\mu_{c^{2}p}^{+})\big{]}\\\
&-2\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{c}^{-}L_{t_{k}}^{-}+2\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{cp}^{-}\\\
0=\frac{\partial f}{\partial
L_{t_{k}}^{-}}&=2{\pi^{-}_{{t_{k+1}}}}(\alpha^{1-}_{{t_{k+1}}}\mu_{c^{2}}^{-}-\mu_{c}^{-})L_{t_{k}}^{-}+{\pi^{-}_{{t_{k+1}}}}\big{[}\mu_{cp}^{-}-{h}_{{t_{k+1}}}^{1-}\mu_{c}^{-}+\alpha^{1-}_{{t_{k+1}}}(-2\mu_{c}^{-}I_{t_{k}}-2\mu_{c^{2}p}^{-})\big{]}\\\
&-2\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{c}^{-}L_{t_{k}}^{+}+2\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{-}\mu_{cp}^{+}.\end{split}$
(A-28)

Solving for $L_{t_{k}}^{+}$ and $L_{t_{k}}^{-}$, we get that the unique
stationary points ${L}_{{t_{k}}}^{+,*}$ and ${L}_{{t_{k}}}^{-,*}$ are given by

$\displaystyle\begin{split}L_{{t_{k}}}^{+,*}&={}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}I_{t_{k}}+{}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}},\\\
L_{{t_{k}}}^{-,*}&=-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}I_{t_{k}}-{}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}},\end{split}$
(A-29)

as pointed out in Eq. (22). To prove that the stationary point
$\Big{(}{L}_{{t_{k}}}^{+,*},{L}_{{t_{k}}}^{-,*}\Big{)}$ is the maximum of $f$,
we apply the second derivative test. Indeed,

$\left(\frac{\partial^{2}f}{\partial\left(L_{t_{k}}^{\pm}\right)^{2}}\right)=2\pi_{t_{k+1}}^{\pm}(\alpha^{1\pm}_{{t_{k+1}}}\mu_{c^{2}}^{\pm}-\mu_{c}^{\pm})<0,$

because, due to Lemma 2, we have $\alpha_{t_{k}}<0$ and, thus,
$\alpha_{t_{k}}^{1\pm}=\mathbb{E}[\alpha_{t_{k}}\,|\,\mathcal{F}_{t_{k-1}},\mathbbm{1}^{\pm}_{t_{k}}=1]\leq{}0$.
It remains to show that

$\displaystyle B_{k}$
$\displaystyle=\left(\frac{\partial^{2}f}{\partial\left(L_{t_{k}}^{+}\right)^{2}}\right)\left(\frac{\partial^{2}f}{\partial\left(L_{t_{k}}^{-}\right)^{2}}\right)-\left(\frac{\partial^{2}f}{\partial
L_{t_{k}}^{+}\partial L_{t_{k}}^{-}}\right)^{2}$
$\displaystyle=4{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mu_{c^{2}}^{+}-\mu_{c}^{+})(\alpha^{1-}_{{t_{k+1}}}\mu_{c^{2}}^{-}-\mu_{c}^{-})-4\big{[}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{{t_{k+1}}}\mu_{c}^{+}\mu_{c}^{-}\big{]}^{2}>0.$
(A-30)

However, $B_{k}$ above is just $-4\gamma_{t_{k}}$, with $\gamma_{t_{k}}$
defined as in (A-8), and it was shown in the proof of Lemma 2 that
$\gamma_{t_{k}}<0$ (see ((A-13)). The proof is now complete. ∎

Proof of Theorem 4.

Throughout, ${W}_{t_{i}},{I}_{t_{i}}$, for $i=k,\dots,N+1$, are the cash
holding and inventory processes resulting from adopting an admissible
placement strategy ${L}_{t_{i}}^{\pm}$, $i=k,\dots,N$. Similarly, for
$i=k+1,\dots,N+1$, ${W}_{t_{i}}^{*},{I}_{t_{i}}^{*}$ are the resulting cash
holding and inventory processes, starting from time $t_{k}$ at the initial
states $W_{t_{k}},{I_{t_{k}}}$, when setting
$L_{t_{i}}^{\pm}={L}_{t_{i}}^{\pm,{*}}$. First note that, for an arbitrary
admissible placement strategy $L_{t_{i}}^{\pm}$,
$\\{v(t_{i},S_{t_{i}},W_{t_{i}},I_{t_{i}})\\}_{i=k,\dots,N+1}$ is a
supermartingale since

$\displaystyle\mathbbm{E}\big{[}v(t_{i+1},S_{t_{i+1}},W_{t_{i+1}},I_{t_{i+1}})|\mathcal{F}_{t_{i}}\big{]}$
$\displaystyle\leq\sup_{\widehat{L}^{\pm}_{t_{i}}}\mathbbm{E}\big{[}v(t_{i+1},S_{t_{i+1}},\widehat{W}_{t_{i+1}},\widehat{I}_{t_{i+1}})|\mathcal{F}_{t_{i}}\big{]}$
$\displaystyle=v(t_{i},S_{t_{i}},W_{t_{i}},I_{t_{i}}),$ (A-31)

where above $\widehat{W}_{t_{i+1}}$ and $\widehat{I}_{t_{i+1}}$ are the
time-$t_{i+1}$ cash holding and inventory for an arbitrary admissible
placement strategy $\widehat{L}_{t_{i}}^{\pm}$ when
$\widehat{W}_{t_{i}}={W}_{t_{i}}$ and $\widehat{I}_{t_{i}}={I}_{t_{i}}$. The
equation in (A-31) follows from (16) and Corollary 3. That is,
$\alpha_{t_{i}},h_{t_{i}},g_{t_{i}}$ are picked in order for (A-31) to hold
true.

From the supermartingale condition, we then have that

$\displaystyle\begin{split}v(t_{k},S_{t_{k}},W_{t_{k}},I_{t_{k}})&\geq\sup_{{(L^{\pm}_{t_{i}})_{k\leq
i\leq N}}}\mathbb{E}[v(T,S_{T},W_{T},I_{T})|\mathcal{F}_{t_{k}}]\\\
&=\sup_{{(L^{\pm}_{t_{i}})_{k\leq i\leq
N}}}\mathbb{E}[W_{T}+S_{T}I_{T}-\lambda I_{T}^{2}|\mathcal{F}_{t_{k}}]\\\
&=V_{t_{k}}.\end{split}$ (A-32)

The first equality in Eq. (A-32) holds because
$v(T,S_{T},W_{T},I_{T})=W_{T}+S_{T}I_{T}-\lambda I_{T}^{2}$ by the terminal
conditions $\alpha_{T}=-\lambda,g_{T}=0,h_{T}=0$.

Next we prove that $v(t_{k},S_{t_{k}},W_{t_{k}},I_{t_{k}})\leq V_{t_{k}}$. To
this end, recall from (16) and Corollary 3 that ${L}_{t_{i}}^{\pm,{*}}$ are
chosen so that

$v(t_{i},S_{t_{i}},{{W}_{t_{i}}^{*}},{{I}_{t_{i}}^{*}})=\mathbb{E}[v(t_{i+1},S_{t_{i+1}},{{W}^{*}_{t_{i+1}},{I}^{*}_{t_{i+1}}})|\mathcal{F}_{t_{i}}],$

for all $i=k,\dots,{N}$. Hence, recalling that we set
${W}^{*}_{t_{k}}=W_{t_{k}}$ and ${I}^{*}_{t_{k}}=I_{t_{k}}$, by induction,

$\displaystyle
v(t_{k},S_{t_{k}},W_{t_{k}},I_{t_{k}})=v(t_{k},{S_{t_{k}},{W}^{*}_{t_{k}},{I}^{*}_{t_{k}}})$
$\displaystyle=\mathbb{E}[v(t_{N+1},S_{t_{N+1}},{{W}^{*}_{t_{N+1}},{I}^{*}_{t_{N+1}}})|\mathcal{F}_{t_{k}}]$
$\displaystyle=\mathbb{E}[{{W}^{*}_{T}+S_{T}{I}^{*}_{T}-\lambda({I}^{*}_{T})^{2}}|\mathcal{F}_{t_{k}}].$

It also trivially follows that

$\mathbb{E}{[{W}^{*}_{T}+S_{T}{I}^{*}_{T}-\lambda({I}^{*}_{T})^{2}}|\mathcal{F}_{t_{k}}]\leq{}\sup_{{(L^{\pm}_{t_{i}})_{k\leq
i\leq N}}}\mathbb{E}[W_{T}+S_{T}I_{T}-\lambda
I_{T}^{2}|\mathcal{F}_{t_{k}}]=V_{t_{k}}.$

We then conclude that $v(t_{k},S_{t_{k}},W_{t_{k}},I_{t_{k}})\leq{}V_{t_{k}}$,
which combined with (A-32) implies that
$v(t_{k},S_{t_{k}},W_{t_{k}},I_{t_{k}})={}V_{t_{k}}$. ∎

###  A.2 Proofs of Section 2.3: Optimal Strategy for a General Midprice

Proof of Theorem 5.

For simplicity, we write $\rho_{k}^{\pm}=\rho_{t_{k}}^{\pm}$ and
$\psi_{k}^{\pm}=\psi_{t_{k}}^{\pm}$. For future reference, define
$\widetilde{h}_{t_{k+1}}^{0}$, $\widetilde{h}_{t_{k+1}}^{1\pm}$, and
$\widetilde{h}_{t_{k+1}}^{1,1}$ using the notation (3). Let us start by
writing the optimization problem (15) in terms of the ansatz (28):

$\displaystyle W_{t_{k}}+\alpha_{{t_{k}}}I_{t_{k}}^{2}+$ $\displaystyle
S_{t_{k}}I_{t_{k}}+\widetilde{h}_{{t_{k}}}I_{t_{k}}=\sup_{L^{\pm}_{t_{k}}\in\mathcal{A}_{t_{k}}}\mathbb{E}[W_{t_{k+1}}+\alpha_{{t_{k+1}}}I_{t_{k+1}}^{2}+S_{t_{k+1}}I_{t_{k+1}}+\widetilde{h}_{{t_{k+1}}}I_{t_{k+1}}\,|\,\mathcal{F}_{t_{k}}].$
(A-33)

We will prove the result by backwards induction. Consider the following
statements:

  * (i)

For $\delta\in\\{+,-\\}$, we have:

$\displaystyle\mathbb{E}\big{[}\widetilde{h}_{t_{k+1}}\mathbbm{1}_{t_{k+1}}^{\delta}c_{t_{k+1}}^{\delta}\big{|}\mathcal{F}_{t_{k}}\big{]}$
$\displaystyle=\widetilde{h}_{t_{k+1}}^{1\delta}\pi_{t_{k+1}}^{\delta}\mu_{c}^{\delta},$
(A-34)
$\displaystyle\mathbb{E}\big{[}\widetilde{h}_{t_{k+1}}\mathbbm{1}_{t_{k+1}}^{\delta}c_{t_{k+1}}^{\delta}p_{t_{k+1}}^{\delta}\big{|}\mathcal{F}_{t_{k}}\big{]}$
$\displaystyle=\widetilde{h}_{t_{k+1}}^{1\delta}\pi_{t_{k+1}}^{\delta}\mu_{cp}^{\delta}.$
(A-35)

  * (ii)

The optimal controls $\widetilde{L}_{{t_{k}}}^{\pm,*}$ that solve (A-33) under
dynamics (11) with terminal conditions $\alpha_{t_{N+1}}=-\lambda$ and
$\widetilde{h}_{t_{N+1}}=0$ are given by

$\displaystyle\widetilde{L}_{{t_{k}}}^{+,*}={}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}I_{t_{k}}+{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}},\qquad\widetilde{L}_{{t_{k}}}^{-,*}=-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}I_{t_{k}}-^{\scaleto{(2)}{5pt}}\\!\\!\widetilde{A}^{-}_{{t_{k}}}{+}{}\,{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}},$
(A-36)

where

$\displaystyle{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\pm}_{{t_{k}}}=\frac{\widetilde{h}_{t_{k+1}}^{1\pm}\rho_{k}^{\pm}-\widetilde{h}_{t_{k+1}}^{1\mp}\psi_{k}^{\pm}}{2\gamma_{t_{k}}},\qquad{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\pm}_{{t_{k}}}={}^{\scaleto{(3)}{5pt}}\\!A^{\pm}_{{t_{k}}}\pm\frac{\rho_{k}^{\pm}-\psi_{k}^{\pm}}{2\gamma_{t_{k}}}\Delta_{t_{k}}^{t_{k}},$
(A-37)

whereas ${}^{\scaleto{(1)}{5pt}}\\!A_{{t_{k}}}^{\pm}$ and
${}^{\scaleto{(3)}{5pt}}\\!A^{\pm}_{{t_{k}}}$ are the same as in Theorem 1.

  * (iii)

The random variables $\widetilde{h}_{{t_{k}}}$ satisfy the following iterative
equation,

$\displaystyle\begin{split}\widetilde{h}_{{t_{k}}}=\widetilde{h}^{0}_{{t_{k+1}}}&+\sum_{\delta=\pm}\pi^{\delta}_{t_{k+1}}\Bigg{\\{}2\Big{[}\alpha^{1\delta}_{{t_{k+1}}}\mu_{c^{2}}^{\delta}-\mu_{c}^{\delta}\Big{]}\Big{[}{}^{\scaleto{(1)}{5pt}}\\!A^{\delta}_{{t_{k}}}\big{(}{\delta\;{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}+{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}}\big{)}\Big{]}+2\alpha^{1\delta}_{{t_{k+1}}}\mu_{c}^{\delta}\big{(}{\delta\,{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}+{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}}\big{)}\\\
&\qquad\qquad\qquad-2\delta\alpha^{1\delta}_{{t_{k+1}}}\mu_{cp}^{\delta}+\delta{}^{\scaleto{(1)}{5pt}}\\!A^{\delta}_{{t_{k}}}\Big{(}\mu_{cp}^{\delta}+\delta\widetilde{h}^{1\delta}_{{t_{k+1}}}\mu_{c}^{\delta}-2\alpha^{1\delta}_{{t_{k+1}}}\mu_{c^{2}p}^{\delta}\Big{)}\Bigg{\\}}\\\
&-2\alpha^{1,1}_{{t_{k+1}}}\pi_{t_{k+1}}(1,1)\mu_{c}^{+}\mu_{c}^{-}\Bigg{[}{}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}\Big{(}{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}\Big{)}-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}\Big{(}{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}\Big{)}\\\
&\qquad+{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}\frac{\mu_{cp}^{+}}{\mu_{c}^{+}}-{}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}\frac{\mu_{cp}^{-}}{\mu_{c}^{-}}\Bigg{]}+\Delta_{t_{k}}^{t_{k}}\Big{(}{}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}\pi_{t_{k+1}}^{+}\mu_{c}^{+}+{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}\pi_{t_{k+1}}^{-}\mu_{c}^{-}+1\Big{)}.\end{split}$
(A-38)

while

$\displaystyle\begin{split}\widetilde{g}_{{t_{k}}}&=\widetilde{g}_{{t_{k+1}}}^{\;0}+\sum_{\delta=\pm}{\pi^{\delta}_{t_{k+1}}}\Bigg{\\{}\Big{(}\alpha_{{t_{k+1}}}^{1\delta}\mu_{c^{2}}^{\delta}-\mu_{c}^{\delta}\Big{)}\Big{(}{{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}+\delta\,{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}}\Big{)}^{2}+\alpha_{{t_{k+1}}}^{1\delta}\mu_{c^{2}p^{2}}^{\delta}-\delta\,\widetilde{h}_{{t_{k+1}}}\mu_{cp}^{\delta}\\\
&\quad\qquad\qquad\qquad\qquad+\Big{(}\mu_{cp}^{\delta}+\delta\widetilde{h}_{{t_{k+1}}}\mu_{c}^{\delta}-2\alpha_{{t_{k+1}}}^{1\delta}\mu_{c^{2}p}^{\delta}\Big{)}\Big{(}{{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}+{(\delta\,{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}})}\Big{)}\Bigg{\\}}\\\
&\qquad\qquad-2\alpha_{{t_{k+1}}}^{1,1}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{c}^{-}\Bigg{[}\Big{(}{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}\Big{)}\Big{(}{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}\Big{)}\\\
&\quad\qquad\qquad\qquad\qquad-\frac{\mu_{cp}^{+}}{\mu_{c}^{+}}\Big{(}{{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{-}_{{t_{k}}}}\Big{)}-\frac{\mu_{cp}^{-}}{\mu_{c}^{-}}\Big{(}{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{+}_{{t_{k}}}\Big{)}+\frac{\mu_{cp}^{+}\mu_{cp}^{-}}{\mu_{c}^{-}\mu_{c}^{+}}\Bigg{]}\\\
&\qquad\qquad+\Delta_{t_{k}}^{t_{k}}\bigg{[}\Big{(}{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}+{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}\Big{)}\pi_{t_{k+1}}^{+}\mu_{c}^{+}-\Big{(}{}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\delta}_{{t_{k}}}\Big{)}\pi_{t_{k+1}}^{-}\mu_{c}^{-}-\pi_{t_{k+1}}^{+}\mu_{cp}^{+}+\pi_{t_{k+1}}^{-}\mu_{cp}^{-}\bigg{]}\end{split}$
(A-39)

  * (iv)

Equation (29) holds true (at time $t_{k}$) and the random variables
$\rho_{k}^{\pm}$ and $\psi_{k}^{\pm}$, as defined in Equations (17), are
$\mathcal{F}_{{t_{k}}}-$measurable while the random variables $\xi_{k+1}$
given by Eq. (30) are $\mathcal{F}_{{t_{k+1}}}-$ measurable.

To start, note that the statement (i) is immediate for $k=N$ due to the
terminal condition on $\widetilde{h}$. The strategy that we will employ to
finish the proof is the following: we will show that if statement (i) holds
true for $k=j,j+1,\dots,N$, then statements (ii), (iii), and (iv) will hold
true for $k=j$. In the final step, we prove that the statement (i) holds true
for $k=j-1$ if (i)-(iv) holds for $k=j$.

Let us start with the first step described in the previous paragraph. Assume
that statement (i) is true for $k=j,j+1,\dots,N$. By substituting the values
of $W_{t_{j+1}}$ and $I_{t_{j+1}}$ given by Equation (11)-(12) into the
optimization problem (A-33) with $k=j$ we get,

$\displaystyle\begin{split}V_{t_{j}}=\sup_{L_{t_{j}}^{\pm}\in\mathcal{A}_{t_{j}}}\mathbb{E}\Bigg{\\{}&\sum_{\delta=\pm}\mathbbm{1}_{t_{j+1}}^{\delta}\big{[}-c_{t_{j+1}}^{\delta}(L_{t_{j}}^{\delta})^{2}+(c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta}-\delta
c_{t_{j+1}}^{\delta}S_{t_{j}})L_{t_{j}}^{\delta}+\delta
c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta}S_{t_{j}}\big{]}\\\
&+\alpha_{{t_{j+1}}}\bigg{\\{}I_{t_{j}}^{2}+\sum_{\delta=\pm}\mathbbm{1}_{t_{j+1}}^{\delta}\Big{[}(c_{t_{j+1}}^{\delta})^{2}(L_{t_{j}}^{\delta})^{2}+\big{(}2\delta
I_{t_{j}}c_{t_{j+1}}^{\delta}-2(c_{t_{j+1}}^{\delta})^{2}p_{t_{j+1}}^{\delta}\big{)}L_{t_{j}}^{\delta}\\\
&\qquad\qquad\quad\qquad\qquad\qquad+(c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta})^{2}-2\delta
I_{t_{j}}c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta}\Big{]}\\\
&\qquad\qquad+2\mathbbm{1}_{t_{j+1}}^{+}\mathbbm{1}_{t_{j+1}}^{-}c_{t_{j+1}}^{+}c_{t_{j+1}}^{-}(-L_{t_{j}}^{+}L_{t_{j}}^{-}+p_{t_{j+1}}^{+}L_{t_{j}}^{-}+p_{t_{j+1}}^{-}L_{t_{j}}^{+}-p_{t_{j+1}}^{+}p_{t_{j+1}}^{-})\bigg{\\}}\\\
&+S_{t_{j+1}}\bigg{[}I_{t_{j}}+\sum_{\delta=\pm}\mathbbm{1}_{t_{j+1}}^{\delta}(-\delta
c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta}+\delta
c_{t_{j+1}}^{\delta}L_{t_{j}}^{\delta})\bigg{]}\\\
&+\widetilde{h}_{{t_{j+1}}}I_{t_{j}}+\sum_{\delta=\pm}\mathbbm{1}_{t_{j+1}}^{\delta}(-\delta{\widetilde{h}_{{t_{j+1}}}}c_{t_{j+1}}^{\delta}p_{t_{j+1}}^{\delta}+\delta{\widetilde{h}_{{t_{j+1}}}}c_{t_{j+1}}^{\delta}L_{t_{j}}^{\delta})+{\color[rgb]{0.333,0.036,0.631}\definecolor[named]{pgfstrokecolor}{rgb}{0.333,0.036,0.631}\widetilde{g}_{{t_{j+1}}}}\Bigg{|}\mathcal{F}_{t_{j}}\Bigg{\\}}.\end{split}$
(A-40)

Computing the conditional expectations on the right-hand side using statement
(i) and the techniques used in the proof of Theorem 1, we get that

$\displaystyle\begin{split}&\alpha_{t_{j}}I_{t_{j}}^{2}+S_{t_{j}}I_{t_{j}}+\widetilde{h}_{t_{j}}I_{t_{j}}+\widetilde{g}_{t_{j}}\\\
&=\sup_{L_{t_{j}}^{\pm}\in\mathcal{A}_{t_{j}}}\
\Bigg{[}\sum_{\delta=\pm}{\pi^{\delta}_{t_{j+1}}}\bigg{\\{}(\alpha^{1\delta}_{t_{j+1}}\mu_{c^{2}}^{\delta}-\mu_{c}^{\delta})(L_{t_{j}}^{\delta})^{2}+\big{[}\mu_{cp}^{\delta}+\delta\mu_{c}^{\delta}\big{(}\widetilde{h}^{1\delta}_{t_{j+1}}+\Delta_{t_{j}}^{t_{j}}\big{)}+\alpha^{1\delta}_{t_{j+1}}(2\delta\mu_{c}^{\delta}I_{t_{j}}-2\mu_{c^{2}p}^{\delta})\big{]}L_{t_{j}}^{\delta}\\\
&\qquad\qquad\qquad\qquad\quad+\alpha^{1\delta}_{t_{j+1}}(\mu_{c^{2}p^{2}}^{\delta}-2\delta\mu_{cp}^{\delta}I_{t_{j}})-\delta\mu_{cp}^{\delta}\big{(}\widetilde{h}^{1\delta}_{t_{j+1}}+\Delta_{t_{j}}^{t_{j}}\big{)}\bigg{\\}}\\\
&\qquad\qquad\qquad+2\alpha^{1,1}_{t_{j+1}}{\pi_{t_{j+1}}(1,1)}\Big{(}\mu_{cp}^{+}\mu_{c}^{-}L_{t_{j}}^{-}+\mu_{c}^{+}\mu_{cp}^{-}L_{t_{j}}^{+}-\mu_{c}^{+}\mu_{c}^{-}L_{t_{j}}^{+}L_{t_{j}}^{-}-\mu_{cp}^{+}\mu_{cp}^{-}\Big{)}\\\
&\qquad\qquad\qquad+\alpha^{0}_{t_{j+1}}I_{t_{j}}^{2}+I_{t_{j}}S_{t_{j}}+\widetilde{h}^{0}_{t_{j+1}}I_{t_{j}}+\Delta_{t_{j}}^{t_{j}}I_{t_{j}}+\widetilde{g}^{\;0}_{t_{j+1}}\Bigg{]}.\end{split}$
(A-41)

Then, since the function of the right-hand side is a quadratic function of the
controls $L_{t_{j}}^{\pm}$, it can be shown, as in the proof Theorem 1, that
the optimal controls are the ones given in statement (ii) with $k=j$.
Furthermore, after plugging in the optimal controls; equating the coefficients
of $I_{t_{j}}$ on both sides; and the independent terms on both sides, we
obtain statement (iii).

To obtain statement (iv), notice that by plugging the expression for
${}^{\scaleto{(2)}{5pt}}\\!A^{\pm}_{t_{j}}$, given by Equation (18), into the
Equation (20), we can rewrite $h_{t_{j}}$ as:

$\displaystyle h_{t_{j}}$
$\displaystyle=\mathbb{E}(h_{t_{j+1}}\xi_{j+1}|\mathcal{F}_{t_{j}})+Z_{j}$
$\displaystyle=\mathbb{E}(h_{t_{N+1}}\prod_{u=j}^{N}\xi_{u+1}|\mathcal{F}_{t_{k}})+\sum_{u=j}^{N}\mathbb{E}(Z_{u}\prod_{i=j+1}^{u}\xi_{i}|\mathcal{F}_{t_{j}})$
(A-42)

where $\\{Z_{k}\\}_{k=1}^{N}$ is a collection of
$\mathcal{H}_{t_{k}}^{\varpi}-$measurable random variables. In the same
fashion, by plugging the quantities defined in Equation (A-37) into Equation
(A-38), we can rewrite $\widetilde{h}_{{t_{j}}}$ as:

$\displaystyle\widetilde{h}_{{t_{j}}}$
$\displaystyle=\mathbb{E}\big{[}(h_{t_{j+1}}+\Delta_{t_{j}}^{t_{j}})\xi_{j+1}\big{|}\mathcal{F}_{t_{j}}\big{]}+Z_{j}=h_{t_{u}}+\sum_{u=j}^{N}\Delta_{t_{u}}^{t_{j}}\mathbb{E}\left[\prod_{i=j+1}^{u+1}\xi_{i}|\mathcal{F}_{t_{j}}\right].$
(A-43)

Next, by substituting the value of $\widetilde{h}_{{t_{j}}}$ in Equation
(A-43) into the term
${}^{\scaleto{(2)}{5pt}}\\!\widetilde{A}^{\pm}_{{t_{j}}}$, as defined in
Equation (A-37), and then plugging this equivalent formula along with
${}^{\scaleto{(3)}{5pt}}\\!\widetilde{A}^{\pm}_{{t_{u}}}$, as given by
Equation (A-37) into Equation (A-36), we obtain Equation (29). To verify the
measurability of $\rho_{j}^{\pm}$ and $\psi_{j}^{\pm}$, notice that by
definition (see Equations (8)-(6)), the random variables
$\pi^{\pm}_{t_{j+1}},\pi_{t_{j+1}}(1,1)$ and $\mu_{c^{m}p^{n}}^{\pm}$ are
$\mathcal{F}_{j}$-measurable. Also, by definition, (see Equation (3)) the
random variables $\\{\alpha_{t_{j+1}}^{0}\\},\\{\alpha_{t_{j+1}}^{1\pm}\\}$
and $\\{\alpha_{t_{j+1}}^{1,1}\\}$ are also $\mathcal{F}_{t_{j}}$-measurable.
All of this implies that $\\{\rho_{j}^{\pm}\\}$ and $\\{\psi_{j}^{\pm}\\}$ are
$\mathcal{F}_{t_{j}}-$measurable. However, due to the presence of the random
variables $\mathbbm{1}_{t_{j+1}}^{+},\mathbbm{1}_{t_{j+1}}^{-}$, the random
variables $\xi_{j+1}$ cannot be $\mathcal{F}_{t_{j}}$-measurable but they are
certainly $\mathcal{F}_{t_{j+1}}-$measurable.

Finally, all what is left is to prove that if statements (i)-(iv) hold true
for $k=j+1,j+2,\ldots,N$, then statement (i) holds true for $k=j$. Indeed,
since Equations (A-34) and (A-35) hold for $k=j+1,\dots,N$, then Equation
(A-43) holds for $k=j+1$. That is,

$\widetilde{h}_{t_{j+1}}=h_{t_{j+1}}+\sum_{i=j+1}^{N}\Delta_{t_{i}}^{t_{j+1}}\mathbb{E}(\prod_{u=j+2}^{i+1}\xi_{u}|\mathcal{F}_{t_{j+1}}).$

Multiplying both sides by $\mathbbm{1}_{t_{j+1}}^{\delta}c_{t_{j+1}}^{\delta}$
and taking the conditional expectation with respect to $\mathcal{F}_{t_{j}}$,
we have that

$\displaystyle\mathbb{E}\big{[}\widetilde{h}_{t_{j+1}}\mathbbm{1}_{t_{j+1}}^{\delta}c_{t_{j+1}}^{\delta}\big{|}\mathcal{F}_{t_{j}}\big{]}$
$\displaystyle=\mathbb{E}\Bigg{\\{}\left[h_{t_{j+1}}+\sum_{i=j+1}^{N}\Delta_{t_{i}}^{t_{j+1}}\mathbb{E}\left.\left(\prod_{u=j+2}^{i+1}\xi_{u}\right|\mathcal{F}_{t_{j+1}}\right)\right]\mathbbm{1}_{t_{j+1}}^{\delta}c_{t_{j+1}}^{\delta}\Bigg{|}\mathcal{F}_{t_{j}}\Bigg{\\}}$
$\displaystyle=h_{t_{j+1}}^{1\delta}\pi_{t_{j+1}}^{\delta}\mu_{c}^{\delta}+\pi_{t_{j+1}}^{\delta}\mu_{c}^{\delta}\sum_{i=j+1}^{N}\Delta_{t_{i}}^{t_{j}}\mathbb{E}\Bigg{[}\mathbb{E}\bigg{(}\prod_{u=j+2}^{i+1}\xi_{u}\bigg{|}\mathcal{F}_{t_{j+1}}\bigg{)}\Bigg{|}\mathcal{F}_{t_{j}},\mathbbm{1}_{t_{j+1}}^{\delta}=1\Bigg{]}$
$\displaystyle=\widetilde{h}_{t_{j+1}}^{1\delta}\pi_{t_{j+1}}^{\delta}\mu_{c}^{\delta}.$

where we used that
$\mathbb{E}\left[\prod_{u=j+2}^{i+1}\xi_{u}|\mathcal{F}_{t_{j+1}}\right]$ is
an $\mathcal{H}_{t_{j+1}}^{\varpi}$-measurable random variable. Thus, we have
that Equation (A-34) holds for $k=j$. Similarly we can prove Equation (A-35)
holds for $k=j$, which concludes the proof. ∎

###  A.3 Proofs of Section 2.4: Admissibility of the Optimal Strategy

Proof of Proposition 6. Let us define the numerators of
${}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}$ and
${}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}$ in (18) as $\beta_{t_{k}}^{\pm}$
and $\eta_{t_{k}}^{\pm}$, respectively. First, we will prove the result when
the price process is a martingale. In this case, all we need to show is that

$L_{{t_{k}}}^{+,*}+L_{{t_{k}}}^{-,*}=({}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}})I_{t_{k}}+({}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}})+({}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}})>0.$

First we prove that, under the Condition (2) (Equation (34)) and Condition (4)
of Proposition 6, $\alpha_{t_{k+1}}^{1+}=\alpha_{t_{k+1}}^{1-}$. Indeed, by
Eqs. (A-10) and (A-11),

$\displaystyle\alpha_{t_{k+1}}^{1+}$
$\displaystyle=\alpha_{t_{k+1}}^{1,1}\dfrac{\pi_{t_{k+1}}(1,1)}{\pi_{t_{k+1}}^{+}}+\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=1,\mathbbm{1}_{t_{k+1}}^{-}=0\big{]}\left(1-\dfrac{\pi_{t_{k+1}}(1,1)}{\pi_{t_{k+1}}^{+}}\right),$
$\displaystyle\alpha_{t_{k+1}}^{1-}$
$\displaystyle=\alpha_{t_{k+1}}^{1,1}\dfrac{\pi_{t_{k+1}}(1,1)}{\pi_{t_{k+1}}^{-}}+\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=0,\mathbbm{1}_{t_{k+1}}^{-}=1\big{]}\left(1-\dfrac{\pi_{t_{k+1}}(1,1)}{\pi_{t_{k+1}}^{-}}\right).$

Now, recall that $\alpha_{t_{k+1}}$ is
$\mathcal{H}_{t_{k+1}}^{\varpi}$-measurable (see Theorem 1), and, by
assumption, $\alpha_{t_{k+1}}$ depends on $\mathbbm{1}_{t_{k+1}}^{+}$ and
$\mathbbm{1}_{t_{k+1}}^{-}$ only through
$\mathbbm{1}_{t_{k+1}}^{+}+\mathbbm{1}_{t_{k+1}}^{-}$. This means that
$\alpha_{t_{k+1}}=\Phi\big{(}\mathbbm{1}_{t_{k+1}}^{+}+\mathbbm{1}_{t_{k+1}}^{-},\mathbbm{1}_{t_{k}}^{\pm},\ldots,\mathbbm{1}_{t_{k+2-\varpi}}^{\pm}\big{)}$
for some function $\Phi$. It follows that

$\displaystyle\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=1,\mathbbm{1}_{t_{k+1}}^{-}=0\big{]}$
$\displaystyle=\Phi\big{(}1+0,\mathbbm{1}_{t_{k}}^{\pm},\ldots,\mathbbm{1}_{t_{k+2-\varpi}}^{\pm}\big{)}=\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=0,\mathbbm{1}_{t_{k+1}}^{-}=1\big{]}.$

Since $\pi_{t_{k+1}}^{+}=\pi_{t_{k+1}}^{-}$, we then conclude that
$\alpha_{t_{k+1}}^{1+}=\alpha_{t_{k+1}}^{1-}$. By doing the same computations
as in the set of Equations A-9, we can obtain the analogous relations to
Equations (A-10) and (A-11) for $h_{t_{k+1}}$ and these can be used to show
that $h_{t_{k}}^{1+}=h_{t_{k}}^{1-}$ in the same way as above.

Let $\alpha_{t_{k+1}}^{1}:=\alpha_{t_{k+1}}^{1\pm}$ and
$h_{t_{k+1}}^{1}:=h_{t_{k+1}}^{1\pm}$. Since
$\pi_{t_{k+1}}^{+}=\pi_{t_{k+1}}^{-}$, it follows that

$\displaystyle\beta_{t_{k}}^{+}-\beta_{t_{k}}^{-}$
$\displaystyle={\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1}_{{t_{k+1}}})^{2}(\mu_{c}^{+}\mu_{c^{2}}^{-}-\mu_{c}^{-}\mu_{c^{2}}^{+})-\alpha^{1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{t_{k+1}}\mu_{c}^{-}\mu_{c}^{+}(\pi_{t_{k+1}}^{-}\mu_{c}^{-}-\pi_{t_{k+1}}^{+}\mu_{c}^{+})$
$\displaystyle=0.$ $\displaystyle\eta_{t_{k}}^{+}-\eta_{t_{k}}^{-}$
$\displaystyle={\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}\alpha^{1}_{{t_{k+1}}}h_{t_{k+1}}^{1}(\mu_{c}^{+}\mu_{c^{2}}^{-}-\mu_{c}^{-}\mu_{c^{2}}^{+})-h^{1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{t_{k+1}}\mu_{c}^{-}\mu_{c}^{+}(\pi_{t_{k+1}}^{-}\mu_{c}^{-}-\pi_{t_{k+1}}^{+}\mu_{c}^{+})$
$\displaystyle=0,$

implying that
${}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}=0$
and
${}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}}=0$.
Thus, it suffice to show that
${}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}>0$.
By assumption (4) of Proposition 6,
$\mu_{cp}^{\pm}=\mu_{c}^{\pm}\mu_{p}^{\pm}$ and
$\mu_{c^{2}p}^{\pm}=\mu_{c^{2}}^{\pm}\mu_{p}^{\pm}$ and, thus, we can write
${}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}$
as

$\displaystyle{}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}=\frac{1}{2\gamma_{t_{k}}}\left(\mu_{p}^{+}\Gamma^{+}\big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\big{)}+\mu_{p}^{-}\Gamma^{-}\big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\big{)}\right),$

where

$\displaystyle\Gamma^{+}(\mathbf{x},\mathbf{y})$
$\displaystyle=\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}(\alpha^{1}_{{t_{k+1}}}\mathbf{y}-\mu_{c}^{-})(\mu_{c}^{+}-2\alpha^{1}_{{t_{k+1}}}\mathbf{x})+2\big{[}\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{c}^{-}\big{]}^{2}-{\pi^{+}_{{t_{k+1}}}}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{{t_{k+1}}}(\mu_{c}^{+})^{2}\mu_{c}^{-},$
$\displaystyle\Gamma^{-}(\mathbf{x},\mathbf{y})$
$\displaystyle=\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}(\alpha^{1}_{{t_{k+1}}}\mathbf{x}-\mu_{c}^{+})(\mu_{c}^{-}-2\alpha^{1}_{{t_{k+1}}}\mathbf{y})+2\big{[}\alpha^{1,1}_{{t_{k+1}}}{\pi_{t_{k+1}}(1,1)}\mu_{c}^{+}\mu_{c}^{-}\big{]}^{2}-{\pi^{-}_{{t_{k+1}}}}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{{t_{k+1}}}(\mu_{c}^{-})^{2}\mu_{c}^{+}.$

Recall from (A-13) that $\gamma_{t_{k}}<0$ and, thus, it remains to show that
the numerator
$N({}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}})=\mu_{p}^{+}\Gamma^{+}\big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\big{)}+\mu_{p}^{-}\Gamma^{-}\big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\big{)}$
is also negative. Since $\Gamma^{+}$ is a linear function of $\mathbf{y}$,
$\mu_{c^{2}}^{-}\geq(\mu_{c}^{-})^{2}$ and, by Lemma 2,
$\alpha^{1}_{t_{k}+1}<0$, we have that

$\left.\frac{\partial}{\partial\mathbf{y}}\Gamma^{+}(\mathbf{x},\mathbf{y})\right|_{\mathbf{x}=\mu_{c^{2}}^{+}}=\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}\alpha^{1}_{t_{k+1}}(\mu_{c}^{+}-2\alpha^{1}_{t_{k+1}}\mu_{c^{2}}^{+})<0.$

From here, it follows that for every $\mathbf{x}\in\mathbb{R}$,
$\Gamma^{+}\Big{(}\mathbf{x},\mu_{c^{2}}^{-}\Big{)}\leq\Gamma^{+}\Big{(}\mathbf{x},(\mu_{c}^{-})^{2}\Big{)}$.
Similarly,

$\frac{\partial}{\partial\mathbf{x}}\Gamma^{+}(\mathbf{x},(\mu_{c}^{-})^{2})=-2\alpha^{1}_{t_{k+1}}\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}\big{[}\alpha^{1}_{t_{k+1}}(\mu_{c}^{-})^{2}-\mu_{c}^{-}\big{]}<0,$

implying that
$\Gamma^{+}\Big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\Big{)}\leq\Gamma^{+}\Big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\Big{)}$.
Note

$\displaystyle\Gamma^{+}\Big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\Big{)}$
$\displaystyle=\mu_{c}^{+}\mu_{c}^{-}\bigg{\\{}2\Big{[}\underbrace{(\alpha^{1,1}_{t_{k+1}}\pi_{t_{k+1}}(1,1))^{2}-(\alpha^{1}_{t_{k+1}})^{2}\pi^{+}_{t_{k+1}}\pi^{-}_{t_{k+1}}}_{\text{(I)}}\Big{]}\mu_{c}^{+}\mu_{c}^{-}+\underbrace{2\alpha^{1}_{t_{k+1}}\pi^{+}_{t_{k+1}}\pi^{-}_{t_{k+1}}\mu_{c}^{+}}_{\text{(II)}}$
$\displaystyle\qquad\qquad\quad+\pi^{+}_{t_{k+1}}\Big{[}\underbrace{\alpha^{1}_{t_{k+1}}\pi^{-}_{t_{k+1}}\mu_{c}^{-}-\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)\mu_{c}^{+}}_{\text{(III)}}\Big{]}-\pi^{+}_{t_{k+1}}\pi^{-}_{t_{k+1}}\bigg{\\}}.$

By Equation (A-12),
$(\alpha^{1,1}_{t_{k+1}}\pi_{t_{k+1}}(1,1))^{2}\leq(\alpha^{1}_{t_{k+1}})^{2}\pi^{+}_{t_{k+1}}\pi^{-}_{t_{k+1}}$,
implying that the term (I) above is negative. Further, by Lemma 2, we know
that $\alpha_{t_{k+1}}<0$ and, thus, the term (II) above is also negative.
Finally, by using our assumption (33) and Eq. (A-12), we can conclude that the
term (III) above also negative. It follows that
$\Gamma^{+}\Big{(}\mu_{c^{2}}^{+},\mu_{c^{2}}^{-}\Big{)}\leq\Gamma^{+}\Big{(}(\mu_{c}^{+})^{2},(\mu_{c}^{-})^{2}\Big{)}<0$.
Similarly, we can show that
$\Gamma^{-}\Big{(}\mu_{c^{2}}^{+},(\mu_{c}^{-})^{2}\Big{)}<0$ and conclude
that
$N{({}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}})}<0$.

For the general midprice dynamics, we need to recall Eqs. (17). Then, under
our conditions (1)-(4), we have trivially that $\rho_{k}^{+}=\rho_{k}^{-}$ and
$\psi_{k}^{+}=\psi_{k}^{-}$. It then follows from Eq. (29) that
$\widetilde{L}_{{t_{k}}}^{+,*}+\widetilde{L}_{{t_{k}}}^{-,*}={L}_{{t_{k}}}^{+,*}+{L}_{{t_{k}}}^{-,*}>0$,
as we have already shown. This concludes the general dynamics of the midprice
process. ∎

Proof of Corollary 7. The proof follows along the same lines as the one of
Proposition 6. As before, since $\alpha_{t_{k+1}}$ depends on
$\mathbbm{1}_{t_{k}}^{+}$ and $\mathbbm{1}_{t_{k}}^{-}$ only through
$\mathbbm{1}_{t_{k}}^{+}+\mathbbm{1}_{t_{k}}^{-}$,

$\alpha_{t_{k+1}}^{1,0}=\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=1,\mathbbm{1}_{t_{k+1}}^{-}=0\big{]}=\mathbb{E}\big{[}\alpha_{t_{k+1}}|\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=0,\mathbbm{1}_{t_{k+1}}^{-}=1\big{]}=\alpha_{t_{k+1}}^{0,1},$

and, thus, recalling (A-10)-(A-11), it follows that
$\alpha_{t_{k+1}}^{1+}=\alpha_{t_{k+1}}^{1-}$ when $\pi_{t_{k+1}}(1,1)=0$.
Similarly, $h_{t_{k+1}}^{1+}=h_{t_{k+1}}^{1-}$ and we can denote
$\alpha_{t_{k+1}}^{1}:=\alpha_{t_{k+1}}^{1\pm}$ and
$h_{t_{k+1}}^{1}:=h_{t_{k+1}}^{1\pm}$. Since $\pi_{t_{k+1}}(1,1)=0$,
$\mu_{c}^{+}=\mu_{c}^{-}$ and $\mu_{c^{2}}^{+}=\mu_{c^{2}}^{-}$, it also
follows that $\beta_{t_{k}}^{+}-\beta_{t_{k}}^{-}=0$ and
$\eta_{t_{k}}^{+}-\eta_{t_{k}}^{-}=0$, implying that
${}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}=0$
and
${}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}-{}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}}=0$.
We are only left to prove that
${}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}>0$,
but as the denominator is negative, we only need to prove that its numerator,
denoted by
$N({}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}})$,
is also negative. However, under our conditions, we have that

$\displaystyle
N({}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}})$
$\displaystyle=\pi^{+}_{{t_{k+1}}}\pi^{-}_{{t_{k+1}}}\big{(}\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}}-\mu_{c}\big{)}\bigg{[}\big{(}\mu_{cp}^{+}-2\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}p}^{+}\big{)}+\big{(}\mu_{cp}^{-}-2\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}p}^{-}\big{)}\bigg{]},$

which is negative because $\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}}-\mu_{c}<0$ (since
$\alpha^{1}_{t_{k+1}}\leq 0$), and each of the terms inside the brackets above
are positive. For the general midprice dynamics notice that our conditions and
the definitions (17) readily imply that $\rho_{k}^{+}=\rho_{k}^{-}$ and
$\psi_{k}^{+}=\psi_{k}^{-}=0$, and the proof follows as in the proof of
Proposition 6. ∎

Proof of Lemma 8. The proof will done by backwards induction. We only give the
details for $\\{\alpha_{t_{k}}\\}_{k=1}^{N+1}$ (the proof for
$\\{h_{t_{k}}\\}_{k=1}^{N+1}$ is very similar). To start, note that at time
$k=N+1$, $\alpha_{t_{N+1}}=-\lambda$ and, hence, it trivially satisfies the
condition. For the inductive step, assume that the lemma holds for $j=k+1$. We
will now proceed to prove that $\alpha_{t_{k}}$ depends on
$\boldsymbol{e}_{t_{k}}^{+},\boldsymbol{e}_{t_{k}}^{-}$ only thorugh
$\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}$. By Equation (A-7),
$\alpha_{t_{k}}=\alpha_{t_{k+1}}^{0}+N_{k}/\gamma_{t_{k}}$ with

$\displaystyle N_{k}$
$\displaystyle=(\alpha_{t_{k+1}}^{1+}\mu_{c}^{+}\pi_{t_{k+1}}^{+})^{2}\pi_{t_{k+1}}^{-}(\alpha_{t_{k+1}}^{1-}\mu_{c^{2}}^{-}-\mu_{c}^{-})+(\alpha_{t_{k+1}}^{1-}\mu_{c}^{-}\pi_{t_{k+1}}^{-})^{2}\pi_{t_{k+1}}^{+}(\alpha_{t_{k+1}}^{1+}\mu_{c^{2}}^{+}-\mu_{c}^{+})$
(A-44)
$\displaystyle\quad-2\alpha_{t_{k+1}}^{1+}\alpha_{t_{k+1}}^{1-}{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}\pi_{t_{k+1}}(1,1)\alpha^{1,1}_{{t_{k+1}}}(\mu_{c}^{+}\mu_{c}^{-})^{2},$
$\displaystyle\gamma_{t_{k}}$
$\displaystyle=\big{[}{\pi_{t_{k+1}}(1,1)}\alpha^{1,1}_{{t_{k+1}}}\mu_{c}^{+}\mu_{c}^{-}\big{]}^{2}-{\pi^{+}_{{t_{k+1}}}}{\pi^{-}_{{t_{k+1}}}}(\alpha^{1+}_{{t_{k+1}}}\mu_{c^{2}}^{+}-\mu_{c}^{+})(\alpha^{1-}_{{t_{k+1}}}\mu_{c^{2}}^{-}-\mu_{c}^{-}).$

By assumption, $\pi_{t_{k+1}}^{+}$, $\pi_{t_{k+1}}^{-}$, and
$\pi_{t_{k+1}}(1,1)$ are functions of
$\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}$. Then, we only need to
show that $\alpha_{t_{k+1}}^{0}$, $\alpha_{t_{k+1}}^{1+}$,
$\alpha_{t_{k+1}}^{1-}$, and $\alpha_{t_{k+1}}^{1,1}$ have the same property.
By the induction hypothesis, we have that $\alpha_{t_{k+1}}$ is of the form:

$\alpha_{t_{k+1}}=\Phi(\boldsymbol{e}_{t_{k+1}}^{+}+\boldsymbol{e}_{t_{k+1}}^{-})=\Phi\big{(}\mathbbm{1}_{t_{k+1}}^{+}+\mathbbm{1}_{t_{k+1}}^{-},\ldots,\mathbbm{1}_{t_{k+2-\varpi}}^{+}+\mathbbm{1}_{t_{k+2-\varpi}}^{-}\big{)},$

for a function $\Phi:\\{0,1,2\\}^{\varpi}\to\mathbb{R}$. Thus, using (A-4) and
(6),

$\displaystyle\alpha_{t_{k+1}}^{0}$
$\displaystyle=\mathbb{E}[\alpha_{t_{k+1}}\,|\,\mathcal{F}_{t_{k}}]$
$\displaystyle=\sum\limits_{u=0}^{1}\sum\limits_{v=0}^{1}\Phi\left(u+v,\mathbbm{1}_{t_{k}}^{+}+\mathbbm{1}_{t_{k}}^{-}\ldots,\mathbbm{1}_{t_{k+2-\varpi}}^{+}+\mathbbm{1}_{t_{k+2-\varpi}}^{-}\right)\mathbb{P}\left[\mathbbm{1}_{t_{k+1}}^{+}=u,\mathbbm{1}_{t_{k+1}}^{-}=v\,\Big{|}\,\mathcal{H}_{t_{k}}^{\varpi}\right],$

implying that $\alpha_{t_{k+1}}^{0}$ is a function of
$\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}$ in light of (9). We
can similarly deal with $\alpha_{t_{k+1}}^{1+}$, $\alpha_{t_{k+1}}^{1-}$, and
$\alpha_{t_{k+1}}^{11}$ using the expressions in (A-6). For instance,

$\displaystyle\alpha_{t_{k+1}}^{1+}$
$\displaystyle=\mathbb{E}[\alpha_{t_{k+1}}\,|\,\mathcal{F}_{t_{k}},\mathbbm{1}_{t_{k+1}}^{+}=1]$
$\displaystyle=\sum_{\ell\in\\{0,1\\}}\Phi(1+\ell,\mathbbm{1}_{t_{k}}^{+}+\mathbbm{1}_{t_{k}}^{-},\dots,\mathbbm{1}_{t_{k+2-\varpi}}^{+}+\mathbbm{1}_{t_{k+2-\varpi}}^{-})\mathbb{P}\big{[}\mathbbm{1}_{t_{k+1}}^{-}=\ell\big{|}\mathcal{F}_{t_{j}},\mathbbm{1}_{t_{k+1}}^{+}=1\big{]}$
$\displaystyle=\sum_{\ell\in\\{0,1\\}}\Phi(1+\ell,\mathbbm{1}_{t_{k}}^{+}+\mathbbm{1}_{t_{k}}^{-},\dots,\mathbbm{1}_{t_{k+2-\varpi}}^{+}+\mathbbm{1}_{t_{k+2-\varpi}}^{-})\frac{\mathbb{P}\big{[}\mathbbm{1}_{t_{k+1}}^{+}=1,\mathbbm{1}_{t_{k+1}}^{-}=\ell\big{|}\mathcal{H}_{t_{j}}^{\varpi}\big{]}}{\mathbb{P}\big{[}\mathbbm{1}_{t_{k+1}}^{+}=1\big{|}\mathcal{H}_{t_{k}}^{\varpi}\big{]}},$

which now is evident that is a function of
$\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}$. ∎

###  A.4 Proofs of Section 2.5: Inventory Analysis of the optimal strategy

Proof of Proposition 9. For simplicity, we omit ‘*’ when referring to the
optimal strategies $L_{{t_{k}}}^{\pm,*}$. By Eq. (11) and Corollary 3, we have
that

$I_{t_{k+1}}=I_{t_{k}}-\mathbbm{1}_{t_{k+1}}^{+}c_{t_{k+1}}^{+}(p_{t_{k+1}}^{+}-L_{t_{k}}^{+})+\mathbbm{1}_{t_{k+1}}^{-}c_{t_{k+1}}^{-}(p_{t_{k+1}}^{-}-L_{t_{k}}^{-})$

with
$L_{{t_{k}}}^{+}={}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}I_{t_{k}}+{}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}$
and
$L_{{t_{k}}}^{-}=-\,{}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}I_{t_{k}}-{}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}}+{}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}$.
Under the condition $\pi_{t_{k+1}}(1,1)=0$, the coefficients simplify to

${}^{\scaleto{(1)}{5pt}}\\!A^{\pm}_{{t_{k}}}=\frac{\mu_{c}^{\pm}\alpha^{1\pm}_{{t_{k+1}}}}{\mu_{c}^{\pm}-\alpha^{1\pm}_{{t_{k+1}}}\mu_{c^{2}}^{\pm}},\quad{}^{\scaleto{(2)}{5pt}}\\!A^{\pm}_{{t_{k}}}=\frac{h^{1\pm}_{{t_{k+1}}}}{2}\frac{\mu_{c}^{\pm}}{\mu_{c}^{\pm}-\alpha^{1\pm}_{{t_{k+1}}}\mu_{c^{2}}^{\pm}},\quad{}^{\scaleto{(3)}{5pt}}\\!A^{\pm}_{{t_{k}}}=\frac{\alpha^{1\pm}_{{t_{k+1}}}}{2}\frac{\mu_{cp}^{\pm}-2\alpha^{1\pm}_{{t_{k+1}}}\mu_{c^{2}p}^{\pm}}{\mu_{c}^{\pm}-\alpha^{1\pm}_{{t_{k+1}}}\mu_{c^{2}}^{\pm}}.$

As explained in the proof of Corollary 7,
${}^{\scaleto{(1)}{5pt}}\\!A^{+}_{{t_{k}}}={}^{\scaleto{(1)}{5pt}}\\!A^{-}_{{t_{k}}}$,
${}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}={}^{\scaleto{(2)}{5pt}}\\!A^{-}_{{t_{k}}}$,
$\alpha_{t_{k+1}}^{1+}=\alpha_{t_{k+1}}^{1-}$, and
$h_{t_{k+1}}^{1+}=h_{t_{k+1}}^{1-}$. In particular, we also have
${}^{\scaleto{(3)}{5pt}}\\!A^{+}_{{t_{k}}}={}^{\scaleto{(3)}{5pt}}\\!A^{-}_{{t_{k}}}$.
Hereafter, we omit $\pm$ in the superscripts of
${}^{\scaleto{(\ell)}{5pt}}\\!A^{\pm}_{{t_{k}}}$, $\alpha_{t_{k+1}}^{1\pm}$,
$h_{t_{k+1}}^{1\pm}$, etc. Furthermore, since
$\pi_{t_{k+1}}=\pi^{+}_{t_{k+1}}=\pi^{-}_{t_{k+1}}$, we can see that (20)
simplifies to

$\displaystyle h_{{t_{k}}}$
$\displaystyle=h^{0}_{{t_{k+1}}}+2\pi_{t_{k+1}}\Bigg{\\{}2\Big{(}\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}}-\mu_{c}\Big{)}\;{}^{\scaleto{(1)}{5pt}}\\!A_{{t_{k}}}{}^{\scaleto{(2)}{5pt}}\\!A_{{t_{k}}}+2\alpha^{1}_{{t_{k+1}}}\mu_{c}{}^{\scaleto{(2)}{5pt}}\\!A^{\delta}_{{t_{k}}}+{}^{\scaleto{(1)}{5pt}}\\!A_{{t_{k}}}h^{1}_{{t_{k+1}}}\mu_{c}\Bigg{\\}}$
$\displaystyle=h^{0}_{{t_{k+1}}}+2\pi_{t_{k+1}}\frac{\alpha^{1}_{{t_{k+1}}}\mu_{c}^{2}}{\mu_{c}-\alpha^{1}_{{t_{k+1}}}\mu_{c^{2}}}h^{1}_{{t_{k+1}}},$

which, combined with the condition $h_{t_{N+1}}=0$, implies that
$h_{{t_{k}}}\equiv 0$ and, thus,
${}^{\scaleto{(2)}{5pt}}\\!A^{+}_{{t_{k}}}\equiv 0$, for all $k$.

We can then write

$I_{t_{k+1}}=\xi_{t_{k+1}}+\Big{(}1+\eta_{t_{k+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{{t_{k}}}\Big{)}I_{t_{k}},$

where

$\displaystyle\eta_{t_{k+1}}$
$\displaystyle:=\mathbbm{1}_{t_{k+1}}^{+}c_{t_{k+1}}^{+}+\mathbbm{1}_{t_{k+1}}^{-}c_{t_{k+1}}^{-},$
$\displaystyle\xi_{t_{k+1}}$
$\displaystyle:=(\mathbbm{1}_{t_{k+1}}^{+}c_{t_{k+1}}^{+}-\mathbbm{1}_{t_{k+1}}^{-}c_{t_{k+1}}^{-}){}^{\scaleto{(3)}{5pt}}\\!A_{{t_{k}}}-(\mathbbm{1}_{t_{k+1}}^{+}c_{t_{k+1}}^{+}p_{t_{k+1}}^{+}-\mathbbm{1}_{t_{k+1}}^{-}c_{t_{k+1}}^{-}p_{t_{k+1}}^{-}).$

Since $\mathbb{E}[\xi_{t_{k+1}}|\mathcal{F}_{t_{k}}]=0$ and
$\mathbb{E}[\eta_{t_{k+1}}|\mathcal{F}_{t_{k}}]=2\pi_{t_{k+1}}\mu_{c}$, we can
then conclude (36)-(i).

For (36)-(ii), first note that, by simple induction, we have that, when
$I_{0}=0$,

$\displaystyle
I_{t_{k+1}}=\sum_{j=1}^{k+1}\xi_{t_{j}}\prod_{\ell=j}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}}),$
(A-45)

under the usual convention that $\prod_{\ell=k+1}^{k}=1$. Next, we claim that,
for any $i$, there is a function $\Phi_{i}:\\{0,1,2\\}^{\varpi}\to\mathbb{R}$
such that

$\displaystyle\mathbb{E}\left[\left.\prod_{\ell=i}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right|\mathcal{F}_{t_{i}}\right]=\Phi_{i}\left(\boldsymbol{e}_{t_{i}}^{+}+\boldsymbol{e}_{t_{i}}^{-}\right).$
(A-46)

Indeed, as shown in the proof of Lemma 8, $\alpha_{t_{k+1}}^{1\pm}$ are
functions of $\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}$ and,
thus,
${}^{\scaleto{(1)}{5pt}}\\!A_{t_{k}}=\Psi_{k}(\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-})$,
for some function $\Psi_{k}$. Then, by conditioning on
$\mathbbm{1}_{t_{k+1}}^{+}$ and $\mathbbm{1}_{t_{k+1}}^{-}$,

$\mathbb{E}\left[\left.(1+\eta_{t_{k+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{k}})\right|\mathcal{F}_{t_{k}}\right]=1+2\mu_{c}\pi_{t_{k+1}}\Psi_{k}(\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-})=:\Phi_{k}\left(\boldsymbol{e}_{t_{k}}^{+}+\boldsymbol{e}_{t_{k}}^{-}\right).$

Suppose (A-46) holds for $i=j+1$. Then,

$\displaystyle\mathbb{E}\left[\left.\prod_{\ell=j}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right|\mathcal{F}_{t_{j}}\right]$
$\displaystyle=\mathbb{E}\left[\left.(1+\eta_{t_{j+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{j}})\mathbb{E}\left[\left.\prod_{\ell=j+1}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right|\mathcal{F}_{t_{j+1}}\right]\right|\mathcal{F}_{t_{j}}\right]$
$\displaystyle=\mathbb{E}\left[\left.(1+\eta_{t_{j+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{j}})\Phi_{j+1}\left(\boldsymbol{e}_{t_{j+1}}^{+}+\boldsymbol{e}_{t_{j+1}}^{-}\right)\right|\mathcal{F}_{t_{j}}\right]$
$\displaystyle=2(1+2\mu_{c}\Psi_{j}(\boldsymbol{e}_{t_{j}}^{+}+\boldsymbol{e}_{t_{j}}^{-}))\Phi_{j+1}(1,\mathbbm{1}_{t_{j}}^{+}+\mathbbm{1}_{t_{j}}^{-},\dots,\mathbbm{1}_{t_{j+2-\varpi}}^{+}+\mathbbm{1}_{t_{j+2-\varpi}}^{-})\pi_{t_{j+1}}$
$\displaystyle\quad+\Phi_{j+1}(0,\mathbbm{1}_{t_{j}}^{+}+\mathbbm{1}_{t_{j}}^{-},\dots,\mathbbm{1}_{t_{j+2-\varpi}}^{+}+\mathbbm{1}_{t_{j+2-\varpi}}^{-})(1-2\pi_{t_{j+1}}),$

which is clearly of the form
$\Phi_{j}\left(\boldsymbol{e}_{t_{j}}^{+}+\boldsymbol{e}_{t_{j}}^{-}\right)$.
This shows the validity of (A-46).

We are now ready to show the result:

$\displaystyle\mathbb{E}\left[\left.\xi_{t_{j}}\prod_{\ell=j}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right|\mathcal{F}_{t_{j-1}}\right]$
$\displaystyle=\mathbb{E}\left[\left.\xi_{t_{j}}\mathbb{E}\left[\left.\prod_{\ell=j}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right|\mathcal{F}_{t_{j}}\right]\right|\mathcal{F}_{t_{j-1}}\right]$
$\displaystyle=\mathbb{E}\left[\left.\xi_{t_{j}}\Phi_{j}\left(\boldsymbol{e}_{t_{j}}^{+}+\boldsymbol{e}_{t_{j}}^{-}\right)\right|\mathcal{F}_{t_{j-1}}\right]$
$\displaystyle=\left(\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=1,\mathbbm{1}_{t_{j}}^{-}=0\right]+\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=0,\mathbbm{1}_{t_{j}}^{-}=1\right]\right)$
$\displaystyle\quad\quad\times\Phi_{j}(1,\mathbbm{1}_{t_{j-1}}^{+}+\mathbbm{1}_{t_{j-1}}^{-},\dots,\mathbbm{1}_{t_{j+1-\varpi}}^{+}+\mathbbm{1}_{t_{j+1-\varpi}}^{-})\pi_{t_{j}}$
$\displaystyle\quad+\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=0,\mathbbm{1}_{t_{j}}^{-}=0\right]\Phi_{j}(0,\mathbbm{1}_{t_{j-1}}^{+}+\mathbbm{1}_{t_{j-1}}^{-},\dots,\mathbbm{1}_{t_{j+1-\varpi}}^{+}+\mathbbm{1}_{t_{j+1-\varpi}}^{-})(1-2\pi_{t_{j+1}}),$

which, recalling that
$\xi_{t_{j}}:=(\mathbbm{1}_{t_{j}}^{+}c_{t_{j}}^{+}-\mathbbm{1}_{t_{j}}^{-}c_{t_{j}}^{-}){}^{\scaleto{(3)}{5pt}}\\!A_{t_{j-1}}-(\mathbbm{1}_{t_{j}}^{+}c_{t_{j}}^{+}p_{t_{j}}^{+}-\mathbbm{1}_{t_{j}}^{-}c_{t_{j}}^{-}p_{t_{j}}^{-})$,
equals to $0$ because
$\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=0,\mathbbm{1}_{t_{j}}^{-}=0\right]=0$
and

$\displaystyle\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=1,\mathbbm{1}_{t_{j}}^{-}=0\right]$
$\displaystyle=\mathbb{E}\left[\left.c^{+}_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=1\right]{}^{\scaleto{(3)}{5pt}}\\!A_{t_{j-1}}-\mathbb{E}\left[\left.c^{+}_{t_{j}}p^{+}_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=1\right]$
$\displaystyle=\mu_{c}{}^{\scaleto{(3)}{5pt}}\\!A_{t_{j-1}}-\mu_{cp}$
$\displaystyle\mathbb{E}\left[\left.\xi_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{+}=0,\mathbbm{1}_{t_{j}}^{-}=1\right]$
$\displaystyle=-\mathbb{E}\left[\left.c^{-}_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{-}=1\right]{}^{\scaleto{(3)}{5pt}}\\!A_{t_{j-1}}+\mathbb{E}\left[\left.c^{-}_{t_{j}}p^{-}_{t_{j}}\right|\mathcal{F}_{t_{j-1}},\mathbbm{1}_{t_{j}}^{-}=1\right]$
$\displaystyle=-\mu_{c}{}^{\scaleto{(3)}{5pt}}\\!A_{t_{j-1}}+\mu_{cp}.$

We then conclude that
$\mathbb{E}\left[\xi_{t_{j}}\prod_{\ell=j}^{k}(1+\eta_{t_{\ell+1}}{}^{\scaleto{(1)}{5pt}}\\!A_{t_{\ell}})\right]=0$
and, thus, in light of the representation (A-45), we conclude that
$\mathbb{E}\left[I_{t_{k+1}}\right]=0$, for any $k$. ∎

###  A.5 Proofs of Section 2.6: Running inventory penalization

Proof of Theorem 10. Plugging the ansatz (40) into 39, we arrive to the
equation

$\displaystyle
W_{t_{k}}+{}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k}}}I_{t_{k}}^{2}+S_{t_{k}}I_{t_{k}}+{}^{\scaleto{(\phi)}{5pt}}\\!h_{{t_{k}}}I_{t_{k}}+{}^{\scaleto{(\phi)}{5pt}}\\!g_{{t_{k}}}$
(A-47)
$\displaystyle\quad=\sup_{L^{\pm}_{t_{k}}\in\mathcal{A}_{t_{k}}}\mathbb{E}\Big{[}W_{t_{k+1}}+{}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}I_{t_{k+1}}^{2}+S_{t_{k+1}}I_{t_{k+1}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{}^{\scaleto{(\phi)}{5pt}}\\!h_{{t_{k+1}}}I_{t_{k+1}}}+{}^{\scaleto{(\phi)}{5pt}}\\!g_{{t_{k+1}}}-\phi
I_{t_{k+1}}^{2}\,\Big{|}\,\mathcal{F}_{t_{k}}\Big{]}.$
$\displaystyle\quad=\sup_{L^{\pm}_{t_{k}}\in\mathcal{A}_{t_{k}}}\mathbb{E}\Big{[}W_{t_{k+1}}+({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}-\phi)I_{t_{k+1}}^{2}+S_{t_{k+1}}I_{t_{k+1}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{}^{\scaleto{(\phi)}{5pt}}\\!h_{{t_{k+1}}}I_{t_{k+1}}}+{}^{\scaleto{(\phi)}{5pt}}\\!g_{{t_{k+1}}}\,\Big{|}\,\mathcal{F}_{t_{k}}\Big{]}.$

The above expression has the same structure as (16), but with
$\alpha_{{t_{k+1}}}$ replaced by
${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}-\phi$. This is the reason why
the results of Theorem 1 and Corollary 3 follow along the same steps as in
Theorem 1 and Corollary 3 with the term $\alpha_{{t_{k+1}}}$ (and its
variants) replaced by ${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}-\phi$.
The proof of Lemma 2 also follows though it needs a bit more of care. Indeed,
this proof is heavily based on the inequalities (A-12), (A-18), (A-26), which
still hold for ${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}^{1\pm}$ and
${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}^{1,1}$, but we need them to
hold for ${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}^{1\pm}-\phi$ and
${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{{t_{k+1}}}^{1,1}-\phi$. Indeed, for
(A-12), since $\pi_{t_{k+1}}(1,1)\leq\pi_{t_{k+1}}^{\pm}$, then
$-\phi\pi_{t_{k+1}}^{\pm}\leq-\phi\pi_{t_{k+1}}(1,1)$, which together with
${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1\pm}\pi_{t_{k+1}}^{\pm}\leq{}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)$
implies
$({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1\pm}-\phi)\pi_{t_{k+1}}^{+}\leq({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1,1}-\phi)\pi_{t_{k+1}}(1,1)$.
The proof of (A-18) is similar. For the second inequality in (A-26), since
$\pi_{t_{k+1}}^{+}+\pi_{t_{k+1}}^{-}-\pi_{t_{k+1}}(1,1)\leq 1$, we have
$-\phi\pi_{t_{k+1}}(1,1)\leq{}-\phi\pi_{t_{k+1}}^{+}-\phi\pi_{t_{k+1}}^{-}+\phi$,
which, together with
${}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1,1}\pi_{t_{k+1}}(1,1)\leq{}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1+}\pi_{t_{k+1}}^{+}+{}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1-}\pi_{t_{k+1}}^{-}-\alpha_{t_{k+1}}^{0}$,
implies

$({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1,1}-\phi)\pi_{t_{k+1}}(1,1)\leq({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1+}-\phi)\pi_{t_{k+1}}^{+}+({}^{\scaleto{(\phi)}{5pt}}\\!\alpha_{t_{k+1}}^{1-}-\phi)\pi_{t_{k+1}}^{-}-(\alpha_{t_{k+1}}^{0}-\phi).$

The rest of the proof follows the same arguments as in the proof of Lemma 2. ∎

## References

  * Adrian _et al._ (2020) Adrian, T., Capponi, A., Fleming, M., Vogt, E. & Zhang, H. (2020).  Intraday market making with overnight inventory costs.  _Journal of Financial Markets_ 50, 100564.
  * Avellaneda & Stoikov (2008) Avellaneda, M. & Stoikov, S. (2008).  High-frequency trading in a limit order book.  _Quantitative Finance_ 8(3), 217–224.
  * Barger & Lorig (2018) Barger, W. & Lorig, M. (2019).  Optimal liquidation under stochastic price impact.  _International Journal of Theoretical and Applied Finance_ , 22(02):1850059.
  * Bäuerle & Rieder (2011) Bäuerle, N. & Rieder, U. (2011).  _Markov Decision Processes with Application to Finance_.  Springer New York.
  * Becherer et al. (2018) Becherer, D., Bilarev, T., & Frentrup, P. (2018).  Optimal liquidation under stochastic liquidity.  _Finance and Stochastics_ , 22, 39-68.
  * Bergault & Guéant (2021) Bergault, P. & Guéant, O. (2021).  Size matters for OTC market makers: general results and dimensionality reduction techniques.  _Mathematical Finance_ , 31(1):279-322.
  * Bradfield (1979) Bradfield, J. (1979).  A formal dynamic model of market making.  _Journal of Financial and Quantitative Analysis_ 14, 275–291.
  * Bhattacharya & Saar (2021) Bhattacharya, A. & Saar, G. (2021).  Limit order markets under asymmetric information.  _SSRN 3688473_.
  * Capponi _et al._ (2021) Capponi, A., Figueroa-López, J. & Yu, C. (2021).  Market Making with Stochastic Liquidity Demand: Simultaneous Order Arrival and Price Change Forecasts.  _arXiv preprint arXiv:2101.03086_.
  * Cartea _et al._ (2021) Cartea, Á., Jaimungal, S. & Sánchez-Betancourt, L. (2021).  Latency and Liquidity Risk.  _International Journal of Theoretical and Applied Finance_ , 24(6).
  * Cartea _et al._ (2015) Cartea, Á., Jaimungal, S. & Penalva, J. (2015).  _Algorithmic and High-Frequency Trading_.  Cambridge University Press.
  * Cartea & Jaimungal (2015) Cartea, Á. & Jaimungal, S. (2015).  Risk metrics and fine tuning of high-frequency trading strategies.  _Mathematical Finance_ 25(3), 576–611.
  * Cartea _et al._ (2014) Cartea, A., Jaimungal, S. & Ricci, J. (2014).  Buy low, sell high: A high frequency trading perspective.  _SIAM Journal on Financial Mathematics_ 5(1), 415–444.
  * Cartea & Jaimungal (2015) Cartea, Á., Donnelly, R., & Jaimungal, S. (2017).  Algorithmic trading with model uncertainty.  _SIAM Journal on Financial Mathematics_ , 8(1):635–671.
  * Cetin & Waelbroeck (2023) Cetin, U., & Waelbroeck, H. (2023).  Power laws in market microstructure.  _Frontiers of Mathematical Finance_ , 2(1), 56-98.
  * Cvitanic & Kirilenko (2010) Cvitanic, J. & Kirilenko, A. A. (2010).  High frequency traders and asset prices.  _Working paper. Preprint available at SSRN 1569067_ .
  * Fodra & Phan (2015) Fodra, P. & Pham, H. (2015).  High frequency trading and asymptotics for small risk aversion in a Markov renewal model.  _SIAM Journal on Financial Mathematics_ , 6(1):656-684.
  * Fodra & Phan (2015) Fodra, P. & Pham, H. (2015).  Semi-markov model for market microstructure.  _Quantitative Finance_ , 22(3):261-295.
  * Gao & Wang (2020) Gao, X. & Wang, X. (2020).  Optimal market making in the presence of latency.  _Quantitative Finance_ , 20(9).
  * Guéant _et al._ (2013) Guéant, O., Lehalle, C.-A. & Fernandez-Tapia, J. (2013).  Dealing with the inventory risk: a solution to the market making problem.  _Mathematics and Financial Economics_ 7(4), 477–507.
  * Guéant (2016) Guéant, O. (2016).  _The Financial Mathematics of Market Liquidity: From optimal execution to market making_ , volume 33. CRC Press.
  * Guilbaud & Phan (2013) Guilbaud, F. & Pham, H. (2013).  Optimal high-frequency trading with limit and market orders.  _Quantitative Finance_ 13(1): 79-94.
  * Guilbaud & Phan (2015) Guilbaud, F. & Pham, H. (2015).  Optimal high-frequency trading in a pro rata microstructure with predictive information.  _Mathematical Finance_ 25(3): 545-575.
  * Hernández-Lerma & Lasserre (1996) Hernández-Lerma, O. & Lasserre, J. (1996).  _Discrete-Time Markov Control Processes_.  Springer New York.
  * Hendershott & Menkveld (2014) Hendershott, T. & Menkveld, A. J. (2014).  Price pressures.  _Journal of Financial Economics_ 114(3), 405–423.
  * Ho & Stoll (1981) Ho, T. & Stoll, H. R. (1981).  Optimal dealer pricing under transactions and return uncertainty.  _Journal of Financial Economics_ 9(1), 47–73.
  * Nyström et al. (2014) Nyström, K., Ould Aly, S. M., & Zhang, C. (2014)  Market making and portfolio liquidation under uncertainty.  _International Journal of Theoretical and Applied Finance_ , 17(5).
  * O’Hara & Oldfield (1986) O’Hara, M. & Oldfield, G. (1986).  The microeconomics of market making.  _Journal of Financial and Quantitative Analysis_ 21(4), 2603–2619.