# Optimal execution and speculation
with trade signals

Peter Bank  Institut für Mathematik, Technische Universität Berlin, Berlin,
Germany Álvaro Cartea  Oxford-Man Institute of Quantitative Finance,
University of Oxford, Oxford, UK  Mathematical Institute, University of
Oxford, Oxford, UK  Laura Körber  Institut für Mathematik, Technische
Universität Berlin, Berlin, Germany Oxford-Man Institute of Quantitative
Finance, University of Oxford, Oxford, UK

###### Abstract

We propose a price impact model where changes in prices are purely driven by
the order flow in the market. The stochastic price impact of market orders and
the arrival rates of limit and market orders are functions of the market
liquidity process which reflects the balance of the demand and supply of
liquidity. Limit and market orders mutually excite each other so that
liquidity is mean reverting. We use the theory of Meyer-$\sigma$-fields to
introduce a short-term signal process from which a trader learns about
imminent changes in order flow. In this setting, we examine an optimal
execution problem and derive the Hamilton–Jacobi–Bellman (HJB) equation for
the value function. The HJB equation is solved numerically and we illustrate
how the trader uses the signal to enhance the performance of execution
problems and to execute speculative strategies.

Mathematical Subject Classification (2022)

91B70, 93E20

Keywords

Optimal execution, speculation, trade signal, Meyer $\sigma$-field, stochastic
optimal control

##  1 Introduction

††P. Bank and L. Körber are supported by Deutsche Forschungsgemeinschaft
through IRTG 2544. L. Körber is grateful to the Oxford-Man-Institute of
Quantitative Finance for their generous hospitality. We are also grateful to
participants at the Mathematical Finance Seminar at Imperial College, the
German Probability and Statistics Days in Essen, the SIAM Conference on
Financial Mathematics and Engineering in Philadelphia and the Women in
Mathematical Finance Conference at Rutgers University.

In this paper, we propose a reduced-form model where the evolution of prices
is determined by the order flow in the market. In our model, changes in the
asset’s price are caused by market order flow, which is modelled as a pure
jump process. Thus, we do not require exogenous dynamics for the evolution of
the unaffected fundamental price as is generally proposed in standard models
including Almgren and Chriss [2000] and Obizhaeva and Wang [2013]. Here, the
price impact of a market order depends on the market’s liquidity which is
given by the difference between the volume of posted limit orders and the
volume of market orders and cancellations. When this difference is high, i.e.,
when the market is liquid, the price impact of the next market order is low;
and when this difference is low, the price impact is high. Our model may be
viewed as a permanent price impact model with finite _market depth_ similar to
that in Huberman and Stanzl [2004]. However, our impact specification depends
on liquidity and we do not require an exogenous fundamental price.

For simplicity, we do not distinguish between the volumes posted on the bid or
ask side of the book, a model extension we leave for future research. This
allows us to specify the _market’s tightness_ with a constant spread. The
_market’s resilience_ is captured by letting the arrival rates of market
orders, cancellations and limit orders depend on the liquidity in the market:
when the level of liquidity is low, the arrival rate of further market orders
is low and the arrival rate of limit orders is high, and vice-versa when
liquidity is high, see Cartea et al. [2020] who provide empirical evidence of
this effect based on data from the Nasdaq stock exchange. As a consequence,
resilience is an endogenous feature in our model as opposed to models such as
the one by Obizhaeva and Wang [2013] where resilience is an exponential
relaxation of price impact towards zero.

We introduce a trader who uses market orders to complete an execution
programme, where her objective is to maximize expected utility of terminal
wealth and the trader does not provide liquidity to the book. She receives a
private signal about the arrival of other limit and market orders and uses it
to execute informed trades ahead of other orders arrive in the book.

In our model, the trader’s market orders affect the price of the asset and the
liquidity in the market in the same way as the _external_ orders sent by other
market participants. Due to the resilience of the book in our model, the
trader’s orders will trigger liquidity provision which may incentivise pump-
and-dump schemes. In such cases, to ensure a well-functioning market, we
introduce a circuit breaker that imposes a minimum liquidity requirement for
the market to continue operating so that the profitability of pump-and-dump
schemes is limited. When a liquidity taking order exhausts liquidity below the
minimum liquidity requirement, the circuit breaker is triggered, trading is
halted, and positions are executed in an auction where the price is randomly
drawn from a Gaussian distribution.

In a market with a circuit breaker, the trader’s optimal execution problem is
well-posed and the value function is non-degenerate. The optimization problem
is a singular stochastic control problem when trading is continuous and it
results in an impulse control problem when trading at a minimum lot size. We
give a direct prove of continuity for the value function which is non-trivial
here because of the unbounded state-space and controls in a setting with
Hawkes-like jump dynamics. We apply the dynamic programming approach to derive
the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) for both
continuous and discrete trading. As a novelty, our HJBQVI contains a double
integral straddling a sup-operator that yields the optimal signal-based trade.

Mathematically, the trader’s information flow with signal is modelled as a
Meyer-$\sigma$-field, see Lenglart [1980]. The technique of
Meyer-$\sigma$-fields was first applied in stochastic optimization by El
Karoui [1981] in the context of optimal stopping. Bank and Besslich [2020]
apply the theory of Meyer-$\sigma$-fields in a singular stochastic control
problem which is solved with convex analysis tools. In Merton’s optimal
investment problem, Bank and Körber [2022] use a Meyer-$\sigma$-field to
incorporate a short-term signal about jumps in the price of the asset and use
dynamic programming methods to solve for the optimal investment strategy.

We use simulations to study the trader’s execution strategies. The trader uses
signals about liquidity taking orders to optimize the times of execution and
the trading volume. Specifically, upon receiving the signal of an imminent
arrival of a liquidity taking order, the trader may submit a market order
before the external order arrives and impacts the price. As the bid-ask spread
widens, a signal about liquidity provision becomes irrelevant for the
execution programme of a trader because there is no incentive to execute a
market order before liquidity increases in the book, so price impact of market
orders decreases.

For a narrow bid-ask spread, we find that the trader uses the signal about
liquidity provision to start speculative roundtrip trades when liquidity is
low and completes the roundtrip after liquidity has recovered upon receiving a
signal about a liquidity taking order.

Trading on information from signals increases the average terminal wealth and
it also increases the variance of the distribution of terminal wealth. In
other words, to extract value from the signal, the trader uses strategies that
increase the expected gain and that increase the risk of the financial
performance of the execution programme.

There exists a broad range of literature on informed trading. Important
seminal works are those by Kyle [1985] and Back [1992]. More recent work on
optimal trading with market signals is in Cartea and Jaimungal [2016] who
examine optimal execution with a general Markovian signals and derive closed
form optimal strategies; see also Casgrain and Jaimungal [2019] who
incorporate latent factors. Similarly, Lehalle and Neuman [2019] and Neuman
and Voß [2022] study optimal trading with signals when there is transient
price impact and Belak et al. [2018] use non-Markovian finite variation
signals. For a market maker, Cartea and Wang [2020] show how to use a signal
of the trend of the price of an asset in optimal liquidity provision
strategies; similarly Lehalle and Mounjid [2017] study optimal market maker
strategies with a signal on liquidity imbalance. More recently, Cartea et al.
[2022] use signatures of the market to generate signals and Cartea and
Sánchez-Betancourt [2022] study how a broker provides liquidity to informed
and uninformed traders.

The feature that the trader’s market orders influence the arrival rate of
other limit and market orders in the same way as external orders is a novelty
compared with the existing literature on stochastic optimal control with
Hawkes processes. For instance, Alfonsi and Blanc [2016] solve an optimal
execution problem in a modified version of the model in Obizhaeva and Wang
[2013] where the external order flow is modelled as a Hawkes process that is
not influenced by the trader. Similarly, Cartea et al. [2018] study a
framework where the fill rates of the trader’s controlled limit order flow are
driven by an external, uncontrolled Hawkes process. The work of Cayé and
Muhle-Karbe [2014] proposes a modification of the Almgren and Chriss [2000]
model where the past orders of the trader have a self-exciting effect on the
price impact. In Horst et al. [2020], the trader influences the base intensity
of the mutually exciting external market order dynamics so that the order flow
of the trader influences the intensity process in a different way from the
external order flow.

The remainder of this paper is organized as follows: Section 2 presents the
market model and introduces a trader who receives signals on the order flow.
Section 3 examines the problem of optimal investment and execution in a market
with a circuit breaker and illustrates the performance of the optimal strategy
with trading signals. Results and proofs are collected in the Appendix.

##  2 The model

In this section, we present a model of stock prices where innovations in
prices are driven by the flow of market orders. First, we introduce a market
model where the arrival of market orders and limit orders is driven by a
marked Poisson point process. The arrival rate of orders is a function of the
liquidity in the market. At times of high liquidity, market orders arrive more
frequently, while at times of low liquidity, the arrival rate of limit orders
increases. This feature of the model introduces resilience in the supply and
demand of liquidity. Next, we introduce a strategic trader who receives
signals about order flow and executes market orders. In our model, all orders
arriving in the market, external or those of the trader, have the same effect
on the dynamics of liquidity and prices.

###  2.1 The uncontrolled model

In the following, we present a model for the price dynamics $(P_{t})_{t\geq
0}$ of a single asset where price changes are driven by the market order flow
$(M_{t})_{t\geq 0}$. Market liquidity $\lambda_{t}$ measures the capacity of
the market to fill an incoming market order at every point in time $t\geq 0$.
The change in liquidity is the difference between the volume of incoming and
cancellations of limit orders $(L_{t})_{t\geq 0}$ and the volume of incoming
market orders. For simplicity, we assume that the liquidity of the buy and
sell side are the same. Specifically, the liquidity process
$(\lambda_{t})_{t\geq 0}$ satisfies the dynamics

$\displaystyle d\lambda_{t}$ $\displaystyle=dL_{t}-|dM_{t}|,$ (1)

where the limit order flow satisfies

$\displaystyle dL_{t}$
$\displaystyle=\int_{E\times\mathbb{R}_{+}}\mathbbm{1}_{\\{y\leq
g(\lambda_{t-})\\}}\rho(e)N(dt,de,dy),$ (2)

and the market order flow satisfies

$\displaystyle dM_{t}$
$\displaystyle=\int_{E\times\mathbb{R}_{+}}\mathbbm{1}_{\\{y\leq
f(\lambda_{t-})\\}}\eta(e)N(dt,de,dy).$ (3)

Here, $N$ is a a marked Poisson point process on $[0,\infty)\times
E\times\mathbb{R}_{+}$ with compensator $dt\otimes\nu(de)\otimes dy$ where we
assume $\nu(E)=1$. The mappings $\eta,\rho\in L^{1}(\nu)$ determine the volume
of the order associated with a mark $e$ that the point process $N$ sets in the
Polish mark space $E$, and $\nu(de)$ is the probability distribution for this
mark.111For a measurable function $f:E\rightarrow\operatorname*{\mathbb{R}}$
and a measure $\mu$ on $E$, the notation $f\in L^{p}(\mu)$ means that $f$ is
measurable with $\int_{E}|f(e)|^{p}\mu(de)<\infty$. When $\rho(e)>0$, a new
limit order of size $\rho(e)$ is posted, and $\rho(e)<0$ corresponds to a
cancellation of limit orders of size $|\rho(e)|$. Similarly, $\eta(e)>0$
corresponds to a buy market order of size $\eta(e)$ and $\eta(e)<0$ is a sell
market order of size $|\eta(e)|$. Modelling the dynamics of limit and market
orders with the same marked Poisson point process allows us to specify the
dependence structure between the arrival of orders in a tractable way. To
exclude the simultaneous arrival of limit orders and market orders (which
would be cumbersome for bookkeeping), we assume $\eta(e)\rho(e)=0$ for all
$e\in E$.

The functions $f$ and $g$ are of at most linear growth, i.e.,

$\displaystyle|f(\lambda)|+|g(\lambda)|\leq
c_{1}+c_{2}|\lambda|,\quad\lambda\in\mathbb{R},$ (4)

for finite constants $c_{1},c_{2}\geq 0$, so the market dynamics
$(\lambda,L,M)$ admit a unique solution.

######  Lemma 2.1.

Suppose the functions $f$ and $g$ are of at most linear growth, i.e., they
satisfy (4). Then, the market dynamics (1), (2), (3) admit a unique solution
$(\lambda,L,M)$.

For a proof, see the Appendix.

The continuous, strictly positive functions $f$ and $g$ specify how the
arrival rates of the orders depend on the liquidity $\lambda$ of the market.
We assume that $f$ is increasing and $g$ is decreasing, and assume

$\displaystyle\int_{E}\rho(e)\nu(de)>0$ (5)

to introduce resilience of liquidity provision in a market with the Hawkes-
like liquidity dynamics in (1). More precisely, as liquidity decreases, fewer
market orders will arrive and limit orders will arrive more frequently — and
vice versa when liquidity increases, where (5) guarantees that liquidity
provision dominates cancellations of limit orders.

Next, we specify the price dynamics of the asset as

$\displaystyle dP_{t}=I(\Delta_{t}M,\lambda_{t-}),$ (6)

with the price impact function $I(\cdot\,,\cdot)$ defined by

$\displaystyle
I(\Delta,\lambda):=\operatorname*{sgn}(\Delta)\int_{0}^{|\Delta|}\iota(\lambda-z)dz,\quad\lambda\in\mathbb{R},$
(7)

where $\operatorname*{sgn}(\,\cdot\,)$ denotes the sign function. Here, the
function $I(\Delta,\lambda)$ describes the price impact of a market order of
size $\Delta\in\mathbb{R}$ that arrives when the liquidity of the market is
$\lambda$. The sign of $I(\Delta,\lambda)$ is determined by that of $\Delta$:
buy orders increase the price and sell orders decrease it. The function
$\iota:\mathbb{R}\rightarrow\mathbb{R}_{+}$ is non-increasing to ensure that
market orders have less impact when liquidity is high. Similarly, the absolute
value of the function $I$ is increasing in the size $|\Delta|$ because,
everything else being equal, as the volume of market orders increases, so does
the impact of the order on the price of the asset. Finally, definition (7)
makes the price dynamics (6) consistent when orders are split, i.e.,
$I(\Delta,\lambda)=I(\Delta_{1},\lambda)+I(\Delta_{2},\lambda-|\Delta_{1}|)$
for $\Delta_{1}+\Delta_{2}=\Delta$ with
$\operatorname*{sgn}(\Delta_{1})=\operatorname*{sgn}(\Delta_{2})$.

Our model exhibits a direct link between price volatility and market liquidity
because price volatility is determined by the arrival rate and the size of
price changes which both depend on the liquidity in the market. More
precisely, market liquidity affects the arrival rate of market orders, i.e.,
the arrival rate of price changes, through the dynamics as in (3) and the size
of price changes in (6) through the price impact function $I$ as in (7). For
instance, when liquidity in the market decreases, the arrival rate of market
orders decreases and the size of the price impact of market orders increases.
The next lemma makes this link between volatility and liquidity explicit and
states an elasticity condition under which the volatility of prices decreases
with market liquidity.

######  Lemma 2.2.

The predictable quadratic variation of the price $P$ is given by

$\displaystyle d\langle P\rangle_{t}=\sigma^{2}(\lambda_{t-})dt,$ (8)

where

$\displaystyle\sigma(\lambda):=\left(f(\lambda)\int_{E}I(\eta(e),\lambda)^{2}\nu(de)\right)^{1/2}.$
(9)

The function in (9) is strictly decreasing in $\lambda$ under the elasticity
condition

$\displaystyle
0<\left.\frac{\partial_{\lambda}f(\lambda)}{f(\lambda)}\middle/\middle(-\frac{\partial_{\lambda}\mathbb{I}^{2}(\lambda)}{\mathbb{I}^{2}(\lambda)}\right)<1,$
(10)

where
$\mathbb{I}(\lambda):=\big{(}\int_{E}I(\eta(e),\lambda)^{2}\nu(de)\big{)}^{1/2}$
is the $L^{2}$-norm of the price impact size from market orders at liquidity
level $\lambda$.

For a proof, see the Appendix.

The elasticity condition compares the relative change in the arrival rate $f$
of market orders with the absolute value of the relative change in the impact
size of market orders under the $L^{2}$-norm. Thus, condition (10) says that
when liquidity decreases, the price impact increases at a higher rate than the
arrival rate of market orders decreases. Hence, price volatility rises with
low liquidity because the increase of the price impact of market orders at low
liquidity compensates for the decrease in the arrival rate of market orders.

###  2.2 The controlled model with trading signal

Next, we introduce a trader who follows an optimal execution programme. For
simplicity, we assume that the trader sends market orders and does not provide
liquidity to the market.

To describe the trader’s information structure, we introduce the (right-
continuous) filtration

$\displaystyle\mathcal{F}_{t}=\sigma(N([0,s]\times
E\times[0,y]),s\in[0,t],E\in\mathcal{E},y\in\mathbb{R}_{+}),\quad t\geq 0$

generated by the point process $N$. The usual information framework of the
predictable information flow $\mathcal{P}(\mathcal{F})$ contains only past
information about the order flow and a trader with information
$\mathcal{P}(\mathcal{F})$ can only trade after the arrival of external limit
and market orders are revealed. Thus, she can only react to market events
after they have happened.

Here, however, the trader receives a short-term signal about imminent changes
in the order flow. She uses this information to anticipate jumps in liquidity
and in prices, and executes signal-based trades before the external order
arrives in the market. The signal is given by the process

$\displaystyle Z_{t}=\int_{[0,t]\times
E\times\mathbb{R}_{+}}z(e,y)N(ds,de,dy),\quad t\geq 0,$ (11)

where $N$ is the same marked Poisson point process that also drives the
external order flows (2) and (3). The function $z\in L^{1}(\nu(de)\otimes dy)$
determines what the trader learns about any mark $(e,y)$ set by $N$. We
introduce the Meyer-$\sigma$-field (cf.[Lenglart, 1980, Def. 2])

$\displaystyle\Lambda:=\mathcal{P}(\mathcal{F})\vee\sigma(Z),$ (12)

which adds the information $\sigma(Z)$ of the signal $Z$ to the predictable
information $\mathcal{P}(\mathcal{F})$.

The trader’s strategy is described by a process $(C_{t})_{t\geq 0}$ of locally
bounded variation starting at $C_{0-}=0$ and whose changes represent changes
in the trader’s inventory so that at any time $t$ the net number of shares
sold or bought up to this moment is given by $C_{t}$. Trades can be of any
size, i.e., $D=\mathbb{R}$, or — in line with market practice — a multiple of
a minimal lot size $\delta>0$, so
$D=\\{...,-2\delta,-\delta,0,\delta,2\delta,...\\}$. The set of admissible
controls is therefore

$\displaystyle\mathcal{C}=\Big{\\{}C\;\Lambda\text{-measurable of bounded
total variation, with $C_{0-}=0$, and }\Delta^{l}C,\Delta^{r}C\text{
D-valued}\Big{\\}}.$ (13)

Here, $\Delta^{l}_{t}C=C_{t}-C_{t-}$ and $\Delta^{r}_{t}C=C_{t+}-C_{t}$
denote, respectively, the ‘left’ and the ‘right’ jumps of $C$ at time $t\geq
0$. They correspond to the proactive signal-based trades and reactive state-
based trades we introduce shortly.

The following lemma provides a representation of the trader’s strategy and
clarifies how to use the signal $\Delta_{t}Z$ in her strategy.

######  Lemma 2.3.

A process $C$ is a $\Lambda$-measurable process of locally bounded variation
if and only if it admits the decomposition

$\displaystyle C_{t}$ $\displaystyle=C^{c}_{t}+\sum_{0\leq s\leq
t}\Delta^{l}_{s}C+\sum_{0\leq s<t}\Delta^{r}_{s}C,$ (14)

where $C^{c}$ is a continuous, adapted process of bounded variation,
$\Delta^{r}C$ is an adapted process of bounded variation, and where

$\displaystyle\Delta^{l}_{t}C$ $\displaystyle=\int_{\\{t\\}\times
E\times\mathbb{R}_{+}}\Gamma_{s}(z(e,y))N(ds,de,dy)$ (15)

for some $\mathcal{P}(\mathcal{F})\otimes\mathcal{B}(\mathbb{R})$\- measurable
field $\Gamma$ satisfying both $\Gamma_{s}(0)=0$ and the integrability
condition

$\displaystyle\int_{[0,t]\times
E\times\mathbb{R}_{+}}|\Gamma_{s}(z(e,y))|N(ds,de,dy)<\infty\quad
a.s.\qquad\text{for all }t\geq 0.$ (16)

Thus, upon observing a signal $\Delta_{t}Z$, the trader sends the order
$\Delta^{l}_{t}C=\Gamma^{l}_{t}(\Delta_{t}Z)$ before the full information
about external orders becomes known to all market participants and its effect
materialises, i.e., liquidity and prices change as a result. Hence, we call
$\Delta^{l}_{t}C$ the _signal-based trade_.

In contrast, the right jump $\Delta^{r}_{t}C$ is the trader’s action after the
arrival of an external order in the market and incorporates the full
information about the post-shock market state, i.e., the market state after
the arrival of external orders. In addition, $\Delta^{r}_{t}C$ can also
represent an order sent out of the trader’s own volition, typically motivated
by the state of her execution programme. Therefore, we call $\Delta^{r}_{t}C$
the _state-based trade_.

######  Remark 2.4.

In our model, the controlled orders $\Delta^{l}_{t}C$, $\Delta^{r}_{t}C$,
$dC^{c}_{t}$ affect market liquidity $\lambda$, and thus affect the arrival
rate of external shocks, in the same way as the external limit orders, market
orders and cancellations. This is a key difference between our model and those
proposed in the existing literature on stochastic control for Hawkes
processes, see e.g. Alfonsi and Blanc [2016], Cartea et al. [2018], and Horst
et al. [2020].

Next, we introduce the controlled dynamics of the state process

$\displaystyle
S^{C}_{t}:=(\lambda^{C}_{t},Q^{C}_{t},P^{C}_{t},X^{C}_{t}),\quad t\geq 0,$

starting at $S^{C}_{0-}:=(\lambda,q,p,x)$. Here, $\lambda^{C}_{t}$ is the
current level of market liquidity, $Q^{C}_{t}$ is the trader’s current
inventory, $P^{C}_{t}$ is the current asset price, and $X^{C}_{t}$ is the
trader’s cash process resulting from her strategy $C$.

When no external market and limits orders arrive in the market and when there
are no jumps in the trader’s strategy, the state process satisfies the
continuous dynamics

$\displaystyle
dS^{C}_{t}=(-|dC^{c}_{t}|,dC^{c}_{t},\iota(\Lambda^{C}_{t})dC_{t},-P^{C}_{t}dC^{c}_{t}-\zeta|dC^{c}_{t}|)\text{
when }\Delta^{l}_{t}C=\Delta^{r}_{t}C=0,N(\\{t\\}\times
E\times\mathbb{R}_{+})=0,$ (17)

where $\zeta\geq 0$ is the half-spread quoted in the book.

Next, we describe how the state changes when there are jumps in the order
flow. A jump can result, for instance, from the trader sending an order of
size $\Delta>0$. Thus, the asset price changes by $I(\Delta,\lambda)$ where
$\lambda$ denotes the pre-trade liquidity level, see (6). When the trader
sends an order of size $\Delta>0$, the price impact from her trade affects the
trader’s post-trade cash position in a nonlinear way. To describe this, it is
convenient to introduce the function

$\displaystyle\Xi(\Delta,\lambda):=\int\limits_{0}^{|\Delta|}I\left(z,\lambda\right)dz,$
(18)

which reflects the impact costs of the trade $\Delta$ when liquidity is
$\lambda$. So the total cost of buying $\Delta\geq 0$ shares at price $p$ is
$(p+\zeta)\Delta+\Xi(\Delta,\lambda)$; similarly, the revenue from selling
$\Delta\geq 0$ shares is $(p-\zeta)\Delta-\Xi(\Delta,\lambda)$. As in the
definition of $I$ in (7), the function $\Xi$ is consistent with order
splitting, i.e.,
$\Xi(\Delta,\lambda)=\Xi(\Delta_{1},\lambda)+\Xi(\Delta_{2},\lambda-|\Delta_{1}|)$
for $\Delta_{1}+\Delta_{2}=\Delta$ with
$\operatorname*{sgn}(\Delta_{1})=\operatorname*{sgn}(\Delta_{2})$.

Finally, the precise timing when market orders, limit orders, and
cancellations arrive is important and not interchangeable. This is because the
different orders arrive when the level of liquidity is $\lambda^{C}_{t-}$,
$\lambda^{C}_{t-}-|\Gamma_{t}(\Delta_{t}Z)|$ and $\lambda^{C}_{t}$,
respectively, and so the impact on the price of the asset varies and the
effect on the trader’s cash process is different, too. We explain this step-
by-step and refer to Figure 1 for an illustration.

Pre-tradestate$S^{C}_{t-}$Post-shockstate$S^{C}_{t}$Post-
tradestate$S^{C}_{t+}$Signal-based trade$\Delta^{l}_{t}C$External
orders$\Delta_{t}M^{C},\Delta_{t}L^{C}$State-based
trade$\Delta^{r}_{t}C$Signal $\Delta_{t}Z$ informs trader

Figure 1: Evolution of state processes

#### From pre-trade state $S^{C}_{t-}$ to post-shock state $S^{C}_{t}$

We assume the pre-trade state $\mathbf{s}_{-}:=S^{C}_{t-}$ is

$\displaystyle\mathbf{s}_{-}=(\boldsymbol{\lambda}_{-},\mathbf{q}_{-},\mathbf{p}_{-},\mathbf{x}_{-}):=(\lambda^{C}_{t-},Q^{C}_{t-},P^{C}_{t-},X^{C}_{t-}).$
(19)

The impending external limit and market orders at time $t$ are

$\displaystyle\rho:=\Delta_{t}L^{C}$ $\displaystyle=\int_{\\{t\\}\times
E\times\operatorname*{\mathbb{R}}_{+}}\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{s-})\\}}\rho(e)N(ds,de,dy),$ (20)
$\displaystyle\eta:=\Delta_{t}M^{C}$ $\displaystyle=\int_{\\{t\\}\times
E\times\mathbb{R}_{+}}\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{s-})\\}}\eta(e)N(ds,de,dy).$ (21)

The trader receives private information about $\Delta_{t}M^{C}$ or
$\Delta_{t}L^{C}$ from the signal ${z:=\Delta_{t}Z}$ and thus trades the
quantity $\gamma:=\Gamma_{t}(z)$. Market liquidity updates to

$\displaystyle\boldsymbol{\lambda}(\gamma,\eta,\rho;\mathbf{s}_{-}):=\boldsymbol{\lambda}_{-}-|\gamma|-|\eta|+\rho,$
(22)

and the trader’s inventory becomes

$\displaystyle\mathbf{q}(\gamma,\eta,\rho;\mathbf{s}_{-}):=\mathbf{q}_{-}+\gamma.$
(23)

Due to price impact, the price changes to

$\displaystyle\mathbf{p}(\gamma,\eta,\rho;\mathbf{s}_{-}):=\mathbf{p}_{-}+I\left(\gamma,\boldsymbol{\lambda}_{-}\right)+I\left(\eta,\boldsymbol{\lambda}_{-}-|\gamma|\right),$
(24)

where the market order of size $\eta$ arrives when liquidity is
$\boldsymbol{\lambda}_{-}-|\gamma|$. The trader’s cash position becomes

$\displaystyle\mathbf{x}(\gamma,\eta,\rho;\mathbf{s}_{-}):=\mathbf{x}_{-}-\mathbf{p}_{-}\gamma-\zeta|\gamma|-\Xi(\gamma,\boldsymbol{\lambda}_{-}).$
(25)

Thus, we define the state-update function $\mathfrak{s}$ as

$\displaystyle\mathfrak{s}(\gamma,\eta,\rho;\mathbf{s}_{-}):=(\boldsymbol{\lambda},\mathbf{q},\mathbf{p},\mathbf{x})(\gamma,\eta,\rho;\mathbf{s}_{-}),$
(26)

to write the post-shock state as

$\displaystyle
S^{C}_{t}:=\mathfrak{s}(\Gamma_{t}(\Delta_{t}Z),\Delta_{t}M^{C},\Delta_{t}L^{C};S^{C}_{t-}).$
(27)

#### From post-shock state $S^{C}_{t}$ to post-trade state $S^{C}_{t+}$

After the realised external shock and the post-shock state $S^{C}_{t}$ become
fully known to the trader, she executes the state-based trade
$\Delta^{r}_{t}C$ and the post-trade state is defined as

$\displaystyle S^{C}_{t+}:=\mathfrak{s}(\Delta^{r}_{t}C,0,0;S^{C}_{t}),$ (28)

where $\mathfrak{s}$ is as in (26).

Immediate roundtrip trades are not profitable in our model because the price
impact of every order includes the effect of previous orders on liquidity. For
instance, consider buying $\Delta$ shares and immediately selling $\Delta$
shares at some time $t\in[0,T]$. Assuming no other orders arrive in between,
the price impact $I(-\Delta,\lambda-\Delta)$ of the sell trade includes the
liquidity depleting effect of the first leg of the roundtrip trade. Due to the
monotonicity of $I$ in the liquidity component $\lambda$, the change in price
as a result of selling $\Delta$ shares is higher than the change in price when
first buying the same amount of shares. Consequently, the revenue received
from selling the shares is less than the initial purchase cost; hence,
instantaneous roundtrips result in a loss.

Moreover, the resilience in market liquidity encourages market participants to
wait for a recovery in liquidity before sending another market order because
low liquidity in the market leads to higher execution costs and increases the
arrival frequency of orders that provide liquidity. Therefore, there is an
incentive to split large orders into child orders to minimize execution costs,
i.e., minimize slippage.

##  3 The optimal investment and execution problem

In this section, we show how a trader who receives the private signal $Z$
executes a position with market orders over some finite time horizon $[0,T]$.
The trader’s performance criterion is the expected utility of terminal wealth.

The self-exciting nature of our system requires special care to avoid blow-
ups. Thus, we introduce a lower bound on liquidity so that liquidity taking
orders (i.e., market orders and limit order cancellations) are limited by the
supply of liquidity in the market. The lower bound on liquidity can be
interpreted as a circuit breaker that is imposed by the exchange to ensure a
well-functioning market. As we show, the circuit breaker ensures that the
value function of the optimization problem is non-degenerate. Finally, we
illustrate the market model with signals and showcase the performance of the
optimal strategy.

###  3.1 A circuit breaker for liquidity

In practice, we observe that markets are temporarily shut down if prices
become too volatile or prices undergo an abrupt change exceeding a predefined
level; see, e.g., Chen et al. [2023]. We use the connection between high
volatility in prices and low market liquidity from Lemma 2.2 to impose a
circuit breaker for price volatility by introducing a lower bound for
liquidity:

$\displaystyle\lambda^{C}_{t}\geq\underline{\lambda},\quad t\geq 0.$ (29)

We refer to $\underline{\lambda}$ as the _liquidity trigger_ that activates
the circuit breaker when a market participant executes an order that depletes
liquidity beyond $\underline{\lambda}$. In this case, the market order or the
cancellation of a limit order that reduces liquidity to the level
$\underline{\lambda}$ is partially executed with the available liquidity in
the market $\lambda_{t}-\underline{\lambda}$ and the market is shut down.
Orders that are sent thereafter cannot be executed. In the following, we
denote by $\tau^{C}\in[0,\infty]$ the stopping time when the circuit breaker
is triggered, where $\tau^{C}=\infty$ corresponds to the case when liquidity
remains above $\underline{\lambda}$ for the entire time horizon.

When the trader’s inventory at market shutdown is $Q^{C}_{\tau^{C}+}\neq 0$,
she executes the outstanding inventory at a price determined in an auction.
For simplicity, we assume that the auction price is $P^{C}_{\tau^{C}+}+\sigma
Y$, where $Y\sim N(0,1)$, $\sigma>0$ and $P^{C}_{\tau^{C}+}$ is the price when
trading was halted. Clearly, as the trader’s level of risk aversion increases,
the incentives to avoid a market shutdown are stronger.

For notational simplicity, we denote
$\tilde{S}^{C}_{t}:=(\lambda^{C}_{t},Q^{C}_{t},P^{C}_{t},X^{C}_{t})$ the state
process in the market with circuit breaker and $\tilde{\mathcal{C}}$ the class
of admissible strategies $C$ with circuit breaker, see Section 4.1 below for a
detailed description.

Finally, for a trader with trading horizon $[0,T]$, the terminal wealth
$\tilde{X}^{C}_{T}$ is the cash position $X^{C}_{T}$ plus the cash from
executing any remaining position $Q^{C}_{T}$ at time $T$, i.e.,

$\displaystyle\tilde{X}^{C}_{T}:=X^{C}_{T}+P^{C}_{T}Q^{C}_{T}+\sigma
Y\operatorname*{sgn}(Q^{C}_{T})(|Q^{C}_{T}|-(\lambda^{C}_{T}-\underline{\lambda})^{+})^{+}-\zeta|Q^{C}_{T}|-\Xi(-Q^{C}_{T},\lambda^{C}_{T}),$
(30)

where $\sigma
Y\operatorname*{sgn}(Q^{C}_{T})(|Q^{C}_{T}|-(\lambda^{C}_{T}-\underline{\lambda})^{+})^{+}$
is the additional cash from executing at the auction price when the circuit
breaker has been triggered before $T$ or when $|Q^{C}_{T}|$ exceeds the
available liquidity at time $T$.

The lower bound (29) requires a sufficient supply of liquidity in the market
and thus ensures that market orders and cancellations cannot deplete liquidity
in an excessive way. This leads to an upper bound for the expected trade
volume of liquidity taking orders, including the trader’s market orders. Here,
we denote by $\tilde{M}^{C}$ the external market order flow in the market with
circuit breaker, by $\tilde{L}^{C,+}$ the posted limit order flow and by
$\tilde{L}^{C,-}$ the cancellations of limit orders, see Section 4.1 below.

######  Lemma 3.1.

Consider a market with circuit breaker and liquidity trigger (29) and suppose
the integrability condition ${\int_{E}|\rho(e)|^{n}\nu(de)<\infty}$, for some
$n\in\mathbb{N}$. Then, for admissible trading strategies
$C\in\tilde{\mathcal{C}}$, the $n$-th moment of the total variation of the
liquidity taking order flows is uniformly bounded from above:

$\displaystyle\mathbb{E}\left[\left(V_{[0,T]}(Q^{C})+V_{[0,T]}(\tilde{M}^{C})+V_{[0,T]}(\tilde{L}^{C,-})\right)^{n}\right]$
$\displaystyle\leq(n+1)(\lambda_{0}-\underline{\lambda})^{n}+(n+1)(Tg(\underline{\lambda}))^{n}\nu(E)^{n-1}\int_{E}\rho^{+}(e)^{n}\nu(de),$

where $\rho^{+}(e)=\max(\rho(e),0)$.

Hence, the expected total variation of the trader’s inventory is uniformly
bounded over a finite trading horizon $[0,T]$.

###  3.2 Well-posedness of the optimization problem

The trader maximises the expected utility from terminal wealth over the
trading horizon $[0,T]$, so her performance criterion is

$\displaystyle J(C):=\mathbb{E}[U_{\alpha}(\tilde{X}^{C}_{T})],$ (31)

for any admissible control $C\in\tilde{\mathcal{C}}$, and the utility function
is

$\displaystyle U_{\alpha}(x)=\begin{cases}-\exp(-\alpha x),&\alpha>0,\\\
\hfill x,&\alpha=0,\end{cases}$ (32)

where $\alpha\geq 0$ is the risk aversion parameter. The terminal cash
position is denoted by $\tilde{X}^{C}_{T}$, including any trades that are
required to complete the execution programme at the terminal time $T$.

The value function is

$\displaystyle
v(T,s):=\max_{C\in\tilde{\mathcal{C}}}\operatorname*{\mathbb{E}}\left[U_{\alpha}(\tilde{X}^{C}_{T})\right],$
(33)

with $\tilde{X}^{C}_{T}$ as in (30) and for
$S^{C}_{0-}=s=(\lambda,q,p,x)\in\mathbb{S}$, where
$\mathbb{S}:=\operatorname*{\mathbb{R}}^{4}$.

Next, we assume that all moments of $\eta$ are finite

$\displaystyle\int_{E}|\eta(e)|^{n}\nu(de)<\infty\quad\text{ for all
}n\in\mathbb{N},$ (34)

and that $\rho$ has the following finite exponential moments

$\displaystyle\int_{E}\exp(n|\rho(e)|)\nu(de)<\infty,\quad\int_{E}\exp(n|\rho(e)|^{2})\nu(de)<\infty\quad\text{
for all }n\in\mathbb{N}.$ (35)

######  Proposition 3.2.

Assume conditions (29), (34), and (35) hold. Then, the value function $v$ in
(33) is non-degenerate, i.e., ${v<U_{\alpha}(+\infty)}$, where
$U_{\alpha}(+\infty)=0$ for $\alpha>0$ respectively
$U_{\alpha}(+\infty)=+\infty$ for $\alpha=0$.

Now, for $f$, $g$, $\iota$ Lipschitz continuous, the following theorem shows
continuity of the value function $v$ as in (33).

######  Theorem 3.3 (Continuity of the value function).

Assume conditions (29), (34), and (35) hold, and assume $f$, $g$, $\iota$ are
Lipschitz continuous. Then, the value function $v$ in (33) is continuous in
$(T,s)=(T,\lambda,q,p,x)\in[0,\infty)\times[\underline{\lambda},\infty)\times\mathbb{R}\times\operatorname*{\mathbb{R}}\times\operatorname*{\mathbb{R}}\times\operatorname*{\mathbb{R}}$.

For the proofs of Proposition 3.2 and Theorem 3.3, see the Appendix.

Finally, in the Appendix, we show that one can use standard techniques from
dynamic programming and the theory of Hamilton-Jacobi-Bellman equations to
derive numerical approximations of solutions to the trader’s problem.

###  3.3 Performance of signal-based trading strategies

In the following, we implement a numerical scheme to solve the HJB (67) from
Subsection 4.2 below and illustrate the trader’s signal-based strategy in an
optimal acquisition problem — Subsection 4.3 provides details of the numerical
scheme.

In our benchmark, the trader receives a private signal about limit orders,
market orders, and cancellations of limit orders with probability $p$.
Compared to a trader without the signal, the trader with the signal optimizes
her execution times and trading volumes by splitting the parent order into
fewer and larger child orders than the trader without signal. She executes her
orders when a signal about a liquidity taking order arrives. Signals about
liquidity provision are not relevant for the execution programme of a risk
averse trader because there is no incentive to execute before the liquidity
providing order increases the liquidity in the book and price impact of market
orders decreases.

Speculative trades are not profitable in the benchmark due to the size of the
bid-ask spread. Whereas when the bid-ask spread is narrow, speculation based
on signals is more profitable and the trader generally initiates speculative
trades upon receiving a signal about liquidity provision when the level of
liquidity is currently low. After waiting for liquidity to improve and the
cost of price impact to decline, she unwinds the speculative position upon
receiving a signal about the imminent arrival of a liquidity taking order.

#### Parameter specification — Benchmark case.

In the simulations, we consider six types of external market orders: buy
orders of one, two, and three lots; and sell orders of one, two and three
lots. Similarly, limit orders and their cancellations are of size one, two,
and three lots.

We choose the arrival rates $f$ and $g$ as

$\displaystyle
f(\lambda)=\theta_{f}\exp(\kappa_{f}\,\lambda)\quad\text{and}\quad
g(\lambda)=\theta_{g}\exp(-\kappa_{g}\,\lambda),$ (36)

where $\theta_{f}=20$, $\theta_{g}=40$ and $\kappa_{f}=\kappa_{g}=0.01$, i.e.,
if liquidity $\lambda$ is at level 0, the market expects to receive 20
external market orders, 30 limit orders, and 10 cancellations of limit orders
over the trading horizon of length $T=1$.

The price impact function is given by

$\iota(\lambda)=\theta_{\iota}+\kappa_{\iota}\lambda,$

where $\theta_{\iota}=0.01$ and $\kappa_{\iota}=-0.0002$. With these parameter
values, the elasticity condition (10) holds.

For numerical reasons, we introduce an upper bound $\overline{\lambda}$ for
the liquidity parameter $\lambda$ so that we obtain a bounded domain
$[\underline{\lambda},\overline{\lambda}]$ for $\lambda$, where
$\underline{\lambda}$ is the lower bound from (29). We set the lower bound at
$\underline{\lambda}=-40$ and the upper bound at $\overline{\lambda}=40$, such
that $\iota(\lambda)\geq 0$ for every
$\lambda\in[\underline{\lambda},\overline{\lambda}]$.

The volatility of the auction price in (30) is $\sigma=0.3$. The bid-ask
spread is set to one cent, i.e. $2\zeta=0.01$.

Finally, the trader’s risk aversion is $\alpha=0.1$, see (32).

#### Signal design.

The signal is given by the variable $\Delta_{t}Z$. A value of $\Delta_{t}Z=1$
signals an incoming limit order and a value of $\Delta_{t}Z=-1$ signals
liquidity taking, i.e., an incoming market order or the cancellation of a
limit order. With fixed probability $\hat{p}\in[0,1]$, the signal alerts the
trader of an imminent order and so an external order has a $1-\hat{p}$ chance
of taking the trader by surprise.

Thus, the signal informs the trader about the sign of the imminent change in
liquidity, but does not provide any information about the size of an order,
i.e., the trader anticipates the direction, but not the magnitude of changes
in liquidity. Particularly, in the case of a signal $\Delta_{t}Z=-1$ about
liquidity taking, the trader does not know whether a buy market order, a sell
market order, or the cancellation of a limit order will be posted.

Note that definition (11) allows for a huge variety of conceivable signals and
the above specification is considered to simplify the implementation.

#### Value function and certainty equivalent.

We implement the numerical scheme from subsection 4.3 to solve the HJB (67).
Figure 3 shows the value function $v$ as in (33) for $p=x=0$ and when the
probability of receiving a signal is $\hat{p}=0.2$.

Figure 2: Value function $v$ as in (33) for $p=x=0$.

Figure 3: Certainty Equivalent for a trader who receives a signal with
probability $\hat{p}=0.2$ compared to a trader without signal for spread
$0.01$.

The value function is symmetric about $q=0$ due to the symmetry of (7) and
(18) with respect to buy and sell orders. Hence, the expected utility of
terminal wealth for an acquisition or liquidation programme is the same —
everything else being equal.

Moreover, the value function is increasing in the liquidity $\lambda$ because
execution costs decrease as the liquidity in the market increases. When
liquidity approaches the liquidity trigger $\underline{\lambda}$, the value
function is the steepest because of the additional risk of entering an auction
through a market shutdown.

Similarly, the value function is particularly steep for large values of $|q|$
because a large execution programme is linked to higher risk in price and
liquidity changes.

Figure 3 illustrates the signal specific certainty equivalent of the value
function when the probability of receiving a signal is $\hat{p}=0.2$. It
corresponds to the additional amount of initial wealth that is needed when the
trader does not receive a signal to achieve the same expected utility as in
the case with signal, i.e., the certainty equivalent $CE$ is such that

$\displaystyle
v_{\hat{p}=0}(t,q,\lambda,p,x+CE)=v_{\hat{p}=0.2}(t,q,\lambda,p,x).$ (37)

We rearrange (37) to write

$\displaystyle
CE=-\frac{1}{\gamma}\log\left(\frac{v_{p=0.2}(t,q,\lambda,p,x)}{v_{p=0}(t,q,\lambda,p,x)}\right).$
(38)

The certainty equivalent is symmetric about $q$ and does not depend on $p$ or
$x$. Moreover, we see that the signal is worth the most for large $|q|$, i.e.,
the signal is more valueable for larger values of inventory. At most, the
trader can make up to four cents from the information of the signal which
corresponds to four times the bid-ask spread, see Figure 3.

The certainty equivalent is decreasing in the liquidity component $\lambda$
because as liquidity decreases, the execution times become more relevant as
price impact is more costly.

Moreover, the certainty equivalent slightly increases towards the liquidity
trigger $\underline{\lambda}=-40$. Here, the signal warns the trader of
potentially reaching the lower bound $\underline{\lambda}$ and reduces the
risk of executing a final trade in an auction.

For liquidity greater than twenty, the certainty equivalent is zero, i.e., the
signal does not add any value because, with and without a signal, a trader
will immediately complete the execution programme.

#### Benchmark — The signal optimises times of execution and trading volume.

Figure 4: Optimal acquisition of eight lots. Pathwise plot when trader does
not receive a signal and with spread 0.01. Figure 5: Optimal acquisition of
eight lots. Pathwise plot when trader receives signal with probability
$\hat{p}=0.2$ and with spread 0.01.

Figure 6: Optimal acquisition of eight lots. Performance of trader without
signal compared to a trader who receives a signal with probability
$\hat{p}\in\\{0,1,0.2,0.3,0.4\\}$ and with spread 0.01.
Left: Distribution of terminal wealth. Right: Signal-Sharpe-ratio.

In the benchmark with spread 0.01, we look at the optimal acquisition problem
where the trader acquires eight lots of the asset over the time horizon
$[0,T]$, so the initial inventory is $q=-8$. The trader can buy and sell any
multiple of lots and can execute speculative trades by trading away from the
target of buying only eight lots over the trading window.

When there is no private signal, the trader sends child orders of at most two
lots when liquidity is sufficiently high, see Figure 4. However, when the
trader receives private signals, she times her execution times with those of
the signal with information about liquidity taking orders and optimizes the
trading volume of her trades, see Figure 5 (simulated with same seed as that
for results in Figure 4). Here, she executes her acquisition programme by
sending two child orders of sizes three and five lots upon receiving a signal
about liquidity taking when the level of liquidity is high. When liquidity is
low during $t\in[0.15,0.7]$, the trader does not act upon the signal and
prefers to wait for liquidity to increase. Note that only the signals about
liquidity taking orders are relevant in her execution programme.

#### More signals lead to less execution costs and riskier strategies.

Figure 6 illustrates the performance of signal-based strategies depending on
the probability of receiving a signal.

Figure 6 (left) shows the distribution of terminal wealth for different
probabilities of receiving a signal compared to the distribution without
signal. When the probability of receiving a signal increases, the densities
shift to the right, i.e., the signal increases the expected terminal wealth by
optimizing the execution. On the other hand, as the probability of receiving a
signal increases, also the variance of terminal wealth increases because the
trader takes more risks due to the additional information through the private
signal. More precisely, knowing that she will be informed about liquidity
shocks through the signal, the trader waits longer for liquidity to recover
from shocks before trading to minimize price impact costs. However, this is
linked to more risk because unsignaled external orders can still arrive which
increases the variance of the trader’s terminal wealth.

To quantify the value of the private signal $Z$, we introduce the Signal-
Sharpe-ratio (SSR)

$\displaystyle SSR(Z):=\frac{\bar{X}(Z)-\bar{X}(0)}{\sigma(Z)},$ (39)

where

$\displaystyle\bar{X}(Z):=\frac{1}{n_{sim}}\sum_{j=1}^{n_{sim}}\tilde{X}^{j}_{T}(Z),\text{
}\bar{X}(0):=\frac{1}{n_{sim}}\sum_{j=1}^{n_{sim}}\tilde{X}^{j}_{T}(0)\text{
and
}\sigma(Z):=\sqrt{\frac{1}{n_{sim}}\sum_{j=1}^{n_{sim}}(\tilde{X}^{j}_{T}(Z)-\bar{X}(Z))^{2}}.$

Here, $\tilde{X}^{j}_{T}(Z)$ resp. $\tilde{X}^{j}_{T}(Z0)$ denote the terminal
wealth for scenario $j\in\\{1,...,n_{sim}\\}$ with signal $Z$ and without
signal.

Precisely, the SSR is the excess return $\bar{X}(Z)-\bar{X}(0)$ for a trader
with signal $Z$ compared to a trader without signal weighted by the risk
$\sigma(Z)$ of a trader with signal $Z$.

Figure 6 (right) shows the SSR as a function of the probability $p$ of
receiving a signal. The SSR increases up to $\hat{p}=0.2$ and slowly decreases
thereafter because the increase in variance dominates the increase in expected
terminal wealth which leads to a non-monotonicity in Figure 6 (left). Note
that the mean-variance-optimization is not the objective of the trader who
maximizes expected utility of terminal wealth, see (33).

#### Signal is more valuable for speculation as bid-ask spread narrows.

Next, consider a bid-ask spread of size 0.002, i.e., the bid-ask spread is one
fifth of that in the benchmark. In this case, the signal incentivises
speculative trades because, everything else being equal, the costs to execute
roundtrip trades are lower.

The speculative roundtrip trades start after receiving a signal on liquidity
provision, i.e., when the trader knows through the signal that the speculation
can be unwound at better liquidity, see Figure 7. After triggering liquidity
provision through her own market order, the trader waits for liquidity to
arrive in the market to unwind the speculative position with better liquidity
upon receiving a signal about a liquidity taking order.

Figure 7: Optimal acquisition of eight lots. Pathwise plot when trader
receives a signal with probability $\hat{p}=0.2$ and with spread $0.002$.

Next, we study the number of paths with speculative trades among
$n_{sim}=100,000$ simulated paths with bid-ask spread 0.01 and 0.002 in the
acquisition ($q=-8$) and pure speculation ($q=0$) scenario where the paths for
wide and narrow spreads are simulated with the same seed.222We say that a path
contains a speculative trade if the trader trades away from the target of
buying eight lots over the time horizon $[0,1]$ by trading both buy and sell
market orders.

When the spread is wide, i.e., when the spread is $0.01$, there is no
speculation because roundtrip trades are too costly. On the other hand, when
the spread is narrow, i.e., when the spread is $0.002$, there is speculation
in about $21\%$ of paths in the acquisition problem ($q=-8$) and in about
$11\%$ of paths in the pure speculation scenario ($q=0$). There are more
speculative paths in the execution example because the trader’s execution of
trades triggers liquidity provision and speculative trades become profitable.
Especially, when the trader trades towards a position close to zero very early
in the trading window, i.e., completes the acquisition problem early on, the
remaining time horizon is long enough so that a speculative roundtrip starting
close to zero is profitable; and vice versa if the trader completes the
acquisition at a later point in the trading window.

Finally, without signal, there is no speculation for both the wide and the
narrow bid-ask spread.

Figure 8: Optimal acquisition of eight lots. Distribution of terminal wealth
when trader receives a signal with probability $\hat{p}=0.2$ with a spread of
$0.01$ and $0.002$.

Figure 9: Certainty equivalent for a small spread of $0.002$ minus the
certainty equivalent for a spread of $0.01$.

Figure 9 shows the distribution of terminal wealth in the optimal acquisition
problem of $q=-8$ when the trader receives a private signal with probability
$p=0.2$, and the bid-ask spread is 0.01 and 0.002. With a narrow spread, the
terminal wealth is, on average, higher because the trader pays less spread and
profits from speculative trades. The terminal wealth increases the most in the
interval $[-801,-800.5]$, see Figure 9, because low level of liquidity, which
cause smaller values of terminal wealth for a spread of $0.01$ due to price
impact costs, can lead to profitable roundtrips when the spread is at $0.002$.
Similarly, the variance of terminal wealth decreases because in general, the
trader executes more orders to bring her inventory to zero and only deviates
from this target in about $21\%$ of cases.

Similarly, Figure 9 illustrates the certainty equivalent as in (38) for spread
0.002 minus the certainty equivalent as in (38) for spread 0.01. The certainty
equivalent improves when liquidity is low and when the trader’s inventory is
close to zero which is when speculative trades are executed.

Figure 10: Optimal acquisition of eight lots. Distribution of terminal wealth
for trader who receives a signal on liquidity taking (LT) with probability
$\hat{p}=0.2$ vs a trader who receives a signal on liquidity provision (LP)
with probability $\hat{p}=0.2$; spread is 0.01.

#### Signal on liquidity taking orders vs signal on liquidity providing
orders.

Finally, we compare the performance of the strategy of a trader who receives a
signal about the arrival of liquidity taking orders against the strategy of a
trader who receives a signal about the arrival of liquidity provision.

The trader with a signal on liquidity taking uses the signal to optimize the
execution times and trading volumes which increases the terminal wealth and
its variance, see Figure 10. Whereas, similarly to Figure 5, the trader does
not use the signal about liquidity provision for her execution so her terminal
wealth coincides with that of the trader who receives no signal, see Figure
10.

##  4 Appendix

In this Appendix, we present the definition of the state process in the market
with circuit breaker, the derivation of the HJB, the numerical scheme, and
other more detailed model specifications. Moreover, we also give the proofs
that were skipped in the main part of the paper.

###  4.1 State process in the market with circuit breaker

First, recall the specification of the state process
$S^{C}_{t}=(\lambda^{C},Q^{C}_{t},P^{C}_{t},X^{C}_{t})$ and the set of
admissible strategies $\mathcal{C}$ as defined in Section 2.2 for the market
without circuit breaker. When introducing a circuit breaker, we denote the new
state process by
$\tilde{S}^{C}_{t}=(\tilde{\lambda}^{C},\tilde{Q}^{C}_{t},\tilde{P}^{C}_{t},\tilde{X}^{C}_{t})$
and the new set of admissible strategies by $\tilde{\mathcal{C}}$ which are
defined as follows:

First, the set of admissible strategies in the market with circuit breaker is

$\displaystyle\tilde{\mathcal{C}}=\Big{\\{}$ $\displaystyle
C_{t}\in\mathcal{C}:\;\Delta^{l}_{t}C\text{ is
}D(\lambda^{C}_{t-})\text{-valued},\;\Delta^{r}_{t}C\text{ is
}D(\lambda^{C}_{t})\text{-valued},\text{ and }C_{t}=C_{\tau^{C}+}\text{ for
}t>\tau^{C}\Big{\\}},$ (40)

where $\mathcal{C}$ is as in (13) and where the stopping time $\tau^{C}$
describes the point in time when the circuit breaker is triggered, i.e.,

$\displaystyle\tau^{C}:=\inf\Big{\\{}t\geq
0:\min\\{\lambda^{C}_{t-}-|\Delta^{l}_{t}C|,\lambda^{C}_{t}\\}<\underline{\lambda}\Big{\\}},$
(41)

with $\inf\emptyset=+\infty$. Moreover, we denote the set of admissible
actions for liquidity level $\lambda$ as

$\displaystyle
D(\lambda):=D\cup\\{\lambda-\underline{\lambda},\underline{\lambda}-\lambda\\}.$
(42)

Here, the enlargement by
$\\{\lambda-\underline{\lambda},\underline{\lambda}-\lambda\\}$ ensures that
the trader always has the possibility to deplete the available liquidity in
the market without activating the circuit breaker, even if this deviates from
trading in multiples of a lot size $\delta>0$.

Note that in (40) the condition $C_{t}=C_{\tau^{C}+}\text{ for }t>\tau^{C}$
ensures that the circuit breaker cannot be triggered by the trader’s
continuous trades $dC^{c}$, but only by a signal-based trade
$\Delta^{l}_{\tau^{C}}C$, a state-based trade $\Delta^{r}_{\tau^{C}}C$ or by
an external market order or a cancellation of limit orders.

Next, we define the operator $\Upsilon$ for market liquidity $\lambda$ and
trade size $\Delta\in\mathbb{R}$ as

$\displaystyle\Upsilon(\Delta,\lambda):=\begin{cases}\hfill\Delta,&\text{if
}\lambda-|\Delta|\geq\underline{\lambda},\\\
\operatorname*{sgn}(\Delta)(\lambda-\underline{\lambda})^{+},&\text{if
}\lambda-|\Delta|<\underline{\lambda}.\end{cases}$ (43)

It returns $\Delta$ if the liquidity is sufficient to fill the order $\Delta$,
but it returns $\lambda-\underline{\lambda}$ if $\Delta$ depletes liquidity
below the level $\underline{\lambda}$ and zero if the circuit breaker has
already been triggered. Here, we denote $(x)^{+}:=\max(x,0)$ for
$x\in\operatorname*{\mathbb{R}}$.

Thus, we denote
$\tilde{S}^{C}_{t}:=(\tilde{\lambda}^{C}_{t},\tilde{Q}^{C}_{t},\tilde{P}^{C}_{t},\tilde{X}^{C}_{t})$
the state process in the market with with circuit breaker which starts in
$\tilde{S}^{C}_{0-}:=(\tilde{\lambda}^{C}_{0-},\tilde{Q}^{C}_{0-},\tilde{P}^{C}_{0-},\tilde{X}^{C}_{0-})$
and which updates according to

$\displaystyle\tilde{S}^{C}_{t}:=\tilde{\mathfrak{s}}(\tilde{\lambda}^{C}_{t-},\tilde{Q}^{C}_{t-},\tilde{P}^{C}_{t-},\tilde{X}^{C}_{t-}),\quad\tilde{S}^{C}_{t+}:=\tilde{\mathfrak{s}}(\tilde{\lambda}^{C}_{t},\tilde{Q}^{C}_{t},\tilde{P}^{C}_{t},\tilde{X}^{C}_{t}),\qquad\text{
for }0\leq t\leq\tau^{C}$ (44) $\displaystyle\text{ and
}\tilde{S}^{C}_{t}=\tilde{S}^{C}_{\tau^{C}+},\qquad\text{ for }t>\tau^{C},$
(45)

where, similarly to $\mathfrak{s}$ as in Section 2.2, the state update
function $\tilde{\mathfrak{s}}$ is defined as

$\displaystyle\tilde{\mathfrak{s}}(\Delta,\eta,\rho;\textbf{s}_{-}):=(\boldsymbol{\lambda},\textbf{q},\textbf{p},\textbf{x}),$
(46) with
$\displaystyle\boldsymbol{\lambda}=\begin{cases}\boldsymbol{\lambda}_{-}-|\Delta|+\mathbbm{1}_{\\{\boldsymbol{\lambda}_{-}-|\Delta|\geq\underline{\lambda}\\}}(-|\eta|+\rho),&\text{
if }\boldsymbol{\lambda}_{-}\geq\underline{\lambda},\\\
\hfill\boldsymbol{\lambda}_{-},&\text{ if
}\boldsymbol{\lambda}_{-}<\underline{\lambda},\\\ \end{cases}$ (47)
$\displaystyle\textbf{q}=\textbf{q}_{-}+\Upsilon(\Delta,\boldsymbol{\lambda}_{-})$
(48)
$\displaystyle\textbf{p}=\textbf{p}_{-}-I\left(\Upsilon(\Delta,\boldsymbol{\lambda}_{-}),\boldsymbol{\lambda}_{-}\right)+\mathbbm{1}_{\\{\boldsymbol{\lambda}_{-}-|\Delta|\geq\underline{\lambda}\\}}I\left(\Upsilon(\eta,\boldsymbol{\lambda}_{-}-|\Delta|),\boldsymbol{\lambda}_{-}-|\Delta|\right)$
(49)
$\displaystyle\textbf{x}=\textbf{x}_{-}-\textbf{p}_{-}\Upsilon(\Delta,\boldsymbol{\lambda}_{-})-\zeta|\Upsilon(\Delta,\boldsymbol{\lambda}_{-})|-\Xi(\Upsilon(\Delta,\boldsymbol{\lambda}_{-}),\boldsymbol{\lambda}_{-}).$
(50)

Here, $\tilde{\mathfrak{s}}(\Delta,\eta,\rho;\textbf{s}_{-})$ coincides with
$\mathfrak{s}(\Delta,\eta,\rho;\textbf{s}_{-})$ when
$\boldsymbol{\lambda}_{-},\boldsymbol{\lambda}\geq\underline{\lambda}$. Else,
the order that pushes $\boldsymbol{\lambda}$ below $\underline{\lambda}$ is
executed only partially via the function $\Upsilon$ as in (43) and trading is
halted for $\boldsymbol{\lambda}$ below $\underline{\lambda}$.
Hence, as long as the circuit breaker has not been triggered yet, the state
process in the market with and without circuit breaker coincide so that
$\tilde{S}^{C}_{t}=S^{C}_{t}$ for $0\leq t<\tau^{C}$.

Note that for simplicity of notation, we will in the following write
$\tilde{S}^{C}_{t}=(\lambda^{C}_{t},Q^{C}_{t},P^{C}_{t},X^{C}_{t})$ for the
components of the state process in the market with circuit breaker.

Finally, we define the external market order flow $\tilde{M}^{C}_{t}$ in the
market with circuit breaker as

$\displaystyle\tilde{M}^{C}_{t}$ $\displaystyle=\int_{[0,t]\times
E\times\operatorname*{\mathbb{R}}_{+}}\Upsilon(\eta(e),\lambda^{C}_{s-}-|\Gamma_{s}(z(e,y))|)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{s-})\\}}N(ds,de,dy).$ (51)

Similarly, the limit orders in the market with circuit breaker are

$\displaystyle\tilde{L}^{C,+}_{t}$ $\displaystyle=\int_{[0,t]\times
E\times\operatorname*{\mathbb{R}}_{+}}\rho^{+}(e)\mathbbm{1}_{\\{\lambda^{C}_{s-}-|\Gamma_{s}(z(e,y))|\geq\underline{\lambda}\\}\cap\\{y\leq
g(\lambda^{C}_{s-})\\}}N(ds,de,dy),$ (52)

and the cancellations of limit orders in the market with circuit breaker are

$\displaystyle d\tilde{L}^{C,-}_{t}$ $\displaystyle=\int_{[0,t]\times
E\times\operatorname*{\mathbb{R}}_{+}}\Upsilon(\rho^{-}(e),\lambda^{C}_{s-}-|\Gamma_{s}(z(e,y))|)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{s-})\\}}N(ds,de,dy).$ (53)

Here, the function $\Upsilon$ in (51) and (53) as well as the indicator in
(52) ensure that trading is halted after circuit breaker activation and that
the order flow remains constant after $\tau^{C}$.

###  4.2 The Hamilton–Jacobi–Bellman equation

In the following, we derive the HJB equation that is satisfied by the value
function $v$ as in (33). Particularly, we investigate for which conditions the
value process $V^{C}_{t}:=v(T-t,\tilde{S}^{C}_{t+})$, $t\in[0,T]$ with
arbitrary, but fixed time horizon $T$, satisfies super-martingale dynamics for
every admissible $C\in\tilde{\mathcal{C}}$ and martingale dynamics for some
optimal strategy $C^{*}\in\tilde{\mathcal{C}}$.

Recall the decomposition of admissible strategies from Lemma 2.3 and suppose
that the value function $v(T,\lambda,q,p,x)$ is sufficiently smooth to apply
Itô’s formula so that we write at least formally for $0\leq
t\leq\tau^{C}\wedge T$

$\displaystyle dV^{C}_{t}=-\frac{\partial v}{\partial
T}(T-t,S^{C}_{t-})dt+\left(\frac{\partial v}{\partial
p}(T-t,S^{C}_{t-})\iota(\lambda)-\frac{\partial v}{\partial
x}(T-t,S^{C}_{t-})P^{C}_{t-}+\frac{\partial v}{\partial
q}(T-t,S^{C}_{t-})\right)d\tilde{C}^{c}_{t}$ (54)
$\displaystyle+\left(\frac{\partial v}{\partial
x}(T-t,S^{C}_{t-})\zeta-\frac{\partial
v}{\partial\lambda}(T-t,S^{C}_{t-})\right)|d\tilde{C}^{c}_{t}|$
$\displaystyle+\int\limits_{E\times\operatorname*{\mathbb{R}}_{+}}\Big{[}v\\!\left(T\\!-\\!t,\tilde{\mathfrak{s}}\left(\Gamma_{t}(z(e,y)),\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)\\!-\\!v\\!\left(T\\!-\\!t,S^{C}_{t-}\right)\Big{]}\bar{N}(dt,de,dy)$
$\displaystyle+\int\limits_{E\times\operatorname*{\mathbb{R}}_{+}}\Big{[}v\\!\left(T\\!-\\!t,\tilde{\mathfrak{s}}\left(\Gamma_{t}(z(e,y)),\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\\!\right)\\!-\\!v\\!\left(T\\!-\\!t,S^{C}_{t-}\right)\\!\Big{]}(\nu(de)\otimes
dy)dt$
$\displaystyle+\left[v\left(T-t,\tilde{\mathfrak{s}}(\Delta^{r}_{t}\tilde{C},0,0;S^{C}_{t})\right)-v\left(T-t,S^{C}_{t}\right)\right],$

where $\bar{N}(dt,de,dy)=N(dt,de,dy)-dt\otimes\nu(de)\otimes dy$ is the
compensated Poisson measure and $\partial v/\partial T$ denotes the derivative
of $v$ with respect to the first component.

For $V^{C}$ to be a supermartingale for _any_ admissible $C$, the impulse
condition

$\displaystyle\sup\limits_{\Delta\in
D(\lambda)}\Big{\\{}v(T,\tilde{\mathfrak{s}}(\Delta,0,0;s))-v(T,s)\Big{\\}}\leq
0,\quad(T,s)\in[0,\infty)\times[\underline{\lambda},\infty)\times\mathbb{R}^{3},$
(55)

must be satisfied. We decompose the $\nu(de)\otimes dy)$-integral in (54) into

$\displaystyle\int\limits_{\hskip
30.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=0\\}}\left[v\\!\left(T-t,\tilde{\mathfrak{s}}\left(0,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)\\!-\\!v\\!\left(T-t,S^{C}_{t-}\right)\right]\nu(de)\otimes
dy$ (56) $\displaystyle+\int\limits_{\hskip
30.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)\neq
0\\}}\Big{[}v\\!\left(T\\!-\\!t,\tilde{\mathfrak{s}}\left(\Gamma_{t}(z(e,y)),\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)\\!-\\!v\\!\left(T\\!-\\!t,S^{C}_{t-}\right)\Big{]}\nu(de)\otimes
dy,$ (57)

where by Lemma 2.3 $\Gamma_{t}(z(e,y))=0$ in (56) where we have $z(e,y)=0$.
For (57), we use the desintegration

$\nu(de)\otimes
dy=\int_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}K(\bar{z};de\otimes
dy)\mu(d\bar{z}),$

with $\mu=(\nu\otimes Leb)\circ(z)^{-1}$ to estimate the expression in (57) as
follows:

$\displaystyle\int\limits_{\hskip
44.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)\neq
0\\}}\left[v\\!\left(T\\!-\\!t,\tilde{\mathfrak{s}}\left(\Gamma_{t}(z(e,y)),\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)\\!-\\!v\\!\left(T\\!-\\!t,S^{C}_{t-}\right)\right]\nu(de)\otimes
dy$
$\displaystyle=\int\limits_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}\int\limits_{\hskip
30.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=\bar{z}\\}}\Big{[}v\left(T-t,\tilde{\mathfrak{s}}\left(\Gamma_{t}(\bar{z}),\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)$ $\displaystyle\hskip
156.49014pt-v\left(T-t,S^{C}_{t-}\right)\Big{]}K(\bar{z};de\otimes
dy)\mu(d\bar{z})$
$\displaystyle\leq\int\limits_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}\sup_{\gamma\in
D(\lambda)}\int\limits_{(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=\bar{z}\\}}\Big{[}v\left(T-t,\tilde{\mathfrak{s}}\left(\gamma,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)$ $\displaystyle\hskip
199.16928pt-v\left(T-t,S^{C}_{t-}\right)\Big{]}K(\bar{z};de\otimes
dy)\mu(d\bar{z}).$

Finally, we collect all the $dt$-terms from the dynamics of $dV^{C}_{t}$ and
have the tight upper bound

$\displaystyle-\frac{\partial v}{\partial T}(T-t,S^{C}_{t-})$
$\displaystyle+\\!\\!\\!\\!\int\limits_{(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=0\\}}\left[v\left(T-t,\tilde{\mathfrak{s}}\left(0,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)-v\left(T-t,S^{C}_{t-}\right)\right](\nu(de)\otimes
dy)$
$\displaystyle+\\!\\!\\!\\!\int\limits_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}\sup_{\gamma\in
D(\lambda)}\int\limits_{(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=\bar{z}\\}}\Big{[}v\left(T-t,\tilde{\mathfrak{s}}\left(\gamma,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda^{C}_{t-})\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda^{C}_{t-})\\}};S^{C}_{t-}\right)\right)$ (58) $\displaystyle\hskip
170.71652pt-v\left(T-t,S^{C}_{t-}\right)\Big{]}K(\bar{z};de\otimes
dy)\mu(d\bar{z}).$

To satisfy the martingale optimality principle, the terms in (58) must be
smaller than or equal to zero for any admissible strategy
$C\in\tilde{\mathcal{C}}$ and zero for some (optimal) strategy $C^{*}$. With
(55), we thus obtain the following HJBQVI,

$\displaystyle\max\bigg{\\{}-\frac{\partial v}{\partial
T}(T,s)+\int\limits_{(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=0\\}}\left[v\left(T,\tilde{\mathfrak{s}}\left(0,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda)\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda)\\}};s\right)\right)-v\left(T,s\right)\right]\nu(de)\otimes dy$
$\displaystyle\quad+\int\limits_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}\sup_{\gamma\in
D(\lambda)}\int\limits_{\hskip
30.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=\bar{z}\\}}\left[v\left(T,\tilde{\mathfrak{s}}\left(\gamma,\eta(e)\mathbbm{1}_{\\{y\leq
f(\lambda)\\}},\rho(e)\mathbbm{1}_{\\{y\leq
g(\lambda)\\}};s\right)\right)\\!-\\!v\left(T,s\right)\right]K(\bar{z};de\otimes
dy)\mu(d\bar{z});$ $\displaystyle\quad\sup\limits_{\Delta\in
D(\lambda)}\Big{\\{}v(T,\tilde{\mathfrak{s}}(\Delta,0,0;s))-v(T,s)\Big{\\}}\bigg{\\}}=0,$
(59)

for
$(T,s)=(T,\lambda,q,p,x)\in[0,\infty)\times[\underline{\lambda},\infty)\times\mathbb{R}\times\operatorname*{\mathbb{R}}\times\operatorname*{\mathbb{R}}$.
Due to the circuit breaker being activated when liquidity falls below
$\underline{\lambda}$, the value function evaluated in
$\lambda<\underline{\lambda}$ is

$\displaystyle
v(T,\lambda,q,p,x)=U_{\alpha}\left(x+pq-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right),$

$\text{for
}(T,q,p,x)\in[0,\infty)\times\mathbb{R}\times\operatorname*{\mathbb{R}}\times\operatorname*{\mathbb{R}}$,
and the initial condition for $s=(\lambda,q,p,x)\in\mathbb{S}$ with zero time
to go $T=0$ is

$\displaystyle
v(0,s)=\begin{cases}U_{\alpha}\left(x+pq-\zeta|q|-\Xi(q,\lambda)\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}((|q|-(\lambda-\underline{\lambda}))^{+})^{2}\right),&\text{
if }\lambda\geq\underline{\lambda},\\\
U_{\alpha}\left(x+pq-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right),&\text{
if }\lambda<\underline{\lambda}.\end{cases}$ (60)

To reduce the dimensions of the HJB equation, we use the Ansatz

$\displaystyle
v(T,s)=\begin{cases}w(T,\lambda,q)|U_{\alpha}(x+pq)|,&\alpha>0,\\\ \hfill
w(T,\lambda,q)+x+pq,&\alpha=0,\end{cases}\qquad\forall(T,s)=(T,\lambda,q,p,x)\in[0,\infty)\times\mathbb{S}.$
(61)

For simplicity, we introduce the notations

$\displaystyle\gamma^{\lambda}$ $\displaystyle:=\Upsilon(\gamma,\lambda),$
(62) $\displaystyle\eta^{\lambda,\gamma}(e,y)$
$\displaystyle:=\Upsilon(\eta(e),\lambda-|\gamma^{\lambda}|)\mathbbm{1}_{\\{y\leq
f(\lambda)\\}},$ (63) $\displaystyle\rho^{\lambda,\gamma}(e,y)$
$\displaystyle:=\Big{(}\rho^{+}(e)\mathbbm{1}_{\\{\lambda-|\gamma^{\lambda}|\geq\underline{\lambda}\\}}+\Upsilon(\rho^{-}(e),\lambda-|\gamma^{\lambda}|)\Big{)}\mathbbm{1}_{\\{y\leq
g(\lambda)\\}},$ (64) $\displaystyle\lambda^{\lambda,\gamma}(e,y)$
$\displaystyle:=\hfill\lambda-|\gamma|-|\eta(e)|\mathbbm{1}_{\\{y\leq
f(\lambda)\\}}+\rho(e)\mathbbm{1}_{\\{y\leq g(\lambda)\\}},$ (65)
$\displaystyle\Delta^{\lambda}$ $\displaystyle:=\Upsilon(\Delta,\lambda).$
(66)

This allows us to write the reduced HJB for $w(T,\lambda,q)$ with
$(T,\lambda,q)\in[0,\infty)\times[\underline{\lambda},\infty)\times\operatorname*{\mathbb{R}}$
when $\alpha>0$ as

$\displaystyle\max$ $\displaystyle\bigg{\\{}-\frac{\partial w}{\partial
T}(T,\lambda,q)+\int\limits_{(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=0\\}}\bigg{[}w\Big{(}T,\lambda^{\lambda,0}(e,y),q\Big{)}\Big{|}U_{\alpha}\Big{(}I(\eta^{\lambda,0}(e,y),\lambda)q\Big{)}\Big{|}-w(T,\lambda,q)\bigg{]}\nu(de)\otimes
dy$ (67)
$\displaystyle+\int\limits_{z(E\times\operatorname*{\mathbb{R}}_{+})\setminus\\{0\\}}\sup_{\gamma\in
D(\lambda)}\bigg{\\{}\int\limits_{\hskip
50.0pt(E\times\operatorname*{\mathbb{R}}_{+})\cap\\{z(e,y)=\bar{z}\\}}\bigg{[}w\Big{(}T,\lambda^{\lambda,\gamma}(e,y),q+\gamma^{\lambda}\Big{)}\cdot$
$\displaystyle\hskip
85.35826pt\Big{|}U_{\alpha}\Big{(}\\!\\!-\zeta|\gamma^{\lambda}|-\Xi(\gamma^{\lambda},\lambda)+\Big{(}I\left(\gamma^{\lambda},\lambda\right)+I\big{(}\eta^{\lambda,\gamma}(e,y),\lambda-|\gamma^{\lambda}|\big{)}\Big{)}(q+\gamma^{\lambda})\Big{)}\Big{|}$
$\displaystyle\hskip 113.81102pt-w(T,\lambda,q)\bigg{]}K(\bar{z};de\otimes
dy)\bigg{\\}}\mu(d\bar{z});$ $\displaystyle\sup\limits_{\Delta\in
D(\lambda)}\Big{\\{}w(T,\lambda,q,\Delta)-w(T,\lambda,q)\Big{\\}}\bigg{\\}}=0.$

For $\lambda<\underline{\lambda}$, the function $w(T,\lambda,q)$ takes the
value

$\displaystyle
w(T,\lambda,q)=U_{\alpha}\left(-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right),\quad(T,q)\in[0,\infty)\times\mathbb{R},$
(68)

and the initial condition for
$(\lambda,q)\in\operatorname*{\mathbb{R}}\times\operatorname*{\mathbb{R}}$
with zero time to go is

$\displaystyle
w(0,\lambda,q)=\begin{cases}U_{\alpha}\left(-\zeta|q|-\Xi(q,\lambda)\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}((|q|-(\lambda-\underline{\lambda}))^{+})^{2}\right),&\text{
if }\lambda\geq\underline{\lambda},\\\
U_{\alpha}\left(-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(\frac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right),&\text{
if }\lambda<\underline{\lambda}.\end{cases}$ (69)

Similarly, one obtains the reduced HJB for $w$ when $\alpha=0$.
The suggested optimal strategy can as usual be constructed in feedback form by
determining the maximizer in the suprema of the HJB (67).

###  4.3 The Numerical Scheme

In the following, we present a numerical scheme to approximate the solution of
the dimension reduced HJB equation in (67) for $\alpha>0$; a similar scheme
can be derived for $\alpha=0$.
First, we introduce a discretized, bounded subset of the space
$[0,T]\times\mathbb{R}\times[\underline{\lambda},\infty)$:
For the time dimension $T^{\prime}\in[0,T]$, we take $\mathbb{T}_{\delta T}$
as the grid on $[0,T]$ with step size $\delta T>0$, where $T=N\delta T$ for
some $N_{1}\in\mathbb{N}$.
For the liquidity component $\lambda\in[\underline{\lambda},\infty)$, we
denote $\mathbb{R}_{\delta\lambda}^{\overline{\lambda}}$ the bounded grid on
$[\underline{\lambda}-\delta\lambda,\overline{\lambda}]$ with stepsize
$\delta\lambda>0$ where
$\overline{\lambda}=\underline{\lambda}+N\delta\lambda$ for some
$N_{2}\in\mathbb{N}$. To remain within a bounded domain, we assume $\lambda$
equals $\underline{\lambda}-\delta\lambda$ after the circuit breaker is
triggered and define the numerical scheme below accordingly.
For inventory $q\in\mathbb{R}$, we introduce the bounded grid
$\mathbb{R}_{\delta q}^{\underline{Q},\overline{Q}}$ on
$[\underline{Q},\overline{Q}]$ with step size $\delta q>0$ where
$\underline{Q}=N_{3}\delta q$ and $\overline{Q}=N_{4}\delta_{q}$ for some
constants $N_{3},N_{4}\in\mathbb{Z}$, $N_{3}<N_{4}$.
For the inventory $q$ to remain within the grid $\mathbb{R}_{\delta
q}^{\underline{Q},\overline{Q}}$ and for $\lambda$ to remain above
$\underline{\lambda}-\delta\lambda$, we define the set of actions

$D^{\underline{Q},\overline{Q}}(q,\lambda):=\Big{\\{}n\cdot\delta q\text{ with
}n\in\mathbb{Z}:q+n\,\delta q\in\mathbb{R}_{\delta
q}^{\underline{Q},\overline{Q}}\text{ and
}\lambda-|n\delta|\geq\underline{\lambda}-\delta\lambda\Big{\\}},\qquad
q\in\mathbb{R}_{\delta
q}^{\underline{Q},\overline{Q}},\lambda\in\mathbb{R}_{\delta\lambda}^{\overline{\lambda}}.$

For any function
$h^{T^{\prime}}:\mathbb{R}_{\delta\lambda}^{\overline{\lambda}}\times\mathbb{R}_{\delta
q}^{\underline{Q},\overline{Q}}\rightarrow\mathbb{R}$ with
$T^{\prime}\in\mathbb{T}_{\delta T}$, we introduce the operator

$\displaystyle\mathcal{L}^{\delta
T,\delta\lambda}(q,\lambda,h^{T^{\prime}}):=h^{T^{\prime}}(\lambda,q)+\delta
t\bigg{(}\Delta_{z=0}h^{T^{\prime}}(\lambda,q)+\int\limits_{z(E\times\mathbb{R}_{+})\setminus\\{0\\}}\sup\limits_{\gamma\in
D^{\underline{Q},\overline{Q}}(q,\lambda)}\Delta_{\text{z=}\bar{z}}h^{T^{\prime}}(\lambda,q,\gamma,\psi)\;\;\mu(d\bar{z})\bigg{)},$

where we use the notations

$\displaystyle\Delta_{\text{z=0}}h^{T^{\prime}}(\lambda,q):=\int\limits_{(E\times\mathbb{R}_{+})\cap\\{z(e,y)=0\\}}\Big{[}\bar{h}^{T^{\prime}}\big{(}\lambda^{\lambda,0}(e,y)\vee(\underline{\lambda}-\delta\lambda),q\big{)}\big{|}U_{\alpha}\big{(}I(\eta^{\lambda,0}(e,y),\lambda)q\big{)}\big{|}-h^{T^{\prime}}(\lambda,q)\Big{]}\nu(de)\otimes
dy,$ $\displaystyle\Delta_{\text{z=}\bar{z}}h^{t}(\lambda,q,\gamma):=$
$\displaystyle\int\limits_{\hskip
30.0pt(E\times\mathbb{R}_{+})\cap\\{z(e,y)=\bar{z}\\}}\bigg{[}\bar{h}^{T^{\prime}}\big{(}\lambda^{\lambda,\gamma}(e,y)\vee(\underline{\lambda}-\delta\lambda),q+\gamma^{\lambda}\big{)}\cdot\;$
$\displaystyle\hskip
17.07182pt\Big{|}U_{\alpha}\Big{(}-\zeta|\gamma^{\lambda}|-\Xi(\gamma^{\lambda},\lambda)-\big{(}I(\gamma^{\lambda},\lambda)+I\big{(}\eta^{\lambda,\gamma}(e,y),\lambda-|\gamma^{\lambda}|\big{)}\big{)}(q+\gamma^{\lambda})\Big{)}\Big{|}-h^{T^{\prime}}(\lambda,q)\bigg{]}K(\bar{z};de,dy),$

and we interpolate for
$\lambda\notin\mathbb{R}_{\delta\lambda}^{\overline{\lambda}}$

$\displaystyle\bar{h}^{T^{\prime}}(q,\lambda):=$ $\displaystyle
h^{T^{\prime}}\left(q,\left\lceil\frac{\lambda}{\delta\lambda}\right\rceil\delta\lambda\right)-\left(\left\lceil\frac{\lambda}{\delta\lambda}\right\rceil\delta\lambda-\lambda\right)\left(h^{T^{\prime}}\left(q,\left\lceil\frac{\lambda}{\delta\lambda}\right\rceil\delta\lambda\right)-h^{T^{\prime}}\left(q,\left\lfloor\frac{\lambda}{\delta\lambda}\right\rfloor\delta\lambda\right)\right).$

Here, $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ denotes rounding to the
closest integer below and above. Similarly, we define

$\displaystyle\mathcal{M}^{\delta
T,\delta\lambda}(\lambda,q,h^{T^{\prime}}):=\sup\limits_{\Delta\in
D^{\underline{Q},\overline{Q}}(q,\lambda)}\\!\bigg{\\{}\bar{h}^{T^{\prime}}\big{(}(\lambda-|\Delta|)\vee(\lambda-\delta\lambda),q+\Delta\big{)}\cdot$
$\displaystyle\hskip
142.26378pt\big{|}U\big{(}-\zeta|\Delta^{\lambda}|-\Xi(\Delta^{\lambda},\lambda)-I(\Delta^{\lambda},\lambda)(q+\Delta^{\lambda})\big{)}\big{|}\bigg{\\}}.$

Finally, we use the following numerical scheme to compute an approximate
solution $w$ to (67)

$\displaystyle w^{0}(q,\lambda)$
$\displaystyle:=\begin{cases}U_{\alpha}\left(-\zeta|q|-\Xi(q,\lambda)\right)\exp\left(-\frac{1}{2}\alpha^{2}\sigma^{2}((|q|-(\lambda-\underline{\lambda}))^{+})^{2}\right),&\text{
if }\lambda\geq\underline{\lambda},\\\ \hfill
U_{\alpha}\left(-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(-\frac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right),&\text{
if }\lambda=\underline{\lambda}-\delta\lambda,\\\ \end{cases}$
$\displaystyle\tilde{w}^{T^{\prime}+\delta T}(q,\lambda)$
$\displaystyle:=\mathcal{L}^{\delta
T,\delta\lambda}(q,\lambda,w^{t}),\qquad\lambda\geq\underline{\lambda},$
$\displaystyle w^{T^{\prime}+\delta T}(q,\lambda)$
$\displaystyle:=\max(\tilde{w}^{T^{\prime}+\delta
T}(q,\lambda),\mathcal{M}^{\delta
t,\delta\lambda}(q,\lambda,\tilde{w}^{T^{\prime}+\delta
T})),\qquad\lambda\geq\underline{\lambda},$ $\displaystyle
w^{T^{\prime}+\delta t}(q,\lambda-\delta\lambda)$
$\displaystyle:=U_{\alpha}\left(-\zeta|q|-\Xi(q,\underline{\lambda})\right)\exp\left(-\tfrac{1}{2}\alpha^{2}\sigma^{2}|q|^{2}\right)\quad\text{for}\;T^{\prime}\in\\{0,\delta
t,...,T-2\delta t,T-\delta t\\}.$

###  4.4 Formal Model Specification of Section 3.3

In this part, we link the model specification in Section 3.3 to the notations
in Chapter 2. For simplicity, we only treat the benchmark; the specification
of the remaining settings in Section 3.3 are similar.
First, the space $E$ is given by
$E=\\{-3,-2,-1,1,2,3\\}\times\\{1,2,3\\}\times\\{0,1\\}$. The mappings $\eta$
and $\rho$ are

$\eta(e)=e_{1}(1-e_{3})\quad\text{and}\quad\rho(e)=e_{2}e_{3}.$

The model is driven by the homogenous Poisson point process $N(dt,de,dy)$ with
compensator $dt\otimes\nu(de)\otimes dy$, where $\nu(de)=P^{1}(de_{1})\otimes
P^{2}(de_{2})\otimes P^{3}(de_{3})$ with

$\displaystyle P^{1}(\\{-3\\})=P^{1}(\\{3\\})=0.1,\quad
P^{1}(\\{-2\\})=P^{1}(\\{2\\})=P^{1}(\\{-1\\})=P^{1}(\\{1\\})=0.2,$ (70)
$\displaystyle P^{2}(\\{1\\})=P^{2}(\\{1\\})=0.4,\quad P^{2}(\\{3\\})=0.2,$
(71) $\displaystyle P^{3}(\\{0\\})=P^{3}(\\{1\\})=0.5.$ (72)

The signal $Z_{t}\in(\\{-1,1\\}\times\mathbb{R}_{+})^{\mathbb{N}}$ is given as
the vector $Z_{t}=(Z_{t}^{n})_{n\in\mathbb{N}}$, with
$Z^{n}=(Z^{n,1}_{t},Z^{n,2}_{t})\in\\{-1,1\\}\times\mathbb{R}_{+}$, defined as

$\displaystyle Z^{n}_{t}:=\int_{[0,T]\times
E\times\operatorname*{\mathbb{R}}_{+}}z^{n}(e,y)N(ds,de,dy),$ (73)
$\displaystyle\text{where
}z^{n}(e,y)=(-\mathbbm{1}_{\\{e_{3}=0\\}}+\mathbbm{1}_{\\{e_{3}=1\\}},y\mathbbm{1}_{[n-1,n]}(y)).$
(74)

We disintegrate for
$\bar{z}=(\bar{z}^{n})_{n\in\mathbb{N}}\in(\\{-1,1\\}\times\mathbb{R}_{+})^{\mathbb{N}}$,
with $\bar{z}^{n}=(\bar{z}^{n,1},\bar{z}^{n,2})$,

$\displaystyle\nu(de)\otimes
dy=\int_{z(E\times\operatorname*{\mathbb{R}}_{+})}K(\bar{z};de,dy)\mu(d\bar{z}),$
(75)

where

$\displaystyle\mu(d\bar{z})$
$\displaystyle=\sum_{n\in\mathbb{N}}\mu^{n}(d\bar{z}^{n}),$ (76)
$\displaystyle\mu^{n}(d\bar{z})$ $\displaystyle=(\nu\otimes
Leb_{\big{|}[n-1,n]}\,)\circ z^{-1}(\bar{z}^{n}),$ (77) $\displaystyle
K(\bar{z};de,dy)$
$\displaystyle=\mathbbm{1}_{\\{\bar{z}_{1}=-1\\}}(P^{1}\otimes
P^{2}\otimes\text{Dirac}_{\\{0\\}})(de)\otimes\text{Dirac}_{\sum_{n\in\mathbb{N}}\bar{z}^{n,2}}(dy)$
(78) $\displaystyle\quad+\mathbbm{1}_{\\{\bar{z}_{1}=1\\}}(P^{1}\otimes
P^{2}\otimes\text{Dirac}_{\\{1\\}})(de)\otimes\text{Dirac}_{\sum_{n\in\mathbb{N}}\bar{z}^{n,2}}(dy).$
(79)

Here, $\bar{z}_{1}$ and $\bar{z}_{2}$ denote the first and second component of
the signal.

###  4.5 Proofs

#### Lemma 2.1 \- The uncontrolled model dynamics admit a unique solution

###### Proof of Lemma 2.1.

It suffices to rule out that the mutually-exciting dynamics lead to blow-ups.
To this end, we prove that the expectation of the total variation
$V_{[0,t]}(\lambda)$ of the liquidity process $\lambda$ on $[0,t]$ is bounded
for each $t\in[0,T]$.
Let $c_{1},c_{2}$ be the constants as in (4). We use that $\eta,\rho\in
L^{1}(\nu)$ to bound the expected variation of $\lambda$ for every $t\in[0,T]$
from above

$\displaystyle\mathbb{E}[V_{[0,t]}(\lambda)]$
$\displaystyle\leq\mathbb{E}\left[\int_{[0,t]\times
E\times\mathbb{R}_{+}}\\!\\!\mathbbm{1}_{\\{y\leq
g(\lambda_{s-})\\}}|\rho(e)|N(ds,de,dy)+\int_{[0,t]\times
E\times\mathbb{R}_{+}}\\!\\!\mathbbm{1}_{\\{y\leq
f(\lambda_{s-})\\}}|\eta(e)|N(ds,de,dy)\right]$
$\displaystyle=\int_{E}|\rho(e)|\nu(de)\mathbb{E}\left[\int_{0}^{t}g(\lambda_{s})ds\right]+\int_{E}|\eta(e)|\nu(de)\mathbb{E}\left[\int_{0}^{t}f(\lambda_{s})ds\right]$
$\displaystyle\leq\left(\int_{E}|\rho(e)|\nu(de)+\int_{E}|\eta(e)|\nu(de)\right)\left(Tc_{1}+c_{2}T|\lambda_{0}|+\mathbb{E}\left[\int_{0}^{t}V_{[0,s]}(\lambda)ds\right]\right).$

Apply Gronwall’s inequality to obtain the uniform upper bound for all
$t\in[0,T]$

$\displaystyle\mathbb{E}[V_{[0,t]}(\lambda)]$ $\displaystyle\leq$
$\displaystyle\left(\int_{E}|\rho(e)|\nu(de)+\int_{E}|\eta(e)|\nu(de)\right)\left(Tc_{1}+c_{2}T|\lambda_{0}|\right)\exp\left(T\int_{E}|\rho(e)|\nu(de)+T\int_{E}|\eta(e)|\nu(de)\right).$

∎

#### Lemma 2.2 — Link between price volatility and liquidity of the market

###### Proof.

The quadratic variation of the price process has dynamics

$\displaystyle d[P]_{t}=$ $\displaystyle\int_{E}\mathbbm{1}_{\\{y\leq
f(\lambda_{t-})\\}}I(\eta(e),\lambda_{t-})^{2}N(dt,de,dy),\quad[P]_{0}=0.$
(80)

Hence, the dynamics of its predictable compensator coincide with (8). The
derivative of the expression in (9) with respect to $\lambda$ is negative if
condition (10) holds; this proves the monotonicity claim. ∎

#### Lemma 2.3 — Decomposition of $C$

###### Proof of Lemma 2.3.

It is easy to see that $C$ of the form (14) is $\Lambda$-measurable and has
bounded expected total variation.
To prove the reverse, we have to show that for a $\Lambda$-measurable process
$C$ of bounded expected variation, there exists a
$\mathcal{P}(\mathcal{F})\otimes\mathcal{B}(\mathbb{R})$-measurable field
$\Gamma$ which satisfies the integrability condition (16) and vanishes at zero
such that (15) holds.
From a monotone class argument, see [Bank and Körber, 2022, Lemma 2.2], we
know that there exists a
$\mathcal{P}(\mathcal{F})\otimes\mathcal{B}(\mathbb{R})$-measurable field
$\Gamma$ such that $\Delta^{l}_{t}C=\Gamma_{t}(\Delta_{t}Z)$, $t\in[0,T]$.
Next, because $C$ is of bounded variation, we conclude that $\Gamma_{t}(0)=0$
for all $t\in[0,T]$. Moreover, $\Gamma$ satisfies (16) because $C$ is of
bounded total variation and $\Delta Z_{s}=0$ for all but countably many times.
We finish the proof by

$\displaystyle\Delta^{l}_{t}C=\Gamma_{t}(\Delta_{t}Z)=\int_{E\times\mathbb{R}_{+}}\Gamma_{t}(z(e,y))N(dt,de,dy),\quad
t\in[0,T].$ (81)

∎

#### An upper bound for the available liquidity in the market with circuit
breaker

Let us note that

$\bar{V}(T):=\lambda_{0}-\underline{\lambda}+\int_{[0,T]\times
E\times\mathbb{R}_{+}}\mathbbm{1}_{\\{y\leq
g(\underline{\lambda})\\}}|\rho(e)|N(dt,de,dy)$ (82)

yields an upper bound for the liquidity to arrive at the market over the time
period $[0,T]$. Here, $g(\underline{\lambda})$ is an upper bound for
$g(\lambda)$ because we have $\lambda\geq\underline{\lambda}$ and because the
function $g$ is decreasing.

#### Lemma 3.1 \- Finite expected $n$-th moment of the total variation of
liquidity taking orders in the market with circuit breaker

###### Proof of Lemma 3.1.

Let $\rho\in L^{n}(\nu)$ for some $n\in\mathbb{N}$. First, use that market
orders and cancellations are executed as long as liquidity remains greater
than $\underline{\lambda}$ to write

$\displaystyle\underline{\lambda}\leq\lambda_{T}=\lambda_{0}-V_{[0,T]}(Q^{C})-V_{[0,T]}(\tilde{M}^{C})-V_{[0,T]}(\tilde{L}^{C,-})+V_{[0,T]}(\tilde{L}^{C,+}),$
(83)

with $\tilde{M}^{C}$, $\tilde{L}^{C,+}$, $\tilde{L}^{C,-}$ as in (51), (52)
and (53). By rearranging (83), we find the uniform upper bound

$\displaystyle
V_{[0,T]}(Q^{C})+V_{[0,T]}(\tilde{M}^{C})+V_{[0,T]}(\tilde{L}^{C,-})$
$\displaystyle\leq\lambda_{0}-\underline{\lambda}+V_{[0,T]}(\tilde{L}^{C,+})$
$\displaystyle\leq´\lambda_{0}-\underline{\lambda}+\int_{[0,T]\times
E\times\mathbb{R}_{+}}\mathbbm{1}_{\\{y\leq
g(\underline{\lambda})\\}}|\rho(e)|N(dt,de,dy)=\overline{V}(T),$

with $\bar{V}(T)$ as in (82). We apply the Cauchy Schwarz inequality and
finish the proof:

$\displaystyle\mathbb{E}\left[\left(V_{[0,T]}(Q^{C})+V_{[0,T]}(\tilde{M}^{C})+V_{[0,T]}(\tilde{L}^{C,-})\right)^{n}\right]\leq\mathbb{E}\left[\overline{V}(T)^{n}\right]$
$\displaystyle\leq(n+1)(\lambda_{0}-\underline{\lambda})^{n}+(n+1)(T\nu(E)g(\underline{\lambda}))^{n-1}\mathbb{E}\left[\int_{[0,T]\times
E\times[0,g(\underline{\lambda})]}|\rho(e)|^{n}N(dt,de,dy)\right]$
$\displaystyle=(n+1)(\lambda_{0}-\underline{\lambda})^{n}+(n+1)(Tg(\underline{\lambda}))^{n}\nu(E)^{n-1}\int_{E}|\rho(e)|^{n}\nu(de).$

∎

#### Preliminaries for the proofs of Proposition 3.2 and Theorem 3.3

Here, we provide an overview of estimates that are used to prove Lemma 3.1,
Proposition 3.2 and Theorem 3.3. First, with $\bar{V}$ as in (82) and by Lemma
3.1, we conclude that for every admissible strategy $Q\in\mathcal{Q}$

$\displaystyle V_{[0,T]}(Q^{C})\leq\overline{V}(T)\quad\text{ and
}\quad\sup_{t\in[0,T]}|Q^{C}_{t}|\leq|q|+\overline{V}(T).$ (84)

Because $\iota$ is decreasing, we have for every $\lambda\in\mathbb{R}$ and
$\Delta\in\mathbb{R}$

$|I(\Delta,\lambda)|\leq\iota(\underline{\lambda})|\Delta|,\quad|\Xi(\Delta,\lambda)|\leq\iota(\underline{\lambda})|\Delta|^{2}.$
(85)

Thus, we conclude

$\displaystyle
V_{[0,T]}(P^{C})\leq\iota(\underline{\lambda})\overline{V}(T)\quad\text{ and
}\quad\sup_{t\in[0,T]}|P^{C}_{t}|\leq|p|+\iota(\underline{\lambda})\overline{V}(T).$
(86)

Next, for $\lambda_{1},\lambda_{2}>\underline{\lambda}$ and
$\Delta,\Delta_{1},\Delta_{2}\in\mathbb{R}$, we suppose Lipschitz continuity
of $\iota$ with Lipschitz constant $c_{\iota}$ to bound the difference of $I$
as in (7) from above

$\displaystyle|I(|\Delta|,\lambda_{1})-I(|\Delta|,\lambda_{2})|\leq
c_{\iota}|\Delta||\lambda_{1}-\lambda_{2}|,$ (87)
$\displaystyle|I(|\Delta_{1}|,\lambda_{1})-I(|\Delta_{2}|,\lambda_{2})|\leq
c_{\iota}\min(|\Delta_{1}|,|\Delta_{2}|)|\lambda_{1}-\lambda_{2}|+\iota(\underline{\lambda})|\Delta_{1}-\Delta_{2}|.$

Similarly, we have for $\Xi$ as in (18)

$\displaystyle|\Xi(\Delta,\lambda_{1})-\Xi(\Delta,\lambda_{2})|\leq
c_{\iota}|\Delta|^{2}|\lambda_{1}-\lambda_{2}|,$ (88)
$\displaystyle|\Xi(\Delta_{1},\lambda_{1})-\Xi(\Delta_{2},\lambda_{2})|\leq
c_{\iota}\min(|\Delta_{1}|,|\Delta_{2}|)|\lambda_{1}-\lambda_{2}|+\iota(\underline{\lambda})\max(|\Delta_{1}|,|\Delta_{2}|)|\Delta_{1}-\Delta_{2}|.$

######  Lemma 4.1.

Let $\rho\in L^{2}(\nu)$ and assume starting values $(\lambda,q,p,x)$ for
$(\lambda^{C},Q^{C},P^{C},X^{C})$. There exists a uniform constant
$c=c(T,\lambda,q,p,x)>-\infty$ that depends continuously on its arguments such
that

$\displaystyle\inf_{C\in\tilde{\mathcal{C}}}\mathbb{E}\left[U_{\alpha}(\tilde{X}^{C}_{T})\right]\geq
c.$ (89)

###### Proof.

Let $(\lambda,q,p,x)$ be a set of starting values for
$(\lambda^{C},Q^{C},P^{C},X^{C})$. For simplicity, we denote by
$c=c(T,q,\lambda,p,x)>0$ a generic constant that depends continuously on
$T,q,\lambda,p,x$ and that may change from line to line.
First, with (84) and (86), we write for the cash position $X^{C}_{T}$ from
trading over the time horizon $[0,T]$:

$\displaystyle X^{C}_{T}$ $\displaystyle\geq
x-\sup_{t\in[0,T]}|P^{C}_{t}|V_{[0,T]}(Q^{C})-\zeta
V_{[0,T]}(Q^{C})-\iota(\underline{\lambda})V_{[0,T]}(Q^{C})^{2}.$ (90)

Similarly, we have for the cash from completing the execution programme at
time $T$:

$\displaystyle\qquad-P^{C}_{T}Q^{C}_{T}-\sigma
Y\operatorname*{sgn}(Q^{C}_{T})(|Q^{C}_{T}|-(\lambda^{C}_{T}-\underline{\lambda})^{+})^{+}-\zeta|Q^{C}_{T}|-\Xi(|Q^{C}_{T}|,\lambda_{T})$
(91)
$\displaystyle\geq-(\sup_{t\in[0,T]}|P^{C}_{t}|+\sigma|Y|+\zeta)(|q|+V_{[0,T]}(Q^{C}))-\iota(\underline{\lambda})(|q|+V_{[0,T]}(Q^{C}))^{2}.$

We aggregate (90) and (91) and apply the estimates from (84) and (86) to bound
$\tilde{X}^{C}_{T}$ from below

$\tilde{X}^{C}_{T}\geq-c(|Y|+\overline{V}(T)+|Y|\overline{V}(T)+\overline{V}(T)^{2}).$
(92)

For $\alpha=0$, the claim follows using Lemma 3.1 for $n=2$.
For $\alpha>0$, we use the monotonocity of $U_{\alpha}$ together with (92) to
estimate

$\displaystyle\mathbb{E}[U_{\alpha}(\tilde{X}^{C}_{T})]$
$\displaystyle\geq-\mathbb{E}\bigg{[}\exp\Big{(}\alpha
c(|Y|+\overline{V}(T)+|Y|\overline{V}(T)+\overline{V}(T)^{2})\Big{)}\bigg{]}$
(93) $\displaystyle=-\mathbb{E}\bigg{[}\mathbb{E}\Big{[}\exp\Big{(}\alpha
c(|Y|+|Y|\overline{V}(T))\Big{)}\Big{|}\overline{V}(T)\Big{]}\exp\Big{(}\alpha
c(\overline{V}(T)+\overline{V}(T)^{2})\Big{)}\bigg{]}$ (94)
$\displaystyle=-\mathbb{E}\bigg{[}2\Phi(\alpha
c(1+\overline{V}(T)))\exp\Big{(}\frac{\alpha^{2}c^{2}}{2}(1+\overline{V}(T))^{2}\Big{)}\exp\Big{(}\alpha
c(\overline{V}(T)+\overline{V}(T)^{2})\Big{)}\bigg{]},$ (95)

where we use the moment generating function of a folded normal distribution
and where $\Phi$ is the cumulative distribution function of a Gaussian
distribution. Next, use that $\Phi\leq 1$ to write for some constant $c$

$\displaystyle\mathbb{E}[U_{\alpha}(\tilde{X}^{C}_{T})]$
$\displaystyle\geq-c\>\mathbb{E}\bigg{[}\exp\Big{(}c(1+\overline{V}(T)^{2})\Big{)}\bigg{]}$
(96)

To estimate this expresssion, we write

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}c\overline{V}(T)^{2}\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}\left[\exp\Big{(}c\Big{(}(\lambda-\underline{\lambda})+\int_{[0,T]\times
E\times[0,g(\underline{\lambda})]}|\rho(e)|N(dt,de,dy)\Big{)}^{2}\Big{)}\right]$
(97)
$\displaystyle\leq\mathbb{E}\left[\exp\Big{(}3c\Big{(}(\lambda-\underline{\lambda})^{2}+\Big{(}\int_{[0,T]\times
E\times[0,g(\underline{\lambda})]}|\rho(e)|N(dt,de,dy)\Big{)}^{2}\Big{)}\Big{)}\right].$
(98)

Next, we apply the Cauchy Schwarz inequality and have with the Lévy-Khintchine
formula

$\displaystyle\mathbb{E}\left[\exp\Big{(}3c\Big{(}(\lambda-\underline{\lambda})^{2}+\Big{(}\int_{[0,T]\times
E\times[0,g(\underline{\lambda})]}|\rho(e)|N(dt,de,dy)\Big{)}^{2}\Big{)}\Big{)}\right]$
(99)
$\displaystyle\leq\exp\left(3c(\lambda-\underline{\lambda})^{2}\right)\mathbb{E}\left[\exp\Big{(}3cTg(\underline{\lambda})\nu(E)\int_{[0,T]\times
E\times[0,g(\underline{\lambda})]}|\rho(e)|^{2}N(dt,de,dy)\Big{)}\right]$
(100)
$\displaystyle=\exp\left(3c(\lambda-\underline{\lambda})^{2}\right)\exp\left(Tg(\underline{\lambda})\int_{E}\Big{[}\exp(3cTg(\underline{\lambda})\nu(E)|\rho(e)|^{2})-1\Big{]}\nu(de)\right)<\infty,$
(101)

where we use (35). This finishes the proof. ∎

#### Proposition 3.2 \- The value function is non-degenerate

We use Lemma 3.1 to prove that the value function as in (33) is non-
degenerate, i.e., $v<U_{\gamma}(+\infty)$.

###### Proof of Proposition 3.2.

Let $(\lambda,q,p,x)$ be a set of starting values for
$(\lambda^{C},Q^{C},P^{C},X^{C})$.
By the concavity of $U_{\alpha}$ and with Jensen’s inequality, the claim
follows if there exists a uniform, finite constant $c=c(T,q,\lambda,p,x)$ such
that

$\displaystyle\sup_{C\in\tilde{\mathcal{C}}}\mathbb{E}\Big{[}|\tilde{X}^{C}_{T}|\Big{]}\leq
c.$ (102)

By the triangle inequality, it is hence sufficient to prove

$\displaystyle\sup_{C\in\tilde{\mathcal{C}}}\Big{(}\mathbb{E}\left[|X^{C}_{T}|\right]+\mathbb{E}\left[|P^{C}_{T}||Q^{C}_{T}|\right]+\mathbb{E}\left[|\sigma
Y||Q^{C}_{T}|\right]+\mathbb{E}\left[\zeta|Q^{C}_{T}|\right]+\mathbb{E}\left[\Xi(|Q^{C}_{T}|,\lambda_{T})\right]\Big{)}\leq
c.$ (103)

For simplicity, we denote by $c=c(T,q,\lambda,p,x)>0$ a generic constant that
depends continuously on $T,q,\lambda,p,x$ and that my change from line to
line. We apply Lemma 3.1 with $n=1,2$ to write

$\displaystyle\mathbb{E}\left[V_{[0,T]}(Q^{C})+V_{[0,T]}(M^{C})\right]+\mathbb{E}\left[\left(V_{[0,T]}(Q^{C})+V_{[0,T]}(M^{C})\right)^{2}\right]\leq
c.$ (104)

Next, by (84) and (104), we know for every $C\in\tilde{\mathcal{C}}$

$\displaystyle\mathbb{E}[|Q^{C}_{T}|]+\mathbb{E}[|Q^{C}_{T}|^{2}]\leq c.$
(105)

Similarly, by (85), (84) and by independence of $Y$, we write

$\displaystyle\mathbb{E}[\sigma|Y||Q^{C}_{T}|]+\mathbb{E}[\zeta|Q^{C}_{T}|]+\mathbb{E}[|\Xi(|Q^{C}_{T}|,\lambda_{T})|]\leq
c.$ (106)

With (86) and (104), we have

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}|P^{C}_{t}|\right]+\mathbb{E}\left[\sup_{t\in[0,T]}|P^{C}_{t}|^{2}\right]\leq
c.$ (107)

Next, we use the Cauchy Schwarz inequality together with (105) and (107) to
write

$\displaystyle\mathbb{E}[|P^{C}_{T}||Q^{C}_{T}|]\leq\mathbb{E}[|P^{C}_{T}|^{2}]^{\frac{1}{2}}\mathbb{E}[|Q^{C}_{T}|^{2}]^{\frac{1}{2}}\leq
c.$

Finally, by (84), (85), (104), (107), and the Cauchy Schwarz inequality, we
have

$\displaystyle\mathbb{E}[|X^{C}_{T}|]$ $\displaystyle\leq
x+\mathbb{E}\left[\sup_{t\in[0,T]}|P^{C}_{t}|V_{[0,T]}(Q^{C})+\zeta
V_{[0,T]}(Q^{C})+\iota(\underline{\lambda})V_{[0,T]}(Q^{C})^{2}\right]\leq c.$

Aggregating the above estimates, we conclude (103). ∎

#### Theorem 3.3 \- The value function is continuous

Proving continuity of the value function is rather involved in the present
problem because neither the state nor the control space is bounded at any
point in time, rendering standard arguments inapplicable. Also the circuit
breaker and the Hawkes-like jump structures pose a challenge for the proof
continuity.

First, consider two sets of starting values
$(\lambda^{\prime},q^{\prime},p,x)$ and
$(\lambda^{\prime\prime},q^{\prime\prime},p,x)$ for
$(\lambda^{C}_{t},Q^{C}_{t},P^{C}_{t},X^{C}_{t})$ and let
$T^{\prime},T^{\prime\prime}\geq 0$. For simplicity and without loss of
generality, we treat the case where p=x=0, a generalization is straight
forward using (61). Let $\varepsilon>0$ and let $C^{\prime}$ be an
$\varepsilon$-optimal strategy for the values
$(T^{\prime},\lambda^{\prime},q^{\prime},p,x)$, and $C^{\prime\prime}$ some
strategy for $(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime},p,x)$.
By concavity of the utility function $U_{\alpha}$, the first Taylor
approximation is an upper bound for the difference of the value functions for
$\alpha>0$:

$\displaystyle\Big{(}w(T^{\prime},\lambda^{\prime},q^{\prime})-w(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})\Big{)}=v(T^{\prime},\lambda^{\prime},q^{\prime},p,x)-v(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime},p,x)$
(108)
$\displaystyle\leq\mathbb{E}\left[U_{\alpha}(\tilde{X}^{C^{\prime}}_{T^{\prime}})\right]+\varepsilon-\mathbb{E}\left[U_{\alpha}(\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})\right]\leq\mathbb{E}\left[\alpha\exp(-\alpha\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})(\tilde{X}^{C^{\prime}}_{T^{\prime}}-\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})\right]+\varepsilon.$
(109)

We rearrange the terms and apply the Cauchy Schwarz inequality to write

$w(T^{\prime},\lambda^{\prime},q^{\prime})-w(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})\leq\exp(\alpha(x+pq))\alpha\mathbb{E}\left[\exp(-2\alpha\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})\right]^{1/2}\mathbb{E}\left[(\tilde{X}^{C^{\prime}}_{T^{\prime}}-\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})^{2}\right]^{1/2}+\varepsilon,$
(110)

From Lemma 4.1, we know that there exists a finite constant $c>0$ depending
continuously on $(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})$
such that

$\mathbb{E}\left[\exp(-2\alpha\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})\right]\leq
c.$

Consequently, we obtain the estimate

$w(T^{\prime},\lambda^{\prime},q^{\prime})-w(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})\leq
c\mathbb{E}\left[(\tilde{X}^{C^{\prime}}_{T^{\prime}}-\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}})^{2}\right]^{1/2}+\varepsilon,$
(111)

where $c>0$ is a finite constant depending continuously on
$(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})$.
For $\alpha=0$, we have the estimate

$\displaystyle
w(T^{\prime},\lambda^{\prime},q^{\prime})-w(T^{\prime\prime},\lambda^{\prime\prime},q^{\prime\prime})$
$\displaystyle\leq\mathbb{E}\left[\tilde{X}^{C^{\prime}}_{T^{\prime}}-\tilde{X}^{C^{\prime\prime}}_{T^{\prime\prime}}\right]+\varepsilon.$
(112)

######  Lemma 4.2 (Continuity in $q$).

Let the conditions of Theorem 3.3 hold. Then, for $T,\lambda$ in a compact
set, the value function $w(T,\lambda,q)$ as in (61) is locally Lipschitz
continuous in $q$, i.e., for $q^{\prime},q^{\prime\prime}$ in a compact set,
we have $|w(T,\lambda,q^{\prime})-w(T,\lambda,q^{\prime\prime})|\leq
c|q^{\prime}-q^{\prime\prime}|$ where the Lipschitz constant $c$ does not
depend of $T,\lambda,q^{\prime},q^{\prime\prime}$.

###### Proof.

For simplicity, we treat the case $\alpha>0$; the case $\alpha=0$ is proven
analogously.
Let $T,\lambda$ be from some compact set and without loss of generality we
consider two different starting values $q^{\prime}$ and $q^{\prime\prime}$
such that $\max(|q^{\prime}|,|q^{\prime\prime}|)\leq c_{max}$ and assume as
above that $p=x=0$.
Let $\varepsilon>0$ and let $C^{\prime}$ be an $\varepsilon$-optimal strategy
for starting value $q^{\prime}$. We define strategy $C^{\prime\prime}$ for
starting value $q^{\prime\prime}$ to copy the trades of $C^{\prime}$, i.e.,
$C^{\prime\prime}:=C^{\prime}$ so that the trader with strategy
$C^{\prime\prime}$ executes the additional amount of
$q^{\prime\prime}-q^{\prime}$ at terminal time $T$.
Consequently, the liquidity processes
$\lambda^{C^{\prime}}$,$\lambda^{C^{\prime\prime}}$, the hitting times
$\tau^{C^{\prime}}$,$\tau^{C^{\prime\prime}}$, the price processes
$P^{C^{\prime}}$, $P^{C^{\prime\prime}}$ and the cash processes
$X^{C^{\prime}}$, $X^{C^{\prime\prime}}$ coincide and we have
$Q^{C^{\prime\prime}}=Q^{C^{\prime}}+(q^{\prime\prime}-q^{\prime})$.
For simplicity, we denote by $c=c(T,\lambda,c_{max})>0$ a generic constant
that depends continuously on $T,\lambda,c_{max}$ and that may change from line
to line. With the Taylor estimate in (111), we have

$\displaystyle w(T,\lambda,q^{\prime})-w(T,\lambda,q^{\prime\prime})$
$\displaystyle\leq
c\>\mathbb{E}\Big{[}\Big{(}X^{C^{\prime}}_{T}+P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}+\sigma
Y\operatorname*{sgn}(Q^{C^{\prime}}_{T})(|Q^{C^{\prime}}_{T}|-(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}-\zeta|Q^{C^{\prime}}_{T}|-\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})$
$\displaystyle\;\>-\\!\left(X^{C^{\prime\prime}}_{T}\\!+P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T}\\!+\sigma
Y\operatorname*{sgn}(Q^{C^{\prime\prime}}_{T})(|Q^{C^{\prime\prime}}_{T}|-(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\\!-\zeta|Q^{C^{\prime\prime}}_{T}|-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T})\right)\Big{)}^{2}\Big{]}^{1/2}\\!\\!+\varepsilon$
$\displaystyle\leq
4c\>\mathbb{E}\Big{[}(P^{C^{\prime}}_{T})^{2}(q^{\prime\prime}-q^{\prime})^{2}+\sigma^{2}Y^{2}(q^{\prime}-q^{\prime\prime})^{2}$
$\displaystyle\quad\qquad+\zeta^{2}(q^{\prime\prime}-q^{\prime})^{2}+\left(\Xi(Q^{C^{\prime}}_{T}+(q^{\prime\prime}-q^{\prime}),\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})\right)^{2}\Big{]}^{1/2}\\!\\!+\varepsilon.$

By (86) and Lemma 3.1 with $n=2$, we have

$\displaystyle\mathbb{E}\Big{[}(P^{C^{\prime}}_{T})^{2}(q^{\prime\prime}-q^{\prime})^{2}\Big{]}\leq(q^{\prime\prime}-q^{\prime})^{2}\left(p^{2}+\iota(\underline{\lambda})^{2}\mathbb{E}\left[\overline{V}(T)^{2}\right]\right)\leq
c(q^{\prime}-q^{\prime\prime})^{2}.$

Similarly, we know
$\mathbb{E}\Big{[}Y^{2}(q^{\prime\prime}-q^{\prime})^{2}\Big{]}\leq
c(q^{\prime\prime}-q^{\prime})^{2}$.
Finally, by (88) and with Lemma (3.1) for $n=2$, we write

$\displaystyle\mathbb{E}\Big{[}\Big{(}\Xi(Q^{C^{\prime}}_{T}+(q^{\prime\prime}-q^{\prime}),\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})\Big{)}^{2}\Big{]}$
$\displaystyle\leq c(q^{\prime}-q^{\prime\prime})^{2}.$

Aggregating the above estimates, we obtain

$\displaystyle w(t,\lambda,q^{\prime})-w(t,\lambda,q^{\prime\prime})\leq
c|q^{\prime}-q^{\prime\prime}|+\varepsilon,$ (113)

for some finite constant $c=c(T,\lambda,c_{max})>0$ that is continuous in the
variables, i.e., does not depend on $T,\lambda$ as long as these are from some
compact set.
Because $\varepsilon$ was chosen arbitrarily and because the local Lipschitz
constant $c$ does not depend on $\varepsilon$, we finish the proof by
exchanging the roles of $q^{\prime}$ and $q^{\prime\prime}$. ∎

######  Lemma 4.3.

Let the conditions of Theorem 3.3 hold. For $T,q$ in a compact set, the
function $w(T,\lambda,q)$ as in (61) is locally $\smash{\frac{1}{2}}$-Hölder
continuous in $\lambda\in[\underline{\lambda},\infty)$, i.e., for
$\lambda^{\prime},\lambda^{\prime\prime}$ in a compact set with
$|\lambda^{\prime}-\lambda^{\prime\prime}|<1$, we have
$|w(T,\lambda^{\prime},q)-w(T,\lambda^{\prime\prime},q)|\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{1/2}$, where $c$ does not depend
on $T,q,\lambda^{\prime},\lambda^{\prime\prime}$.

###### Proof.

The argument for $\alpha=0$ being similar, we treat the case $\alpha>0$.
Let $T,q$ be fixed and let
$\lambda^{\prime},\lambda^{\prime\prime}>\underline{\lambda}$ be two starting
values, where we can assume without loss of generality that
$\max(|\lambda^{\prime}|,|\lambda^{\prime\prime}|)\leq c_{max}$ and as before
that $p=x=0$.
For simplicity, we denote by $c=c(T,c_{max},q)>0$ a generic constant that
depends continuously on $T,c_{max},q$ and that may change from line to line.
For the starting value $\lambda^{\prime}$, we fix $\varepsilon>0$ and choose
$C^{\prime}$ an $\varepsilon$-optimal strategy with predictable field
$\Gamma^{\prime}_{s}$, and impulses $\Delta^{r}_{s}\tilde{C}^{\prime}$, see
Lemma 2.3. We denote $(\lambda^{C^{\prime}}_{s})_{s\in[0,T]}$ the respective
liquidity processes for starting value $\lambda^{\prime}$ and strategies
$C^{\prime}$.
Next, we denote by $C^{\prime\prime}$ a strategy for starting value
$\lambda^{\prime\prime}$ with the corresponding liquidity process
$(\lambda^{C^{\prime\prime}}_{s})_{s\in[0,T]}$, that is defined by the
predictable field $\Gamma^{\prime\prime}_{s}$ and impulses
$\Delta^{r}_{s}C^{\prime\prime}$ that are as follows:

Strategy $C^{\prime\prime}$ copies the trades of $C^{\prime}$ as long as
$\lambda^{C^{\prime}}$ and $\lambda^{C^{\prime\prime}}$ remain above the lower
bound $\underline{\lambda}$, i.e.,

$\displaystyle\Gamma^{\prime\prime}_{s}:=\Gamma^{\prime}_{s},\qquad\text{ if
}\lambda^{C^{\prime}}_{s-}-|\Gamma^{\prime}_{s}(\Delta_{s}Z)|\geq\underline{\lambda}\text{
and
}\lambda^{C^{\prime\prime}}_{s-}-|\Gamma^{\prime}_{s}(\Delta_{s}Z)|\geq\underline{\lambda}$
(114)

and similarly for the impulse $\Delta^{r}_{s}C^{\prime\prime}$. If the trade
of $C^{\prime}$ would trigger the circuit breaker for the trader with strategy
$C^{\prime\prime}$, i.e., if
$\lambda^{C^{\prime\prime}}-|\Gamma^{\prime}_{s}(\Delta_{s}Z)|<\underline{\lambda}$,
but $C^{\prime}$ does not trigger it, $C^{\prime\prime}$ depletes the
available liquidity $\lambda^{C^{\prime\prime}}_{s-}-\underline{\lambda}$
without triggering the circuit breaker:

$\displaystyle\Gamma^{\prime\prime}_{s}:=\operatorname*{sgn}(\Gamma^{\prime}_{s})(\lambda^{C^{\prime\prime}}_{s-}-\underline{\lambda}),\qquad\text{
if
}\lambda^{C^{\prime}}_{s-}-|\Gamma^{\prime}_{s}|\geq\underline{\lambda},\text{
and
}\lambda^{C^{\prime\prime}}_{s-}-|\Gamma^{\prime}_{s}|<\underline{\lambda},$
(115)

and similarly for the impulse $\Delta^{r}_{s}C^{\prime\prime}$. Note that this
is an admissible action by (40).
Finally, in the case where the trader with strategy $C^{\prime}$ triggers the
circuit breaker, the trader with $C^{\prime\prime}$ triggers the circuit
breaker as well, i.e.,

$\displaystyle\Gamma^{\prime\prime}_{s}:=\operatorname*{sgn}(\Gamma^{\prime}_{s})(\lambda^{C^{\prime\prime}}_{s-}-\underline{\lambda}+\delta^{*}),\qquad\text{
if }\lambda^{C^{\prime}}_{s-}-|\Gamma^{\prime}_{s}|<\underline{\lambda},$
(116)

where we w.l.o.g. set $\delta^{*}=\delta$ when the tarder trades in multiples
of some lot size $\delta$ and $\delta^{*}=1$ for continuous trading. The
impulse $\Delta^{r}_{s}C^{\prime\prime}$ is defined analogously.
The respective hitting times are $\tau^{C^{\prime}}$ and
$\tau^{C^{\prime\prime}}$, the inventory processes are $Q^{C^{\prime}}$,
$Q^{C^{\prime\prime}}$, the price processes are $P^{C^{\prime}}$,
$P^{C^{\prime\prime}}$, and the cash processes are $X^{C^{\prime}}$,
$X^{C^{\prime\prime}}$.

Next, we consider the set $A$ where the external shocks in
$\lambda^{C^{\prime}}_{s}$ and $\lambda^{C^{\prime\prime}}_{s}$ are different:

$\displaystyle A:=\Bigg{\\{}\int_{[0,\tau^{C^{\prime}}]\times
E\times\mathbb{R}_{+}}\Big{[}$
$\displaystyle\mathbbm{1}_{\\{f(\lambda^{C^{\prime}}_{s-})\wedge
f(\lambda^{C^{\prime\prime}}_{s-})<y\leq f(\lambda^{C^{\prime}}_{s-})\vee
f(\lambda^{C^{\prime\prime}}_{s-})\\}}$ (117)
$\displaystyle\qquad+\mathbbm{1}_{\\{g(\lambda^{C^{\prime}}_{s-})\wedge
g(\lambda^{C^{\prime\prime}}_{s-})<y\leq g(\lambda^{C^{\prime}}_{s-})\vee
g(\lambda^{C^{\prime\prime}}_{s-})\\}}\Big{]}N(ds,de,dy)\geq 1\Bigg{\\}}.$

By definition of $C^{\prime\prime}$, we conclude that on its complement
$A^{c}$, both liquidity processes $\lambda^{C^{\prime}}_{s}$ and
$\lambda^{C^{\prime}}_{s}$ reach the lower bound at the same time, i.e., we
have
$\mathbbm{1}_{A^{c}}\tau^{C^{\prime}}=\mathbbm{1}_{A^{c}}\tau^{C^{\prime\prime}}$.
Similarly, we have

$\displaystyle\mathbbm{1}_{A^{c}}|\lambda^{C^{\prime}}_{s+}-\lambda^{C^{\prime\prime}}_{s+}|\leq\mathbbm{1}_{A^{c}}|\lambda^{C^{\prime}}_{s-}-\lambda^{C^{\prime\prime}}_{s-}|\leq|\lambda^{\prime}-\lambda^{\prime\prime}|,\qquad
s\in[0,\tau^{C^{\prime}}).$ (118)

For the general case, using (118) and the Lipschitz continuity of $f$ and $g$,
we have for $t\leq T$

$\displaystyle\mathbb{E}\left[\sup_{s\in[0,t\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]}|\lambda_{s-}^{C^{\prime}}-\lambda_{s-}^{C^{\prime\prime}}|\right]$
$\displaystyle\leq|\lambda^{\prime}-\lambda^{\prime\prime}|+\mathbb{E}\bigg{[}\int_{[0,t\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]\times
E\times\mathbb{R}_{+}}\Big{[}\mathbbm{1}_{\\{f(\lambda^{C^{\prime}}_{s-})\wedge
f(\lambda^{C^{\prime\prime}}_{s-})<y\leq f(\lambda^{C^{\prime}}_{s-})\vee
f(\lambda^{C^{\prime\prime}}_{s-})\\}}|\eta(e)|$ $\displaystyle\hskip
85.35826pt+\mathbbm{1}_{\\{g(\lambda^{C^{\prime}}_{s-})\wedge
g(\lambda^{C^{\prime\prime}}_{s-})<y\leq g(\lambda^{C^{\prime}}_{s-})\vee
g(\lambda^{C^{\prime\prime}}_{s-})\\}}|\rho(e)|\Big{]}N(ds,de,dy)\bigg{]}$
$\displaystyle\leq|\lambda^{\prime}-\lambda^{\prime\prime}|+\mathbb{E}\bigg{[}\int_{[0,t\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]}\Big{(}|f(\lambda^{C^{\prime}}_{s-})-f(\lambda^{C^{\prime\prime}}_{s-})|\int_{E}|\eta(e)|\nu(de)$
$\displaystyle\hskip
139.41832pt+|g(\lambda^{C^{\prime}}_{s-})-g(\lambda^{C^{\prime\prime}}_{s-})|\int_{E}|\rho(e)|\nu(de)\Big{)}ds\bigg{]}$
$\displaystyle\leq|\lambda^{\prime}-\lambda^{\prime\prime}|+c\Big{(}\int_{E}|\eta(e)|\nu(de)+\int_{E}|\rho(e)|\nu(de)\Big{)}\mathbb{E}\bigg{[}\int_{[0,t\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]}|\lambda^{C^{\prime}}_{s-}-\lambda^{C^{\prime\prime}}_{s-}|ds\bigg{]}$
$\displaystyle\leq|\lambda^{\prime}-\lambda^{\prime\prime}|+c\Big{(}\int_{E}|\eta(e)|\nu(de)+\int_{E}|\rho(e)|\nu(de)\Big{)}\int_{[0,t]}\mathbb{E}\bigg{[}\sup_{r\in[0,s\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]}|\lambda_{r-}^{C^{\prime}}-\lambda_{r-}^{C^{\prime\prime}}|\bigg{]}ds.$

Applying Gronwall’s inequality, we obtain

$\displaystyle\mathbb{E}\left[\sup_{s\in[0,T\wedge\tau^{C^{\prime}}\wedge\tau^{C^{\prime\prime}}]}|\lambda_{s-}^{C^{\prime}}-\lambda_{s-}^{C^{\prime\prime}}|\right]\leq
c^{*}|\lambda^{\prime}-\lambda^{\prime\prime}|,$ (119)

for some finite constant $c^{*}>0$ that is independent of $T$, $q$,
$\lambda^{\prime}$, and $\lambda^{\prime\prime}$ for
$T,q,\lambda^{\prime},\lambda^{\prime\prime}$ from a compactum.
With the Taylor estimate (111), we write with $A$ as in (117)

$\displaystyle w(T,\lambda^{\prime},q)-w(T,\lambda^{\prime\prime},q)$
$\displaystyle\leq
c\mathbb{E}\bigg{[}\Big{(}X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}+P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T}+\zeta(|Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T}|)+(\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T}))$
$\displaystyle\;+\sigma
Y\big{(}\operatorname*{sgn}(Q^{C^{\prime}}_{T})(|Q^{C^{\prime}}_{T}|\\!-\\!(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}\\!\\!-\operatorname*{sgn}(Q^{C^{\prime\prime}}_{T})(|Q^{C^{\prime\prime}}_{T}|\\!-\\!(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\big{)}\Big{)}^{2}\\!\Big{(}\mathbbm{1}_{A}+\mathbbm{1}_{A^{c}}\Big{)}\bigg{]}^{1/2}\\!\\!\\!+\varepsilon.$

For the expectation over $A$, we use the Cauchy Schwarz inequality to obtain
the upper bound

$\displaystyle\mathbb{E}\bigg{[}\Big{(}X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}+P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T}+\zeta(|Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T}|)+(\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T}))$
(120) $\displaystyle\quad+\sigma
Y\big{(}\operatorname*{sgn}(Q^{C^{\prime}}_{T})(|Q^{C^{\prime}}_{T}|-(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}-\operatorname*{sgn}(Q^{C^{\prime\prime}}_{T})(|Q^{C^{\prime\prime}}_{T}|-(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\big{)}\Big{)}^{4}\bigg{]}^{1/2}\operatorname*{\mathbb{E}}\Big{[}\mathbbm{1}_{A}\Big{]}^{1/2}$

We apply Lemma 3.1 for $n=8$ to conclude

$\displaystyle\mathbb{E}\bigg{[}\Big{(}X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}+P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T}+\zeta(|Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T}|)+(\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T}))$
$\displaystyle\quad+\sigma
Y\big{(}\operatorname*{sgn}(Q^{C^{\prime}}_{T})(|Q^{C^{\prime}}_{T}|-(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}-\operatorname*{sgn}(Q^{C^{\prime\prime}}_{T})(|Q^{C^{\prime\prime}}_{T}|-(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\big{)}\Big{)}^{4}\bigg{]}\leq
c.$

Next, we use Markov’s inequality, the Lipschitz continuity of $f$ and $g$ and
(119) to write

$\displaystyle\operatorname*{\mathbb{E}}\Big{[}\mathbbm{1}_{A}\Big{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\\!\int_{[0,T]\times
E\times\mathbb{R}_{+}}\\!\\!\\!\\!\\!\\!\\![\mathbbm{1}_{\\{f(\lambda^{C^{\prime}}_{s-})\wedge
f(\lambda^{C^{\prime\prime}}_{s-})<y\leq f(\lambda^{C^{\prime}}_{s-})\vee
f(\lambda^{C^{\prime\prime}}_{s-})\\}}\\!+\\!\mathbbm{1}_{\\{g(\lambda^{C^{\prime}}_{s-})\wedge
g(\lambda^{C^{\prime\prime}}_{s-})<y\leq g(\lambda^{C^{\prime}}_{s-})\vee
g(\lambda^{C^{\prime\prime}}_{s-})\\}}]N(ds,de,dy)\Bigg{]}$
$\displaystyle=\nu(E)\mathbb{E}\left[\int_{[0,T]}\Big{[}|f(\lambda^{C^{\prime}}_{s-})-f(\lambda^{C^{\prime\prime}}_{s-})|+|g(\lambda^{C^{\prime}}_{s-})-g(\lambda^{C^{\prime\prime}}_{s-})|\Big{]}ds\right]$
$\displaystyle\leq
c\mathbb{E}\left[\sup_{s\in[0,T]}|\lambda^{C^{\prime}}_{s-}-\lambda^{C^{\prime\prime}}_{s-}|\right]\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|.$

Consequently, (120) is bounded from above by
$c|\lambda^{\prime}-\lambda^{\prime\prime}|^{1/2}$.

Next, on the set $A^{c}$, recall that
$\tau^{C^{\prime}}=\tau^{C^{\prime\prime}}$ and that we know with (118)

$\displaystyle\mathbbm{1}_{A^{c}}|\lambda^{C^{\prime}}_{s}-\lambda^{C^{\prime\prime}}_{s}|\leq|\lambda^{\prime}-\lambda^{\prime\prime}|\quad\text{
for }s\in[0,T\wedge\tau^{C^{\prime}}].$ (121)

Moreover, by definition of $C^{\prime\prime}$, the differences of the trades
in $\tilde{C}^{\prime}$ and $\tilde{C}^{\prime\prime}$ sum up to at most
$|\lambda^{C^{\prime}}_{s}-\lambda^{C^{\prime\prime}}_{s}|$ so that with (121)
we have

$\mathbbm{1}_{A^{c}}\bigg{(}\sum_{0\leq s\leq
T}|\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s-})-\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s-})|+\sum_{0\leq
s<T}|\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s})-\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s})|\bigg{)}\leq|\lambda^{\prime}-\lambda^{\prime\prime}|.$
(122)

With (84), (121), (122) and (87), we conclude

$\displaystyle\mathbbm{1}_{A^{c}}\sup_{s\in[0,T]}|P^{C^{\prime}}_{s}-P^{C^{\prime\prime}}_{s}|^{2}\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{2}\left(1+\overline{V}^{2}\right).$
(123)

Now, use the triangle inequality to estimate

$\displaystyle\mathbb{E}[\mathbbm{1}_{A^{c}}|X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}|^{2}]$
$\displaystyle\leq\mathbb{E}\bigg{[}\mathbbm{1}_{A^{c}}\Big{(}\sup_{s\in[0,T]}|P^{C^{\prime}}_{s}-P^{C^{\prime\prime}}_{s}|\overline{V}$
$\displaystyle\quad+(\sup_{s\in[0,T]}|P^{C^{\prime}}_{s}|+\zeta)\sum_{0\leq
s\leq
T}\left(|\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s-})\\!-\\!\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s-})|+|\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s})\\!-\\!\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s})|\right)$
$\displaystyle\quad+\sum_{0\leq s\leq
T}|\Xi(\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s-}),\lambda^{C^{\prime}}_{s-})-\Xi(\Upsilon(\Delta^{l}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s-}),\lambda^{C^{\prime\prime}}_{s-})|$
$\displaystyle\quad+\sum_{0\leq
s<T}|\Xi(\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime},\lambda^{C^{\prime}}_{s}),\lambda^{C^{\prime}}_{s})-\Xi(\Upsilon(\Delta^{r}_{s}\tilde{C}^{\prime\prime},\lambda^{C^{\prime\prime}}_{s}),\lambda^{C^{\prime\prime}}_{s})|\Big{)}^{2}\bigg{]}.$

With (88), (119), (121),(122), and Lemma 3.1 for $n=2,4$, we thus have

$\displaystyle\mathbb{E}[\mathbbm{1}_{A^{c}}|X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}|^{2}]$
$\displaystyle\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{2}\mathbb{E}\Big{[}\overline{V}(T)^{2}+\overline{V}(T)^{4}\Big{]}\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{2},$ (124)

With Lemma 3.1 for $n=4$ and (122), we write

$\displaystyle\mathbbm{1}_{A^{c}}|Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T}|^{4}\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{4},$ (125)
$\displaystyle\mathbbm{1}_{A^{c}}(|Q^{C^{\prime}}_{T}|^{4}+|Q^{C^{\prime\prime}}_{T}|^{4})\leq
c(q^{4}+\overline{V}(T)^{4}),$ (126)

and by (85) and (87), we know

$\displaystyle\mathbbm{1}_{A^{c}}|P^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}|^{4}\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{4}(1+\overline{V}(T)^{4}),$ (127)
$\displaystyle\mathbbm{1}_{A^{c}}(|P^{C^{\prime}}_{t}|^{4}+|P^{C^{\prime\prime}}_{t}|^{4})\leq
c(p^{4}+\overline{V}(T)^{4}).$ (128)

Hence, we have with the Cauchy Schwarz inequality

$\displaystyle\mathbb{E}\Big{[}\mathbbm{1}_{A^{c}}(P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T})^{2}$
$\displaystyle\qquad+\mathbbm{1}_{A^{c}}\mathbbm{1}_{\\{\tau^{C^{\prime}}\leq
T\\}}\sigma^{2}Y^{2}\big{(}(|Q^{C^{\prime}}_{T}|-(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}-(|Q^{C^{\prime\prime}}_{T}|-(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\big{)}^{2}\Big{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\mathbbm{1}_{A^{c}}\Big{[}\max(Q^{C^{\prime}}_{T},Q^{C^{\prime\prime}}_{T})^{2}(P^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T})^{2}\\!+\max(P^{C^{\prime}}_{T},P^{C^{\prime\prime}}_{T})^{2}(Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T})^{2}$
$\displaystyle\hskip 56.9055pt+\mathbbm{1}_{\\{\tau^{C^{\prime}}\leq
T\\}}\sigma^{2}Y^{2}(Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T})^{2}+\mathbbm{1}_{\\{\tau^{C^{\prime}}=T\\}\cap\\{\lambda^{C^{\prime}}_{T}\geq\underline{\lambda},\lambda^{C^{\prime}}_{T}\geq\underline{\lambda}\\}}\sigma^{2}Y^{2}(\lambda^{C^{\prime}}_{T}-\lambda^{C^{\prime\prime}}_{T})^{2}\Big{]}\bigg{]}$
$\displaystyle\leq
c(|\lambda^{\prime}-\lambda^{\prime\prime}|^{2}+|\lambda^{\prime}-\lambda^{\prime\prime}|^{4}),$

where we use Lemma 3.1 for $n=2,4$ and where
$\\{\tau^{C^{\prime}}=T\\}\cap\\{\lambda^{C^{\prime}}_{T}\geq\underline{\lambda},\lambda^{C^{\prime}}_{T}\geq\underline{\lambda}\\}$
accounts for the case when the circuit breaker is triggered by the final
execution at terminal time $T$ and not through a possible signal-base trade
$\Delta^{l}_{T}C$. Similarly, by (88), Lemma 3.1 for $n=2$ and (122), we have

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{A^{c}}\Big{(}\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T}))-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T}))\Big{)}^{2}\right]\leq
c|\lambda^{\prime}-\lambda^{\prime\prime}|^{2}.$ (129)

Finally, we aggregate the above estimates to obtain the upper bound

$\displaystyle\mathbb{E}\bigg{[}\Big{(}X^{C^{\prime}}_{T}-X^{C^{\prime\prime}}_{T}+P^{C^{\prime}}_{T}Q^{C^{\prime}}_{T}-P^{C^{\prime\prime}}_{T}Q^{C^{\prime\prime}}_{T}+\zeta(|Q^{C^{\prime}}_{T}-Q^{C^{\prime\prime}}_{T}|)+(\Xi(Q^{C^{\prime}}_{T},\lambda^{C^{\prime}}_{T})-\Xi(Q^{C^{\prime\prime}}_{T},\lambda^{C^{\prime\prime}}_{T}))$
$\displaystyle\quad+\sigma
Y\big{(}\operatorname*{sgn}(Q^{C^{\prime}}_{T})(|Q^{C^{\prime}}_{T}|-(\lambda^{C^{\prime}}_{T}-\underline{\lambda})^{+})^{+}-\operatorname*{sgn}(Q^{C^{\prime\prime}}_{T})(|Q^{C^{\prime\prime}}_{T}|-(\lambda^{C^{\prime\prime}}_{T}-\underline{\lambda})^{+})^{+}\big{)}\Big{)}^{2}\mathbbm{1}_{A^{c}}\bigg{]}^{1/2}$
$\displaystyle\leq
c(|\lambda^{\prime}-\lambda^{\prime\prime}|^{2}+|\lambda^{\prime}-\lambda^{\prime\prime}|^{4}).$

Together with the estimate for (120), we have

$\displaystyle w(T,\lambda^{\prime},q)-w(T,\lambda^{\prime\prime},q)\leq
c(|\lambda^{\prime}-\lambda^{\prime\prime}|+|\lambda^{\prime}-\lambda^{\prime\prime})^{4}|^{1/2}+\varepsilon.$
(130)

We use that $\varepsilon$ was chosen arbitrarily and that $c$ does not depend
on $\varepsilon$ and exchange the roles of $\lambda^{\prime}$ and
$\lambda^{\prime\prime}$ to conclude

$\displaystyle|w(T,\lambda^{\prime},q)-w(T,\lambda^{\prime\prime},q)|\leq
c(|\lambda^{\prime}-\lambda^{\prime\prime}|+|\lambda^{\prime}-\lambda^{\prime\prime}|^{4})^{1/2}.$

Here, for $T,q$ in some compactum, the constant $c=c(T,c_{max},q)$ does not
depend on the values of $T,q$ because it is continuous in its variables. This
finishes the proof. ∎

Next, we observe monotonicity of the value function with respect to $t$.

######  Lemma 4.4.

Let the conditions of Theorem 3.3 hold. The function $w(T,\lambda,q)$ as in
(61) is monotonously increasing in the remaining time horizon $T$.

###### Proof.

Let $0\leq T^{\prime}\leq T^{\prime\prime}$. Let $C^{\prime}$ be some strategy
for time horizon $T^{\prime}$. We define the strategy $C^{\prime\prime}$ for
time horizon $T^{\prime\prime}$ as
$\Gamma^{\prime\prime}_{s}:=\Gamma^{\prime}_{s}$ for $s\in[0,T^{\prime}]$ and
$\Delta^{r}_{s}C^{\prime\prime}:=\Delta^{r}_{s}C^{\prime\prime}$ for
$s\in[0,T^{\prime})$. At time $T^{\prime}$, the trader with strategy
$C^{\prime\prime}$ executes her remaining position through an impulse trade of
$\Delta^{r}_{T^{\prime}}C^{\prime\prime}:=-Q^{C^{\prime\prime}}_{T^{\prime}}$.
Both strategies result in the same utility from terminal wealth so that we
conclude

$\displaystyle w(T^{\prime},\lambda,q)\leq w(T^{\prime\prime},\lambda,q).$

∎

######  Lemma 4.5.

Let the conditions of Theorem 3.3 hold. Then, for $\lambda,q$ in a compact
set, the value function $w(T,\lambda,q)$ as in (33) is locally Lipschitz
continuous in $T$, i.e., for $T^{\prime}$, $T^{\prime\prime}$ in a compact
set, we have $|w(T^{\prime},\lambda,q)-w(T^{\prime\prime},\lambda,q)|\leq
c|T^{\prime}-T^{\prime\prime}|$, where the Lipschitz constant $c$ does not
depend on $T^{\prime},T^{\prime\prime},\lambda,q$.

###### Proof.

For simplicity, we prove the claim for $\alpha>0$; the case $\alpha=0$ is
treated analogously.
Let $\lambda,q$ be starting values from some compact set, let
$0<T^{\prime\prime},T^{\prime}\leq\hat{T}$. Without loss of generality, we
consider the case where $T^{\prime\prime}\leq T^{\prime}$ and assume as before
that $p=x=0$.
For simplicity, we denote by $c=c(\hat{T},\lambda,q)>0$ a generic constant
that depends continuously on $\hat{T},\lambda,q$ and that may change from line
to line.
For some arbitrary $\varepsilon>0$ and for time to go $T^{\prime}$, let
$C^{\prime}$ be a $\varepsilon$-optimal strategy. By a density argument, it
suffices to prove the claim for $C^{\prime}$ which only changes in jumps,
i.e., for which $(C^{\prime})^{c}=0$.

For the time horizon $T^{\prime\prime}$, we define the strategy
$C^{\prime\prime}$ by $C^{\prime\prime}_{s}:=C^{\prime}_{s}$ for
$s\in[0,T^{\prime\prime}]$ so that the trader with strategy $C^{\prime\prime}$
copies the trades of strategy $C^{\prime}$. At time $T^{\prime\prime}$, the
trader with strategy $C^{\prime\prime}$ executes the remaining position
$Q^{C^{\prime\prime}}_{T^{\prime\prime}}=Q^{C^{\prime}}_{T^{\prime\prime}}$
all at once, while the trader with strategy $C^{\prime}$ liquidates the same
position over the time horizon $[T^{\prime\prime},T^{\prime}]$. With the
monotonicity from Lemma 4.4, we have

$0\leq|w(T^{\prime},\lambda,q)-w(T^{\prime\prime},\lambda,q)|=w(T^{\prime},\lambda,q)-w(T^{\prime\prime},\lambda,q)\leq\mathbb{E}\left[U_{\alpha}(\tilde{X}_{T^{\prime}}^{C^{\prime}})\right]+\varepsilon-\mathbb{E}\left[U_{\alpha}(\tilde{X}_{T^{\prime\prime}}^{C^{\prime\prime}})\right].$
(131)

We start by recalling the definition from (30)

$\displaystyle\tilde{X}_{T^{\prime}}^{C^{\prime}}$
$\displaystyle=X_{T^{\prime\prime}}^{C^{\prime}}+\sum_{T^{\prime\prime}<s\leq
T^{\prime}}\left(-P^{C^{\prime}}_{s-}\Delta^{l}_{s}Q^{C^{\prime}}-\zeta|\Delta^{l}_{s}Q^{C^{\prime}}|-\Xi(\Delta^{l}_{s}Q^{C^{\prime}},\lambda^{C^{\prime}}_{s-})\right)$
(132) $\displaystyle+\sum_{T^{\prime\prime}\leq
s<T^{\prime}}\left(-P^{C^{\prime}}_{s}\Delta^{r}_{s}Q^{C^{\prime}}-\zeta|\Delta^{r}_{s}Q^{C^{\prime}}|-\Xi(\Delta^{r}_{s}Q^{C^{\prime}},\lambda^{C^{\prime}}_{s})\right)$
(133)
$\displaystyle+P^{C^{\prime}}_{T^{\prime}}Q^{C^{\prime}}_{T^{\prime}}-\zeta\left|Q^{C^{\prime}}_{T^{\prime}}\right|-\Xi\left(Q^{C^{\prime}}_{T^{\prime}},\lambda^{C^{\prime}}_{T^{\prime}}\right)+\sigma
Y\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime}})\left(\left|Q^{C^{\prime}}_{T^{\prime}}\right|-(\lambda^{C^{\prime}}_{T^{\prime}}-\underline{\lambda})^{+}\right)^{+}.$
(134)

0$Q_{T^{\prime\prime}}^{C^{\prime}}$$Q_{T^{\prime}}^{C^{\prime}}$${T^{\prime\prime}}$${T^{\prime}}$

$\rightarrow$

0$Q_{T^{\prime\prime}}^{C^{\prime}}$$Q_{T^{\prime}}^{C^{\prime}}$${T^{\prime\prime}}$${T^{\prime}}$

$\rightarrow$

$\dots$

$\rightarrow$

0$Q_{T^{\prime\prime}}^{C^{\prime}}$$Q_{T^{\prime}}^{C^{\prime}}$${T^{\prime\prime}}$${T^{\prime}}$

Figure 11: Example trajectory of the trader’s inventory; stepwise separation
of roundtrips from monotone strategy.

Next, as illustrated in Figure 11, we note that we can decompose the strategy
$C^{\prime}$ that executes the position $Q^{C^{\prime}}_{T^{\prime\prime}}$
over the time interval $[T^{\prime\prime},T^{\prime}]$ into a strategy that
executes the position $Q^{C^{\prime}}_{T^{\prime\prime}}$ in a monotone way
(red dashed line) and into roundtrip trades (gray areas). Here, we consider a
roundtrip trade to be a trade which is reversed by (parts of) the next one.
Particularly, we know that the trades that are part of the monotone strategy
are of the opposite sign as $Q^{C^{\prime}}_{T^{\prime\prime}}$ and sum up to
$-Q^{C^{\prime}}_{T^{\prime\prime}}$. Moreover, for a roundtrip trade that is
started at some time $s\in[T^{\prime\prime},T^{\prime}]$, we know that its
trade is of the same sign as the current inventory $Q^{C^{\prime}}_{s}$
respectively any sign if $Q^{C^{\prime}}_{s}=0$.

To bound $\tilde{X}_{T^{\prime}}^{C^{\prime}}$ pathwise from above, we start
by considering the best possible price development over the time horizon
$[T^{\prime\prime},T^{\prime}]$ with respect to a trader’s market order of
sign $\pm$. On the one hand, the favorable price impacts can come from
external market orders that trade in the opposite direction, i.e., that are of
sign $\mp$. The absolute value of the price impact of such market orders is
bounded by

$\displaystyle|I(\mp
V_{[T^{\prime\prime},T^{\prime}]}(\tilde{M}^{C^{\prime}}),\underline{\lambda})|\leq\iota(\underline{\lambda})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{M}^{C^{\prime}}).$
(135)

On the other hand, favorable price impacts can come from the trader’s own
roundtrip trades. Particularly, this is the case if the price impact from
unwinding a roundtrip trade is in absolute value less than the price impact of
the trade starting the roundtrip. For a trader’s market order of sign $\pm$
that is sent at some point in time $t\in[T^{\prime\prime},T^{\prime}]$, this
means that her own roundtrip from before time $t$ have a favorable price
impact if

$\displaystyle\Big{|}I\Big{(}\mp\sum_{\Delta^{l/r}_{s}Q^{C^{\prime}}\in\mathcal{R}_{[T^{\prime\prime},t]}}|\Delta^{l/r}_{s}Q^{C^{\prime}}|,\;\lambda-\sum_{\Delta^{l/r}_{s}Q^{C^{\prime}}\in\mathcal{R}_{[T^{\prime\prime},t]}}|\Delta^{l/r}_{s}Q^{C^{\prime}}|+V_{[T^{\prime\prime},t]}(L^{C^{\prime},+})\Big{)}\Big{|}$
(136)
$\displaystyle\leq\Big{|}I\Big{(}\pm\sum_{\Delta^{l/r}_{s}Q^{C^{\prime}}\in\mathcal{R}_{[T^{\prime\prime},t]}}|\Delta^{l/r}_{s}Q^{C^{\prime}}|,\lambda\Big{)}\Big{|},$
(137)

where we denote by $\mathcal{R}_{[T^{\prime\prime},t]}$ the set of trades
$\Delta^{l/r}_{s}Q^{C^{\prime}}$, $s\in[T^{\prime\prime},t]$, that start a
roundtrip which is unwound before time $t$, i.e., the size of such a
roundtrip. Because by definition (7), the price impact function
$|I(\Delta,\lambda)|$ is decreasing in $\lambda$ and symmetric in $\Delta$,
inequality (136) is equivalent to

$\displaystyle\sum_{\Delta^{l/r}_{s}Q^{C^{\prime}}\in\mathcal{R}_{[T^{\prime\prime},t]}}|\Delta^{l/r}_{s}Q^{C^{\prime}}|\leq
V_{[T^{\prime\prime},t]}(L^{C^{\prime},+}).$ (138)

Consequently, the favorable price changes due to roundtrip trades are linearly
bounded by the liquidity provision over the time interval
$[T^{\prime\prime},T^{\prime}]$

$\displaystyle|I(\mp
V_{[T^{\prime\prime},t]}(\tilde{L}^{C^{\prime},+}),\underline{\lambda})|\leq\iota(\underline{\lambda})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{L}^{C^{\prime},+}).$
(139)

Hence, when considering the monotone part of $C^{\prime}$, we know that the
corresponding $P^{C^{\prime}}$ terms in (133) are pathwise bounded from above
by

$\displaystyle\left(P^{C^{\prime}}_{T^{\prime\prime}}+I(\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime\prime}})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{M}^{C^{\prime}}),\underline{\lambda})+I(\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime\prime}})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{L}^{C^{\prime},+}),\underline{\lambda})\right)Q^{C^{\prime}}_{T^{\prime\prime}}$
(140)
$\displaystyle\leq\left(P^{C^{\prime}}_{T^{\prime\prime}}+\iota(\underline{\lambda})\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime\prime}})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{M}^{C^{\prime}})+\iota(\underline{\lambda})\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime\prime}})V_{[T^{\prime\prime},T^{\prime}]}(\tilde{L}^{C^{\prime},+})\right)Q^{C^{\prime}}_{T^{\prime\prime}},$
(141)

where with (135) and (139) we assume favorable price impacts for trades of
sign $-\operatorname*{sgn}(Q^{C^{\prime}}_{T^{\prime\prime}})$.

Next, we consider the terms in (133) that correspond to roundtrip trades in
$C^{\prime}$. For this we consider a toy example of a roundtrip over some time
interval $[\tau_{1},\tau_{2}]\subset[T^{\prime\prime},T^{\prime}]$ that starts
with a trade of size $\Delta$ at time $\tau_{1}$ at price $p_{1}$ and at
liquidity level $\lambda_{1}$. The profit $x(\Delta)$ from the roundtrip is
bounded from above by

$\displaystyle x(\Delta)$ $\displaystyle\leq-
p_{1}\Delta-\zeta|\Delta|-\Xi(\Delta,\lambda_{1})-\Xi(\Delta,\lambda_{1}-|\Delta|)+\Xi(\Delta,\lambda_{1}-|\Delta|)$
(142)
$\displaystyle\quad+\left(p_{1}+I(\Delta,\lambda_{1})+I(-\operatorname*{sgn}(\Delta)V_{[\tau_{1},\tau_{2}]}(\tilde{M}),\underline{\lambda})+I(-\operatorname*{sgn}(\Delta)V_{[\tau_{1},\tau_{2}]}(\tilde{L}^{C^{\prime},+}),\underline{\lambda})\right)\Delta$
(143)