Periodic strategies in optimal execution with multiplicative price impact

TL;DR

This paper investigates optimal trading strategies considering multiplicative price impacts, modeling decision times with a Poisson process, and proves that the optimal approach has a barrier structure based on remaining shares and asset price level.

Contribution

It introduces a novel model for optimal execution with multiplicative price impact and characterizes the optimal strategy as a barrier policy depending on shares and price.

Findings

Optimal strategy has a barrier form.

Strategy depends only on remaining shares and asset price.

Expected net gains are maximized under this policy.

Abstract

In this work we study the optimal execution problem with multiplicative price impact in algorithm trading, when an agent holds an initial position of shares of a financial asset. The inter-selling-decision times are modelled by the arrival times of a Poisson process. The criterion to be optimised consists in maximising the expected net present value of gains of the agent, and it is proved that an optimal strategy has a barrier form, depending only on the number of shares left and the level of asset price.

Equations

$$Y_t:=y-{\xi }_t^{\mathrm{s}}+{\xi }_t^{\mathrm{b}}\ge 0,\text{for t≥0,}$$

$$\xi_{0}^{{\rm s}}=\xi_{0}^{{\rm b}}=0,\quad\mathbb{E}\big{[}\operatorname{e}^{4\lambda\xi_{t+}^{{\rm b}}}\big{]}<\infty\quad\text{and}\quad\displaystyle\lim_{t\to\infty}Y_{t}=0.$$

$$X_t^0=\mu X_t^0\mathrm{d}t+\sigma X_t^0\mathrm{d}{W}_t,X_0^0=x>0,$$

$$X_t=X_t^0\exp \{-\lambda ({\xi }_t^{\mathrm{s}}-{\xi }_t^{\mathrm{b}})\},$$

$$d{X}_t=\mu X_t\mathrm{d}t+\sigma X_t\mathrm{d}{W}_t-\lambda (X_t{\circ }_{\mathrm{s}}\mathrm{d}{\xi }_t^{\mathrm{s}}-X_t{\circ }_{\mathrm{b}}\mathrm{d}{\xi }_t^{\mathrm{b}}),$$

$$\begin{array}{rl}{X}_t{\circ }_{\mathrm{s}}\mathrm{d}{\xi }_t^{\mathrm{s}}& =X_t\mathrm{d}({\xi }^{\mathrm{s}}{)}_t^{\mathrm{c}}+\frac{1}{\lambda }{X}_t(1-e^{-\lambda \Delta {\xi }_t^{\mathrm{s}}})=X_t\mathrm{d}({\xi }^{\mathrm{s}}{)}_t^{\mathrm{c}}+X_t{\int }_0^{\Delta {\xi }_t^{\mathrm{s}}}{e}^{-\lambda u}\mathrm{d}u,\\ X_t{\circ }_{\mathrm{b}}\mathrm{d}{\xi }_t^{\mathrm{b}}& =X_t\mathrm{d}({\xi }^{\mathrm{b}}{)}_t^{\mathrm{c}}+\frac{1}{\lambda }{X}_t(e^{\lambda \Delta {\xi }_t^{\mathrm{b}}}-1)=X_t\mathrm{d}({\xi }^{\mathrm{b}}{)}_t^{\mathrm{c}}+X_t{\int }_0^{\Delta {\xi }_t^{\mathrm{b}}}{e}^{\lambda u}\mathrm{d}u,\end{array}$$

$${\xi }_t^{\mathrm{s}}={\int }_0^t{\nu }_s^{\mathrm{s}}\mathrm{d}{N}_s^{\gamma }\text{and}{\xi }_t^{\mathrm{b}}={\int }_0^t{\nu }_s^{\mathrm{b}}\mathrm{d}{N}_s^{\gamma },$$

$$\{\begin{array}{ll}{\nu }_t^{\mathrm{s}}=0,& \text{if}{\nu }_t^{\mathrm{b}}>0,\\ {\nu }_t^{\mathrm{b}}=0,& \text{if}{\nu }_t^{\mathrm{s}}>0,\\ {\nu }_t^{\mathrm{s}}={\nu }_t^{\mathrm{b}}=0,& \text{otherwise}.\end{array}$$

$$\begin{split}X_{t}\circ_{{\rm s}}{\rm d}\xi_{t}^{{\rm s}}&=\dfrac{1}{\lambda}X_{t}(1-\operatorname{e}^{-\lambda\nu^{{\rm s}}_{t}}){\rm d}N_{t}^{\gamma}=X_{t}\bigg{(}\int_{0}^{\nu^{{\rm s}}_{t}}\operatorname{e}^{-\lambda u}{\rm d}u\bigg{)}{\rm d}N_{t}^{\gamma},\\ X_{t}\circ_{{\rm b}}{\rm d}\xi_{t}^{{\rm b}}&=\dfrac{1}{\lambda}X_{t}(\operatorname{e}^{\lambda\nu_{t}^{{\rm b}}}-1){\rm d}N_{t}^{\gamma}=X_{t}\bigg{(}\int_{0}^{\nu_{t}^{{\rm b}}}\operatorname{e}^{\lambda u}{\rm d}u\bigg{)}{\rm d}N_{t}^{\gamma}.\end{split}$$

$${\int }_0^{\infty }\left(X_t{\circ }_{\mathrm{s}}\mathrm{d}{\xi }_t^{\mathrm{s}}-X_t{\circ }_{\mathrm{b}}\mathrm{d}{\xi }_t^{\mathrm{b}}-C_{\mathrm{s}}\mathrm{d}{\xi }_t^{\mathrm{s}}-C_{\mathrm{b}}\mathrm{d}{\xi }_t^{\mathrm{b}}\right),$$

$$J_{x,y}(\xi^{{\rm s}},\xi^{{\rm b}})\raisebox{0.4pt}{$:$}=\mathbb{E}_{x}\biggr{[}\int_{0}^{\infty}\operatorname{e}^{-\delta t}\left(X_{t}\circ_{{\rm s}}{\rm d}\xi^{{\rm s}}_{t}-X_{t}\circ_{{\rm b}}{\rm d}\xi^{{\rm b}}_{t}-C_{{\rm s}}{\rm d}\xi_{t}^{{\rm s}}-C_{{\rm b}}{\rm d}\xi_{t}^{{\rm b}}\right)\biggl{]},$$

$$J_{x,y}({\xi }^{\mathrm{s}},{\xi }^{\mathrm{b}})\le J_{x,y}({\xi }^{\mathrm{s}\ast },{\xi }^{\mathrm{b}\ast }),\text{for all}({\xi }^{\mathrm{s}},{\xi }^{\mathrm{b}})\in \mathcal{A}(y).$$

$$u_0(x,y)=\underset{({\xi }^{\mathrm{s}},{\xi }^{\mathrm{b}})\in \mathcal{A}(y)}{\sup }{J}_{x,y}({\xi }^{\mathrm{s}},{\xi }^{\mathrm{b}}).$$

$$u(x,y)\raisebox{0.4pt}{$:$}=\sup_{(\xi^{{\rm s}},0)\in\mathcal{A}(y)}J_{x,y}(\xi^{{\rm s}},0)=\sup_{\xi^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)}\mathbb{E}_{x}\biggr{[}\int_{0}^{\infty}\operatorname{e}^{-\delta t}\left(X_{t}\circ_{{\rm s}}{\rm d}\xi^{{\rm s}}-C_{{\rm s}}{\rm d}\xi_{t}^{{\rm s}}\right)\biggl{]}.$$

$$\begin{array}{rl}d{X}_t& =\mu X_t\mathrm{d}t+\sigma X_t\mathrm{d}{W}_t-\lambda (X_t{\circ }_{\mathrm{s}}\mathrm{d}{\xi }_t^{\mathrm{s}}-X_t{\circ }_{\mathrm{b}}\mathrm{d}{\xi }_t^{\mathrm{b}}),\\ d{\overline{X}}_t& =\mu {\overline{X}}_t\mathrm{d}t+\sigma {\overline{X}}_t\mathrm{d}{W}_t-\lambda {\overline{X}}_t{\circ }_{\mathrm{s}}\mathrm{d}{\bar{\xi }}_t^{\mathrm{s}},\end{array}$$

$${\int }_0^{t\wedge {\bar{\tau }}_m}$$

$$=$$

$$-{\int }_0^{t\wedge {\bar{\tau }}_m}{e}^{-\delta s}(C_{\mathrm{s}}\mathrm{d}{\xi }_s^{\mathrm{s}}+C_{\mathrm{b}}\mathrm{d}{\xi }_s^{\mathrm{b}})+\frac{\sigma }{\lambda }{\int }_0^{t\wedge {\bar{\tau }}_m}{e}^{-\delta s}{X}_s^0{e}^{-\lambda ({\xi }_s^{\mathrm{s}}-{\xi }_s^{\mathrm{b}})}\mathrm{d}{W}_s$$

$$\le$$

$$-{\int }_0^{t\wedge {\bar{\tau }}_m}{e}^{-\delta s}{C}_{\mathrm{s}}\mathrm{d}{\bar{\xi }}_s^{\mathrm{s}}+\frac{\sigma }{\lambda }{\int }_0^{t\wedge {\bar{\tau }}_m}{e}^{-\delta s}{X}_s^0{e}^{-\lambda ({\xi }_s^{\mathrm{s}}-{\xi }_s^{\mathrm{b}})}\mathrm{d}{W}_s$$

$$=$$

$$\mathbb{E}_{x}\bigg{[}\int_{0}^{t\wedge\bar{\tau}_{m}}\operatorname{e}^{-\delta t}\left(X_{t}\circ_{{\rm s}}{\rm d}\xi^{{\rm s}}_{t}-X_{t}\circ_{{\rm b}}{\rm d}\xi^{{\rm b}}_{t}-C_{{\rm s}}{\rm d}\xi_{t}^{{\rm s}}-C_{{\rm b}}{\rm d}\xi_{t}^{{\rm b}}\right)\bigg{]}\\ \leq\mathbb{E}_{x}\bigg{[}\int_{0}^{t\wedge\bar{\tau}_{m}}\operatorname{e}^{-\delta s}(\overline{X}_{s}\circ_{s}{\rm d}\bar{\xi}^{{\rm s}}_{s}-C_{{\rm s}}{\rm d}\bar{\xi}_{s}^{{\rm s}})\bigg{]}.$$

$$J_{x,y}({\xi }^{\mathrm{s}},{\xi }^{\mathrm{b}})\le J_{x,y}({\bar{\xi }}^{\mathrm{s}},0)\le u(x,y).$$

$${\nu }_F(T_i):=Y_{T_i}\wedge \frac{1}{\lambda }(\ln X_{T_i}-\ln F{)}^{+},$$

$$u_{\mathrm{b}}(x,y)=\underset{{\xi }^{\mathrm{s},F}\in {\mathcal{A}}_{\mathrm{b}}^{\mathrm{s}}(y)}{\sup }{J}_{x,y}({\xi }^{\mathrm{s},F},0),$$

$$\{\begin{array}{ll}\mathcal{L}w(x,y)+\underset{0\le l\le y}{\max }\{\gamma G(x,y,l;w)\}=0,& \text{for all}x>0\text{and}y>0,\\ w(x,y)=0,& \text{for all}x>0\text{and}y=0,\end{array}$$

$$\begin{cases}\mathcal{L}v(x,y)+\gamma G\biggr{(}x,y,\biggl{[}y\wedge\dfrac{1}{\lambda}\ln(x/F_{\gamma})^{+}\biggr{]};v\biggl{)}=0,&\text{for all}\ x>0\ \text{and}\ y>0,\\ v(x,y)=0,&\text{for all}\ x>0\ \text{and}\ y=0,\end{cases}$$

$$\mathcal{L}f(x,y)$$

$$G(x,y,l;f)$$

$$\mathcal{L}v(x,y)=0.$$

$${\mathcal{L}}_{\gamma }v(x,y)$$

$${\mathcal{L}}_{\gamma }v(x,y):=\frac{1}{2}{\sigma }^2{x}^2{v}_{xx}(x,y)+\mu x{v}_x(x,y)-(\delta +\gamma )v(x,y).$$

$$\displaystyle\mathcal{L}_{\gamma}v(x,y)+\gamma\biggl{[}\dfrac{1}{\lambda}(1-\operatorname{e}^{-\lambda y})x-C_{{\rm s}}y\biggr{]}=0.$$

Full text

$*$  Department of Probability and Statistics, Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico. Email: dher@cimat.mx.

${\ddagger}$  National Research University Higher School of Economics, Shabolovkaya 31, building G, C.P. 115162, Moscow, Russia. Email: hmoreno@hse.ru.

$\dagger$  Department of Probability and Statistics, Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico. Email: jluis.garmendia@cimat.mx.

1. Introduction

In this work we are interested in the problem of finding optimal execution strategies for a financial market impact model where transactions can have a permanent effect. The analysis of this problem has practical and mathematical motivations, and has been studied from different perspectives. Nowadays the use of algorithmic trading to execute large book orders has given rise to important questions on the best way to execute the position, in order to decrease the negative effect on the shift of asset price, and also obtain the better performance of the criteria to be optimised. In general, existence of optimal strategies can not be guaranteed, and clearly depend on the structure of the market model as well as on the parameters involved in its description.

In any market impact model, it is crucial to describe the way that order execution algorithms will be generated. Despite the analytical tractability of the classical continuous time models, these are unfortunately not implementable in practice. On the other hand, while the models with discrete execution decision times are ideal, they lack analytical tractability, and numerical methods are required to solve them. Recently, with the aim of developing a more realistic yet analytically tractable model, random discrete execution times have been considered. Random observations are suggested in various economic literatures. See, for example, the discussion in the introduction of [27] motivated by rational inattention (see [29]) in macroeconomics literature. See also the discussions given in [11, 21] for real option problems with random intervention times, and [13, 17] for applications in optimal stopping problems and Bermudan look-back option pricing. On this regard, an important motivation for considering the Poissonian interarrival model is its potential applications in approximating the constant interarrival time cases. It is known, in the mathematical finance literature, that randomization techniques (see, e.g. [12]) are efficient in approximating constant maturity problems with those with Erlang-distributed maturities. In particular, for short maturity cases, it is known empirically that accurate approximations can be obtained by simply replacing the constant with an exponential random variable [22].

In this note we propose a random clock, attached to the jumps of a Poisson process, for the times when the execution decisions will take place. This is a new instrument that may represent advantages from the implementation perspective, since the randomness provided by the random clock included in the execution strategy introduce an additional unpredictable structure to the strategies. The empirical justification of this model can be approached from the following perspective, related to market micro-structural factors. It is well known that the dynamics and the volume of the trades interact with the evolution of the market liquidity of an asset. The design of each portfolio is based on the information arriving from the order flow of buys and sells decisions of other investors, but not on who is behind each decision. This suggest that a strategic sequential trading to execute a large book order is “event based”, represented by the new information inferred regarding the value of the asset from the composition and existence of trades from the market participants. Thus, we are proposing that the arrival process is linked with market parameters, like liquidity, volume, market depth and order flows. Interestingly, this view point reflects the fact, well understood in high frequency trading for instance, that time has a different meaning when we are operating an algorithmic trading strategy using cycles depending on the amount of information received, instead of the measurements based on chronological time.

The benchmark models assume that either trading can be done in continuous time or there are constant intervals of time at which the portfolio is balanced. Neither of them has practical reasons to be sustainable, besides their analytical tractability, since investors continuously gain information about the trading environment. In the model proposed in this paper we are allowing a random clock of time in which new information is processed, based on the evolution of the main factors of the market. This is a good example of algorithmic trading which does not necessarily occur at a high frequency rate, but leaving this possibility to execute the position with frequent sells when the parameter of the Poisson process is manipulated to do so. Of course, it also helps to include asynchronous transactions to hide as much as possible the strategy followed.

In this note we are assuming that the agent holds a large position and, as typically happen, we expect that any selling strategy will lead to decrease of prices. When the agent is not active the model adopted is a standard geometric Brownian motion with drift, following the work by Guo and Zervos [18]. Another important element in the model is related with the manner to quantify the revenues received by each selling strategy. On this regard, the criterion will be calculated as the net present value of the difference between the gains of the selling strategy and the transaction costs associated.

In contrast with the multiplicative impact model presented here, in the seminal papers of Almgren and Chriss [4], [5] and Almgren [3], the execution strategies are assumed to be absolutely continuous functions of time, having a price impact acting in an additive way; Bertsimas and Lo [11] also made fundamental contributions considering a discrete random walk model. In our case the strategies are described as Lebegue-Stieltjes integrals with respect to the paths of the Poisson process, named periodic strategies, in analogy with the terminology used in insurance models when dividend payment decisions are taken; see, for instance, [7, 8, 30, 31].

More recent contributions to the theory of optimal execution found in the literature include Huberman and Stanzl [20], He and Mamaysky [19], Gatheral, Schied, and Slynko [16], Obizhaeva and Wang [24], Almgren and Lorenz [6], Engle and Ferstenberg [14], Schied and Schöneborn [27], Alfonsi, Fruth, and Schied [1] [2], Schied, Schöneborn, and Tehranchi [28], Predoiu, Shaikhet, and Shreve [25], and Lokka [23].

In order to find an optimal strategy over the set of periodic strategies, we restrict our analysis to the set of periodic barrier strategies. This class of barrier strategies are very easy to be implemented, since selling decisions are taken observing if the price of the stock lies above a certain fixed level $F$ and the remaining number of shares. Then, the first step consists in finding the optimal barrier strategy that maximises the performance criteria. This is done by solving the Hamilton-Jacobi-Bellman (HJB) equation associated to this problem, which allows us to obtain an explicit form of the value function for this restricted problem. Imposing a suitable smoothness condition on the value function we obtain the explicit value of the barrier $F_{\gamma}$ associated to the optimal strategy. This strategy can be described as follows: If the stock price is below a critical level $F_{\gamma}$ at a selling time, then it is optimal not to sell any shares. However, if the stock price lies above the level $F_{\gamma}$ when the random clock rings, it is optimal either to sell all available shares or liquidate a fraction of the position that will have as a consequence that the stock price decreases. A Verification Lemma is used to proved that the original optimisation problem within the periodic strategies can be solved implementing only barrier strategies.

The rest of the paper is organised as follows. In Section 2, we review the underlying model for the stock price with multiplicative price impact and provide the performance criterion, as well as the formulation of the optimal execution problem with periodic strategies. In Section 3, we obtain an explicit form for the solution of the HJB equation associated to the value function over the set of periodic barrier strategies. A Verification Theorem is provided (Theorem 3.5), showing an explicit form for the optimal (or $\varepsilon$-optimal) periodic strategy, under appropriate conditions on the parameters of the model. Finally, we defer the proofs of some technical lemmas to the Appendix.

2. Market impact model

In this section we describe the optimal execution model, based on the paper by Guo and Zervos [18]. Let us fix a filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathcal{P})$ satisfying the usual conditions and carrying a standard $(\mathcal{F}_{t})$-Brownian motion $W$ and an independent Poisson process $N^{\gamma}$. We consider an agent holding an initial position of $y$ shares of a financial asset, which has to be sold maximising the expected gains. The information available to the agent is enclosed in the filtration $\mathcal{F}_{t}$.

The trading strategies are denoted by the duple $(\xi_{t}^{{\rm s}},\xi_{t}^{{\rm b}})$, which represents the total number of shares that the investor has sold and bought up to time $t$, respectively. Then, the total number of shares held by the agent at time $t$ are given by

[TABLE]

where, $\xi^{{\rm s}}$, $\xi^{{\rm b}}$ are $(\mathcal{F}_{t}$)-adapted increasing càglàd processes such that

[TABLE]

Although not reflected in $(\xi^{{\rm s}},\xi^{{\rm b}})$, there is the restriction that the agent cannot sell and buy shares at the same time. The set of admissible strategies satisfying the previous conditions is denoted by $\Xi(y)$.

The stock price observed by the agent, independently of the actions of other market participants, is modelled by the geometric Brownian motion $X^{0}$ with drift

[TABLE]

where $\sigma\in\mathbb{R}$ and $\mu>0$ are constants. Let us suppose that the agent is implementing a strategy $(\xi^{{\rm s}},\xi^{{\rm b}})\in\Xi(y)$. Hence, when the agent decides to sell or buy some number of shares of the asset at time $t$, we assume that there is an impact in the price, described as a multiplicative factor, namely, the resulting price $X_{t}$ is assumed to have the form

[TABLE]

for some positive constant $\lambda$ describing the permanent impact on the price, and $X^{0}_{t}$ is the solution to (2.3). More specifically, following [18], the controlled process dynamics can be described as the solution of the following stochastic differential equation

[TABLE]

where

[TABLE]

and the processes $(\xi^{{\rm s}})^{{\rm c}}$ and $(\xi^{{\rm b}})^{{\rm c}}$ are the continuous part of $\xi^{{\rm s}}$ and $\xi^{{\rm b}}$, respectively. The pair $(X_{t},Y_{t})$ is referred as the state process associated to the strategy $(\xi^{{\rm s}},\xi^{{\rm b}})$.

One of the main differences between the model introduced by Guo and Zervos [18] with the approach presented in this paper consists in presenting a different framework to execute the initial position $y$. While Guo and Zervos assume that at each time $t\geq 0$ the agent should decide the number of shares to sell or buy, in this paper we assume that selling or buying can only occur at some (typically random) time points, modelled by the jump times of an independent Poisson process $(N^{\gamma}_{t}\;:\;t\geq 0)$, with rate $\gamma>0$. More precisely, the selling and buying strategies are given by

[TABLE]

where $\nu^{{\rm s}}_{t},\ \nu^{{\rm b}}_{t}$ are $\mathcal{F}_{t}$ adapted processes, representing the number of shares sold and bought at time $t$, respectively. Since the agent cannot sell and buy at the same time, the following condition holds

[TABLE]

Within this context, selling-buying shares are necessarily done at discrete periods of time (there cannot be continuous selling and buying) since selling-buying decisions can only occur when the process $N^{\gamma}$ has jumps. The set of selling-buying decision times is denoted as $\mathcal{T}=\{T_{1},T_{2},\dots\}$, and the quantities $T_{k}-T_{k-1}$, $k\geq 0$, are the inter-selling-buying-decision times, which are exponentially distributed with mean $1/\gamma$. The number of shares sold and bought at each decision time $T_{j}$ are denoted by $\nu^{{\rm s}}_{T_{j}}$ and $\nu^{{\rm b}}_{T_{j}}$ , respectively, with $\Theta^{{\rm s},{\rm b}}=\{(\nu^{{\rm s}}_{T_{1}},\nu^{{\rm s}}_{T_{2}},\dots),(\nu^{{\rm b}}_{T_{1}},\nu^{{\rm b}}_{T_{2}},\dots)\}$ representing a selling-buying strategy via (2.7); the subset of strategies $(\xi^{{\rm s}},\xi^{{\rm b}})\in\Xi(y)$ which can be represented as in (2.7) is denoted by $\mathcal{A}(y)$. For those strategies $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$, the operator defined in (2.6) can be written as

[TABLE]

Let $C_{{\rm s}},C_{{\rm b}}$ be positive constants representing the transaction cost associated with the sell and buy of shares, respectively. Then, the gains associated with each strategy $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$ is given by

[TABLE]

and the agent’s objective is to maximise the expected net present value of gains

[TABLE]

over the set $\mathcal{A}(y)$. The parameter $\delta>0$ is the discount factor, and we assume that $\delta>\mu$ in order to avoid arbitrage opportunities, as described in [18, Proposition 3.4]. Given an initial condition $(x,y)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$, we say that $(\xi^{{\rm s}*},\xi^{{\rm b}*})\in\mathcal{A}_{(}y)$ is an optimal strategy if and only if

[TABLE]

The value function of this stochastic control problem is defined as

[TABLE]

2.1. Regularity property of the model

A remarkable property of our model is the regularity [15], which is understood as the requirement that, first, the optimisation problem (2.10) has an optimal solution and, second, as the initial position $y$ is positive, it should be expected that the optimal execution strategy does not involve buying decisions along the time needed to liquidate the initial position. In the rest of this section we elaborate along the second condition, while the first one will be approached in the next section.

Consider strategies where the agent only sell shares, i.e., strategies of the form $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$ such that $\xi^{{\rm b}}\equiv 0$. The subset of strategies $(\xi^{{\rm s}},0)\in\mathcal{A}(y)$ is denoted by $\mathcal{A}^{{\rm s}}(y)$, whose elements are represented only by $\xi^{{\rm s}}$, and the value function $u(x,y)$ for this problem is given by

[TABLE]

It is clear that $u(x,y)\leq u_{0}(x,y)$, since $\mathcal{A}^{{\rm s}}(y)\subset\mathcal{A}(y)$; the equality between these value functions is established in the next result.

Proposition 2.1**.**

Let $u_{0}$ and $u$ be the value functions given in (2.10) and (2.11), respectively. Then, $u=u_{0}$.

In order to prove this result we need to use a technical tool described in the next lemma, which proof is presented in the Appendix.

Lemma 2.2**.**

For each $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$, there exists $\bar{\xi}^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)$ such that $\xi^{{\rm s}}_{t}-\xi^{{\rm b}}_{t}\leq\bar{\xi}^{{\rm s}}_{t}\leq\xi^{{\rm s}}_{t}$ for all $t\geq 0$.

Proof of Proposition 2.1.

Recall that we only need to show the inequality $u_{0}(x,y)\leq u(x,y)$. Let $(\xi^{{\rm s}},\xi^{{\rm b}})$ be in $\mathcal{A}(y)$ and consider the strategy $\bar{\xi}^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)$ as in Lemma 2.2. Now, we consider the processes $X_{t}=X^{0}_{t}\exp\{-\lambda(\xi^{{\rm s}}_{t}-\xi^{{\rm b}}_{t})\}$ and $\overline{X}_{t}=X^{0}_{t}\exp\{-\lambda\bar{\xi}^{{\rm s}}_{t}\},$ which satisfy the following SDEs,

[TABLE]

respectively, where $X_{t}\circ_{{\rm s}}{\rm d}\xi_{t}^{{\rm s}}$, $X_{t}\circ_{{\rm b}}{\rm d}\xi_{t}^{{\rm b}}$ and $\overline{X}_{t}\circ_{{\rm s}}{\rm d}\bar{\xi}_{t}^{{\rm s}}$ are given as in (2.8). Applying integration by parts in $\operatorname{e}^{-\delta(t\wedge\bar{\tau}_{m})}X_{t\wedge\bar{\tau}_{m}}$, where \bar{\tau}_{m}\raisebox{0.4pt}{:}=\inf\{t>0:X^{0}_{t}>m\}, and using (2.12), it follows

[TABLE]

Note that $\displaystyle\int_{0}^{t\wedge\bar{\tau}_{m}}\operatorname{e}^{-\delta s}X_{s}^{0}(\operatorname{e}^{-\lambda(\xi^{{\rm s}}_{s}-\xi^{{\rm b}}_{s})}-\operatorname{e}^{-\lambda\bar{\xi}_{s}^{{\rm s}}}){\rm d}B_{s}$ is a square-martingale, since by hypothesis

$\mathbb{E}[\operatorname{e}^{4\lambda\xi^{{\rm b}}_{(t\wedge\bar{\tau}_{m})+}}]<\infty$, whose expected value is zero; see more in [18, p. 293]. Then, taking expected value in (2.13), it yields

[TABLE]

Then, letting $t,m\rightarrow\infty$ and using the Dominated Convergence Theorem, we get

[TABLE]

Since the above inequality holds for each $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$, we conclude that $u_{0}(x,y)\leq u(x,y)$, completing the proof of this result. ∎

3. HJB equations and optimal execution

The remainder of this paper is devoted to the presentation of an optimal solution to the execution problem (2.11), paying particular attention to the structure of this strategy. Roughly, we look for optimal selling strategies with a simple structure, facilitating its adaptability from the practical view point. One of the main results of this work establishes the existence of an optimal strategy $\hat{\xi}^{{\rm s}}$, which has a barrier form in the state space. A barrier strategy is described in terms of the remaining number of shares to be sold and the level of the observed price at each period of time. These are compared with a mark, which is decided from the beginning and depends on a non-negative constant $F$, referred hereafter as a periodic barrier.

More precisely, fixing a periodic barrier $F>0$, the number of shares to be sold in the $i$-th arrival time $T_{i}$ of the Poisson process $N^{\gamma}$, is given by

[TABLE]

where $(X_{T_{i}},Y_{T_{i}})$ is the position of the state process at the arrival time $T_{i}$ and (\ln X_{T_{i}}-\ln F)^{+}\raisebox{0.4pt}{:}=\max\{0,(\ln X_{T_{i}}-\ln F)\}. This type of strategies is denoted by $\xi^{{\rm s},F}$, and the set of these strategies is defined by $\mathcal{A}^{{\rm s}}_{{\rm b}}(y)$, which is clearly a subset of $\mathcal{A}^{{\rm s}}(y)$. The corresponding value function for this set of strategies is defined as

[TABLE]

with $J_{x,y}$ as in (2.9).

In order to relate the above value function $u_{{\rm b}}$ with the original one $u$, defined in (2.11), it is convenient to provide a brief description of the approach to be followed. As a first step, we shall solve the problem (3.2) using dynamic programming techniques. Noting that the periodic strategies are described by a single parameter, the crucial point consists in proving that there is a parameter $F_{\gamma}$, defined below by (3.11), for which there exists a smooth solution $v$ to the HJB equation (3.4) associated with $u_{{\rm b}}$; see Proposition 3.2. As a second step, using the parameter $F_{\gamma}$ a periodic strategy $\xi^{{\rm s},F_{\gamma}}$ can be built using the expression (3.1). Finally, using the HJB equation associated with the original value function $u$ described in (3.3), in the last step it is proved that $v$ solves that equation, and that $\xi^{{\rm s},F_{\gamma}}$ is optimal for $u$ within the set $\mathcal{A}^{{\rm s}}(y)$, concluding that $u_{{\rm b}}=u$; see the Verification Theorem 3.7.

Remark 3.1*.*

By standard dynamic programming arguments, it is well known that the value functions $u$ and $u_{{\rm b}}$, are associated to HJB equations, which are given, respectively, by

[TABLE]

[TABLE]

where $F_{\gamma}$ is a positive constant which will be determined later on. Here, the operators $\mathcal{L}$ and $G$ are defined by

[TABLE]

3.1. Construction and regularisation of the solution $v$ to the HJB equation (3.4)

Observe that we can simplify the HJB equation (3.4) according with the following three different scenarios:

• (i)

When $x<F_{\gamma}$, this restriction corresponds to the waiting region $\mathcal{W}$ because the price is too low for selling any shares to be optimal, and therefore (3.4) takes the form

[TABLE]

• (ii)

When $F_{\gamma}\leq x<F_{\gamma}\operatorname{e}^{\lambda y}$, the agent takes and intermediate position of selling \mathbb{Y}(x)\raisebox{0.4pt}{:}=\dfrac{1}{\lambda}\ln(x/F_{\gamma}) assets. Now since $\operatorname{e}^{-\lambda\mathbb{Y}(x)}=\dfrac{F_{\gamma}}{x}$, (3.4) can be written as

[TABLE]

where

[TABLE]

• (iii)

Finally, when $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$, we have that the asset price is sufficiently high and the corresponding decision is to execute the complete set of assets available, and then (3.4) is reduced to

[TABLE]

We obtain explicit solutions for each one of the three regions, which are described in (3.10). A quite important issue in the form of the solutions proposed below is the smoothness at the boundary of each region, which derives in obtaining an explicit form of the general solution. The proof of the following result is given in the Appendix.

Proposition 3.2**.**

The HJB equation (3.4) has a solution $v$, which belongs to $\operatorname{C}^{2,1}(\mathbb{R}_{+}\times\mathbb{R}_{+})$ and is given by

[TABLE]

where \eta\raisebox{0.4pt}{:}=\delta-\mu,

[TABLE]

and b\raisebox{0.4pt}{:}=\dfrac{1}{2}\sigma^{2}-\mu. The constants $n$, $m_{\gamma}$ are the positive and negative solutions to

[TABLE]

respectively.

Remark 3.3*.*

The fact that $\delta>\mu$ implies that the solution $n$ to (3.14) satisfies that $n>1$, for all $\sigma\in\mathbb{R}$.

3.2. Equivalence between the HJB equations

The rest of this paper is dedicated to verify that the strategy given in (3.1), with barrier $F_{\gamma}$, defined above in (3.11), is optimal within the set of strategies $\mathcal{A}^{{\rm s}}(y)$, and that the function $v$ given in (3.10), satisfies the HJB equation (3.3). To this end we need the following technical results, whose proof is given in the Appendix in order to introduce in a succinct way their main consequences.

Lemma 3.4**.**

Let $a_{\gamma}$ be defined by

[TABLE]

where $\gamma>0$. Then, for each $\gamma>0$, $1<a_{\gamma}<\dfrac{n}{n-1}$, and it satisfies the following asymptotic limits

[TABLE]

Proposition 3.5**.**

Let $v$ be as in (3.10). Then, for each $(x,y)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$ the following identity holds

[TABLE]

with $G$ defined as in (3.6).

Remark 3.6*.*

Note that putting together Propositions 3.2 and 3.5, it follows immediately that $v$ is a solution to the HJB equation (3.3).

Next result identifies the solution of the HJB equation (3.10) with the value function $u$, providing also an optimal strategy within the set $\mathcal{A}^{{\rm s}}(y)$.

Theorem 3.7** (Verification Theorem).**

Consider the periodic optimal execution problem formulated in Section 2 and the function $v$ defined by (3.10). Then, $v$ agrees with the value function $u$ of the periodic stochastic control problem. In particular,

[TABLE]

Furthermore, define the strategy $\xi^{{\rm s}*}_{t}=\displaystyle\int_{0}^{t}\nu_{s}^{*}dN^{\gamma}_{s}$ with

[TABLE]

where $F_{\gamma}$ is as in (3.11), $\mathcal{T}=\{T_{i}\}_{i=1}^{\infty}$ is the set of arrival times of the Poisson process $N^{\gamma}$, and $(X^{*},Y^{*})$ is the state process associated with the liquidation strategy $\xi^{{\rm s}*}$. Then, the following statements hold

• (i)

If $\mu-\frac{1}{2}\sigma^{2}\geq 0$, then $\xi^{{\rm s}*}$ is an optimal periodic liquidation strategy.

• (ii)

If $\mu-\frac{1}{2}\sigma^{2}<0$, then $\xi^{{\rm s}*}$ is not an optimal periodic liquidation strategy. So if we define

[TABLE]

then $\{\xi^{{\rm s}*j}\}_{j=1}^{\infty}$ is a sequence of $\varepsilon$-optimal periodic strategies.

Proof.

Take $(x,y)\in\mathbb{R}_{+}^{*}\times\mathbb{R}_{+}$ an initial condition, $\xi^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)$, described by the following selling strategy $\Theta=\{\nu_{T_{1}},\nu_{T_{2}},\dots\}$, where $\mathcal{T}=\{T_{1},T_{2},\dots\}$ is the set of selling times, and $(\tau_{m})_{m\in\mathbb{N}}$ the sequence of stopping times defined by $\tau_{m}:=\inf\{t>0:X_{t}>m\}$. Using Itô-Tanaka-Meyer’s formula and the left continuity of the processes $X$ and $Y$, we can see

[TABLE]

where M_{t\wedge\tau_{m}}\raisebox{0.4pt}{:}=\sigma\displaystyle\int_{0}^{t\wedge\tau_{m}}\operatorname{e}^{-\delta s}X_{s}w_{x}(X_{s},Y_{s}){\rm d}W_{s}. On the other hand

[TABLE]

with $G$ as in (3.17), $H_{t\wedge\tau_{m}}:=\displaystyle\int_{0}^{t\wedge\tau_{m}}\operatorname{e}^{-\delta s}G(X_{s-},Y_{s-},\nu_{s};v){\rm d}\widetilde{N}^{\gamma}_{s}$ and $\widetilde{N}$ is the compensated Poisson process. Hence, putting these pieces together and observing that

[TABLE]

we have

[TABLE]

From (3.10) it is not difficult to see that there exists a positive constant $K$ such that,

[TABLE]

hence it follows that the processes $(M_{t\wedge\tau_{m}};t\geq 0)$ and $(H_{t\wedge\tau_{m}};t\geq 0)$ are zero-mean $\mathbb{P}$-martingales. Then, taking expectations in (3.20) we obtain

[TABLE]

Now, from the expression of the process $X_{t}$ in (2.4) and recalling that $\delta>\mu$, we have that

[TABLE]

Using (2.1) we note the following

[TABLE]

Then, letting $m,t\rightarrow\infty$ in (3.22), using (3.23), (3.24) and Monotone Convergence Theorem, we obtain that

[TABLE]

Taking the maximum over all $\xi^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)$ we conclude that $u(x,y)\leq v(x,y)$. Let $(X^{*},Y^{*})$ be the state process associated with the liquidation strategy $\xi^{{\rm s}*}$, given by $\xi^{{\rm s}*}_{t}=\displaystyle\int_{0}^{t}\nu^{*}_{s}{\rm d}N^{\gamma}_{s}$, with $\nu^{*}$ as in (3.18). Note that $\xi^{{\rm s}}$ is admissible as long as $\displaystyle\lim_{t\to\infty}\xi^{{\rm s}}_{t}=y$. We can easily check using (2.4) that this indeed the case if and only if $\mu-\frac{1}{2}\sigma^{2}\geq 0$ since $\displaystyle\limsup_{t\rightarrow\infty}X_{t}^{0}=\infty$. Proceeding in a similar way as in (3.20), we get that

[TABLE]

where $\bar{\tau}_{m}:=\inf\{t>0:X^{*}_{t}>m\}$. Now, from the construction of $v$, we know that it is the solution to (3.4). Therefore, we have that

[TABLE]

Hence, we obtain that

[TABLE]

Then, taking expectations in the previous identity it follows that

[TABLE]

Now, letting $m,t\rightarrow\infty$ in (3.2), using (3.23), (3.24) and Monotone Convergence Theorem, we get

[TABLE]

which implies the result. For the case when $\mu-\frac{1}{2}\sigma^{2}<0$, let us take $(\overline{X}^{*},\overline{Y}^{*})$ as the state process associated with the strategy $\xi^{{\rm s}*j}$ given by (3.19). We can check that $\xi^{{\rm s}*j}$ has payoff

[TABLE]

where $\tilde{\tau}_{j}=\inf\{T_{i}>0:T_{i}>j\}$. Also, we see that (3.2) is satisfied, if $t=j$, $(X^{*},Y^{*})$ is replaced by $(\overline{X}^{*},\overline{Y}^{*})$, and $\tau_{m}$ with the stopping time $\bar{\tau}_{m}=\inf\{t>0:\overline{X}^{*}_{t}>m\}$. Then, letting $m\to\infty$ in (3.2), it follows

[TABLE]

Now, applying (3.28) in (3.27),

[TABLE]

Therefore noting that the right-hand side of this expression converges to $v(x,y)$ as $j\to\infty$ allow us to establish that $\{\xi^{{\rm s}*j}\}_{j=1}^{\infty}$ is a sequence of $\varepsilon$-optimal strategies. ∎

Remark 3.8*.*

Notice that as a consequence of Proposition 3.2 and Lemma 3.4, we obtain that

[TABLE]

Recall that $F_{\gamma},C_{\gamma},A_{\gamma}$ are given in (3.11)–(3.13), respectively. Hence, straightforward computations yield

[TABLE]

These asymptotic limits allow us to recover the value function for the case of singular strategies for the optimal execution problem obtained by Guo and Zervos in Proposition 5.1 of [15].

Appendix. Proofs of some technical results

Proof of Lemma 2.2.

Let us take $(\xi^{{\rm s}},\xi^{{\rm b}})\in\mathcal{A}(y)$ and $\Theta^{{\rm s},{\rm b}}=\{(\nu^{{\rm s}}_{T_{1}},\nu^{{\rm s}}_{T_{2}},\dots),(\nu^{{\rm b}}_{T_{1}},\nu^{{\rm b}}_{T_{2}},\dots)\}$ the selling-buying strategy associated with $(\xi^{{\rm s}},\xi^{{\rm b}})$. Let $\mathcal{T}^{{\rm s}}\subset\mathcal{T}$ the subset of decision times whose elements $\kappa_{i}$ are given in the following way

[TABLE]

From here, we see

[TABLE]

for $i\in\{1,2,3,\dots\}$, with $\kappa_{0}=0$. Let $\alpha_{i,j}$ be defined as

[TABLE]

We construct $\bar{\nu}^{{\rm s}}$ in the following way:

1. (i)

If $T_{j}\notin\mathcal{T}^{{\rm s}}$, then $\bar{\nu}_{T_{j}}^{{\rm s}}=0$. 2. (ii)

If $T_{j}=\kappa_{1}$, then, \bar{\nu}^{{\rm s}}_{\kappa_{1}}\raisebox{0.4pt}{:}=\alpha_{1,1}. 3. (iii)

If $T_{j}=\kappa_{2}$,

[TABLE] 4. (iv)

If $T_{j}=\kappa_{3}$,

[TABLE] 5. (v)

Recursively, if $T_{j}=\kappa_{i}$, with $i\in\{3,4,\dots\}$,

[TABLE]

Now, we take the selling strategy $\bar{\xi}^{{\rm s}}$ as

[TABLE]

and define \overline{Y}_{t}\raisebox{0.4pt}{:}=y-\bar{\xi}^{{\rm s}}_{t} for all $t\geq 0$. Note that $\bar{\xi}^{{\rm s}}_{t}=\xi^{{\rm s}}_{t}=0$, when $0\leq t<\kappa_{1}$. Then, $\overline{Y}_{t}=y-\bar{\xi}^{{\rm s}}_{t}\geq 0$ and

[TABLE]

Let us take $t\in[\kappa_{1},\kappa_{2})$. If $\bar{\nu}^{{\rm s}}_{\kappa_{1}}>0$, it means that the agent sold $\bar{\nu}^{{\rm s}}_{\kappa_{1}}$ shares which are a fraction (or the total) of his $y$ initial shares. Otherwise, he only sold a fraction (or the total) of $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{0<T_{k}\leq\kappa_{1}\}}$ and $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{0<T_{k}\leq\kappa_{1}\}}-\nu^{{\rm s}}_{\kappa_{1}}$ are accumulated for the next occasion when the agent decides to sell. Then, $\overline{Y}_{t}=y-\bar{\nu}^{{\rm s}}_{\kappa_{1}}\geq 0$. On the other hand,

[TABLE]

Let us take $t\in[\kappa_{2},\kappa_{3})$. If $\bar{\nu}^{{\rm s}}_{\kappa_{2}}>0$, at the same way that the above case, we have that the agent sold $\bar{\nu}^{{\rm s}}_{\kappa_{2}}$ shares which are a fraction (or the total) of $y-\bar{\nu}_{\kappa_{1}}^{{\rm s}}$. Otherwise, he only sold a fraction (or the total) of $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{\kappa_{1}<T_{k}\leq\kappa_{2}\}}$, or, a fraction of $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{0<T_{k}\leq\kappa_{2}\}}-\nu^{{\rm s}}_{\kappa_{1}}$. Then, $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{\kappa_{1}<T_{k}\leq\kappa_{2}\}}-\nu^{{\rm s}}_{\kappa_{2}}$, or, $\displaystyle\sum_{k=1}^{\infty}\nu^{{\rm b}}_{T_{k}}1_{\{0<T_{k}\leq\kappa_{2}\}}-(\nu^{{\rm s}}_{\kappa_{1}}+\nu^{{\rm s}}_{\kappa_{2}})$ are accumulated for the next occasion when the agent decides to sell. Then, $\overline{Y}_{t}=y-(\bar{\nu}^{{\rm s}}_{\kappa_{1}}+\bar{\nu}^{{\rm s}}_{\kappa_{2}})\geq 0$. On the other hand,

[TABLE]

Recursively, we can see that if $\kappa_{i-1}\leq t<\kappa_{i}$, with $i\in\{3,4,\dots\}$, then, $\overline{Y}_{t}=y-\displaystyle\sum_{\iota=1}^{i-1}\bar{\nu}^{{\rm s}}_{\kappa_{\iota}}\geq 0$ and

[TABLE]

Therefore, by the previously seen, we conclude $\bar{\xi}^{{\rm s}}\in\mathcal{A}^{{\rm s}}(y)$ and $\xi^{{\rm s}}_{t}-\xi^{{\rm b}}_{t}\leq\bar{\xi}^{{\rm s}}_{t}\leq\xi^{{\rm s}}_{t}$ for all $t\geq 0$. ∎

Proof of Proposition 3.2. (Construction of (3.10)).

The proof of this result shall be given in two parts. In the first part, by smooth fit arguments, it is constructed the function $v$ which is a solution to the HJB equation (3.4). In the second part, we prove that $v$ is in $\operatorname{C}^{2,1}(\mathbb{R}_{+}\times\mathbb{R}_{+})$. Let $x<F_{\gamma}$ and consider Eq. (3.7). In this case, the only solution that remains bounded as $x\downarrow 0$ is given by

[TABLE]

where $n$ is the positive solution to (3.14). In order to find the form of the function $A_{1}(y)$ that appears above, we study the behaviour of the solution $v(x,y)=A_{1}(y)x^{n}$ along the boundary $x=F_{\gamma}$. Now we look for a solution that is continuous differentiable at the boundary $x=F_{\gamma}$. Evaluating (A.2) on the left hand side of the equality in (3.8), and recalling that $\mathbb{Y}(x)=\dfrac{1}{\lambda}\ln(x/F_{\gamma})$, we obtain

[TABLE]

By differentiating with respect to $x$, we get that

[TABLE]

In order for the solution to be continuously differentiable at the boundary, we take $x=F_{\gamma}$ in (A.3), and note that

[TABLE]

where the equality follows since (3.8) holds in $x=F_{\gamma}$. The above equation is equivalent to the following ODE for $A_{1}$,

[TABLE]

The solution of this equation is given by

[TABLE]

which implies that, when $x<F_{\gamma}$, the solution to the HJB equation (3.4) is given by

[TABLE]

Now we look for the solution of the HJB equation within the region $F\leq x<F_{\gamma}\operatorname{e}^{\lambda y}$. Since

[TABLE]

Eq. (3.8) is given by

[TABLE]

In order to find the solution to this equation, we look first at the following set of equations,

[TABLE]

The solutions to the previous equations are given, respectively, by

[TABLE]

with $\eta=\delta-\mu$ and C\raisebox{0.4pt}{:}=\displaystyle\dfrac{\gamma}{\lambda(\delta+\gamma)}\biggl{(}\dfrac{bC_{{\rm s}}}{\delta+\gamma}+C_{{\rm s}}\ln F_{\gamma}-F\biggr{)}. Hence, the solution to (A.5) that remains bounded for large values of $\gamma$ is given by

[TABLE]

for some function $A_{2}:\mathbb{R}_{+}\longrightarrow\mathbb{R}$. Recall that $m_{\gamma}$ is the negative solution to (3.15). Since $u$ satisfies $u(F_{\gamma}-,y)=u(F_{\gamma}+,y)$, we conclude that for each $F_{\gamma}\leq x<F_{\gamma}\operatorname{e}^{\lambda y}$, the solution $u$ to the equation (3.4), has the following expression

[TABLE]

where $A_{\gamma}$ is as in (3.12). Finally, in order to obtain the value of the optimal barrier $F_{\gamma}$, look for a solution $v$ such that $v_{x}$ is continuous at $x=F_{\gamma}$. Since $v_{x}(F_{\gamma}-,y)=v_{x}(F_{\gamma}+,y)$, we get

[TABLE]

This implies that $F_{\gamma}$ is given as in (3.11).

Now, let us find a general solution to (3.4) for the region $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$. We have that a particular solution to (3.9) is given by

[TABLE]

Then, the solution to the equation (3.9) that remains bounded for large values of $\gamma$ is given by

[TABLE]

for some function $A_{3}(y):\mathbb{R}_{+}\longrightarrow\mathbb{R}$. Finally, to find the expressions for the function $A_{3}(y)$ involved in (A.7), we ask that $v$ is continuous at $x=F_{\gamma}\operatorname{e}^{\lambda y}$. Then, since $v(F_{\gamma}\operatorname{e}^{\lambda y}-,y)=v(F_{\gamma}\operatorname{e}^{\lambda y}+,y)$, it is not difficult to check that

[TABLE]

Therefore, for each $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$, the solution $u$ has the following expression

[TABLE]

From (A.4), (A.6), (A.8) and since $v(x,0)=0$, we conclude that the solution $v$ to the HJB equation (3.4) is given by (3.10). ∎

Now, we shall proceed to verify that $v$, given in (3.10), belongs to $\operatorname{C}^{2,1}(\mathbbm{R}^{+}\times\mathbbm{R}^{+})$.

Proof of Proposition 3.2. (Smooth of (3.10)).

Note that by construction, it is sufficient to show that $v$ is $\operatorname{C}^{2,1}$ at $x=F_{\gamma}$ and $x=F_{\gamma}\operatorname{e}^{\lambda y}$, respectively, since $v\in\operatorname{C}^{2,1}((\mathbbm{R}^{+}\times\mathbbm{R}^{+})\setminus\mathcal{A})$, where

[TABLE]

Smooth fit at the variable $y$. Using (3.10), it is easy to see that $v_{y}(F_{\gamma}-,y)=v_{y}(F_{\gamma}+,y)$. Implying that $v_{y}$ is continuous at $x=F_{\gamma}$. Calculating first derivative in (3.10), it can be verified that

[TABLE]

From (3.11)–(3.12), it can be verified that

[TABLE]

Then, by (A.9)–(A.10), it yields that $v_{y}(F_{\gamma}\operatorname{e}^{\lambda y}-,y)=F_{\gamma}-C_{{\rm s}}=v_{y}(F_{\gamma}\operatorname{e}^{\lambda y}+,y)$. Therefore $v_{y}$ is continuous at $x=F_{\gamma}\operatorname{e}^{\lambda y}$.

Smooth fit at the variable $x$. We will show that $v_{xx}$ is continuous on $\mathbbm{R}^{+}\times\mathbbm{R}^{+}$. We will first verify that $v_{x}$ is continuous at $x=F_{\gamma}\operatorname{e}^{\lambda y}$. From (3.10), it follows that

[TABLE]

Then, using (A.10), we get that

[TABLE]

We now show that $v_{xx}$ is continuous on $\mathbbm{R}^{+}\times\mathbbm{R}^{+}$ using the fact that $v_{x}$ is continuous on $\mathbbm{R}^{+}\times\mathbbm{R}^{+}$. Since $v_{x}$ is continuous at $x=F_{\gamma}$, from (3.7) and (3.8), we have

[TABLE]

If $x=F_{\gamma}\operatorname{e}^{\lambda y}$, using (3.8) and (3.9), it follows that

[TABLE]

Hence (A.11) and (A.12), we conclude that $v_{xx}$ is continuous on $\mathbbm{R}^{+}\times\mathbbm{R}^{+}$. ∎

Proof of Lemma 3.4.

First, recall that $m_{\gamma}$ was defined as the negative solution of (3.15), and observe that $m_{\gamma}\underset{\gamma\rightarrow 0}{\longrightarrow}m_{0}$, where $m_{0}$ is the negative solution to (3.14). Letting $\gamma\rightarrow 0$ in (3.16), it is easy to see that $a_{\gamma}\underset{\gamma\rightarrow 0}{\longrightarrow}1$. On the other hand, letting $\gamma\rightarrow\infty$ in (3.16), it can be verified that

[TABLE]

Since $n$ is the positive solution to (3.14), it yields that

[TABLE]

Therefore, from (A.13), (A.14), we conclude that $a_{\gamma}\rightarrow\dfrac{n}{n-1}$, if $\gamma\rightarrow\infty$. Now we shall prove that $\displaystyle 1<a_{\gamma}<\frac{n}{n-1}$, for all $\gamma>0$. In order to prove this result we first note that by (A.14),

[TABLE]

On the other hand, we have for each $\gamma>0$,

[TABLE]

since $m_{\gamma}<0$, with $\gamma>0$. (A.15), (A.16) imply

[TABLE]

Therefore using (3.16), (A.17), it follows that $1<a_{\gamma}$. In order to prove the remaining inequality we just note that using (3.16) it is enough to show that

[TABLE]

Note that

[TABLE]

since $\delta-n\mu>0$. Similarly using (A.14) and the fact that $\eta=\delta-\mu$ we obtain that

[TABLE]

Therefore using (A.19), (A.20), and the fact that $m_{\gamma}<0$ we obtain that (A.18) holds and hence $\displaystyle a_{\gamma}<\frac{n}{n-1}$. ∎

Proof of Proposition 3.5.

In order to show that (3.17) holds, it is enough to prove that

[TABLE]

where $\mathbb{Y}(x)=\dfrac{1}{\lambda}\ln(x/F_{\gamma})$.

Maximum on the first zone. Let $x<F_{\gamma}$. Taking first derivatives in (3.10) and evaluating at the point $(x\operatorname{e}^{-\lambda l},y-l)$, we get that

[TABLE]

Then,

[TABLE]

Note that the above expression is negative if and only if

[TABLE]

Taking the first derivative with respect to $x$ on the left hand side of (A.21), we have

[TABLE]

By (3.11) and Lemma 3.4, we know that $x<F_{\gamma}\operatorname{e}^{\lambda l}<\dfrac{nC_{{\rm s}}}{n-1}\operatorname{e}^{\lambda l}$. Then, \dfrac{\partial}{\partial x}\biggl{(}\dfrac{\operatorname{e}^{-\lambda l}x-C_{{\rm s}}}{(x\operatorname{e}^{-\lambda l})^{n}}\biggr{)}>0, which implies that $\dfrac{\operatorname{e}^{-\lambda l}x-C_{{\rm s}}}{(x\operatorname{e}^{-\lambda l})^{n}}$ is non-decreasing with respect to $x$. It yields that

[TABLE]

Showing that $\dfrac{\operatorname{e}^{-\lambda l}F_{\gamma}-C_{{\rm s}}}{(F_{\gamma}\operatorname{e}^{-\lambda l})^{n}}<\dfrac{F_{\gamma}-C_{{\rm s}}}{F_{\gamma}^{n}}$ we obtain (A.21), which is equivalent to see that

[TABLE]

since $F_{\gamma}=C_{{\rm s}}a_{\gamma}$. Taking l^{*}\raisebox{0.4pt}{:}=\dfrac{\ln n}{\lambda(n-1)}, it can be verified that

[TABLE]

Since $a_{\gamma}<\dfrac{n}{n-1}$ and $(n+1)^{n}<n^{n}(n+1)$, with $n>1$, we get that

[TABLE]

This means that (A.22) is satisfied for any $l>l^{*}$. Now, if $l\leq l^{*}$, we shall prove the statement (A.22) by contradiction. Suppose that there exists $0\neq l_{1}\leq l^{*}$ such that

[TABLE]

Since

[TABLE]

we have that $(\operatorname{e}^{-\lambda l^{*}}-\operatorname{e}^{-\lambda nl^{*}})a_{\gamma}\geq 1-\operatorname{e}^{-\lambda nl^{*}}$, which is a contradiction with (A.23). If $l^{*}<l_{1}$ and satisfies that (A.24), we have that

[TABLE]

which contradicts (A.23). Therefore, (A.22) is true for any $l$ and it yields (A.21). We conclude that the maximum on the right hand side of (3.17) is achieved at $l=0$ when $x<F_{\gamma}$.

Maximum on the second zone. Let $F_{\gamma}\leq x<F_{\gamma}\operatorname{e}^{\lambda y}$. Taking first derivatives of $v$ and evaluating $(F_{\gamma},y-\mathbb{Y}(x))$ in them, it follows that

[TABLE]

Then, recalling that $F_{\gamma}=C_{{\rm s}}a_{\gamma}$, where $a_{\gamma}$ is given in (3.16), we get that

[TABLE]

Therefore, $l=\mathbb{Y}(x)$ is a critical point of $G(x,y,l;v)$; recall that the definition of $G$ is given in (3.6). To verify that $l=\mathbb{Y}(x)$ is a maximum of $G(x,y,l;v)$, we need to see that

[TABLE]

Firstly, note that

[TABLE]

Now, taking the second derivatives of $v$ and evaluating $(F_{\gamma},y-\mathbb{Y}(x))$ in them, it follows that

[TABLE]

By (A.25)–(A.26), we get that

[TABLE]

To see that the above expression is negative, we need only to prove that

[TABLE]

which is equivalent to see

[TABLE]

since $-a_{\gamma}\dfrac{\eta(\delta+\gamma)}{(\eta+\gamma)m_{\gamma}^{2}}<0$. Verifying that

[TABLE]

and recalling that $a_{\gamma}$ is given by (3.16), it yields (A.29). We shall show (A.30). Observe that

[TABLE]

is non-increasing with respect to $\gamma>0$ and

[TABLE]

If $b>0$, i.e. $\dfrac{\sigma^{2}}{2}>\mu$, then

[TABLE]

is non-decreasing with respect to $\gamma>0$ and

[TABLE]

From here and by (A.32), it follows (A.30) and therefore we have that (A.27) is negative. If $b\leq 0$, i.e. $\dfrac{\sigma^{2}}{2}\leq\mu$, it can be verified that (A.33) is non-increasing with respect to $\gamma>0$ and

[TABLE]

Defining the function $h(\gamma)$ as

[TABLE]

we can see that

[TABLE]

since (A.32), (A.35) hold. In order to show (A.30), it is enough to prove that $h(\gamma)$ is non-increasing and

[TABLE]

Since $n$ is the positive solution to (3.14) and is bigger than one, it follows that

[TABLE]

this yields that $n\mu<\delta$. Then, applying this in (A.39), we have

[TABLE]

which implies (A.38). Now, taking first derivative in (A.36), we see that

[TABLE]

Using (A.40), it can be shown $-\delta\mu+\dfrac{\delta\mu}{n}-b\eta<0$. This implies that $h(\gamma)$ is a negative non-increasing function. Therefore, it is true (A.29) and we have that (A.27) is negative. Thus the maximum at the right hand side of (3.17) is achieved at $l=\mathbb{Y}(x)$, when $F_{\gamma}\leq x<F_{\gamma}\operatorname{e}^{\lambda y}$.

Maximum on the third zone. Let $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$. Taking the first derivatives of $v$ and evaluating $(x\operatorname{e}^{-\lambda l},y-l)$ in them, it follows that

[TABLE]

Then,

[TABLE]

To see that the above expression is positive, is equivalent to show that

[TABLE]

Observe that from (3.12) and (A.29), it can be verified that $A_{\gamma}<0$. Then, it follows that

[TABLE]

since $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$ and $m_{\gamma}<0$. Then

[TABLE]

Note that $g(l)$ is non-increasing with respect to $l$ and from (3.11)–(3.12), we get that $g(y)=\dfrac{\delta C_{{\rm s}}}{\delta+\gamma}$. Therefore, (A.43) yields that (A.42). Thus, the maximum at the right hand side of (3.17) is achieved at $l=y$, when $x\geq F_{\gamma}\operatorname{e}^{\lambda y}$. ∎

Acknowledgements

The research of D. Hernández-Hernández was partially supported by CONACyT, under grant 254166. H. A. Moreno-Franco acknowledges financial support from CIMAT, CONACyT and HSE. The last one has been funded by the Russian Academic Excellence Project “5-100”.