MD-PhD †# Joe-Elie Salem
email: joeelie.salem@gmail.com

MD Pauline Dureau

MD- PhD Christian Funck-Brentano

Jean-Sébastien Hulot Md-Phd

MD ‡ Maria El-Aissaoui

MD-PhD ‡ Nadia Aissaoui

Saik Urien

MD-PhD ‖ Christophe Faisy

Md-Phd †‖

Effectiveness of heart rate control on hemodynamics in critically ill patients with atrial tachyarrhythmias managed by amiodarone

Keywords: atrial tachyarrhythmias bpm: beats per minute CRRT: continuous renal replacement therapy ECG: electrocardiogram ECV: electrical cardioversion HR: heart rate ICU: intensive care unit IV: intravenous MAP: mean arterial pressure mg: milligram amiodarone, atrial tachyarrhythmia, pharmacology, shock, heart rate

Atrial tachyarrhythmias (AT) are common in intensive care unit (ICU) patients and might contribute to hemodynamic instability if heart rate (HR) is persistently too rapid. We aimed to assess if HR control below 115 or 130 bpm with amiodarone improves hemodynamics in ICU patients with AT. This observational study included 73 ICU patients with disabling AT receiving amiodarone for HR control. A total of 525 changes (mainly within 4-8 h) in mean arterial pressure (MAP) and 167 changes in plasma lactate in response to HR variations above 115 or 130 bpm were analyzed. Epinephrine, sedative drugs, fluid loading, use of diuretics, continuous renal replacement therapy and amiodarone dosing were among covariables assessed.

Univariable analysis showed that HR variations above 115 bpm were poorly correlated to change in MAP (r=0.11, p<0.01). Multivariable analysis showed that changes in MAP were still positively associated to HR variation (p<0.05) and to initiation or termination of epinephrine (p<0.05) or sedatives infusions (p<0.05). Changes in plasma lactate did not correlate to HR variations above 115 bpm. When considering 130 bpm as a threshold, HR variations were not associated to changes in MAP or to changes in plasma lactate. Amiodarone dose was associated to HR decrease but not to MAP or plasma lactate increase.

In ICU patients with AT, strict HR control below 115 bpm or 130 bpm with amiodarone does not improve hemodynamics. A prospective randomized trial assessing strict versus lenient HR control in this setting is needed.

INTRODUCTION

Atrial tachyarrhythmias (AT) (atrial fibrillation, atrial flutter, atrial tachycardia or atrioventricular arrhythmias) are highly prevalent and are associated with a poor outcome in critically ill patients admitted in intensive care units (ICU) [START_REF] Artucio | Cardiac arrhythmias in critically ill patients: epidemiologic study[END_REF][START_REF] Walkey | Optimizing atrial fibrillation management: from ICU and beyond[END_REF] . AT are often associated with a ventricular heart rate (HR) above 120-130 beats per minute (bpm) which contributes to baroreflex impairment, [START_REF] Field | Restoring Sinus Rhythm Improves Baroreflex Function in Patients With Persistent Atrial Fibrillation[END_REF] diastolic dysfunction and may lead to cardiac systolic dysfunction. Such induced ventricular tachycardiomyopathy is associated with hemodynamic instability and increased morbidity or mortality [START_REF] Walkey | Optimizing atrial fibrillation management: from ICU and beyond[END_REF][START_REF] Nerheim | Heart failure and sudden death in patients with tachycardia-induced cardiomyopathy and recurrent tachycardia[END_REF][START_REF] Tseng | Atrial Fibrillation on Intensive Care Unit Admission Independently Increases the Risk of Weaning Failure in Non heart Failure Mechanically Ventilated Patients in a Medical Intensive Care Unit: A Retrospective Case-Control Study[END_REF] . In severely ill patients with AT, a rate or rhythm control strategy with amiodarone is recommended for the treatment or prevention of ventricular tachycardiomyopathy [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] .

Attempts to restore sinus rhythm are frequently unsuccessful in the setting of critically ill patients [START_REF] Arrigo | Disappointing Success of Electrical Cardioversion for New-Onset Atrial Fibrillation in Cardiosurgical ICU Patients[END_REF][START_REF] Mayr | Atrial tachyarrhythmia after cardiac surgery[END_REF][START_REF] Mitrić | Variable use of amiodarone is associated with a greater risk of recurrence of atrial fibrillation in the critically ill[END_REF] , particularly when using electrical cardioversion without antiarrhythmic pretreatment. HR control is thus the preferred strategy in this situation [START_REF] Mitrić | Variable use of amiodarone is associated with a greater risk of recurrence of atrial fibrillation in the critically ill[END_REF][START_REF] Delle Karth | Amiodarone versus diltiazem for rate control in critically ill patients with atrial tachyarrhythmias[END_REF][START_REF] Carlsson | Randomized trial of ratecontrol versus rhythm-control in persistent atrial fibrillation: the Strategies of Treatment of Atrial Fibrillation (STAF) study[END_REF] . In non-ICU patients, guidelines recommend aiming at a HR control between 90-115 bpm during stress situations [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF][START_REF] Van Gelder | Rate Control versus Electrical Cardioversion for Persistent Atrial Fibrillation Study Group. A comparison of rate control and rhythm control in patients with recurrent persistent atrial fibrillation[END_REF] . There are no specific guidelines for patients in severely ill condition but it is commonly advocated that reaching a HR below 115 bpm or a maximum of 130 bpm is advisable [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF][START_REF] Khoo | Acute management of atrial fibrillation[END_REF] . Thus, it is unknown whether exceeding these HR cut-offs, even for short periods of time, is associated with alterations of hemodynamics, i.e. fall of mean arterial pressure (MAP) [START_REF] Rhodes | Surviving Sepsis Campaign: International Guidelines for Management of Sepsis and Septic Shock: 2016[END_REF][START_REF] Thiele | Management of cardiogenic shock[END_REF] or increase of plasma lactate levels [START_REF] Kraut | Lactic acidosis[END_REF] .

In a former study [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] , we modelled the relationship between amiodarone dose and heart rate variation but did not assess the influence of heart rate control on restoration of hemodynamics. The objective of this study was to determine if magnitude of HR variations above 115 or 130 bpm are associated to variations in MAP or plasma lactate.

The analyses took into account variables such as paroxysmal vs persistent AT and others confounding factors relevant to ICU patients [START_REF] Lafont | Modeling for critically ill patients: An introduction for beginners[END_REF] such as epinephrine treatment, sedative or curare drugs infusion, fluid loading, amiodarone dose, use of diuretics or continuous renal replacement therapy (CRRT).

MATERIAL AND METHODS

Patients and Study Design

This observational, cohort study was performed from January 2007 to April 2012 in the 18-bed medical ICU of a tertiary teaching hospital. In accordance with the French legislation on observational studies, approval by an investigational review board was not required. The use of confidential electronically processed patient data was approved by the French National for Data Protection Commission (Commission Nationale de l'Informatique et des Libertés; reference: 1922081). Data were extracted from the files of 73 consecutive critically ill patients who had received at least one dose of amiodarone (Mylan laboratory, Saint-Priest, FRANCE) to treat a disabling AT during their ICU stay (Figure 1). HR and MAP data (average of at least 3 measures over 5 minutes) were collected at the time of the first amiodarone administration in intensive care unit (ICU) and then 4 to 6 times daily over 6 days or until death or ICU discharge. MAP values were obtained through an arterial catheter attached to a fluid-filled pressure transducer system incorporating a flush system, which continuously infused a solution under pressure to maintain patency of the catheter. An attached transducer senses arterial pressure and converted the pressure signal to a waveform on a Phillips® bedside monitor. The waveform reflected pressure generated by the left ventricle during systole. The monitor also displayed automatically numerical MAP values. HR were also displayed on the monitor and computed by automatic QRS detection from continuous acquisition of a digital electrocardiographic signal. Plasma lactate levels were drawn at the discretion of the treating physician without any predefined protocol and were determined by an arterial blood gas analyser (ABL 825, Radiometer, Copenhagen, Denmark). Details concerning modalities of amiodarone administration in this cohort and estimation of pharmacodynamically active amiodarone dose at each time point were estimated as previously described in detail [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] . Briefly, amiodarone could be administered either intravenously (IV) as amiodarone hydrochloride 150mg/3mL or orally as 200 mg tablet(s). Route, frequency of administrations and doses were left at the discretion of the treating physician. Other treatments were administered according to standard guidelines [START_REF] Khoo | Acute management of atrial fibrillation[END_REF] .

Variables collection

The following covariables were recorded at the time of first HR and MAP collection: age, gender, body weight, severity score at ICU admission, diabetes, hypertension, AT history, bilirubinemia, previous amiodarone treatment and type of AT. Paroxysmal and persistent AT were defined as an arrhythmia occurring less or more than one week before the first amiodarone administration in ICU, respectively. Prior chronic and subacute amiodarone treatment before ICU were defined as continuous amiodarone treatment since more than one month or a cumulated amiodarone dose of less than 4 grams in the last month, respectively. Time-dependent covariables potentially interacting with hemodynamic conditions were recorded at the time of each HR, MAP, and plasma lactate collection. These were: body temperature, arterial pH or PaO2, hemoglobin, electrical cardioversion, fluid loading, pharmacodynamically active amiodarone dose or use of other antiarrhythmic drugs, loop diuretics, catecholamine, curare or sedative drugs and need for CRRT.

Fluid loading was defined as administration of more than 0.5 L of saline solution in 30 min. Electrocardiographic acquisition allowing for accurate evaluation of type of rhythm was left at the discretion of the treating physician.

As previously reported [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] , pharmacodynamically active amiodarone dose for each subject was determined at the time of each HR, MAP, and plasma lactate collection.

In brief, amiodarone pharmacokinetics was ascribed to a virtual compartment model including zero or first order input rates. This virtual compartment, A(t), represents the biophase in which amiodarone amount is in equilibrium with the observed effect on heart rate. Pharmacodynamic half-life (KDE) was determined to be 3.33 days. G(t) represents the amiodarone amount in the gut at a given time and ka, the first-order absorption rate, which was fixed to 8. Amiodarone bioavailability (F) was fixed to 0.33.

dA(t)/dt = input -KDE × A(t) (1) input = ka × G(t) × F if administered orally (2)

Statistics

Results are expressed as numbers (%), means ± standard deviation, or medians

Where Y is the dependent variable, X1,..., Xn and β1,..., βn stand for the explanatory variables and the corresponding regression coefficient respectively.  denotes the random effect (it models the differences between the patients). Estimation of pharmacodynamically active amiodarone doses [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] at each time-point were derived using R statistical software (https://www.r-project.org).

Statistical significance was accepted for P<.05.

RESULTS

Study Population and Time-Dependent Observations

Seventy three consecutive patients admitted to ICU received at least one amiodarone dose for AT HR control and were eligible for analysis [Figures 12]. Of these 73 patients, 68 and 62 patients had at least one HR above 115 bpm or 130 bpm, respectively, thereby allowing for analysis of association between concomitant variations of MAP (ΔMAP) and HR (ΔHR), above 115 bpm (n = 525) or 130 bpm (n = 313) (Figure 1). Most of these variations were evaluated during a period of time of 4 to 8 hours (Table 1). Fifty three and 45 patients had at least one HR above 115 bpm and 130 bpm and at least two plasma lactate levels available, respectively, allowing for analysis of association between concomitant variations of plasma lactate (Δlactate)

and HR (ΔHR), above 115 bpm (n = 167) or 130 bpm (n = 106) (Figures 12). Most of these variations were evaluated during a period of time of 6 to 24 h (Table 1). Patients included were severely ill. They were mainly admitted for cardiogenic shock or sepsis, had cardiovascular comorbidities and initially received amiodarone during their ICU stay for heart rate control either IV or mostly orally ( 4]. When considering 130 bpm as threshold, HR variations were not associated to changes in MAP in univariable analysis (Figure 5B). Multivariable analysis showed that variations in MAP were also associated to initiation or termination/weaning of epinephrine (β= 27±9.4 for initiation vs. termination/weaning, P=0.005; β= 16±7.8 for no change vs. termination/weaning, P=0.04) and sedative drugs (β= 27.2±11.9 for termination vs. initiation, P=0.02). Variation of amiodarone pharmacodynamically active amount was also positively associated with variation in MAP (β= 0.01± 0.005, P=0.03). Other co-variables, including variation in HR, were not associated to variation in MAP.

Heart rate variation and hemodynamics

We did not test other HR threshold because too few HR were available above 150 or 200 bpm in a limited number of patients (Table 1). Restoration of sinus rhythm could not be examined in our multivariable analysis because we had too few accurate characterizations of rhythm type evaluated by electrocardiogram (Table 1). However, univariable analysis showed that changes in plasma lactate were not associated to HR variations above 115 bpm (Figure 5C) but tended to be associated to restoration of sinus rhythm (-0.3 vs 0 mmol/L, P= 0.06, Figure 5E). Univariable analysis showed that changes in plasma lactate were not associated to HR variations above 130 bpm (ns;

Figure 5D) or to restoration of sinus rhythm (-0.3 vs 0 mmol/L, P= 0.12, Figure 5F).

DISCUSSION

The present study indicates that, in critically ill patients with AT, acute HR increases above 115 bpm or 130 bpm are not associated with acute deterioration of hemodynamic surrogates, i.e. increase of plasma lactate and decreases of MAP. In this setting, use of epinephrine or absence of sedative drugs are major determinants of MAP increase while restoration of sinus rhythm might improve plasma lactate levels.

Patients with paroxysmal AT had a higher proportion of HR above 130bpm than patients with persistent AT. Thus, patients with paroxysmal AT had a lower proportion of plasma lactate above 2.2mmol/L, as compared to patients with persistent AT. This finding further enhance absence of negative association between increased HR and deleterious hemodynamic response. We also found that amiodarone slows HR without altering hemodynamic markers.

Avoiding Ventricular Tachycardiomyopathy

AT, if persistent and rapid, can lead to tachycardia-induced cardiomyopathy with heart failure [START_REF] Field | Restoring Sinus Rhythm Improves Baroreflex Function in Patients With Persistent Atrial Fibrillation[END_REF] . This condition is reversible, either by HR or rhythm control. Amiodarone is the gold standard medication to prevent or cure this condition in critically ill patients with AT, particularly if kidney failure, ionic disturbances and heart failure are present [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . Whether brief episodes of rapid AT or sinus rhythm exceeding thresholds of 115 or 130 bpm carry a significant risk of tachycardia-induced cardiomyopathy in ICU patients remains unknown. Also, the best target HR and molecule to use are still unknown [START_REF] Tseng | Atrial Fibrillation on Intensive Care Unit Admission Independently Increases the Risk of Weaning Failure in Non heart Failure Mechanically Ventilated Patients in a Medical Intensive Care Unit: A Retrospective Case-Control Study[END_REF][START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] Walkey | Practice Patterns and Outcomes of Treatments for Atrial Fibrillation During Sepsis: A Propensity-Matched Cohort Study[END_REF][START_REF] Don | t Rush to "Block" Atrial Fibrillation in Sepsis[END_REF] .

Amiodarone and Hemodynamics

In this observational study, amiodarone was mainly administered by oral route. In the past, the IV route of administration was known to be associated with transient hypotension and hemodynamic impairment [START_REF] Schwartz | Hemodynamic effects of intravenous amiodarone in patients with depressed left ventricular function and recurrent ventricular tachycardia[END_REF][START_REF] Remme | Hemodynamic effects and tolerability of intravenous amiodarone in patients with impaired left ventricular function[END_REF][START_REF] Souney | PM101: intravenous amiodarone formulation changes can improve medication safety[END_REF] . However, the aqueous form of IV amiodarone used in our patients was not expected to be associated with this specific side effect [START_REF] Salem | Impact of inodilator drugs on echocardiographic assessments of left ventricular filling pressure in patients with decompensated end-stage heart failure[END_REF] . Our results did not show any negative association between the estimated pharmacodynamically active amiodarone dose in organism and surrogates of global hemodynamic status. These data further support the safety of newer IV amiodarone hydrochloride formulation. Beta-blockers and verapamil were exceptionally used in this study because of their negative inotropic effects [START_REF] Walkey | Practice Patterns and Outcomes of Treatments for Atrial Fibrillation During Sepsis: A Propensity-Matched Cohort Study[END_REF] . Digoxin was also very rarely used because of its poor efficacy to rapidly slow HR, important dependency to renal clearance, and significant interaction with amiodarone or ionic disorder that may lead to adverse events [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] .

Epinephrine, Sedative Drugs and Hemodynamics

In our study, we found that epinephrine use was associated with an increase of MAP while sedative drugs had an opposite effect. Epinephrine has alpha-1 and beta adrenergic agonist properties, determining its inotropic and vasopressive effects. In contrast, dobutamine, an inodilator selective beta adrenergic agonist [START_REF] Salem | Impact of inodilator drugs on echocardiographic assessments of left ventricular filling pressure in patients with decompensated end-stage heart failure[END_REF] , was not found to be a major determinant of MAP variation. This might be explained by vasodilatation compensating its inotropic effect. Sedative drugs used in these patients were systematically relying on an association of a morphine derivate (sufentanil) with benzodiazepine (midazolam), which association is well known to decrease MAP [START_REF] Crozier | Total intravenous anaesthesia with sufentanil-midazolam for major abdominal surgery[END_REF] .

Limitations

In our multivariable analysis, there was no significant influence of fluid loading. Its effect on MAP is expected to occur in the short term and not necessarily after a few hours of administration. Since the delay between two MAP collections was of a few hours, the effect of fluid therapy on MAP was possibly not sustained enough to be detected. Neither furosemide administration nor CRRT were identified as significant covariables. We hypothesize that patients treated with these depletive strategies had high pre-therapeutic filling pressures and did not experience hypovolemia and reactional MAP decrease.

In this observational study in critically ill patients, other covariables of interest such as left ventricular ejection fraction, plasma catecholamine levels, oxidative stress measurement, baroreflex and autonomic function analysis, which may influence hemodynamic response were not taken into consideration. However, this study shows for the first time the lack of effectiveness of aggressive acute heart rate control strategy to restore hemodynamics in critically ill patients.

Plasma levels of amiodarone and its metabolite N-desethylamiodarone, were not tested in our analysis because amiodarone therapeutic drug monitoring is not recommended in clinical practice [START_REF] Surgery | Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC)[END_REF][START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the Heart Rhythm Society[END_REF] . It should be emphasized that individual plasma concentrations are poorly predictive of amiodarone pharmacodynamics response [START_REF] Greenberg | Relation between amiodarone and desethylamiodarone plasma concentrations and electrophysiologic effects, efficacy and toxicity[END_REF][START_REF] Deharo | Clinical electrophysiologic effects of a single high oral dose of amiodarone[END_REF] .

Perspectives

Several studies have evaluated the potential benefit of lenient versus strict heart rate control or rhythm control in the context of persistent AT in non-critically ill patients [START_REF] Carlsson | Randomized trial of ratecontrol versus rhythm-control in persistent atrial fibrillation: the Strategies of Treatment of Atrial Fibrillation (STAF) study[END_REF][START_REF] Van Gelder | Rate Control versus Electrical Cardioversion for Persistent Atrial Fibrillation Study Group. A comparison of rate control and rhythm control in patients with recurrent persistent atrial fibrillation[END_REF][START_REF] Van Gelder | Lenient versus strict rate control in patients with atrial fibrillation[END_REF] . These studies did not show any difference in outcome among these strategies.

Recently, Gillinov et al. further showed that strategies of rate or rhythm control to treat postoperative atrial fibrillation were not significantly different in terms of numbers of days of hospitalization, complication rates and rates of persistent atrial fibrillation 60 days after onset. [START_REF] Gillinov | Rate Control versus Rhythm Control for Atrial Fibrillation after Cardiac Surgery[END_REF] Our study is the first to address this issue in critically ill patients. Its findings emphasize the importance of conducting a prospective trial of amiodarone dosing in critically ill patients with mostly "reactional" AT due to severe conditions comparing a lenient versus a strict rate control or even a rhythm control strategy. We recently proposed a tailored amiodarone dosing strategy according to duration of AT (paroxysmal versus persistent) and amiodarone pretreatment or not, which might help achieve a heart rate of 115bpm more conveniently (less systemic side effects, bradycardia and hypotension) than a highly aggressive strategy relying on physician's decisions without any standardized protocol [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] . Paroxysmal AT needed lower doses of amiodarone to achieve the same level of rate control, as compared to persistent AT.

Furthermore, addition of intravenous magnesium and vascular filling improved amiodarone efficacy to slow HR, as opposed to concomitant use of dobutamine [START_REF] Salem | Modeling of Amiodarone Effect on Heart Rate Control in Critically Ill Patients with Atrial Tachyarrhythmias[END_REF] .

CONCLUSION

In critically ill patients with AT, strict heart rate control below 115 bpm or even 130 bpm is not associated with hemodynamic improvement. Our data suggest that lenient heart rate control around 130 bpm might not be deleterious for these patients. A prospective randomized trial assessing strict versus lenient heart rate control or even rhythm control in critically ill patients with AT is needed to further address this issue.

DISCLOSURES

JES, PD, CFB, MEA, NA, SU, CF declare that they have no conflict of interest.

(

  interquartile ranges) as appropriate. Normality was assessed by the D'Agostino-Pearson omnibus normality test. Comparison of continuous variables were analyzed by Mann-Whitney test for non-paired and non-parametric distribution. Comparison of categorical variables were analyzed by Chi-2 test. The correlation (r) between linear variables was assessed by calculating Pearson's for parametric distribution or Spearman's coefficient for non-parametric distribution (Prism 6, GraphPad software®, San Diego, USA). Of note, we had a power of at least 80% to detect a modest correlation (r>0.27) between variation of HR and variation of hemodynamic surrogates (MAP and plasma lactate). Multivariable analysis was performed by using linear mixed effect modeling that takes into account repeated measures (package nlme, Linear and Nonlinear Mixed Effects Models, R statistical software, https://www.r-project.org). The general equation is: Y = intercept + β1*X1 + β2*X2 +…+ βn*Xn + 

  Univariable analysis showed that HR variations above 115 bpm were poorly associated to changes in MAP (r= 0.11±0.04, P.02; Figure5A). The results of multivariable analysis integrated catecholamine, curare or sedative drugs infusion, fluid loading, persistent versus paroxysmal AT, use of diuretics or CRRT and estimated pharmacodynamically active amiodarone dose or its variation. It showed that variations in MAP were associated to initiation or termination/weaning of epinephrine (β= 14.5±6.5 for initiation vs. termination/weaning, P=0.02; β= 10.6±4.9 for no change vs. termination/weaning, P=0.03) and sedative drugs (β= 18.3±8.4 for termination vs. initiation, P=0.03). Variation in HR was still positively associated to variation in MAP (β =0.06±0.03, P=0.03) but contribution of amiodarone pharmacodynamically active amount or other covariables were insignificant to explain variation in MAP.

FIGURE 1 .FIGURE 2 .

 12 FIGURE 1. Flow chart of patients included and analyses performed; when considering 115 bpm (A), or 130 bpm (B) as threshold.

Figure 3 .

 3 Figure 3. HR (bpm) in paroxysmal and persistent AT groups, represented as median and interquartile range (A). HR (bpm), MAP (mmHg) and plasma lactate (mmol/l) in paroxysmal and persistent AT groups, represented as proportion of HR>130bpm (B), MAP<65mmHg (C) and plasma lactate>2.2mmol/L (D). Statistics were performed by Mann-Whitney or Chi-2 tests, as appropriate.

FIGURE 4 .

 4 FIGURE 4. Correlation between variation in heart rate (ΔHR, bpm) and variation in simulated pharmacodynamically (PD) active amiodarone dose (mg).

FIGURE 5 .

 5 FIGURE 5. Correlation between variations of heart rate (ΔHR, bpm) and mean arterial pressure (ΔMAP, mmHg) when considering 115bpm (A) or 130bpm (B) as thresholds. Correlation between concomitant variations of heart rate (ΔHR, bpm) and plasma lactate (ΔLactate, mmol/l) when considering 115bpm (C) or 130bpm (D) as thresholds. Variation of plasma lactate (ΔLactate, mmol/l) as a function of restoration of sinus rhythm (SR restoration) or not (No), when considering 115bpm (E) or 130bpm (F) as thresholds.

Table 2

 2

	paroxysmal than persistent AT (25% vs 19%, p=0.01), proportion of MAP below 65
	mmHg was not different (20% vs 20%, p=ns) and proportion of plasma lactate
	>2.2mmol/L was lower in patients with paroxysmal AT as compared to patients with
	persistent AT (23% vs 43%, p<0.0001) [Figure 3]. Variations in estimated
	pharmacodynamically active dose of amiodarone were correlated to variations in HR
	(r = 0.31, P <.0001) [Figure

). Dobutamine (11%) or epinephrine (61%) were the only IV catecholamine used. CRRT (25%), loop diuretics (49%), sedative drugs (60%) and fluid loading (70%) were frequently used. The main clinical, demographic and biological patient's characteristics and other treatments that could interfere with hemodynamic status are summarized in Table

2

. Forty eight patients (66%) had paroxysmal AT and 25 patients (34%) had persistent AT. Despite a higher proportion of uncontrolled HR collected above 130 bpm in patients with

Table 1 . Heart rate, rhythm and hemodynamic data of the 73 patients

 1 Abbreviations: bpm, beat per minute; HR, Heart rate; MAP, Mean arterial pressure.Statistics: Values are given as the median (interquartile range 25-75) or number.

	Characteristic

ACKNOWLEDGMENTS

Dr Salem had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Dr Salem, Dr Aissaoui and Pr Faisy contributed to the conception and design of the study. Dr Salem, Dr El-Aissaoui and Dr Dureau contributed to data collection. Dr Salem, Dr Urien, Pr Faisy, Pr Funck-brentano and Pr Hulot contributed to analysis and/or interpretation of the data. Dr Salem, Pr Faisy, and Dr Urien contributed to drafting the manuscript. All authors contributed to revision of the manuscript and accepted the submitted version.

We thank Margaux Alazard for help in data collection.

No external funding was used in the preparation of this manuscript.

Conflict of interest: None


Paulwin Graewe
email: graewe@math.hu-berlin.de

Ulrich Horst
email: horst@math.hu-berlin.de

Eric Séré

Smooth Solutions to Portfolio Liquidation Problems under Price-Sensitive Market Impact *

Keywords: AMS Subject Classification: Primary 93E20; secondary 35Q93, 91G80 stochastic optimal control, portfolio liquidation, singular terminal value

We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.

Introduction

Traditional financial market models assume that asset prices follow an exogenous stochastic process and that all transactions can be settled without any impact on market prices. This assumption is appropriate for small investors who trade only a negligible proportion of the average daily trading volume. It is not always appropriate, though, for institutional investors trading large blocks of shares over a short time span.

The analysis of optimal liquidation problems has received considerable attention in the mathematical finance and stochastic control literature in recent years. Starting with the paper of Almgren & Chriss [START_REF] Almgren | Optimal execution of portfolio transactions[END_REF] existence and uniqueness results of optimal liquidation strategies under various market regimes and price impact functions have been established by many authors, including [4, 5, 8, 12-15, 19-21, 31, 32]. One of the main characteristics of stochastic optimization problems arising in portfolio liquidation models is the singular terminal condition of the value function induced by the liquidation constraint. The singularity is already present, yet not immediately visible, in the original price impact model of Almgren & Chriss [START_REF] Almgren | Optimal execution of portfolio transactions[END_REF]. Within their mean variance framework and with arithmetic Brownian motion as the benchmark price process, the objective function is deterministic, and the optimization problem is essentially a classical variational problem where the terminal state constraint causes no further difficulties. However, when considering a geometric Brownian motion as the underlying price process as in Forsyth et al. [START_REF] Forsyth | Optimal trade execution: A mean quadratic variation approach[END_REF], the optimal execution strategies become price-sensitive. One is then faced with a genuine stochastic control problem where the singularity becomes a challenge when determining the value function and applying verification arguments.

Several approaches to overcome this challenge have recently been suggested in the stochastic control literature. Forsyth et al. [START_REF] Forsyth | Optimal trade execution: A mean quadratic variation approach[END_REF] solve the control problem numerically by penalizing open positions at the final time. Ankirchner & Kruse [START_REF] Ankirchner | Price-sensitive liquidation in continuous-time[END_REF] characterize the value function of a Markovian liquidation problem as the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) equation. Their verification argument uses a discrete approximation of the continuous time model. Ankirchner et al. [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF], Graewe et al. [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF], and Horst et al. [START_REF] Horst | A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition[END_REF] consider non-Markovian liquidation problems where the cost functional is driven by general adapted factor processes and the HJB equation solves a BSDE or BSPDE, depending on the dynamics of the factor processes. In all three cases, existence of solutions to the HJB equation is established by analyzing limits of sequences of BS(P)DEs with increasing finite terminal values while the verification argument uses generalized Itô-Kunita formulas [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF], resp., the link between degenerate BSPDEs and forward-backward stochastic differential equations [START_REF] Horst | A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition[END_REF].

A general class of Markovian liquidation problems has been solved in Schied [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF] by means of Dawson-Watanabe superprocess. This approach avoids the use of HJB equations. Instead, it uses a probabilistic verification argument based on log-Laplace functionals of superprocesses that requires sharp upper and lower bounds for the candidate value function.

This paper establishes existence of a smooth solution to a class of Markovian portfolio liquidation problems. While existence of a weakly differentiable solution to our HJB equation can be inferred from [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF] and existence of optimal liquidation strategies can be inferred from, e.g., [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF][START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF], smooth solutions to stochastic portfolio liquidation problems have not yet been established in the literature before. As in [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF] the key is to know the precise asymptotic behavior of the value function at the terminal time. The asymptotics allows us to characterize the HJB equation in terms of a PDE with finite terminal value yet singular nonlinearity, for which existence of a unique smooth solution can be proved using standard fixed point arguments in a suitable function space.

As in [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF][START_REF] Horst | When to cross the spread? Trading in two-sided limit order books[END_REF][START_REF] Kratz | Portfolio liquidation in dark pools in continuous time[END_REF] we allow for simultaneous submission of active orders, i.e., orders for immediate execution at the best available price, to a primary venue, and of passive orders, i.e., orders for future execution, into a dark pool. Dark pools are alternative trading venues that allow investors to reduce market impact and hence trading costs by submitting liquidity that is shielded from public view. Trade execution is uncertain, though, as trades will be settled only if matching liquidity becomes available.

Active orders incur market impact costs while passive orders incur adverse selection (or "slippage") costs. In our model impact, adverse selection and risk costs are driven by a multidimensional time-homogeneous diffusion process. We think of the multi-dimensional diffusion process as describing the joint dynamics of a fundamental or benchmark stock price process and of a stochastic factor process driving the fluctuations in the available liquidity over time. Assuming that all cost terms are of the same order p > 1 allows us to make the standard separation ansatz of expressing the value function as v(t, y)|x| p where y is the state of the factor process and x is the portfolio position. We establish sharp a priori estimates on v at the terminal time. This is similar to [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF] and avoids the stochastic penalization method applied in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF][START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF][START_REF] Horst | A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition[END_REF].

For deterministic cost coefficients the asymptotics of v can be inferred from, e.g., Seidman & Yong [START_REF] Seidman | How violent are fast controls?, II[END_REF] who give the asymptotics of minimal L p -norm controls of multi-dimensional linear systems with terminal state constraint. To the best of our knowledge a corresponding result for stochastic systems is not available in the literature. 1 Our a priori estimates allow us to prove the uniqueness of a continuous viscosity solution of polynomial growth to the HJB equation under continuity and polynomial growth conditions on the cost coefficients. The proof uses a comparison result for parabolic PDEs with possibly infinite terminal values. As a byproduct we obtain that the minimal nonnegative solution to the stochastic HJB equation in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF] is indeed the unique nonnegative solution to their singular BSDE if the coefficients are essentially bounded.

Our main contribution is the proof of the existence of a classical solution to the HJB equation under additional smoothness and boundedness conditions on the cost coefficients. The proof is based on the a priori estimates. They provide us with the precise asymptotic behavior of the solution at the terminal time and thus allow us to reduce the problem of solving the PDE with singular terminal value for v to solving a PDE with finite terminal value, yet degenerate non-linearity. Using Krylov's generalized Itô formula we prove that the classical solution is indeed the value function and that the optimal trading strategies can be given in closed form.

The remainder of this paper is organized as follows. The stochastic control problem is formulated in Section 2. The a priori estimates and the comparison principle that yields uniqueness of a continuous viscosity solution is established in Section 3. Existence and uniqueness of a classical solution to the HJB equation is proven in Section 4. The verification argument is carried out in Section 5. Finally, we show in Section 6 how our uniqueness result extends to the non-Markovian case analyzed in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF].

Notational conventions. We denote by C(R d ) the space of bounded continuous functions. The functions in u ∈ C poly ([0, T ] × R d ) are continuous and for some C > 0 and n ≥ 1,

|u(t, y)| ≤ C(1 + |y| n ), (t, y) ∈ [0, T ] × R d . (1.1)
The space C 1,2 loc ([0, T ]×R d ) denotes the class of the functions u(t, y) that are continuous (possibly unbounded) along with their first derivative in t and their first and second derivative in y. For generic α ∈ (0, 1) and normed vector space E, the functions in C k+α ([0, T ]; E) have α-Hölder continuous derivatives up to the order k. By W 2 q,loc (R d ) we denote the usual Sobolev spaces of all functions that are locally L q -integrable along with their weak first and second order derivative [START_REF] Adams | Sobolev Spaces[END_REF]Definition 1.62]. The parabolic version W 1,2 q,loc ((0, T ) × R d ) is the space of all functions u(t, y) that are locally L q -integrable along with their weak first derivative in t and their weak first and second derivative in y. Whenever the notation T -appears in the definition of a function space we mean the set of all functions whose restrictions satisfy the respective property when T -is replaced by any s < T , e.g.,

C poly ([0, T -] × R d ) = {u : [0, T ) × R d → R : u |[0,s]×R d ∈ C poly ([0, s] × R d ) for all s ∈ [0, T )} and W 1,2 q,loc ((0, T -) × R d ) = {u : (0, T ) × R d → R : u |(0,s)×R d ∈ W 1,2 q,loc ((0, s) × R d ) for all s ∈ (0, T )}. The set of adapted R d -valued processes (Z t ) t∈[0,T ] satisfying E[ T 0 |Z t | q dt] < ∞ is denoted by L q F (0, T ; R d ); the subset of processes with continuous paths satisfying E[sup t∈[0,T ] |Z t | q ] < ∞ is denoted by L q F (Ω; C([0, T ]; R d )).
If not otherwise indicated then • denotes the supremum norm. For arbitrary β > 0 we occasionally write β √ • instead of ( • ) 1/β . All equations are to be understood in the a.s. sense.

Problem formulation, assumptions and main results

We consider the stochastic optimization problem of an investor that needs to close a (large) position of shares within a given time interval [0, T ]. Following Horst & Naujokat [START_REF] Horst | When to cross the spread? Trading in two-sided limit order books[END_REF] and Kratz & Schöneborn [START_REF] Kratz | Portfolio liquidation in dark pools in continuous time[END_REF] the investor may trade in an absolutely continuous manner in a primary exchange and simultaneously place passive block orders into a dark pool. Execution of passive orders is modeled by the jump times of a Poisson process (N t ) t∈[0,T ] with constant intensity θ ≥ 0.

The Poisson process N and an n-dimensional standard Brownian motion (W t ) t∈[0,T ] are defined on a stochastic basis (Ω, F, (F t ) t∈[0,T ] , P) satisfying the usual conditions. In what follows we repeatedly use the independence of N and W .

As the factor process driving trading costs we consider the d-dimensional Itô diffusion The preceding assumption guarantees that (2.1) admits a unique strong solution (Y t,y s ) s∈[t,T ] , for every initial state (t, y) ∈ [0, T ] × R d . Furthermore, see [START_REF] Pardoux | Stochastic Differential Equations, Backward SDEs, Partial Differential Equations[END_REF]Theorem 3.35], the map (s, t, y) → Y t,y s is a.s. continuous and for every n ≥ 2 there exists C n > 0 such that the following moment estimate holds with the convention Y t,y s = y for 0 ≤ s ≤ t:

E[sup s∈[0,T ] |Y t,y s | n ] ≤ C n (1 + |y| n ), (t, y) ∈ [0, T ] × R d . (2.2)
This moment estimate in particular guarantees by Vitali's convergence theorem that functions of the form

(t, y) → E T t f (s, Y t,y s ) ds , (2.3)
with f ∈ C poly ([0, T ] × R d ), belong again to C poly ([0, T ] × R d ).

The stochastic control problem

For any initial time t ∈ [0, T ) and initial position x ∈ R, we denote by A(t, x) the set of all admissible liquidation strategies (ξ, π). Here, ξ = (ξ s ) s∈[t,T ] describes the rates at which the agent trades in the primary market, while π = (π s ) s∈[t,T ] describes the passive orders submitted to the dark pool. A pair of strategies (ξ, π) is admissible if ξ is progressively measurable and π is predictable such that the resulting portfolio process

X ξ,π s = x - s t ξ r dr - s t π r dN r , t ≤ s ≤ T,
satisfies the liquidation constraint X ξ,π T = 0.

(2.4)

The costs associated with an admissible liquidation strategy (ξ, π) are modeled by the cost functional

J(t, y, x; ξ, π) := E T t η(Y t,y s )|ξ s | p + θγ(Y t,y s )|π s | p + λ(Y t,y s )|X ξ,π s | p ds .
The first term of the nonnegative running costs c(y, x, ξ, π) := η(y)|ξ| p + θγ(y)|π| p + λ(y)|x| p describes the temporary price impact at the primary exchange; the second term describes adverse selection costs associated with dark pool trading (see [START_REF] Horst | When to cross the spread? Trading in two-sided limit order books[END_REF][START_REF] Klöck | Price manipulation in a market impact model with dark pool[END_REF] for details) while the third term penalizes slow liquidation. The latter may be interpreted as the p-th power of the Valueat-Risk of the open position (see [START_REF] Ankirchner | Price-sensitive liquidation in continuous-time[END_REF][START_REF] Gatheral | Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework[END_REF] for details).

The value function of the stochastic control problem is defined for each initial state (t, y, x)

∈ [0, T ) × R d × R as V (t, y, x) := inf (ξ,π)∈A(t,x)
J(t, y, x; ξ, π).

(2.5) Assumption 2.2. We assume throughout that p > 12 and put β := 1/(p -1) > 0. We further assume that the cost coefficients satisfy the following conditions:

(i) The coefficients η, γ, λ, 1/η : R d → [0, ∞) are continuous.

(ii) The coefficients η, λ, 1/η are of polynomial growth, i.e., for some n ≥ 1 and C > 0,

η(y) + λ(y) + 1/η(y) ≤ C(1 + |y| n ), y ∈ R d . (2.6)
Example 2.3. Our assumptions on the factor process allow us to capture simultaneously several key determinants of trading costs. The assumptions are satisfied for the arithmetic Brownian motion model

dY 1 t = µ dt + σ dW 1 t
as well as for a mean-reverting process of the form

dY 2 t = f (ν -Y 2 t ) dt + dW 2 t
for a bounded Lipschitz continuous function f . The (logarithmic) price process Y 1 may drive the market risk factor λ while Y 2 may describe stochastic order book heights (stochastic liquidity) and hence drive η.

Assumption 2.4. In order to state our main result (Theorem 2.9), we will need the following additional assumptions:

(A1) σσ * is uniformly positive definite.

(A2) b and σ are bounded.

(A3) η, 1/η, and λ are bounded. In particular, η ≥ κ 0 for some constant κ 0 > 0.

(A4) η is twice continuously differentiable, and Lη is bounded.

Remark 2.5. Condition (A1) and (A2) provide, in particular, the fact that

   D(L) = {u ∈ q≥1 W 2 q,loc (R d ) : u, Lu ∈ C(R d )} L : D(L) ⊂ C(R d ) → C(R d )

Heuristics and the main result

The dynamic programing principle suggests that the value function satisfies the following HJB equation, cf. [34, Theorem 2.2]:

-∂ t V (t, y, x) -LV (t, y, x) -inf ξ,π∈R H(t, y, x, ξ, π, V ) = 0, (t, y, x) ∈ [0, T ) × R d × R, (2.7)
where L := 1 2 tr(σσ * D 2 y ) + b, D y denotes the infinitesimal generator of the factor process, and the Hamiltonian H is given by

H(t, y, x, ξ, π, V ) := -ξ∂ x V (t, y, x) + θ(V (t, y, x -π) -V (t, y, x)) + c(y, x, ξ, π).
The specific structure of our control problem with respect to the state variable x -linear in the control dynamics and of p-th power in the running costs -suggests an ansatz of the form:

V (t, y, x) = v(t, y)|x| p . (2.8)
Recalling that β = 1/(p -1), the proof of Lemma 2.7 below is in fact standard. Before stating the lemma we formulate the different solution concepts for parabolic equations used in this paper.

Definition 2.6. For continuous functions v : [0, T ) × R d → R we use the following solution concepts to parabolic PDEs

-∂ t v(t, y) -H(t, y, v(t, y), D y v(t, y), D 2 y v(t, y)) = 0, (2.9)
where

H : [0, T ) × R d × R × R d × S d → R and S d denotes the set of symmetric d × d matrices. (i) v is a classical solution if v ∈ C 1,2 loc ([0, T ) × R d ) such that (2.9) is satisfied for all (t, y) ∈ [0, T ) × R d . (ii) v is a strong solution if v ∈ W 1,2 1,loc ((0, T ) × R d ) such that (2.9) is satisfied in terms of the weak derivatives of v a.e. in [0, T ) × R d . (iii) v is a viscosity subsolution if for every ϕ ∈ C 1,2 loc ([0, T ) × R d ) such that ϕ ≥ v and ϕ(t, y) = v(t, y) at a point (t, y) ∈ [0, T ) × R d it holds -∂ t ϕ(t, y) -H(t, y, v(t, y), D y ϕ(t, y), D 2 y ϕ(t, y)) ≤ 0. (iv) v is a viscosity supersolution if for every ϕ ∈ C 1,2 loc ([0, T ) × R d ) such that ϕ ≤ v and ϕ(t, y) = v(t, y) at a point (t, y) ∈ [0, T ) × R d it holds -∂ t ϕ(t, y) -H(t, y, v(t, y), D y ϕ(t, y), D 2 y ϕ(t, y)) ≥ 0.
(v) v is a viscosity solution if v is both viscosity sub-and supersolution.

We recall the well-known fact that classical solutions of parabolic PDEs are also viscosity solutions, see, e.g., [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF].

Lemma 2.7. A nonnegative function v : [0, T ) × R d → [0, ∞) is a classical/strong/viscosity (sub-/super-)solution to -∂ t v(t, y) -Lv(t, y) -F (y, v(t, y)) = 0, (2.10)
where

F (y, v) := λ(y) - |v| β+1 βη(y) β + θγ(y)v β γ(y) β + |v| β -θv, (2.11)
if and only if v(t, y)|x| p is a classical/strong/viscosity (sub-/super-)solution to the HJB equation (2.7). In this case the infimum in (2.7) is attained at

ξ * (t, y, x) = v(t, y) β η(y) β x and π * (t, y, x) = v(t, y) β γ(y) β + v(t, y) β x (2.12)
and

H(t, y, x, ξ * (t, y, x), π * (t, y, x), v(•, •)| • | p ) = F (y, v(t, y))|x| p . (2.13)
Proof. When testing the viscosity property of

V = v(•, •)| • | p at a point ( t, ȳ, x) ∈ [0, T ) × R d × R it is sufficient to consider test functions of the form ϕ = φ(•, •)| • | p . (2.14) In fact, let ϕ ≤ V (the case ϕ ≥ V is similar) be an arbitrary test function such that ϕ( t, ȳ, x) = V ( t, ȳ, x). If x = 0, we may define φ(t, y, x) = ϕ(t, y, x) |x| p |x| p . Then, φ ≤ V , φ( t, ȳ, x) = V ( t, ȳ, x), ∂ t φ( t, ȳ, x) = ∂ t ϕ( t, ȳ, x), L φ( t, ȳ, x) = Lϕ( t, ȳ, x). Since V is continuously differentiable in x and ( t, ȳ, x) is a extreme point of both V -φ and V -ϕ we furthermore have that ∂ x φ( t, ȳ, x) = ∂ x V ( t, ȳ, x) = ∂ x ϕ( t, ȳ, x).
Testing at a point with x = 0 is trivial because V (t, y, 0) ≡ 0 and the derivatives ∂ t V (t, y, 0) and LV (t, y, 0) exist and are identically equal to zero. As a result, we may w.l.o.g. restrict ourselves to test functions of the form (2.14). With this observation at hand the statement is then verified by straightforward calculations using that for

V = v(•, •)|•| p with v ≥ 0 the Hamiltonian H is convex in ξ and π.
To guarantee the uniqueness of a viscosity solution to (2.10) we need to impose a suitable terminal condition. Due to the liquidation constraint (2.4), we expect the value function to tend to infinity for any fixed non-trivial portfolio position as t → T . More precisely, when disregarding any adverse selection and risk costs, as well as any scenarios in which passive orders are executed, and using P(no jumps of N in [t, T ]) = e -θ(T -t) , one obtains for any admissible control (ξ, π) ∈ A(t, x),

J(t, y, x; ξ, π) ≥ e -θ(T -t) E T t η(Y t,y s )|ξ s | p ds no jumps of N in [t, T ] . (2.15)
Applying the reverse Hölder inequality to the inner integral of the RHS of (2.15) yields,

J(t, y, x; ξ, π) ≥ e -θ(T -t) E T t η(Y t,y s ) -1 p-1 ds -(p-1) T t |ξ s | ds p no jumps of N in [t, T ] .
Given that no jumps occur, the liquidation constraint (2.4) yields

T t ξ s = x.
Using the independence of Y t,y and N we hence obtain,

J(t, y, x; ξ, π) ≥ e -θ(T -t) E T t η(Y t,y s ) -1 p-1 ds -(p-1) T t ξ s ds p no jumps of N in [t, T ] = E   e -θ(T -t) β T t 1 η(Y t,y s ) β ds   |x| p .
In view of (2.6) and (2.2), we therefore expect that lim t→T v(t, y) = +∞ locally uniformly on R d .

It turns out that this singular terminal condition along with the standing assumptions on the diffusion and cost coefficients already ensures uniqueness of a viscosity solution to the HJB equation. The deterministic closing strategy that liquidates at a constant rate and uses no dark pool incurs the cost

J t, y, x; x T -t , 0 = 1 (T -t) p E T t η(Y t,y s ) + (T -s) p λ(Y t,y s ) ds |x| p ≤ C (T -t) p-1 (1 + |y| n )|x| p , (2.16)
due to estimate (2.2) and Assumption 2.2. As a result, we also expect v(t, •) to satisfy a polynomial growth condition. More precisely, we have the following result; its proof is given in Section 3.

Proposition 2.8. Under Assumptions 2.1 and 2.2 the singular terminal value problem

   -∂ t v(t, y) -Lv(t, y) -F (y, v(t, y)) = 0, (t, y) ∈ [0, T ) × R d , lim t→T v(t, y) = +∞ locally uniformly on R d , (2.17)
with the nonlinearity F given in (2.11) admits at most one nonnegative viscosity solution in

C poly ([0, T -] × R d ).
If such a viscosity solution exists, then it satisfies the following a priori estimates for (t, y)

∈ [0, T ) × R d : E   e -θ(T -t) β T t 1 η(Y t,y s ) β ds   ≤ v(t, y) ≤ 1 (T -t) p E T t η(Y t,y s ) + (T -s) p λ(Y t,y s ) ds . (2.18)
Our main contribution is the proof of the existence of a classical solution to the singular terminal value problem (2.17). This is achieved under assumptions (A1)-(A4). The boundedness away from zero of η, along with the existence and boundedness of Lη will for instance be used to derive the precise asymptotic behavior of the solution near the terminal time. The main result of this paper is:

Theorem 2.9. Under Assumptions 2.1 and 2.2 and the conditions (A1)-(A4), the singular terminal value problem (2.17) admits a nonnegative classical solution

v ∈ C α ([0, T -]; D(L)) ∩ C 1+α ([0, T -]; C(R d )).
The proof of this theorem will be carried out in Section 4 below.

Our last result is a verification theorem; its proof is given in Section 5. The singularity at the terminal time prevents a straightforward application of the standard verification arguments. Instead, we first use Krylov's generalized Itô formula [22, Theorem 2.10.1, p. 122] to establish optimality away from the terminal time and then use the a priori estimate (2.18) to prove optimality on the whole time interval. Here we require that the strong solution is locally L qintegrable along with its weak derivatives for some q > d + 2 to guarantee that the parabolic Sobolev embedding theorem [24, Lemma II.3.3] applies.

Proposition 2.10. Under Assumptions 2.1 and 2.2 and the conditions (A1) and (A3), for some

q > d + 2, let v ∈ W 1,2 q,loc ((0, T -) × R d ) ∩ C poly ([0, T -] × R d
) be a nonnegative strong solution to (2.17). Then, the value function of the control problem (2.5) is given by V (t, y, x) = v(t, y)|x| p , and the optimal control (ξ * , π * ) is given in feedback form by

ξ * s = v(s, Y t,y s ) β η(Y t,y s ) β X * s and π * s = v(s, Y t,y s ) β γ(Y t,y s ) β + v(s, Y t,y s ) β X * s-. (2.19)
In particular, the resulting optimal portfolio process (X * s ) s∈[t,T ] is given by

X * s = x exp - s t v(r, Y t,y r ) β η(Y t,y r ) β dr ∆Nr =0 t<r≤s 1 - v(t, Y t,y r ) β γ(Y t,y r ) β + v(t, Y t,y r ) β . (2.20) Since C α ([0, T -]; D(L)) ∩ C 1+α ([0, T -]; C(R d )) ⊂ W 1,2 q,loc ((0, T -) × R d ) ∩ C poly ([0, T -] × R d )
we immediately obtain the following corollary to Theorem 2.9 and Proposition 2.10.

Corollary 2.11. Assume that the assumptions of Theorem 2.9 hold and let v be the unique classical solution to the singular terminal value problem (2.17). Then the optimal liquidation strategy is given in feedback form by (2.19).

Comparison principle and a priori estimates

In this section we prove Proposition 2.8. The proof is based on the following comparison principle that is itself a consequence of the comparison principle given in the Appendix for viscosity sub-and supersolutions to parabolic equations with finite terminal values and monotone nonlinearities.

Lemma 3.1. Let v, v ∈ C poly ([0, T -] × R d )
be a nonnegative viscosity sub-and a nonnegative viscosity supersolution to (2.17), respectively, such that

lim t→T v(t, y) = +∞ locally uniformly on R d . Then, v ≤ v in [0, T ) × R d .
In particular, there exists at most one nonnegative viscosity solution in

C poly ([0, T -] × R d ) to problem (2.17).
Proof. Due to the time-homogeneity of the PDE in (2.17), viscosity (super-/sub-)solutions stay viscosity (super-/sub-)solutions when shifted in time. The idea is therefore to separate the singularities to have finite values to compare.

More precisely, we define, for any δ > 0, the difference function w By the first order Taylor approximation of F in v at v(t+δ, y), along with

: [0, T -δ) × R d → R by w(t, y) = v(t + δ, y) -v(t, y). A direct computation shows that v → F (•, v) is decreasing on [0, ∞). In fact, both ∂ v F and ∂ 2 v F are nonpositive on R d × [0, ∞). Hence, by Lemma A.2, w is on [0, T -δ) × R d a
∂ v F ≤ 0 and ∂ 2 v F ≤ 0, we obtain that -l(t, y)w(t, y) ≤ -∂ v F (y, v(t + δ, y))w(t, y).
In terms of the continuous coefficient

l(t, y) := ∂ v F (y, v(t + δ, y)) ≤ 0, it follows that w is on [0, T -δ) × R d also a viscosity supersolution to the equation -∂ t w(t, y) -Lw(t, y) -l(t, y)w(t, y) = 0, (3.1)
for which the assumptions of Theorem A.1 hold with µ = 0. Hence, for all 0 < t ≤ s < T -δ and y ∈ R d Theorem A.1 yields

w(t, y) ≥ E w(s, Y t,y s ) exp s t l(r, Y t,y r ) dr , (3.2)
where the RHS is the Feynman-Kac viscosity solution [26, Theorem 3.2] to (3.1) on [0, s] × R d with terminal value w(s, •).

Since v ≥ 0, l ≤ 0, and

E[sup s∈[t,T -δ] v(s, Y t,y s )] < ∞ due to v ∈ C poly ([0, T -δ] × R d ),
v(t + δ, y) -v(t, y) ≥ 0.
Finally, by letting δ → 0 we conclude v -v ≥ 0 on [0, T ) × R d by continuity of v.

The comparison principle establishes the uniqueness statement in Proposition 2.8. It also allows us to establish the a priori estimates (2.18).

Proof of Proposition 2.8. We show that the lower (upper) estimate in (2.18)-denoted in the following by v (respectively, v)-is a viscosity subsolution (supersolution) to (2.17). The assertion then follows from the comparison principle established in the preceding lemma.

First, note that v ∈ C poly ([0, T -] × R d ).
In fact, by Jensen's inequality

0 ≤ v(t, y) = e -θ(T -t) (T -t) 1/β E 1 T -t T t 1 η(Y t,y s ) β ds -1/β ≤ e -θ(T -t) (T -t) (β+1)/β E T t η(Y t,y s ) ds ,
and hence the polynomial growth of v in y uniformly away from the terminal time follows from (2.2) and the polynomial growth of η. The continuity along with the polynomial growth of 1/η guarantees continuity of v, due to Vitali's convergence theorem as pointed out in (2.3).

To establish the subsolution property of v, let ϕ ≥ v be a smooth test function on [0, T ) × R d such that ϕ(t, y) = v(t, y) for some (t, y) ∈ [0, T ) × R d . Moreover, we define the stopping time

τ = inf{s ∈ [t, T ] : |Y t,y s -y| ≥ 1}. By uniqueness Y t,y r = Y s,Y t,y s r
for all t ≤ s ≤ r ≤ T . Hence, by the definition of v and the Markov property of Y t,y ,

v(s ∧ τ, Y t,y s∧τ ) = E   e -θ(T -(s∧τ )) β T s∧τ η(Y t,y r ) -β dr F s∧τ   .
Because ϕ ≥ v, ϕ(t, y) = v(t, y), and the tower rule, it holds for t < s < T ,

0 ≤ E[ϕ(s ∧ τ, Y t,y s∧τ ) -v(s ∧ τ, Y t,y s∧τ )] = E[ϕ(s ∧ τ, Y t,y s∧τ ) -ϕ(t, y)] -E   e -θ(T -(s∧τ )) β T s∧τ η(Y t,y r ) -β dr   + E   e -θ(T -t) β T t η(Y t,y r ) -β dr   .
Dividing by s -t, using Itô's formula and the integral representation of increments of the function r → e -θ(T -r)

β T r η(Y t,y u ) -β du
, and noticing that the stochastic integral being stopped at τ is a true martingale, we obtain

0 ≤ E 1 s -t s∧τ t ∂ t ϕ(r, Y t,y r ) + Lϕ(r, Y t,y r ) dr -E 1 s -t s∧τ t 1 βη(Y t,y r ) β e -θ(T -r) T r η(Y t,y u ) -β du (β+1)/β dr -E 1 s -t s∧τ t θe -θ(T -t) β T r η(Y t,y u ) -β du dr .
By Jensen's inequality, for r ∈ [t, s],

β 1 T r η(Y t,y u ) -β du ≤ β 1 (T -r) 2 T r η(Y t,y u ) β du ≤ 1 β √ T -s sup t≤u≤T η(Y t,y u ).
Since η is of polynomial growth it follows from the equation (2.2) that we can apply the dominated convergence theorem to the third term above when letting s → t. Since |Y t,x r -y| ≤ 1 for r ∈ [t, τ ] and because η -1 is bounded on compact domains similar arguments show that we can apply the dominated convergence theorem also to the second term. As τ > t, the fundamental theorem of calculus yields 0 ≤ ∂ t ϕ(t, y) + Lϕ(t, y) -

1 βη(y) β E      e -θ(T -t)/(β+1) β T t η(Y t,y u ) -β du   β+1    -θv(t, y).
Using Jensen's inequality and β + 1 > 1 we obtain, 0 ≤ ∂ t ϕ(t, y) + Lϕ(t, y) -v(t, y) β+1 βη(y) β -θv(t, y).

Hence, from the definition (2.11) of F and λ, γ, v ≥ 0 it is seen,

-∂ t ϕ(t, y) -Lϕ(t, y) -F (y, v(t, y)) ≤ 0.
Next, we verify the supersolution property of v. Because η and λ are of polynomial growth it

follows v ∈ C poly ([0, T -] × R d ) from (2.
3). Again an application of Itô's formula and Leibniz's rule similar as above yields that for every smooth test function

ϕ ≤ v on [0, T ) × R d such that ϕ(t, y) = v(t, y) for some (t, y) ∈ [0, T ) × R d , 0 ≥ ∂ t ϕ(t, y) + Lϕ(t, y) + η(y) (T -t) p + λ(y) - pv(t, y) T -t .
From the definition (2.11) of F , since v ≥ 0,

-F (y, v(t, y)) ≥ -λ(y) + |v(t, y)| β+1 βη(y) β .
Hence, since p = (β + 1)/β and by setting u(t, y) = (T -t) 1/β v(t, y)/η(y),

-∂ t ϕ(t, y) -Lϕ(t, y) -F (y, v(t, y)) ≥ η(y) (T -t) p - pv(t, y) T -t + |v(t, y)| β+1 βη(y) β = η(y) (T -t) β+1 β 1 -(β+1) β u(t, y) -1 β |u(t, y)| β+1 . (3.4)
Using the fact that the map u → 1 -β+1 β u + 1 β |u| β+1 is nonnegative on R as it attains its minimum at 1, we conclude

-∂ t ϕ(t, y) -Lϕ(t, y) -F (t, v(t, y)) ≥ 0.
We close this section with a further application of the comparison principle. Under the conditions (A3) and (A4) it establishes the precise asymptotic behavior of a viscosity solution at the terminal time. This observation will be the starting point of the next section.

Corollary 3.2. Let v ∈ C poly ([0, T -]×R d
) be a nonnegative viscosity solution to problem (2.17). If the assumptions (A3) and (A4) hold, then v satisfies the following asymptotic behavior:

(T -t) 1/β v(t, y) = η(y) + O(T -t) uniformly in y as t → T . (3.5)
Proof. The statement is proved by identifying a sub-and a supersolution with the desired asymptotics. Due to (A4), the quantity Lη is well-defined and finite, hence δ := κ 0 / Lη > 0.

We verify below that v(t, y)

:= η(y) -Lη (T -t) e θ(T -t) (T -t) 1/β and v(t, y) := η(y) + 1 2 Lη (T -t) (T -t) 1/β + (T -t) λ
are a nonnegative classical sub-and supersolution to (2.17

-∂ t v(t, y) -Lv(t, y) = - η(y) + 1-β 2 Lη (T -t) + βLη(T -t) β(T -t) (β+1)/β + λ . (3.6)
Recalling the definition (2.11) of F , we have since v ≥ 0,

-F (y, v(t, y)) ≥ -λ(y) + |v(t, y)| β+1 βη(y) β .
Next, we apply Bernoulli's inequality in the form (u + v + w) β+1 ≥ u β+1 (1 + v/u) β+1 ≥ u β+1 + (β + 1)u β v for u, v, w ≥ 0 to the term |v(t, y)| β+1 and obtain

-F (y, v(t, y)) ≥ -λ(y) + η(y) β+1 + (β + 1)η(y) β 1 2 Lη (T -t) βη(y) β (T -t) (β+1)/β . (3.7)
Hence, adding (3.6) and (3.7) yields,

-∂ t v(t, y) -Lv(t, y) -F (y, v(t, y)) ≥ λ -λ(y) + Lη -Lη(y) (T -t) 1/β ≥ 0.
Next, we verify the subsolution property of v. By a direct computation

-∂ t v(t, y) -Lv(t, y) = - η(y) + (β -1) Lη (T -t) + βLη(y)(T -t) βe θ(T -t) (T -t) (β+1
u -v) β+1 = u β+1 (1 -v/u) β+1 ≤ u β+1 -u β v for u ≥ v ≥ 0 and obtain -F (y, v(t, y)) ≤ η(y) -Lη (T -t) βe θ(β+1)(T -t) (T -t) (β+1)/β + θv(t, y).
(3.9)

Finally, adding (3.8) and (3.9), and using β > 0 yields,

-∂ t v(t, y) -Lv(t, y) -F (t, v(t, y)) ≤ 0.

Existence of a classical solution

In this section we prove Theorem 2.9 and hence assume throughout that (A1)-(A4) hold. Our existence proof is based on the explicit asymptotic behavior established in Corollary 3.2. It tells us the solution must be of the form

v(T -t, y) = η(y) + ũ(t, y) t 1/β , ũ(t, y) = O(t) uniformly in y as t → 0, (4.1)
where we reversed the time variable as we will do for the rest of this subsection. For reasons that will become clear later, it will be more convenient to choose the following equivalent ansatz:

v(T -t, y) = η(y) t 1/β + u(t, y) t 1+1/β , u(t, y) = O(t 2 ) uniformly in y as t → 0. (4.2)
Plugging the asymptotic ansatz into (2.17) results in a semilinear parabolic equation for u with finite initial condition, but with a singularity in the nonlinearity. This motivates the following lemma.

Lemma 4.1. If for some δ > 0 a function u ∈ C α ([0, δ]; D(L)) ∩ C 1+α ([0, δ]; C(R d )) satisfies |u(t, y)| ≤ tη(y), t ∈ [0, δ], y ∈ R d , (4.3)
and solves the equation

∂ t u(t, y) = Lu(t, y) + tLη(y) + t p λ(y) - η(y) β ∞ k=2 β + 1 k u(t, y) tη(y) k + θt p γ(y)(tη(y) + u(t, y)) β (t p γ(y)) β + |tη(y) + u(t, y)| β -θ(tη(y) + u(t, y)), t > 0 , y ∈ R d , (4.4
)

then a local solution v ∈ C α ([T -δ, T -]; D(L)) ∩ C 1+α ([T -δ, T -]; C(R d )) to problem (2.17) is given by v(t, y) = η(y) (T -t) 1/β + u(T -t, y) (T -t) 1+1/β .
Proof. The statement is verified by plugging the ansatz into (2.17), multiplying by t p = t (β+1)/β , and by using the binomial series for the term

t p |v(T -t, y)| β+1 βη(y) β = η(y) β 1 + u(t, y) tη(y) β+1 = η(y) β ∞ k=0 β + 1 k u(t,

y) tη(y)

k to see that the first two terms of the series cancel out. The growth condition (4.3) guarantees that the binomial series does indeed converge.

Remark 4.2. The reason for choosing the ansatz (4.2) is that the series in (4.4) starts at k = 2, and not at k = 1. This will be crucial for the fixed point argument below. The more straightforward ansatz (4.1) results in the equation

∂ t ũ = Lũ + Lη + t 1 β λ - η βt ũ η + ∞ k=2 β + 1 k ũ η k + θt 1 β γ(η + ũ) β tγ β + |η + ũ| β -θ(η + ũ),
for which we have no analogues to Lemma 4.4 and Lemma 4.5 below, due to the term t -1 ũ.

We will solve equation (4.4) using the semigroup approach for parabolic equations in Banach spaces; we refer to the monograph by Lunardi [START_REF] Lunardi | Analytic Semigroups and Optimal Regularity in Parabolic Problems[END_REF] as the standard reference. To this end, we interpret (4.4) as an evolution equation

u (t) = Lu(t) + f (t, u(t)), t > 0; u(0) = 0, (4.5)
in the Banach algebra U := C(R d ) of bounded continuous functions endowed with the supremum norm • , where the nonlinearity f is given by

f (t, u) = tLη + t p λ - η β ∞ k=2 β + 1 k u tη k + θt p γ(tη + u) β (t p γ) β + |tη + u| β -θ(tη + u).
The general theory suggests to look first for a local mild solution of (4.5). That is, to show there is a fixed point u of the integral operator Γ defined in C([0, δ]; U ) by

Γ(u)(t) = t 0 e (t-s)L f (s, u(s)) ds, 0 ≤ t ≤ δ, (4.6)
if δ > 0 is small enough where {e tL : t ≥ 0} is the analytic semigroup generated by L in U . Regularity of the mild solution u will then follow from analyticity of the semigroup and Hölder continuity of t → f (t, u(t)).

The singular behavior of f near t = 0 prevents us from directly applying general theory. In fact, the operator Γ is not defined on the whole space C([0, δ]; U ), and its domain is not closed with respect to the supremum norm. We overcome these difficulties by carrying out the usual contraction argument with respect to an appropriate weighted norm on C([0, δ]; U ).

In order to guarantee that the function t → f (t, u(t)) behaves well at t = 0 it seems reasonable to restrict the set of potential mild solutions to those functions u ∈ C([0, δ]; U ) such that u(t) = o(t) as t → 0. Yet, there is no nice norm making this set of functions a Banach space. Recalling (3.5) however, we actually expect the slightly stronger condition u(t) = O(t 2 ) as t → 0 to be satisfied. This suggests to view Γ as an operator acting in the space

E = u ∈ C([0, δ]; U ) : u(t) = O(t 2 ) as t → 0 ,
endowed with the weighted norm

u E = sup 0<t≤δ t -2 u(t) .
Lemma 4.3. The vector space E endowed with the norm • E is a Banach space.

The next lemma shows in particular that the integral operator Γ given in (4.6) is well-defined on the closed ball (i) For every u ∈ B E (R), the function f ( • , u( • )) belongs to C([0, δ]; U ). In particular, the operator Γ defined in (4.6) is well defined on B E (R).

B E (κ 0 /δ) := u ∈ E : u E ≤ κ 0 /δ .
(ii) If u ∈ B E (R) ∩ C α ([0, δ]; U ) for some α ∈ (0, 1), then f ( • , u( • )) is α-Hölder continuous, i.e., belongs to C α ([0, δ]; U ).
Proof. For u ∈ B E (R) we consider the functions g : [0, δ] → U and h : [0, δ] × U → U given by

g(t) = ∞ k=2 β + 1 k u(t) tη k and h(t, w) = t p γw β (t p γ) β + |w| β -w,
so that we may decompose f (t, u(t)) in the following way:

f (t, u(t)) = tLη + t p λ -(p -1)ηg(t) + θh(t, tη + u(t)). (4.7)
The assumption δ ≤ κ 0 /R guarantees that the series defining g(t) converges in U since then

u(t) tη ≤ t 2 R tκ 0 ≤ δR κ 0 ≤ 1, t ∈ [0, δ].
In view of (4.7) it will be sufficient to show that g and h( • , • η + u( • )) are continuous, or even α-Hölder continuous if u ∈ C α ([0, δ]; U ). For the latter note that h is continuously differentiable on (0, δ] × U . In fact,

∂ t h(t, w) L(U ) = pt p-1 γw|w| β β (t p γ) β + |w| β β+1 ≤ pt p-1 γw|w| β (t p γ) β+1 + pt p γ|w| β ≤ w t ,
where we used Bernoulli's inequality and β + 1 = pβ, and

∂ w h(t, w) L(U ) = (t p γ) β+1 β (t p γ) β + |w| β β+1 -1 ≤ 1.
Hence, for all 0

≤ t ≤ s ≤ δ, h(t, tη + u(t)) -h(s, sη + u(s)) ≤ tη + u(t) t |t -s| + η |t -s| + u(t) -u(s) ≤ (2 η + κ 0 )|t -s| + u(t) -u(s) .
In order to establish the continuity of g, notice that for every k ≥ 2 and 0

≤ t ≤ s ≤ δ it holds that u(t) tη k - u(s) sη k ≤ u(t) tη k - u(t) sη k + u(t) sη k - u(s) sη k ≤ u(t) k κ k 0 1 t k - 1 s k + 1 s k κ k 0 u(t) -u(s) k-1 l=0 u(t) l u(s) k-1-l ≤ t 2k R k t k s k κ k 0 t k -s k + R k-1 s k κ k 0 u(t) -u(s) k-1 l=0 t 2l s 2(k-1-l) ≤ kδ k-1 R k κ k 0 |t -s| + kδ k-2 R k-1 κ k 0 u(t) -u(s) ≤ kR κ 0 |t -s| + kR κ 2 0 u(t) -u(s) . (4.8)
Using the identity

k β+1 k = (β + 1) β k-1 it follows that g(t) -g(s) ≤ (2 β -1)(β + 1)R κ 0 |t -s| + (2 β -1)(β + 1)R κ 2 0 u(t) -u(s) .
Hence, g is uniformly continuous and even α-Hölder

continuous if u ∈ C α ([0, δ]; U ).
The usual assumption on the nonlinearity to carry out the fixed point argument would be that f (t, u) is locally Lipschitz continuous in u uniformly in t. The next lemma proves an appropriate analogue to this assumption for our singular nonlinearity f . Lemma 4.5. For every R > 0 there exists a constant L > 0 independent of δ ∈ (0,

κ 0 /R] such that f (t, u(t)) -f (t, v(t)) ≤ L u(t) -v(t) , u, v ∈ B E (R), t ∈ [0, δ]. Proof. Let u, v ∈ B E (R)
u(t) tη k - v(t) tη k ≤ kR κ 2 0 u(t) -v(t)
for every k ≥ 2. Hence, using once more that k β+1 k = (β + 1) β k-1 and (β + 1)/β = p we conclude that

f (t, u(t)) -f (t, v(t)) ≤ (p(2 β -1)κ -2 0 R η + θ) u(t) -v(t) .
We are now ready to carry out the fixed point argument and to prove the desired regularity of the fixed point. In view of Lemma 4.1 this then gives us a local solution to the problem (2.17).

Proposition 4.6. Under assumptions (A1) and (A2), there exists a short-time solution

u ∈ C α ([0, δ]; D(L)) ∩ C 1+α ([0, δ]; C(R d ))
to the equation (4.4) that satisfies the growth condition (4.3).

Proof. We prove below that there exists R > 0 and δ ∈ (0, κ 0 /R] such that the operator Γ defined by (4.6) has a fixed point u in B E (R).

In order to see that this (local) mild solution to (4.5) belongs to C α ([0, δ]; D(L))∩C 1+α ([0, δ]; U ) notice first that f ( • , u( • )) ∈ C([0, δ]; U ), due to Lemma 4.4(i). Thus, it follows from [25, Proposition 4.2.1] that Γ maps into C α ([0, δ]; U ), for every α ∈ (0, 1). Since u is a fixed point of Γ it follows from Lemma 4.4(ii) that f ( It remains to prove the existence of a fixed point of the operator Γ. In terms of M = sup 0≤t≤1 e tL L(U ) we claim that one can choose

• , u( • )) ∈ C α ([0, δ]; U ). Moreover, u ∈ E implies that u(0) = Lu(0) + f (0, u (0)
R = 2M ( Lη + λ + θ η ) and δ = min{κ 0 /R, (2M L) -1 , 1},
where L > 0 is the Lipschitz constant given by Lemma 4.5. Since δ ≤ κ 0 /R the operator Γ is well-defined on B E (R), due to Lemma 4.4. To show that Γ is a contraction with respect to

• E , let u, v ∈ B E (R). By the choice of M it holds for every t ∈ [0, δ] that Γ(u)(t) -Γ(v)(t) ≤ tM sup s∈[0,t] f (s, u(s)) -f (s, v(s)) ≤ δM L sup s∈[0,t] u(s) -v(s) ≤ δM Lt 2 u -v E .
Hence,

Γ(u) -Γ(v) E ≤ 1 2 u -v E .
To show that Γ maps B E (R) into itself, note that since δ ≤ 1 and p > 1 one has that s p ≤ s for all s ∈ [0, δ], and so it holds for every t ∈ [0, δ] that

Γ(u)(t) ≤ Γ(u)(t) -Γ(0)(t) + Γ(0)(t) ≤ t 2 R 2 + tM sup s∈[0,t] sLη + s p λ + θs p γsη β (s p γ) β + (sη) β -θsη ≤ t 2 R 2 + tM sup s∈[0,t] {s Lη + s p λ + θs η } ≤ t 2 R.
The operator Γ does therefore map B E (R) contractive into itself. Hence, it has a unique fixed point u in B E (R).

Remark 4.7. In order to apply the above fixed point argument establishing a mild solution to (4.5) it is not needed that L generates an analytic semigroup. We use the analyticity of L to obtain maximal regularity results. We thank an anonymous referee for pointing out that one may possibly drop the additional assumptions (A1) and (A2) on the diffusion coefficients and interpret L as the generator L of the π-continuous Markov transition semigroup induced by Y on C(R d ); see [START_REF] Fabbri | Stochastic Optimal Control in Infinite Dimension[END_REF]Appendix B.5] for a comprehensive overview of the theory of πcontinuous semigroups. It is shown in [START_REF] Priola | The Cauchy problem for a class of Markov-type semigroups[END_REF] (see also [START_REF] Cerrai | Strong solutions of cauchy problems associated to weakly continuous semigroups[END_REF]) that the obtained mild solution is a π-strong approximation of strict solutions u n . However, for the verification argument a πstrong approximation satisfying u n (t, •) ∈ C 2 (R d ) is needed to apply Itô's formula on u n and then use the dominated convergence. This may possibly be achieved as in [START_REF] Cerrai | Strong solutions of cauchy problems associated to weakly continuous semigroups[END_REF] by establishing that {u ∈ C 2 (R d ) : Lu ∈ C(R d )} is a π-core for L. The later is proven in [START_REF] Da Prato | Cores for generators of some markov semigroups[END_REF] under the assumption that b and σ have a continuous and bounded second derivative.

We are now ready to prove Theorem 2.9.

Proof of Theorem 2.9. In view of Lemma 4.1 and Proposition 4.6 there exists a (unique) local classical, and hence mild solution

u ∈ C α ([T -δ, T -]; D(L)) ∩ C 1+α ([T -δ, T -]; C(R d ))
to (2.17). In order to see that the local solution extends to a global solution

v ∈ C α ([0, T -]; D(L)) ∩ C 1+α ([0, T -]; C(R d )) notice first that the functional v → F (•, v(•)) mapping C(R d ) into itself

Verification argument

This section is devoted to the verification argument. Throughout, for some q > d + 2, let

v ∈ W 1,2 q,loc ((0, T -) × R d ) ∩ C poly ([0, T -] × R d )
denote a nonnegative strong solution to (2.17). We recall that by the parabolic Sobolev embedding theorems [24, Lemma II.3.3], for every R > 0 the parabolic Sobolev space W 1,2 q ((0,

T ) × B d (R)) is continuous embedded into C l/2,1+l ([0, T ]×B d (R)) with l = 1-(d+2)/q.
Here, B d (R) denotes the d-dimensional ball of radius R centered at the origin and C l/2,1+l ([0, T ]×B d (R)) denotes the usual parabolic Hölder space of functions u(t, y) on [0, T ] × B d (R) that are l/2-Hölder continuous in t and l-Hölder continuous in y along with their first derivative in y. Hence, we assume from now on that Dv is continuous.

The verification argument is established as follows. We first prove that the candidate optimal strategy (ξ * , π * ) is admissible, and that the resulting portfolio process is monotone. This uses the lower estimate in (2.18). Admissibility does not a priori guarantee that the strategy (ξ * , π * ) generates finite costs, though. This requires an extra argument.

Subsequently, we show that we may w.l.o.g. restrict ourselves to admissible controls that result in a monotone portfolio process. Similar to [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF][START_REF] Kratz | An explicit solution of a nonlinear-quadratic constrained stochastic control problem with jumps: Optimal liquidation in dark pools with adverse selection[END_REF], we then prove the optimality of ξ * and π * in every interval [t, s] with s < T . The upper estimate in (2.18) will be used to show that candidate strategy is optimal on the whole time interval.

Lemma 5.1. The pair of feedback controls (ξ * , π * ) given by (2.19) is admissible. The portfolio process (X * s ) s∈[t,T ] with respect to (ξ * , π * ) is monotone.

Proof. One readily verifies that the portfolio process (X * s ) s∈[t,T ] with respect to the controls ξ * and π * is given by (2.20) and thus is monotone. To show that X * T = 0 we define the random variable ν(ω) := sup t≤r≤T e βθ(T -r) η(Y t,y r ) β E sup r≤u≤T η(Y t,y u ) -β F r that is a.s. finite due to Assumption 2.2 and the moment estimates (2.2). For 0 ≤ t ≤ s < T , using the lower estimate in (2.18) and Jensen's inequality we obtain,

|X * s | ≤ |x| exp - s t v(r, Y t,y r ) β η(Y t,y r ) β dr ≤ |x| exp   - s t 1 η(Y t,y r ) β E   e -θ(T -r) β T r η(Y t,y u ) -β du F r   β dr    ≤ |x| exp   - s t e -βθ(T -r) η(Y t,y r ) -β E T r η(Y t,y u ) -β du F r dr   ≤ |x| exp - 1 ν s t 1 T -r dr = |x| T -s T -t 1/ν s→T -→ 0.
This yields X * T -= 0. Hence, π * T = 0 and so X * T = 0. Lemma 5.2. For every (ξ, π) ∈ A(t, x) there exists ( ξ, π) ∈ A(t, x) with lesser or equal costs such that (X ξ,π s ) s∈[t,T ] is monotone.

Proof. For (ξ, π) ∈ A(t, x) with x ≥ 0 consider the strategy ( ξ, π) given by ξs = ξ s 1 {ξs≥0} 1 {X ξ,π s >0}

and πs = (π s ∧ X ξ,π s-)1 {πs≥0} 1 {X ξ,π s->0} .

By construction it holds that 0

≤ ξs ≤ |ξ s |, 0 ≤ πs ≤ |π s | and 0 ≤ X ξ,π s ≤ |X ξ,π s | for all s ∈ [t, T ]. As a result, (X ξ,π s ) s∈[t,T
] is monotone decreasing and X ξ,π T = 0 by admissibility of (ξ, π). Hence, ( ξ, π) ∈ A(t, x) with less or equal costs than (ξ, π). The case x ≤ 0 is similar.

We denote by Ā(t, x) the set of all admissible controls under which the portfolio process is monotone. For any (ξ, π) ∈ Ā(t, x) with finite costs the expected residual costs vanish as s → T as shown by the following lemma. Next, we apply again Jensen's inequality to obtain,

|X ξ,π s | p ≤ C(T -t) p-1 E T s |ξ r | p dr F s .
Therefore, by the upper estimate in (2.18) and the boundedness of η and λ due to (A2),

E v(s, Y t,y s )|X ξ,π s | p ≤ CE   E T s η(Y t,y r ) + (T -r) p λ(Y t,y r ) dr F s T -s E T s |ξ r | p dr F s   ≤ CE T s |ξ r | p dr .
Letting s → T , we conclude (5.1) by the monotone convergence theorem, where it is used that ξ ∈ L p F (0, T ; R) for any strategy (ξ, π) that has finite costs as η is bounded away from zero under assumption (A2).

The following estimate is key to the verification argument. Together with the preceding lemma it allows us to show that v(

v(s ∧ τ R , Y t,y s∧τ R ). It yields that v(t, y) = v(s ∧ τ R , Y t,y s∧τ R ) + s∧τ R t ∂ t v(r, Y t,y r ) + Lv(r, Y t,y r ) dr - s∧τ R t σ(Y t,y r )Dv(r, Y t,y r ) dW r .
This allows us to apply to v(s ∧ τ R , Y t,y s∧τ R )|X ξ,π s∧τ R | p the classical integration by parts formula for semimartingales [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]Theorem 4.57] in order to obtain

v(t, y)|x| p = v(s ∧ τ R , Y t,y s∧τ R )|X ξ,π s∧τ R | p - s∧τ R t ∂ t v(r, Y t,y r )|X ξ,π r | p + Lv(r, Y t,y r )|X ξ,π r | p -pξ r v(r, Y t,y r ) sgn(X ξ,π r )|X ξ,π r | p-1 + θv(r, Y t,y r )(|X t,x r -π r | p -|X t,x r | p ) dr - s∧τ R t σ(Y t,y r )Dv(r, Y t,y r )|X ξ,π r | p dW r - s∧τ R t v(r, Y t,y r )(|X ξ,π r--π r | p -|X ξ,π r-| p ) d N r ,
where N r = N r -θr denotes the compensated Poisson process. Both v and Dv are continuous and hence bounded on [0, s] × B(y, R). Furthermore, |X ξ,π | ≤ |x| and |π| ≤ |x|, due to the monotonicity of the portfolio process. As a consequence, the above stochastic integrals are true martingales. Hence, recalling (2.13),

v(t, y)|x| p = E v(s ∧ τ R , Y t,y s∧τ R )|X ξ,π s∧τ R | p + E s∧τ R t c(Y t,y r , X ξ,π r , ξ r , π r ) dr -E s∧τ R t (∂ t + L)v(r, Y t,y r )|X ξ,π r | p + H(r, Y t,x r , X ξ,π r , ξ r , π r , v(•, •)| • |) dr ≤ E v(s ∧ τ R , Y t,y s∧τ R )|X ξ,π s∧τ R | p + E s∧τ R t c(Y t,y r , X ξ,π r , ξ r , π r ) dr (5.5) -E s∧τ R t (∂ t + L)v(r, Y t,y r ) + F (Y t,y r , v(r, Y t,y r )) |X ξ,π
E s∧τ R t (∂ t + L)v(r, Y t,y r ) + F (Y t,y r , v(r, Y t,y r )) |X ξ,π r | p dr = 0. Hence, v(t, y)|x| p ≤ E v(s ∧ τ R , Y t,y s∧τ R )|X ξ,π s∧τ R | p + E s∧τ R t c(Y t,y r , X ξ,π r , ξ r , π r ) dr .
Letting R → ∞ the assertion follows from the polynomial growth condition on v and positivity of the cost function c.

We are now ready to carry out the verification argument. In particular (ξ * , π * ) has finite costs. Hence, Lemma 5.3 applies to (ξ * , π * ). Thus,

v(t, y)|x| p = E[v(s, Y t,y s )|X ξ * ,π * s | p ] + E s t c(Y t,y r , X ξ * ,π * r , ξ * r , π * r ) dr -→ J(t, y, x; ξ * , π * ) as s → T.
This shows that the strategy (ξ * , π * ) is indeed optimal.

6 Uniqueness in the non-Markovian framework

Within our Markovian framework we obtained optimal controls in feedback form. Of course, one may as well interpret the cost coefficients as processes η t , γ t and λ t adapted to the filtration generated by the Brownian motion. This has been recently suggested by Ankirchner, Jeanblanc & Kruse [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF], which allowed them to analyze non-Markovian coefficients, while losing the feedback form of the optimal controls.

Disregarding in this section any passive orders and assuming the filtration to be solely generated by the Brownian motion the value function to the control problem consider in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF] is given by

V t (x) := ess inf ξ∈A(t,x) E T t η s |ξ s | p + λ s |X ξ s | p ds F t , (t, x) ∈ [0, T ) × R,
where ξ ∈ L 0 F (t, T ; R) belongs to the set of admissible controls A(t, x) if the state process

X ξ s = x - s t ξ r dr, t ≤ s ≤ T,
satisfies the liquidation constraint X ξ T = 0. In the non-Markovian framework the value function has been related by Peng [START_REF] Peng | Stochastic Hamiltonian-Jacobi-Bellman equations[END_REF] (see also [START_REF] Graewe | A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions[END_REF][START_REF] Horst | A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition[END_REF]) to the BSPDE:

-dV t (x) = inf ξ∈R {-ξ∇V t (x) + η t |ξ| p + λ t |x| p } dt -Ψ t (x) dW t .
A solution to the BSPDE is a pair (V t , Ψ t ) of adapted processes. The ansatz V t (x) = Y t |x| p and Ψ t (x) = Z t |x| p results in the BSDE:

-dY t = λ t - |Y t | β+1 βη β t dt -Z t dW t , 0 ≤ t < T ; lim t→T Y t = +∞. (6.1) Assuming η ∈ L 2 F (0, T ; R + ), η -β ∈ L 1 F (0, T ; R + ), λ ∈ L 2 F (0, T -; R + ), and E[ T 0 (T -t) p λ t dt] < ∞ existence of a minimal nonnegative solution (Y, Z) ∈ L 2 F (Ω; C([0, T -]; R + )) × L 2 F (0, T -; R n )
to (6.1) has been established in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF]. We now show how our uniqueness argument can be applied to establish uniqueness in the BSDE setting when the coefficients are continuous. While the general shifting argument in the proof of our comparison principle fails in a non-time-homogeneous setting, it can be applied to establish the following a priori estimates.

Proposition 6.1. For any nonnegative solution (Y, Z) to (6.1) the following estimates hold:

1 β E T t 1 η β s ds F t ≤ Y t ≤ 1 (T -t) p E T t η s + (T -s) p λ s ds F t =: Y t , 0 ≤ t < T. (6.2)
Proof. The lower estimate has already been established in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF] for the minimal nonnegative solution. To establish the upper estimate first note that (Y t ) t∈[0,T ) is a supersolution to (6. 1). Yet, one can not directly compare Y and Y at the terminal time. As a workaround, similarly as in [START_REF] Popier | Backward stochastic differential equations with singular terminal condition[END_REF], we modify Pardoux's proof of his comparison principle for BSDEs with monotone drivers [START_REF] Pardoux | weak convergence and homogenization of semilinear PDEs[END_REF]Theorem 2.4] by shifting the singularity of Y . That is, for δ > 0 we define (Y t δ ) t∈[0,T -δ) by

Y t δ = 1 (T -δ -t) p E T -δ t η s + (T -δ -s) p λ s ds F t .
These processes are again supersolutions to (6.1) but with the singularity at t = T -δ. Precisely, it holds that Hence, letting s → T , we conclude (6.3) by the monotone convergence theorem, where it is used that ξ ∈ L p F (0, T ; R) for any control ξ that has finite costs as η is bounded away from zero under assumption (A5). Proof. We may again restrict the argument without loss of generality [4, Letting s → T , we conclude Y t |x| p ≤ V t (x) by Corollary 6.2. Thus, Y t ≤ V t (1). But V t (1) is characterized in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF] as the minimal nonnegative solution to (6.1). Hence, Y t = V t (1) is unique.

-

Conclusion

In this paper we proposed a novel approach to establishing smooth solutions to stochastic optimal control problems with singular terminal state constraints in a Markovian framework. Under standard assumptions on the diffusion and cost coefficients we proved that there exists at most one continuous viscosity and hence strong/classical solution to the HJB equation. As a byproduct we obtained a uniqueness theorem in a non-Markovian framework that complements results in [START_REF] Ankirchner | BSDEs with singular terminal condition and control problems with constraints[END_REF]. Our main contribution is the existence of a classical solution under boundedness and differentiability assumptions on the coefficiants. Existence of a viscosity solution is still open. In its present form our comparison principle only applies to continuous sub-and supersolutions, and hence does not allow us to apply Perron's method to establish the existence of a viscosity solution. Our verification argument uses Krylov's generalized Itô formula. As such it applies to strong and classical solutions. It is not hard, though, to extend the verification argument to viscosity solutions.

  ) on [T -δ, T ) × R d , respectively, where nonnegativity follows from η ≥ κ 0 > 0 by (A3). Hence, (3.5) follows from the comparison principle. Specifically, let us fix (t, y) ∈ [T -δ, T ) × R d . To verify the supersolution property of v, we first obtain by a direct computation,

Lemma 4 . 4 .

 44 Let R > 0 and δ ∈ (0, κ 0 /R].

  and t ∈ [0, T ]. The proof of Lemma 4.4 shows that the function h in (4.7) is nonexpanding in the second argument, and estimates similar to those in (4.8) yield

  ) ≡ 0 belongs to the domain of L. Along with the Hölder continuity of f ( • , u( • )) it now follows from [25, Theorem 7.1.10(iv)] that u ∈ C α ([0, δ]; D(L)) ∩ C 1+α ([0, δ]; U ) as desired.

  is continuously differentiable and thus locally Lipschitz continuous. By[START_REF] Lunardi | Analytic Semigroups and Optimal Regularity in Parabolic Problems[END_REF] Corollary 3.1.9], the operator L generates an analytic semigroup in C(R d ). Hence, by[START_REF] Lunardi | Analytic Semigroups and Optimal Regularity in Parabolic Problems[END_REF] Theorem 7.1.2] there exists a mild solutionv ∈ L ∞ (τ, T -δ; C(R d )) to -∂ t v -Lv -F (y, v) = 0 for some 0 ≤ τ < T -δ when imposed at t = T -δ with a terminal value in C(R d ). Due to the a priori estimates established in Corollary 3.2 and [25, Proposition 7.1.8] we may choose τ = 0. This gives us a global mild solution v ∈ L ∞ (0, T -; C(R d )) by pasting v and u at T -δ. In order to verify the desired regularity we recall that F is independent of the time variable. Hence, the regularity follows from [25, Proposition 7.1.10(iv)] if v(T -δ, •) ∈ D(L) and if Lv(T -δ, •) + F (•, v(T -δ, •)) belongs to the real interpolation space D L (α, ∞). The former condition is a part of the assumption, the later is a consequence of [25, Proposition 2.2.12(i)].

Lemma 5 . 3 .+ 2 p- 1 ( 2 p

 53212 Under assumption (A3), for every (ξ, π) ∈ Ā(t, x) with finite costs it holds thatE v(s, Y t,y s )|X ξ,π s | p -→ 0, s → T . (5.1)Proof. The monotonicity of (X ξ,π s ) s∈[t,T ] together with the terminal condition X ξ,π T = 0 implies|x| ≥ |X ξ,π s-| ≥ |X ξ,π s | ≥ T s π r dN r and |X ξ,π s-| ≥ |π s | (5.2)for all t ≤ s ≤ T , and moreover by Jensen's inequality|X ξ,π s | p ≤ 2 p|π r | p dN r ≤ T s (2 p -1)|X ξ,π r-| p dN r .(5.4)As the integrand in (5.4) is bounded by (2 p -1)|x| its jump integral has a true martingale part. Hence, by decomposing the jump integral in (5.4) into its martingale part with respect to the compensated Poisson process N t -θt and its deterministic part we obtain from (5.3),|X ξ,π s | p ≤ 2 p-1 E -1)|X ξ,π r | p θdr F s ,which implies by Gronwall's inequality the existence of a constant C > 0 such that|X ξ,π s | p ≤ CE

Finally

  , by a verification argument analogous to the one in Section 5 we obtainV t (x) = Y t |x| p for any nonnegative solution (Y, Z) ∈ L 2 F (Ω; C([0, T -]; R + )) × L 2 F (0, T -; R n) and conclude: Theorem 6.3. Under assumption (A5) uniqueness holds for problem(6.1) in the class of nonnegative solutions in L 2 F (Ω; C([0, T -]; R + )) × L 2 F (0, T -; R n ).

  We assume throughout that the coefficients b : R d → R d and σ : R d → R d×n are Lipschitz continuous.

	s	s
	Y t,y s = y +	b(Y t,y r ) dr +	σ(Y t,y r ) dW r ,	t ≤ s ≤ T.	(2.1)
	t	t
	Assumption 2.1.

  we can apply Fatou's lemma to the expectation in (3.2) as s → T -δ to obtainw(t, y) ≥ E lim inf lim s→T -δ v(s + δ, •) = +∞ uniformly on compact sets and v ∈ C poly ([0, T -δ] × R d ).The limit inferior in (3.3) is therefore a.s. nonnegative. Hence,

		s→T -δ	w(s, Y t,y s ) exp	t	s	l(r, Y t,y r ) dr .	(3.3)
	The sample paths of (Y t,y s ) s∈[t,T ] are bounded a. s., and hence,
	lim s→T -δ	w(s, Y t,y s ) = lim s→T -δ	v(s + δ, Y t,y s ) -v(s, Y t,y s ) = +∞ a.s.

because

  •, •)| • | p is indeed equal to the value function associated with our control problem. Lemma 5.4. Under assumption (A1), for ever (ξ, π) ∈ Ā(t, x) and s ∈ [t, T ) it holds, v(t, y)|x| p ≤ E v(s, Y t,y s )|X ξ,π s | p + E Proof. Let us denote by B(y, R) the open ball with radius R > 0 centered at y ∈ R d and introduce the first exit time

		s
		c(Y t,y r , X ξ,π r , ξ r , π r ) dr .
		t
	τ R = inf{r ≥ t : Y t,y r	/ ∈ B(y, R)}.
	Since v ∈ W 1,2 q ((t, s) × B(y, R)) and Y t,y is non-degenerated, due to assumption (A1), Krylov's
	generalized Itô formula [22, Theorem 2.10.1] applies to the stopped process

  Since v satisfies (2.10) a.e., since Y t,y is non-degenerated, and because |X ξ,π | ≤ |x|, if follows from Krylov's estimate [22, Theorem 2.4] that

r | p dr .

  Proof of Proposition 2.10. Let (ξ, π) ∈ Ā(t, x). By Lemma 5.4 and Lemma 5.3 letting s → T (assuming w.l.o.g. that (ξ, π) has finite costs) we get v(t, y)|x| p ≤ J(t, y, x; ξ, π).

	Finally note by Lemma 2.7 that equality holds in (5.5) if ξ = ξ * and π = π * . Since v and c are
	both nonnegative this implies that
	v(t, y)|x| p ≥ lim s→T	E	t	s	c(Y t,y r , X ξ * ,π * r	, ξ * r , π * r ) dr = J(t, y, x; ξ * , π * ).	(5.6)

  dY t δ = λ t + η t (T -δ -t) p -R n ) given by the Martingale Representation Theorem with the singular terminal value limWe now consider the difference of Y and Y δ for 0≤ t ≤ s < T -δ: Y r ) -f (r, Y r ) dr F t . Now, letting s → T -δ this yields Y t δ -Y t ≥ 0 by Fatou's lemma. This completes the proof since Y t -Y t = limThe upper estimate in (6.2) may be used to establish the analogue to Lemma 5.3. Here we need to impose the following essentially boundedness assumption on the coefficients:(A5) η, η -1 , (T -•) p λ ∈ L ∞ F (0, T ; R + ). Corollary 6.2. Under assumption (A5), for any solution (Y, Z) ∈ L 2 F (Ω; C([0, T -]; R + )) × L 2 F (0, T -; R n ) to (6.1) and any admissible control ξ ∈ A(t, x) with finite costs it holds that Proof. From the upper estimate in (6.2) and the tower property, E Y s |X ξ s | p F t ≤ E

	pY t T -δ -t δ t→T -δ Y t δ = +∞. g δ (r, Y r δ ) dr -s t p(Y r δ -Y r ) T -δ -r dr + g δ (r, By the solution formula for linear BSDEs: Y t δ -Y t = E Y s δ -Y s + s t f (r, Y r ) dr F t = E Y s δ -Y s -s t s t Y t Y t δ -Y t ≥ E Y s δ -sup t≤s≤T -δ Y s exp -s t p T -δ -r δ→0 Y t δ -Y t by the monotone convergence theorem. δ -Y Therefore, lim s→T E Y s |X ξ s | p F t = 0.	dr F t .	(6.3)

=:g δ (t,Yt δ ) dt -Z t δ dW t , 0 ≤ t < T -δ, for some Z δ ∈ t∈[0,T -δ) L 2 F (0, t;

A calculation as in

(3.4)

verifies that for all 0 ≤ t < T -δ and y ∈ R,

g δ (t, y) ≥ λ t -|y| β+1 βη β t =: f (t, y). t = E (Y s δ -Y s ) exp -s t p T -δ -r dr + s t g δ (r, Y r ) -f (r, Y r ) dr F t . T s {η r + (T -r) p λ r } dr (T -t) p |X ξ s | p F t .

Taking the liquidation constraint into account, we obtain after an application of Jensen's inequality,

E Y s |X ξ s | p F t ≤ E T s {η r + (T -r) p λ r }

dr T -t T s |ξ r | p dr F t ≤ CE T s |ξ r | p dr F t .

  Lemma 1.6] to monotone controls ξ ∈ A(t, x) with finite costs. By Itô-Kunita formula [23, Theorem I.8.1], since Y t |x| p solves the stochastic HJB equation, Y t |x| p ≤ E Y s |X ξ s | p + s t η r |ξ r | p + λ r |X ξ r | p dr F t , t ≤ s < T.

Unlike[START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF] we do not exclude exponents 1 < p < 2, which correspond to root shaped temporary price impact. Almgren et al.[START_REF] Almgren | Equity market impact[END_REF] give empirical evidence for p = 8/5.

We thank seminar participants at various institutions for valuable comments and suggestions. Financial support through the CRC 649 Economic Risk and d-fine GmbH is gratefully acknowledged. Parts of this manuscript were written while Horst was visiting the Hausdorff Research Institute for Mathematics in Bonn and the Center for Interdisciplinary Research at Bielefeld University; grateful acknowledgment is made for hospitality.

A Appendix

In this appendix we present a modification of the comparison result for viscosity solutions given in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]. The original statement [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]Theorem 3.5] concerns the uniqueness of viscosity solutions to systems of semilinear parabolic equations. The related comparison result is mentioned in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]Remark 3.9]. As suggested in [START_REF] Pardoux | Stochastic Differential Equations, Backward SDEs, Partial Differential Equations[END_REF]Remark 6.105], the present scalar formulation covers the case of a monotone (not necessarily Lipschitz continuous) nonlinearity

and consider for given terminal value g : R d → R the following parabolic problem

It is worth mentioning that no uniform continuity type assumptions are needed due to the lack of any gradient dependence of G.

Theorem A.1. Let the assumptions (A6) and (A7) hold, and let u, u ∈ C poly ([0, T ] × R d ) be a viscosity sub-and a viscosity supersolution to (A.1), respectively. Then,

The proof is as in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]. The only modification needed is to linearize the difference G(t, y, u(t, y))-G(t, y, u(t, y)) in terms of l(t, y) := 1 u(t,y) =u(t,y) G(t, y, u(t, y)) -G(t, y, u(t, y)) u(t, y) -u(t, y) .

rather than estimating it by a Lipschitz property, cf. [6, p.78]. The key lemma [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]Lemma 3.7] based on the theorem of sums and the doubling variable technique then reads:

Lemma A.2. The difference w := u -u is a viscosity subsolution to the linear equation

Since l is by (A7) bounded above by µ the supersolution property of the function given in [6, Lemma 3.8] carries over to equation (A.2) so that the rest of the proof matches again with the original reference.