E Pulido-Villena

A.-C Baudoux

I Obernosterer

M Landa

J Caparros

P Catala

C Georges

J Harmand

C Guieu

Microbial food web dynamics in response to a Saharan dust event: results from a mesocosm study in the oligotrophic Mediterranean Sea

The significant impact of dust deposition on heterotrophic bacterial dynamics in the surface oligotrophic ocean has recently been evidenced. Considering the central role of bacteria in the microbial loop, it is likely that dust deposition also affects the structure and the functioning of the whole microbial food web. In the frame of the DUNE project, aiming to estimate the impact of dust deposition on the oligotrophic Mediterranean Sea through mesocosm experiments, the main goal of the present paper was to assess how two successive dust deposition events affect the dynamics of the microbial food web. The first dust seeding delivered new P and N to the amended mesocosms and resulted in a pronounced stimulation of bacterial respiration. It also induced pronounced, but transient, changes in the bacterial community composition. No significant effects were observed on the abundances of viruses and heterotrophic nanoflagellates. The second dust seeding also delivered new P and N to the amended mesocosms, but the effect on the microbial food web was very different. Bacterial respiration remained constant and bacterial abundance decreased. Compositional changes following the second seeding were minor compared to the first one. The decrease in bacterial abundance coincided with an increase in virus abundance, resulting in higher virus : bacteria ratios throughout the second seeding period. Our study shows that dust deposition to the surface oligotrophic ocean may involve important modifications of the trophic links among the components of the microbial food web with presumed consequences on C and nutrient cycling.

Introduction

Nutrient cycles control the strength of the biological carbon pump through which CO 2 is consumed in surface water and transported as sinking organic carbon to the deep sea [START_REF] Balino | Ocean biogeochemistry and global change: JGOFS research highlights 1988-2000[END_REF]. Therefore, understanding what governs nutrient cycles in the ocean and to which extent climate change plays a role in this schema is essential to understand the oceanic carbon cycle and to predict its evolution. Increasing evidence indicates that not only phytoplankton but also heterotrophic bacteria are limited by inorganic nutrients, mainly P, in oligotrophic oceanic systems [START_REF] Rivkin | Inorganic nutrient limitation of oceanic bacterioplankton[END_REF][START_REF] Thingstad | P limitation of heterotrophic bacteria and phytoplankton in the northwest Mediterranean[END_REF][START_REF] Sala | Seasonal and spatial variations in the nutrient limitation of bacterioplankton growth in the northwestern Mediterranean[END_REF][START_REF] Obernosterer | P-limitation of respiration in the Sargasso Sea and uncoupling of bacteria from P-regeneration in size-fractionation experiments[END_REF][START_REF] Zohary | P-limited bacteria but N and P co-limited phytoplankton in the Eastern Mediterranean microcosm experiment[END_REF] thus suggesting that the supply of these resources could explain variability in bacterial activity.

Dust deposition is recognized as a significant source of macro-and micro-nutrients to the surface ocean [START_REF] Jickells | Global iron connections between desert dust, ocean biogeochemistry, and climate[END_REF][START_REF] Mahowald | The global distribution of atmospheric phosphorus sources, concentrations and deposition rates and anthropogenic impacts[END_REF]. The recent effort to understand the role of these inputs on ocean biogeochemistry has focused on the impact on autotrophic productivity, given their potential to generate new production. However, recent studies combining field and experimental approaches have demonstrated significant increase in heterotrophic bacterial abundance and respiration following dust deposition in oligotrophic ecosystems [START_REF] Pulido-Villena | Bacterial response to dust pulses in the western Mediterranean: Implications for carbon cycling in the oligotrophic ocean[END_REF][START_REF] Lekunberri | Effects of a dust deposition event on coastal marine microbial abundance and activity, bacterial community structure and ecosystem function[END_REF][START_REF] Romero | Coastal Mediterranean plankton stimulation dynamics through a dust storm event: An experimental simulation[END_REF]. Moreover, heterotrophic processes appear to be more stimulated by dust pulses compared to autotrophic processes with increasing degree of oligotrophy, the dominant response being modulated by the competition for nutrients between phytoplankton and bacteria [START_REF] Marañón | Degree of oligotrophy controls the response of microbial plankton to Saharan dust[END_REF]. Considering the central role of bacteria in the microbial loop [START_REF] Azam | Microbial control of oceanic carbon flux: The plot thickens[END_REF], it is likely that dust deposition also affects the structure and, importantly, the functioning of the whole microbial food web. To our knowledge, no studies have so far documented the consequences of dust pulses on the dynamics of the microbial food web (including viruses, heterotrophic nanoflagellates) and, thus, the fate of bacterial production after a dust pulse remains unexplored.

Both viruses and heterotrophic nanoflagellates can be significant sources of bacteria mortality in oligotrophic oceanic systems [START_REF] Zhang | Viruses and flagellates sustain apparent richness and reduce biomass accumulation of bacterioplankton in coastal marine waters[END_REF][START_REF] Bonilla-Findji | Viral and Flagellate Control of Prokaryotic Production and Community Structure in Offshore Mediterranean Waters[END_REF][START_REF] Boras | Annual changes of bacterial mortality due to viruses and protists in an oligotrophic coastal (NW Mediterranean[END_REF][START_REF] Boras | Effect of viruses and protists on bacteria in eddies of www[END_REF]. Both loss processes are densitydependent and will thereby be affected by the abundance of heterotrophic bacteria (e.g., [START_REF] Vaqué | Grazing rates on bacteria: the significance of methodology and ecological factors[END_REF]Murray and Jackson, 1994). Dust-induced changes in bacterial dynamics may thus indirectly modify the rates of grazing and viral lysis. In addition, physical interactions of bacteria-virusnanoflagellates with inorganic particles can also shift the relative impact of grazing and viruses on bacterial mortality [START_REF] Salter | The effect of inorganic particle concentration on bacteria-virus-nanoflagellate dynamics[END_REF]. Understanding the relative contribution of the losses due to viruses and heterotrophic nanoflagellates is critical because each of these processes influence the functioning of the microbial C pump in different ways [START_REF] Fuhrman | Marine viruses and their biogeochemical and ecological effects[END_REF]Wilhem and Suttle, 2000). Grazing channels bacterial biomass to the higher trophic levels whereas viral lysis transforms this living biomass into dissolved organic matter that can be readily incorporated by non-infected bacteria. This viral shunt has been shown to result in a net increase in bacterial respiration and decrease in growth efficiency [START_REF] Middelboe | Viral lysis of bacteria: an important source of dissolved amino acids and cell wall compounds[END_REF][START_REF] Motegi | Viral control of bacterial growth efficiency in marine pelagic environments[END_REF]. The dynamics of the microbial food web may thus play a much larger role in the connections between dust and the ocean carbon cycle than previously recognized.

The Mediterranean Sea is a typical LNLC (Low Nutrient Low Chlorophyll) region, particularly well adapted to tackle the question of the role of atmospheric input: it is an oligotrophic, quasi-enclosed basin, which receives a noticeable atmospheric flux of Aeolian soil dust, mainly derived from the Sahara desert, in the form of strong pulses [START_REF] Löye-Pilot | Influence of Saharan dust on the rain acidity and atmospheric input to the Mediterranean[END_REF][START_REF] Guerzoni | The role of atmospheric deposition in the biogeochemistry of the Mediterranean Sea[END_REF]. Moreover, during the stratification period, the Mediterranean Sea exhibits a severe nutrient depletion after the spring bloom, and both bacterioplankton and phytoplankton are strongly P-limited and/or N-P co-limited [START_REF] Thingstad | P limitation of heterotrophic bacteria and phytoplankton in the northwest Mediterranean[END_REF][START_REF] Sala | Seasonal and spatial variations in the nutrient limitation of bacterioplankton growth in the northwestern Mediterranean[END_REF][START_REF] Van Wambeke | Bottom up effects on bacterioplankton growth and composition during summer-autumn transition in the open NW Mediterranean Sea[END_REF][START_REF] Tanaka | Lack of P-limitation of phytoplankton and heterotrophic prokaryotes in surface waters of three anticyclonic eddies in the stratified Mediterranean Sea[END_REF][START_REF] Moutin | Introduction to the Biogeochemistry from the Oligotrophic to the Ultraoligotrophic Mediterranean (BOUM) experiment[END_REF].

The DUNE project aimed at estimating the impact of dust deposition on an oligotrophic system, the Mediterranean Sea, based on mesocosm experiments simulating strong atmospheric inputs of desert dust. In the frame of this project, the main goal of the present paper was to assess how two successive dust deposition events affect the dynamics of the microbial food web in oceanic surface waters under oligotrophic conditions.

Material and methods

Experimental design and sampling

An experimental dust addition into large mesocosms (52 m 3 , 15 m depth) was conducted between 26 June and 9 July 2010 in the frame of the DUNE project (DUNE-R experiment, www.obs-vlfr.fr/LOV/DUNE, Guieu et al. 2014, this issue). The DUNE-R experiment was carried out in the Scandola Nature Reserve (Corsica, France), an oligotrophic site (chl a = 0.1 µg L -1 , Guieu et al., 2014, this issue) protected from human influence. Three replicate mesocosms (hereafter "D-mesocosms") were amended one first time with 41 g of mineral dust mimicking a high but realistic flux of 10 g m -2 [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. A second identical seeding was performed 168 h after the first one in order to test the hypothesis that changes in the biogeochemical conditions of the water column induced by a dust pulse can modify the impact of a successive dust pulse. Three other mesocosms (hereafter, "C-mesocosms") served as controls. A detailed justification of this strategy and a description of the dust used for the experiment can be found in [START_REF] Pulido-Villena | Transient fertilizing effect of dust in P-deficient LNLC surface ocean[END_REF][START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. Briefly, mineral dust deposition was mimicked using the finest fraction (< 20 µm) of alluvial soils collected in a desert soil source area in south Tunisia known to export Aeolian dust to the western Mediterranean [START_REF] Pulido-Villena | Transient fertilizing effect of dust in P-deficient LNLC surface ocean[END_REF]. Dust was subjected to physico-chemical transformations through condensation/evaporation cycles that involved HNO 3 and H 2 SO 4 , mimicking atmospheric transport [START_REF] Pulido-Villena | Transient fertilizing effect of dust in P-deficient LNLC surface ocean[END_REF]. This evapocondensed dust contained on average 0.045 ± 0.015 % of P, 2.31 ± 0.04 % of Fe and 1.19 ± 0.05 % of N, in weight. The amendment of the dust mesocosms in the DUNE-R experiment was performed with the dust mixed with 2 L of ultrapure water in order to mimic a wet deposition event.

Samples from the six mesocosms were taken just before the first dust addition and at 24 h interval times during 7 days starting 4 h after the first amendment. A schematic picture of the sampling strategy can be found in [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. Seawater was also collected outside the mesocosms (hereafter "OUT") at selected times during the experiment in order to check for any contamination inside the mesocosms. Filtered seawater (< 0.2 µm, Sartobran cartridge filter) for dissolved inorganic phosphorus (DIP) analyses and unfiltered seawater for bacterial abundance (BA), bacterial respiration (BR), bacterial community structure, virus abundance (VA), and heterotrophic nanoflagellate abundance (HNFA) was collected using a trace metal-clean Teflon pump system. Samples for DIP, BA and VA were collected at three different depths (0, 5 and 10 m) whereas samples for BR, HNFA and bacterial community structure were collected at 5 m depth.

Dissolved inorganic phosphorus (DIP) analysis

DIP was analyzed by spectrophotometry [START_REF] Murphy | A modified single solution method for the determination of phosphate in natural waters[END_REF][START_REF] Zhang | Automated analysis of nanomolar concentrations of phosphate in natural waters with liquid waveguide[END_REF] using a long waveguide capillary cell (LWCC). In the recent years, a number of studies focusing on the analytical accuracy of the LWCC technique for measuring DIP have demonstrated a good agreement with the MAGIC technique [START_REF] Li | Intercomparison and coupling of magnesium-induced co-precipitation and long-path liquidwaveguide capillary cell techniques for trace analysis of phosphate in seawater[END_REF] and negligible interference with arsenate and silicate [START_REF] Li | Intercomparison and coupling of magnesium-induced co-precipitation and long-path liquidwaveguide capillary cell techniques for trace analysis of phosphate in seawater[END_REF][START_REF] Ma | Reverse flow injection analysis of nanomolar soluble reactive phosphorus in seawater with a long path length liquid waveguide capillary cell and spectrophotometric detection[END_REF]. The used LWCC was 2 m long and the technique achieved a limit of detection (LOD = three times the standard deviation of 10 measurements of the blank) of 1 nmol L -1 . More details on the analytical procedure can be found in [START_REF] Pulido-Villena | Transient fertilizing effect of dust in P-deficient LNLC surface ocean[END_REF].

Enumeration of heterotrophic bacteria, heterotrophic nanoflagellates and viruses

For heterotrophic bacterial abundance (BA), duplicate subsamples (1.8 mL) were fixed with formaldehyde (2 % final concentration), kept at 4 • C for 30 min, quick-frozen in liquid nitrogen and then stored at -80 • C until flow cytometry analysis as described in [START_REF] Lebaron | Does the high nucleic acid content of individual bacterial cells allow us to discriminate between active cells and inactive cells in aquatic systems?[END_REF]. Heterotrophic bacterial counts were performed with the FACSCalibur flow cytometer (Becton Dickinson) equipped with an air-cooled ar-gon laser (488 nm, 15 mW). Bacterial cells were stained with SYBR Green I (Invitrogen -Molecular Probes) at 0.025 % (vol / vol) final concentration for 15 min at room temperature in the dark. Stained bacterial cells were discriminated and enumerated according to their right-angle light scatter (SSC) and green fluorescence measured at 530/30 nm. In a plot of green fluorescence versus red fluorescence, we were able to distinguish photosynthetic from non-photosynthetic bacteria.

The volume analyzed was around 15 µL at low speed. The cell abundance was determined from the flow rate, which was calculated by weighing one tube of Milli-Q water before and after a 5 min run of the cytometer. The flow rate was determined after every five samples. Fluorescent beads (1.002 µm; Polysciences Europe) were systematically added to each analyzed sample as an internal standard.

For the enumeration of heterotrophic nanoflagellates (HNFA), samples (3.8 mL) were fixed with glutaraldehyde (1 %, final concentration), incubated for 30 min at 4 • C, quick-frozen in liquid nitrogen and then stored at -80 • C until flow cytometric analyses following the protocol described in [START_REF] Christaki | Optimized routine flow cytometric enumeration of heterotrophic flagellates using SYBR Green I[END_REF]. HNF counts were performed with the FACSCanto II flow cytometer (Becton Dickinson) equipped with three air-cooled lasers: blue (argon 488 nm), red (633 nm) and violet (407 nm). Staining was performed with SYBR Green I (Invitrogen -Molecular Probes) at 0.05 % (v/v) final concentration for 15-30 min at room temperature in the dark. The volume analyzed was around 1 mL at high speed (around 100 µL min -1 ) for 10 to 30 min to obtain a minimum of 600 events.

For virus counts (VA), samples (1 mL) were fixed with glutaraldehyde (0.5 % final concentration) incubated for 15 min at 4 • C, frozen in liquid nitrogen and stored at -80 • C until flow cytometric analysis according to [START_REF] Brussaard | Optimization of procedures for counting viruses by flow cytometry[END_REF]. Virus counts were performed on a FAC-SCalibur flow cytometer (Becton Dickinson). Numbers were corrected for the blank consisting of TE buffer with autoclaved 0.02 µm filtered seawater in the correct dilution. Data were analyzed using the freeware CY-TOWIN (http://www.sbroscoff.fr/Phyto/index.php?option= com_content&task=view&id=72&Itemid=123).

Bacterial respiration

Samples from 5 m depth of each mesocosm were gently filtered through a 0.8 µm PC filter under low vacuum pressure, and the filtrate was siphoned into a set of six BOD bottles. Three BOD bottles from each mesocosm were immediately fixed with Winkler reagents, and the rest were incubated in the dark in a tap water bath to minimize temperature variations. All the incubated BOD bottles were fixed with Winkler reagents after 24 h. Oxygen concentration was measured through a spectrophotometric approach based on measuring the absorbance at 466 nm of the colored I 2 and I - 3 [START_REF] Labasque | Spectrophotometric Winkler determination of dissolved oxygen: reexamination of critical factors and reliability[END_REF][START_REF] Reinthaler | Fully automated spectrophotometric approach to determine oxygen concentrations in seawater via continuous-flow analysis[END_REF]. The concentration of oxygen was calculated by comparing the absorbance in a sample against standards of known oxygen content made from 0.0467 mol L -1 potassium iodate (KIO 3 ) solutions.

Bacterial diversity

For bacterial diversity, volumes of 500 mL were sampled every 2 days in the mesocosms at 5 m depth. Seawater was filtered through 0.8 µm filters (47 mm polycarbonate (PC) filters, Nuclepore, Whatman), and bacterial cells were collected from the < 0.8 µm filtrate on 0.2 µm PC filters (47 mm PC; Nuclepore, Whatman). Filters were stored at -80 • C until analysis. DNA and RNA extraction, amplification as well as CE-SSCP analysis were performed as described in [START_REF] Laghdass | Effects of Saharan dust on the microbial community during a large in situ mesocosm experiment in the NW Mediterranean Sea[END_REF].

Filters were plunged in lysis buffer (40 mM EDTA, 50 mM Tris, 0.75 M sucrose) and underwent three freeze and thaw cycles using liquid nitrogen and a 65 • C water bath. They were then incubated with a freshly prepared lysozyme solution (2 mg mL -1 final) during 45 min at 37 • C under gentle agitation. Proteinase K (0.2 mg mL -1 final) and SDS (1 % final) were added, and a second incubation (1 h, 55 • C) was performed. DNA and RNA were coextracted using an Allprep DNA/RNA Mini Kit (Qiagen) according to the manufacturer's instructions. To avoid DNA contamination in the RNA fraction, a DNase step was performed during RNA extraction, using the RNase-free DNase set from Qiagen and following the kit's instructions. The quality and concentration of DNA and RNA were checked by agarose gel electrophoresis (1 %), and DNA was stored at -20 • C.

For RNA samples, denaturation was performed by adding reverse primer w34 (5 ′ -TTA CCG CGG CTG CTG GCA C-3 ′ ) (Lee et al., 1996) (1 µM final concentration) to 13 µL of RNA before incubation at 94 • C for 5 min followed by rapid cooling on ice. Reverse transcription was achieved by adding dNTPs (0.8 µM final concentration), MMLV reverse transcriptase and 5X MMLV buffer (Promega) and incubating the samples at 42 • C for 1 h. The cDNA samples were stored at -20 • C.

DNA and cDNA were amplified using the bacterial specific primers w49 forward (5 ′ -ACG GTC CAG ACT CCT ACG GG-3 ′ ) [START_REF] Delbes | Monitoring of activity dynamics of an anaerobic digester bacterial community using 16S rRNA polymerase chain reaction-single-strand conformation polymorphism analysis[END_REF] and reverse w34 (see above), targeting about 200 bp of the V3 region of the 16S rRNA gene. W34 was 5 ′ -labeled with TET to enable detection. Polymerase chain reactions (PCRs) contained 1 µL of DNA or cDNA, 0.3 µM primers, 0.8 µM dNTPs, pfu DNA polymerase (1U final), pfu buffer (1X, Promega), and UVsterilized Milli-Q water (QS 50 µL). The PCR protocol we used was 1 min at 94 • C followed by 25 cycles of 30 s at 94 • C, 30 s at 61 • C and 30 s at 72 • C with a final extension of 10 min at 72 • C. Amplification products were verified by agarose gel electrophoresis (1 %) and their concentrations estimated by comparison with molecular size markers (Smart Ladder, Promega).

For analysis of 16S rRNA genes and transcripts in CE-SSCP, samples were diluted and 1 µL of each dilution was mixed with 18.8 µl of deionized formamide (Hi-Di for-mamideTM; Applied Biosystems) and 0.2 µl of the internal size standard GeneScan-400HD (Rox) (Applied Biosystems). A denaturation step of 5 min at 94 • C was performed, and samples were immediately placed in a water/ice bath for 10 min. Fragments were separated by CE-SSCP as described previously [START_REF] Delbes | Monitoring of activity dynamics of an anaerobic digester bacterial community using 16S rRNA polymerase chain reaction-single-strand conformation polymorphism analysis[END_REF] using an ABI 310 Genetic Analyzer (Applied Biosystems) with electrophoresis at 12 kV and 30 • C for 30 min per sample. The electropherograms were analyzed by the GeneScan software (Applied Biosystems), and the obtained profiles were aligned and normalized using the Rox internal standard with SAFUM [START_REF] Zemb | Safum statistical analysis of SSCP fingerprints using PCA projections, dendrograms and diversity estimators[END_REF]. Operational taxonomic units (OTUs) were defined for peaks reaching at least 150 fluorescence units, that is, three times the value of the base line. Peaks with identical retention times between samples were considered one identical OTU. For the fingerprints obtained from the 16S rRNA gene, we refer in the following to the total community, and for the fingerprints obtained from the 16S rRNA transcripts we refer to the active community.

Statistical analysis

To explore dust-induced changes in the study parameters, we checked for significant differences between the Cmesocosms and the D-mesocosms immediately after each seeding (up to day 2 and up to day 8; pairwise Student's t test) and during the whole first and second seeding periods (up to day 6 and from day 7 to the end, respectively, Student's t test grouped comparison). During the first seeding period, the comparison between C-and D-mesocosms gives information on the effect of the first seeding. During the second seeding period, the comparison between C-and Dmesocosms gives information on the combined effect of both performed seedings. We followed dust-induced changes in the composition of the total and active bacterial community using the following approach. Our fingerprinting approach provided the relative abundances of OTUs, based on their contributions to the total peak area, in three D-mesocosms and three C-mesocosms for each time point. We first tested for each OTU whether its relative abundance was significantly different between treatments at a given time point (Students t test, p < 0.05). For any OTU that was significantly different between treatments, we calculated its mean relative abundance in the D-and C-mesocosms, and then the difference in the relative abundance as D mean -C mean . This difference could be positive -that is the relative abundance of a given OTU is higher in the D-treatment compared to the C-treatment -or negative -that is the relative abundance of a given OTU is lower in the D-treatment compared to the C-treatment. To weigh these relative changes, we expressed them as percent of the total peak area of the control community. Summing all positive or negative changes allowed us to illustrate the overall extent of change induced by dust addition. 3 Results

Biogeochemical characteristics of the study site

A detailed description of the main environmental conditions encountered during the DUNE-R experiment can be found in [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. Briefly, seawater temperature during the whole experiment ranged between 20.0 and 27.3 • C, with a significant increase in surface temperature and well-stratified waters towards the end of both seeding periods. At the beginning of the experiment, chl a averaged 0.07 µg L -1 , nitrate concentration was below the limit of detection of the used technique (30 nM) and dissolved iron averaged 3.3 nM [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF], Table 1). The first dust seeding resulted in a 1.9-fold increase in the chl a concentration in the D-mesocosms compared to the C-mesocosms. The second seeding induced an additional increase, yielding up to a 2.4-fold higher chl a concentration in the D-mesocosms as compared to the C-mesocosms [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. Nitrate concentration increased in the D-mesocosms to 3 µM after the first seeding and then decreased to 0.5 µM by the end of the first seeding period. The second seeding induced a greater increase in nitrate concentration (up to 10 µM) [START_REF] Ridame | Phytoplanktonic response to contrasted Saharan dust deposition events during mesocosm experiments in LNLC environment[END_REF]. The initial iron values before the dust addition varied between 2.2 and 4.4 nmol L -1 inside the mesocosms and outside. The first seeding resulted in a decrease of dissolved iron, whereas the second addition of dust induced dissolution of Fe from the dust particles [START_REF] Wuttig | Impacts of dust deposition on dissolved trace metal concentrations (Mn, Al and Fe) during a mesocosm experiment[END_REF]. Dissolved inorganic phosphorus (DIP) concentration was 5 ± 2 nM (mean value of OUT samples and D-and Cmesocosms at the three depths) before the seeding. Ten hours after the first seeding, depth-averaged DIP concentration increased in the D-mesocosms to 9 ± 1 nM and decreased to initial values (4 ± 1 nM) in the following sampling (23 h after the seeding) and, thereafter, remained constant until the end of the first phase of the experiment (Fig. 1). At the end of the first seeding period, DIP was homogeneous among C-and D-mesocosms and OUT station, averaging 4 ± 2 nM (Fig. 1). Regarding the second seeding, DIP increased significantly in surface and 5 m depth after 5 h. Contrary to the first seeding, DIP in D-mesocosms remained fairly constant and significantly higher than in C-mesocosms until the end of the experiment (Fig. 1). While for the first seeding the increase in DIP was similar among the three mesocosms, the second seeding induced a higher inter-mesocosm variability.

Bacterial abundance and respiration

Before the first seeding, bacterial abundance (BA) in the six mesocosms and in OUT samples averaged 4.7 ± 0.4 10 5 cell mL -1 (all depths). Throughout the experiment, BA was slightly higher in OUT samples than inside the mesocosms at all depths. BA behaved differently after the two conducted dust seedings. One day after the first seeding, there was a significant 1.2-fold difference in BA in D-mesocosms compared to C-mesocosms at the surface (t (4) = 3.8, p < 0.05), whereas no significant differences were observed between Dand C-mesocosms at 5 and 10 m depth. During the rest of the first seeding period, no significant differences in BA between C-and D-mesocosms were observed at any of the study depths. By contrast, during the second seeding period, BA in D-mesocosms showed substantial variability (a decrease was observed in two out of three mesocosms), but it was, on average, significantly lower than in C-mesocosms at the surface (1.1-fold, t (34) = 2.4, p < 0.05), 5 m (1.1-fold, t (34) = 3.9, p < 0.001) and 10 m depth (1.1-fold, t (34) = 2.5, p < 0.01) (Fig. 2).

Bacterial respiration (BR), determined only at 5 m depth, was 1.0 ± 0.2 µmol C L -1 d -1 (average value for mesocosms and OUT sample) before the seeding, and it exhibited similar values between OUT samples and C-mesocosms throughout the experiment. After the first dust amendment, BR was enhanced by 3.5-fold in D-mesocosms compared to C-mesocosms, and it remained significantly higher, on average (t (27) = 6.7, p < 0.001), until the last available sampling point (10 days after the seeding). No additional increase in BR was observed after the second dust amendment (Fig. 3).

Bacterial community structure

We observed a pronounced change in the bacterial community composition 4 days after the first seeding, and changes were minor after the second seeding (Fig. 4a). After 4 days, the relative abundance of 4 OTUs was significantly higher in the D-than in the C-mesocosms, and the relative abundance of 8 OTUs was significantly lower in the D-than in the Cmesocosms. The sum of the dust-induced changes accounted for up to 55 % of the total community 4 days after the first seeding, while it was < 22 % after the second seeding. Dustinduced changes of the active bacterial community revealed a similar pattern following the first seeding with a marked response after 4 days (Fig. 4b). A second remarkable change was detectable at day 13 corresponding to 6 days after the second seeding.

We used the Simpson index to evaluate whether changes in the number of OTUs and their relative contributions occurred in response to dust addition. Our results reveal that dust ad- dition had no effect on the diversity of the total and active bacterial community (Fig. 5). Diversity of the total community decreased during the first 11 days of the experiment, then rapidly increased. This pattern was less pronounced for the diversity of the active community.

Virus and heterotrophic nanoflagellate abundance

Before the first seeding, total virus abundance (VA) in the six mesocosms and in OUT samples averaged 3.5 ± 0.4 10 6 particles mL -1 (all depths, Fig. 6). VA at the station OUT was slightly lower than in C-mesocosms during the course of the experiment. In C-mesocosms, VA remained quite stable except for two steep decreases in abundance observed at day 2 and day 11 (at all depths). No significant differences in VA were observed in D-mesocosms compared to Cmesocosms during the first dust seeding period at any of the study depths. However, during the second seeding period, VA was, on average, significantly higher in D-mesocosms than in C-mesocosms at surface (1.1-fold, t (34) = 2.4, p < 0.05) and 5 m depth (1.1-fold, t (34) = 2.4, p < 0.05) (Fig. 6). The observed higher VA was associated with a concomitant lower in BA in D-mesocosms, resulting in significantly higher virus to bacteria ratio in D-mesocosms than in C-mesocosms during the second seeding period at surface (t (34) = 3.0, p < 0.01), 5 m depth (t (34) = 2.4, p < 0.05) and 10 m depth (t (34) = 2.9, p < 0.01) (Fig. 7).

The abundance of heterotrophic nanoflagellates (HNFA), determined only at 5 m depth, varied during the duration of the experiment between 196 and 1000 cell mL -1 . HNFA was higher in OUT samples than inside the mesocosms throughout the experiment. Similar dynamics were observed for Cand D-mesocosms, and there were no significant differences in HNFA between both treatments at any of the seeding periods (Fig. 8).

Discussion

Effect of dust on inorganic nutrients

The successive dust seedings performed during the DUNE-R experiment induced distinct modifications in the biogeochemical conditions of the study waters. Generally, both seedings resulted in increased concentrations of dissolved inorganic phosphorus (DIP, this study) and nitrate (NO Ridame et al., 2014) concentrations. However, the intensity of these effects was notably different between the seedings, with greater differences between D-and C-mesocosms observed after the second seeding. The homogeneity among mesocosms of the concentrations in DIP and NO - 3 introduced by dust differed also between the first and second seeding periods. While after the first seeding, the effect was similar among the three D-mesocosms, a clear response was only observed in two (D1 and D2) out of the three mesocosms during the second seeding.

At the beginning of the experiment, the DIP and NO - 3 concentrations (5 nM and ≤ 30 nM, respectively) would suggest a N-limitation or N-P co-limitation of the biological activity (N : P ratio ≤ 6). The dust used for the seedings exhibits a total N : P ratio of 55 [START_REF] Guieu | Introduction to project DUNE, a DUst experiment in a low Nutrient, low chlorophyll Ecosystem[END_REF]. Assuming a 100 % dissolution of N [START_REF] Ridame | Phytoplanktonic response to contrasted Saharan dust deposition events during mesocosm experiments in LNLC environment[END_REF] and a 15 % dissolution of P [START_REF] Ridame | Saharan input of phosphate to the oligotrophic water of the open western Mediterranean Sea[END_REF][START_REF] Pulido-Villena | Transient fertilizing effect of dust in P-deficient LNLC surface ocean[END_REF], the seeding would have delivered new N and P at a N : P ratio of 350, and would have pushed the system towards a strong P-limitation. Indeed, the N : P ratio in the D-mesocosms increased to ca. 300 after the first seeding, and it was further enhanced to 500 at the end of the first seeding period and during most part of the second one. N : P ratios decreased to values around 16 at the end of the experiment. The higher delivery of N relative to P through atmospheric deposition has been evoked as responsible, in part, for the high N : P ratios encountered in the Mediterranean Sea, particularly in the eastern basin [START_REF] Krom | Why is the Eastern Mediterranean phosphorus limited?[END_REF][START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N / P ratio[END_REF]. Although estimated N : P ratios in DUNE dust (350) are substantially higher than reported for atmospheric deposition in western (50) and eastern (100) basins [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N / P ratio[END_REF], these results would confirm the contribution of atmospheric deposition in determining N : P ratios in the Mediterranean Sea.

Pronounced response of bacterial respiration

Bacterial respiration revealed the most pronounced response to dust addition, while the responses of heterotrophic bacteria and virus abundances were moderate and restricted to the second seeding period. A strong stimulation of BR by dust pulse has already been reported for the open Mediterranean Sea [START_REF] Pulido-Villena | Bacterial response to dust pulses in the western Mediterranean: Implications for carbon cycling in the oligotrophic ocean[END_REF], and the observed response agrees with previous studies showing clearer responses of metabolic rates and weaker responses in terms of biomass [START_REF] Herut | Response of East Mediterranean surface water to Saharan dust: On-board microcosm experiment and field observations[END_REF][START_REF] Lekunberri | Effects of a dust deposition event on coastal marine microbial abundance and activity, bacterial community structure and ecosystem function[END_REF][START_REF] Marañón | Degree of oligotrophy controls the response of microbial plankton to Saharan dust[END_REF][START_REF] Laghdass | Effects of Saharan dust on the microbial community during a large in situ mesocosm experiment in the NW Mediterranean Sea[END_REF]. Given the N : P ratios encountered in the study site at the beginning of the DUNE-R experiment, heterotrophic bacterial activity would have been presumably limited by the availability of DIP or co-limited by N and P. Indeed, bacterial activity has been generally reported to be P-limited in the Mediterranean Sea (e.g., [START_REF] Thingstad | P limitation of heterotrophic bacteria and phytoplankton in the northwest Mediterranean[END_REF][START_REF] Sala | Seasonal and spatial variations in the nutrient limitation of bacterioplankton growth in the northwestern Mediterranean[END_REF] although recent studies have pointed to a N-P co-limitation [START_REF] Tanaka | Lack of P-limitation of phytoplankton and heterotrophic prokaryotes in surface waters of three anticyclonic eddies in the stratified Mediterranean Sea[END_REF]. In this regard, the first seeding period seems to be characterized by a bottom-up control of bacterial dynamics with an increase in BA and BR just after the dust amendment.

Whether dust addition stimulated directly or indirectly bacterial respiration constitutes a key question for this study. The rapid decrease of DIP introduced by the first seeding and concomitant increase in BA after 2 days suggests a direct effect due to the addition of the limiting nutrient P. However, the extent of increase in bacterial respiration, and the still high bacterial respiration rates recorded during the second seeding, when P was not consumed, would also suggest an indirect effect by phytoplankton. Phytoplankton was stimulated by the first and the second dust addition [START_REF] Ridame | Phytoplanktonic response to contrasted Saharan dust deposition events during mesocosm experiments in LNLC environment[END_REF]. Bacterial respiration could have been stimulated by labile phytoplankton-derived dissolved organic matter (DOM). Besides C, phytoplankton DOM also provides N and P, offering an alternate source of the limiting element to bacteria. The switch to this source could explain the non-utilization of DIP during the second seeding.

Effect of dust on bacterial community composition

The first dust addition induced rapid but transient changes in bacterial community composition, and changes following the second seeding were minor. This pattern reflects a [START_REF] Parada | Viral burst size of heterotrophic prokaryotes in aquatic systems[END_REF]. Grey vertical bars represent the dust seedings.

rather short-term response to the environmental conditions set by the first seeding, such as those related to inorganic nutrient conditions and/or phytoplankton growth. Our results are in line with observations from microcosm studies where the effect of dust was followed over 2-3 days. Dust addition to surface waters of the central Atlantic Ocean stim-ulated Roseobacter and Gammaproteobacteria, and lead to a decrease in SAR11 after 2 days of incubation [START_REF] Marañón | Degree of oligotrophy controls the response of microbial plankton to Saharan dust[END_REF]. Similarly, metabolic rates of Prochlorococcus and a bacterial population dominated by SAR11 were reduced upon dust addition in the North Atlantic Ocean [START_REF] Hill | Differential responses of Prochlorococcus and SAR11-dominated bacterioplankton groups to atmospheric dust inputs in the tropical Northeast Atlantic Ocean[END_REF]. In a microcosm experiment performed in the NW Mediterranean Sea, the dust-induced changes detectable after 3 days were similar to those obtained in P-amended microcosms suggesting that the response was mainly driven by P [START_REF] Lekunberri | Effects of a dust deposition event on coastal marine microbial abundance and activity, bacterial community structure and ecosystem function[END_REF]. In a previous mesocosm study performed in June 2008 at the same site (DUNE1), but with a lower temporal sampling resolution, bacterial communities collected 8 days after the seeding did not reveal any marked differences [START_REF] Laghdass | Effects of Saharan dust on the microbial community during a large in situ mesocosm experiment in the NW Mediterranean Sea[END_REF]). Together, these results indicate that dust addition can selectively stimulate certain bacterial groups, probably by providing specific conditions favorable for their growth. However, the present study demonstrates that these changes are transient, and that consecutive dust deposition events greatly diminish the initial bacterial response.

Divergent responses of viruses and HNF dynamics to dust addition

Slight differences were observed in virus and HNF abundance between OUT samples and inside the mesocosms. This may be caused by enclosure effects, but does not preclude from extracting valid patterns of response of viruses and HNF dynamics to dust addition. Although both heterotrophic bacteria and phytoplankton clearly responded to the first seeding, neither HNF nor viruses appeared to be impacted, at least in terms of abundance. By contrast, the second seeding was associated with moderate but significantly higher virus abundance (VA) while HNF dynamics remained unchanged. Viral dynamics result from the balance between loss and production processes. The observed higher standing stock of virus might result either from a reduced decay rate or a stimulated production of viruses in the D-mesocosms. The latter hypothesis seems more plausible as the observed higher VAs were associated with a concomitant decrease in bacterial abundance (BA). Dust-induced changes in VA (VAdust -VAcontrol) correlate significantly with dust-induced changes in BA (BAdust -BAcontrol) at all depths, during the second seeding period (surface: r = -0.55, p < 0.05; 5 m: r = -0.59, p < 0.01; 10 m: r = -0.65, p < 0.01). Although viral production on heterotrophic bacteria was not monitored, this strong relationship suggests a possible viral mediated control of the bacterial community during the second dust amendment. Theoretical calculation of the % of infected bacteria using the dust-induced changes in VA and a conservative burst size of 20 [START_REF] Parada | Viral burst size of heterotrophic prokaryotes in aquatic systems[END_REF] would actually explain 23 to 90 % (average 45 %) of the observed decrease in bacterial abundance during the second seeding period (Fig. 9). Altogether, these results suggest that the second seeding stimulated viral lysis of bacteria and, thereby, shifted the balance of bottom-up and top-down controls of the bacteria production.

Although few field studies have inferred simultaneous rates of viral lysis and HNF grazing in oligotrophic settings, bacteria mortality due to viruses appears to be a more variable process than HNF grazing [START_REF] Bonilla-Findji | Viral and Flagellate Control of Prokaryotic Production and Community Structure in Offshore Mediterranean Waters[END_REF][START_REF] Boras | Annual changes of bacterial mortality due to viruses and protists in an oligotrophic coastal (NW Mediterranean[END_REF][START_REF] Boras | Effect of viruses and protists on bacteria in eddies of www[END_REF]. For example, a time-series study conducted in a coastal oligotrophic system recorded enhanced rates of viral lysis with increasing bacterial abundance and production whereas concomitant rates of HNF grazing remained constant [START_REF] Boras | Annual changes of bacterial mortality due to viruses and protists in an oligotrophic coastal (NW Mediterranean[END_REF]. Factors responsible for increased viral proliferation are the relative host-virus abundance [START_REF] Murray | Viral dynamics -a model of the effects of size, shape motion and abundance of single-celled planktonic organisms and other particles[END_REF], these hosts' activity [START_REF] Middelboe | Bacterial growth rate and marine virus host dynamics[END_REF] and the induction of prophage (i.e., phages integrated into their host genome), ultimately leading to the lysis of lysogenic bacteria [START_REF] Paul | Prophages in marine bacteria: dangerous molecular time bombs or the key to survival in the seas?[END_REF][START_REF] Williamson | Seasonal variation in lysogeny as depicted by prophage induction in Tampa Bay, Florida[END_REF]. The second dust addition induced an enhanced viral production while bacterial respiration and community composition remained unchanged. This uncoupling between bacterial and viruses responses would favor the hypothesis of an induction of prophages. A significant proportion of inducible lysogenic bacteria was indeed reported in oligotrophic ecosystems [START_REF] Jiang | The abundance of lysogenic bacteria in marine microbial communities as determined by prophage induction[END_REF][START_REF] Paul | Prophages in marine bacteria: dangerous molecular time bombs or the key to survival in the seas?[END_REF] including the Mediterranean Sea [START_REF] Weinbauer | Lysogeny and virus induced mortality of bacterioplankton in surface, deep, and anoxic marine waters[END_REF]. Although the mechanisms that result in prophage induction are not well constrained, several studies suggested that this process might be governed by the changes in host metabolic status [START_REF] Williamson | Seasonal variation in lysogeny as depicted by prophage induction in Tampa Bay, Florida[END_REF][START_REF] Oppenheim | Switches in bacteriophage lambda development[END_REF]. Particularly relevant to our study, [START_REF] Boras | Annual changes of bacterial mortality due to viruses and protists in an oligotrophic coastal (NW Mediterranean[END_REF] evidenced that the addition of P to nutrient-poor waters could be a factor inducing lysogenic cells in natural bacterial communities. In the present study, the addition of P by dust addition could have induced prophage upon the second amendment, but the lack of response of the virus community to the first seeding remains puzzling. Altogether, these results indicate that dust deposition can cause substantial modifications of the trophic interactions within the microbial loop.

Summary and conclusions

Our results confirm previous evidence that heterotrophic bacteria play a crucial role within the connections between dust deposition and ocean carbon cycle. In oligotrophic systems, where C export in the dissolved form can be equal to or higher than particulate C export [START_REF] Carlson | Annual flux of dissolved organic carbon from the euphotic zone in the Northwestern Sargasso Sea[END_REF], the stimulation of bacterial respiration by dust pulses may decrease the amount of DOC susceptible to be exported to depth through winter mixing. In addition, by enhancing bacterial activity, dust deposition may modify bacterial processing of dissolved organic matter and, thus, its residence time in the water column, with consequences on the efficiency of the microbial carbon pump (MCP; [START_REF] Jiao | Microbial production of recalcitrant dissolved organic matter: long-term carbon storage in the global ocean[END_REF]. Process studies devoted to the bacterial transformation of DOM after dust pulses would allow the parameterization of the links between dust deposition and the MCP in the oligotrophic ocean.

Our study also shows that the dust deposition to the surface ocean can shape the trophic links among the components of the microbial food web. If, as indicated by our results, viral lysis is stimulated by dust addition, this would have crucial consequences on the cycling of DOC and the remineralization of nutrients. Indeed, the consequences of such an effect may include the re-mineralization and release of macro-and micro-nutrients to the water column and/or the decrease in the export of particulate C due to the conversion of organic matter from the particulate to the dissolved phase [START_REF] Fuhrman | Marine viruses and their biogeochemical and ecological effects[END_REF][START_REF] Middelboe | Bacterial growth rate and marine virus host dynamics[END_REF][START_REF] Wilhelm | Viruses as regulators of nutrient cycles in aquatic environments[END_REF].

Our study shows that the effects of dust deposition on ocean carbon cycle are not restricted to a simple response of autotrophic phytoplankton and/or heterotrophic bacteria. They may involve important modifications of the trophic links, particularly within the microbial food web, with crucial consequences on C and nutrient cycling.

Table 1 .a

 1 Initial biogeochemical conditions of seawater in the DUNE-R experiment (average for C-, D-mesocosms and OUT samples). DIP: dissolved inorganic phosphorus; DL: detection limit (0.03 µM for NO - 3 ). Values for chlorophyll a, DIP, NO - 3 and dFe data are average concentrations at surface, 5 m and 10 m depth. Value for primary production is the mean rate at 5 m depth.DUNE-RChlorophyll a, µg L -1 0.07 ± 0.02 a Primary production, mg C m -3 d -1 3.89 ± 0From Guieu et al. (2014), b from Ridame et al. (2014), c from Wuttig et al. (2013).

Figure 1 .

 1 Figure 1. Dissolved inorganic phosphorus (nM, mean ± standard deviation of three replicate mesocosms) in D-and C-mesocosms and OUT samples at surface (a), 5 m (b) and 10 m (c) depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 2 .

 2 Figure 2. Bacterial abundance (cell mL -1 , mean ± standard deviation of three replicate mesocosms) in D-and C-mesocosms and OUT samples at surface (a), 5 m (b) and 10 m (c) depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 3 .

 3 Figure 3. Bacterial respiration (µmol C L -1 d -1 , mean ± standard deviation of three replicate mesocosms) in D-and C-mesocosms and OUT samples at 5 m depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 4 .

 4 Figure 4. Changes in the relative abundance of total (a) and active (b) OTUs in response to dust addition at 5 m depth during the DUNE-R experiment. Positive changes indicate higher, and negative changes indicate lower relative abundance in dust than in control mesocosms. Changes in relative abundances are expressed as percent of the total community. Only OTUs with significantly different relative abundances in the dust and control-mesocosms are shown. See Material and Methods for more details.

Figure 5 .

 5 Figure 5. Diversity (Simpson index) of the total (squares) and active (triangles) bacterial community in D-(filled symbols) and Cmesocosms (empty symbols) at 5 m depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 6 .

 6 Figure 6. Virus abundance (particle mL -1 , mean ± standard deviation of three replicate mesocosms) in D-and C-mesocosms and OUT samples at surface (a), 5 m (b) and 10 m (c) depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 7 .

 7 Figure 7. Ratios virus and bacterial abundance (mean ± standard deviation of three replicate mesocosms) in D-and C-mesocosms and OUT samples at surface (a), 5 m (b) and 10 m (c) depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 8 .

 8 Figure 8. Heterotrophic nanoflagellate abundance (cell mL -1 , mean ± standard deviation of three replicate mesocosms) in dustand control-mesocosms and OUT samples at 5 m depth during the DUNE-R experiment. Grey vertical bars represent the dust seedings.

Figure 9 .

 9 Figure 9. Theoretical calculation of viral infected bacterial abundance (cell mL -1 , mean ± standard deviation of three replicate mesocosms) based on the dust-induced changes in VA (VAdust -VAcontrol) at surface (a), 5 m (b) and 10 m (c) depth during the DUNE-R experiment and a conservative burst size of 20[START_REF] Parada | Viral burst size of heterotrophic prokaryotes in aquatic systems[END_REF]. Grey vertical bars represent the dust seedings.

Biogeosciences, 11, 5607-5619, 2014 www.biogeosciences.net/11/5607/2014/

www.biogeosciences.net/11/5607/2014/ Biogeosciences, 11, 5607-5619, 2014

Acknowledgements. This work is a contribution of the DUNE experiment (a DUst experiment in a low Nutrient, low chlorophyll Ecosystem, http://www.obs-vlfr.fr/LOV/DUNE/index.html) and was funded by the LEFE, UPMC and ANR-DUNE under the contract ANR-07-BLAN-0126-01. DUNE was endorsed by the international SOLAS (Surface Ocean -Lower Atmosphere) program in February 2009, (http://www.solas-int.org/activities/project-endorsement.html). We would like to sincerely thank the Réserve naturelle de Scandola, Parc naturel régional de Corse, in particular J. M. Dominici and collaborators, for professionalism and cooperation in the implementation of the field work in the bay of Elbo. We thank all of the participants of the DUNE-2 experiment for their involvement. The service charges for this open access publication have been covered by CNRS-INSU and the Laboratoire d'Océanographie de Villefranche (LOV), CNRS UMR7093.

Edited by: E. Marañón


Naïma El Farouq
email: naima.elfarouq@univ-bpclermont.fr

Pierre Bernhard
email: pierre.bernhard@inria.fr

Proportional Transaction Costs in the Robust Control Approach to Option Pricing: the Uniqueness Theorem

Keywords: Option pricing, viscosity solutions, robust control

We prove the missing uniqueness theorem for the viscosity solution of a quasi-variational inequality related to a minimax impulse control problem modeling the option pricing with proportional transactions costs. This result makes our robust control approach of option pricing in the interval market model essentially complete.

Introduction

In a series of papers [START_REF] Bernhard | Une approche déterministe de l'évaluation d'options. Optimal Control and Partial Differential Equations[END_REF][START_REF] Bernhard | The robust control approach to option pricing and interval models: an overview[END_REF][START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF][START_REF] Bernhard | An impulsive differential game arising in finance with interesting singularities[END_REF] culminating with [START_REF] Bernhard | The Interval Market Model in Mathematical Finance: a Game Theoretic Approach[END_REF], we have developed a robust control, probabilityfree approach of option pricing, for both vanilla and digital european options. We claim that, on the one hand, we have the possibility of constructing a consistent theory of hedging portfolios with either continuous or discrete time trading paradigms, with one and the same market model, and, on the other hand, we can accommodate transaction costs and closing costs in a natural way, with a nontrivial hedging portfolio.

We also argue that the probabilistic description of future securities prices, as used by the classic approaches, imbeds much more information on these prices future histories than does the interval market model of the robust control approach which only assumes bounds on the rate of relative variation. As a consequence, the new approach is less sensitive to modelling uncertainties.

The last missing item in our theory was a uniqueness theorem for the viscosity solution of a particular, highly degenerate, Isaacs quasi-variational inequality (QVI). This degeneracy comes first from the fact that the QVI is of first order, so it cannot enjoy the ellipticity property [START_REF] Bensoussan | Contrôle impulsionnel et inéquations quasi-variationnelles[END_REF]. The other fact is that we assume that the transaction costs are proportional to the amount of transactions, so that the infimum of impulse costs is zero. We will show that the uniqueness of the viscosity solution of this QVI can not be guaranteed. We also recall the general result in [START_REF] El Farouq | Deterministic Minimax Impulse Control[END_REF] where uniqueness of the viscosity solution is proved under the assumption that the infimum of impulse costs is positive. That paper was motivated by our theory of option pricing where we consider that a fixed cost is charged for any transaction in addition to a variable part depending on the amount of the transaction.

In [START_REF] Bernhard | An impulsive differential game arising in finance with interesting singularities[END_REF], the authors replace the minimax impulse control problem with an equivalent problem without impulses. They prove that they have the same value function. In this paper, we deal with the associated differential QVI (DQVI). The technique in [START_REF] El Farouq | Degenerate first-order quasi-variational inequalities: an approach to approximate the value function[END_REF], where the author considers infinite-horizon time control problems where the infimum of impulse costs is zero is, in a large extent, different. The idea is to give a new QVI that admits the same viscosity subsolutions than the original one and to prove that the value function is the unique viscosity solution of the new QVI.

A first direct and different proof of uniqueness of the viscosity solution of our DQVI was presented in the conference paper [START_REF] Bernhard | A robust control approach to option pricing: the uniqueness theorem[END_REF].

In this article, we sketch the overall context and prove the uniqueness sought, with a new proof requiring less regularity on the data. Notice, however, that the present proof does not account for the discontinuous payment digital options, while the indirect argument of [START_REF] Bernhard | Singular surfaces in differential games, an introduction[END_REF] can be extended to that case, thanks to the concept of barrier.

We are obliged to recall the basics of our modelling of the problem to introduce the DQVI. The uniqueness of its viscosity solution is the object of this article. A more detailed exposition can be found in [START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF], where the uniqueness theorem is not proved, and the more recent [START_REF] Bernhard | The Interval Market Model in Mathematical Finance: a Game Theoretic Approach[END_REF], where a complete theory of hedging in the interval market model is developed.

Modeling

The robust control approach to option pricing

In our previous papers, we have introduced and discussed a probability free, robust control approach to option pricing, as follows.

Notation

We use the following notation:

• t ∈ [0, T ]: time, 0 = date when the option is sold, T = exercise time.

• u(t): the time varying, unpredictable, market price of the underlying security.

• v(t): the currency value of the risky part of the hedging portfolio, invested in the underlying.

• τ (t) = u(t)/u(t): action of the market.

• τ -< 0 and τ + > 0: fixed known parameters: lower and upper bounds of τ .

• Ψ = set of measurable functions from [0, T ] to [τ -, τ + ].
• ξ(t): control of the seller of the option, or "trader": the rate of buying (selling if negative) underlying stocks, assumed to be infinitely divisible and liquid.

• t k : control of the trader: time of k-th block buy or sale of underlying stock.

• ξ k : control of the trader: amount of the k-th block buy (if positive) or sale (if negative) of underlying stock.

• Ξ: set of time distributions defined over [0, T ] which are the sum of a measurable function ξ c (•) and a finite number of weighted translated Dirac impulses ξ k δ(t -t k ).

• ϕ ∈ Φ: strategy of the trader: ϕ : Ψ → Ξ : τ (•) → ξ(t) = ϕ(τ (•))(t) nonanticipative strategy including impulses that cause "jumps" ξ k .

• C + > 0 and C -< 0: fixed known parameters: the rates of the proportional transaction costs for buying resp. selling underlying stocks.

• c + ∈ [0, C + ] and c -∈ [C -, 0]: fixed known parameters: same as C + and C -but for closing costs.

•

N (u, v) = max{c ε (-v), u -K + c ε (u -v)}:
terminal cost to the trader, sum of contractual commitment and closing costs. (See notation c ε (a) below.)

• W (t, u, v): Value function of the minimax problem to be solved, giving the premium sought as P (u(0)) = W (0, u(0), 0).

Further, for Z ∈ {τ, C, c} the notation Z ε stands for (Z ε , ε ∈ {-, +}), and for any a ∈ R, Z ε a will mean Z sign(a) a. We also set

τ = max ε |τ ε | , C = min ε |C ε | . (1)

The minimax impulse control problem

This section refers to the results of [START_REF] Bernhard | The robust control approach to option pricing and interval models: an overview[END_REF][START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF][START_REF] Bernhard | An impulsive differential game arising in finance with interesting singularities[END_REF]] that we will need in the sequel. In these articles, we have shown that the premium sought can be obtained as the solution of the following minimax impulse control problem

Dynamics u = τ u , (2)
v = τ v + ξ c , (3) v(t + k ) = v(t - k ) + ξ k . (4)
Equivalently, we may describe v with equation (3) only, including the jumps (4) as impulses in ξ(•). Accordingly, from here on, we let ξ ∈ Ξ be the sum of a measurable function ξ c (•) and a finite number of weighted translated Dirac:

ξ(t) = ξ c (t) + k ξ k δ(t -t k ) .
Criterion The optimum (least safe) premium, or seller's price, is obtained as1

P (u 0 ) = inf ϕ∈Φ sup τ (•)∈Ψ N (u(T ), v(T )) + T 0 -τ (t)v(t) + C ε ξ(t) dt ,
where the time functions u(•) and v(•) are generated from u(0) = u 0 and v(0) = 0 by equations (2),(3) and ξ(•) = ϕ(τ (•)).

Solving the minimax impulse control problem 2.2.1 The related QVI and DQVI

In a classical fashion we introduce its Isaacs value function:

W (t, u, v) = inf ϕ∈Φ sup τ (•)∈Ψ N (u(T ), v(T )) + T t -τ (s)v(s) + C ε ξ(s) ds , (5)
where the dynamics are integrated from

u(t) = u, v(t) = v. Hence the seller's price is P (u 0 ) = W (0, u 0 , 0).
The natural QVI [3, pp 9-10] associated to our problem is

0 = max - ∂W ∂t + min τ ∈[τ -,τ + ] sup ξ∈R -τ ∂W ∂u u + τ 1 - ∂W ∂v v -ξ ∂W ∂v -C ε ξ , W (t, u, v) -min ξ∈R W (t, u, v + ξ) + C ε ξ , ∀(u, v) ∈ R 2 , W (T, u, v) = N (u, v) . (6)
Note that if

∂W ∂v + C -> 0, then by choosing ξ going to -∞, then -ξ ∂W ∂v -C -ξ will go to +∞
and this is in contradiction with QVI [START_REF] Bernhard | Singular surfaces in differential games, an introduction[END_REF]. If ∂W ∂v + C + < 0, then by choosing ξ going to +∞, then -ξ ∂W ∂v -C + ξ will go to +∞ and this is also in contradiction with QVI [START_REF] Bernhard | Singular surfaces in differential games, an introduction[END_REF]. We therefore necessarily have

C -≤ - ∂W ∂v ≤ C + . (7)
It is then easy to see that the sup ξ in this QVI is obtained for ξ = 0. So, QVI ( 6) is equivalent to [START_REF] Bernhard | A robust control approach to option pricing: the uniqueness theorem[END_REF] and the following new QVI

0 = max - ∂W ∂t + min τ ∈[τ -,τ + ] τ - ∂W ∂u u + 1 - ∂W ∂v v , W (t, u, v) -min ξ∈R W (t, u, v + ξ) + C ε ξ , ∀(u, v) ∈ R 2 , W (T, u, v) = N (u, v) .
There are new features in that game, in that, on the one hand, impulse controls are allowed, but, on the other hand, impulse costs have a zero infimum. It is easy to see that any function satisfies the viscosity supersolution inequality and that any constant function is a viscosity solution to this QVI, so we cannot prove that the value function of the impulse minimax problem is the unique viscosity solution of this QVI.

In [START_REF] Bernhard | An impulsive differential game arising in finance with interesting singularities[END_REF], we introduced the so-called "Joshua transformation" that changes the minimax impulse control problem into an equivalent problem without impulse controls. We briefly explain here this transformation and the results obtained in that paper.

A new artificial time s and a new trader's control j ∈ {-1, 0, 1} are introduced. Natural time t becomes a state variable that stops when j = 0. The jumps amplitudes |ξ k | are now the sizes of such s-intervals. Hence, jumps in v have been turned into continuous variations in artificial time. When j = 0, natural time and artificial time have equal variations. By noting with a prime the derivatives with respect to s, and by  = 1 -|j|, the dynamics of this game are

   t =  , u = τ u , v = (τ v + ξ) + j ,
In these dynamics, ξ(s) contains no impulse anymore. Let t(S) = T . The new cost function is

J(0, u(0), v(0), j(•), ξ(•), τ (•)) = S 0 (s)(-τ (s)v(s) + C ε ξ(s)) + C ε j ds + N (u(S), v(S)) .
This differential game is stationary. Therefore, its Value function does not depend on s. In [START_REF] Bernhard | An impulsive differential game arising in finance with interesting singularities[END_REF], we proved the relatively obvious fact that it is the same as the Value function of the original game:

W (0, u(0), v(0)) = inf j(.) inf ξ(•) sup τ (•) J(0, u(0), v(0), j(•), ξ(•), τ (•))
The Isaac's equation of this new problem is the following:

∀(t, u, v) ∈ [0, T [×R 2 , 0 = max j∈{-1,0,1} - ∂W ∂t  + min τ ∈[τ -,τ + ] sup ξ∈R - ∂W ∂v ξ -C ε ξ - ∂W ∂u τ u - ∂W ∂v τ v + j -C ε j + τ v , ∀(u, v) ∈ R 2 , W (T, u, v) = N (u, v) .
Successively replacing j by either 0 (in this case with the same reasoning as above, the sup in ξ is obtained for ξ = 0,) 1 and -1, this equation writes as the following "differential QVI" (DQVI):

∀(t, u, v) ∈ [0, T [×R 2 , 0 = max - ∂W ∂t + min τ ∈[τ -,τ + ] τ - ∂W ∂u u + 1 - ∂W ∂v v , - ∂W ∂v + C + , ∂W ∂v + C -, ∀(u, v) ∈ R 2 , W (T, u, v) = N (u, v) . (8)
Applying standard results of Differential Games (see e.g. [START_REF] Souganidis | Two-player, zero-sum differential games and viscosity solutions[END_REF]) to the Isaacs equation associated to the Joshua-transformed problem, we easily get the following fact:

Theorem 2.1 The function W defined by (5) is a continuous viscosity solution of DQVI [START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF].

But classic results such as [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF][START_REF] Bardi | Numerical methods for pursuit-evasion games via viscosity solutions[END_REF][START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations[END_REF][START_REF] Souganidis | Two-player, zero-sum differential games and viscosity solutions[END_REF] do not apply here to prove the uniqueness of a viscosity solution.

Uniqueness

In [START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF], we show that if the viscosity solution of the DQVI (8) can be proved unique, we have an interesting representation formula for the Value function of the continuous trading problem, a convergence theorem of the Value function W h of a discrete trading version of the problem to the Value function of the continuous trading problem as the step size h vanishes, and a fast algorithm to compute the Value function of the discrete trading problem and the associated optimal trading strategy. Hence the importance of proving that uniqueness.

In order to exploit the technical result of the next section, we need to introduce a modified problem.

Let R be a fixed positive number, and

R ⊂ R + × R be the region u ∈ [0, R], |v| ≤ R.
For the time being, we consider only problems of option hedging where (u(0), v(0)) ∈ R. As a consequence, for these problems, and for all t ∈ [0, T ], we have u(t) ≤ Re τ + T .

Concerning v(•), the control ξ might send it anywhere in R. But we know from the analysis according to the Isaacs-Breakwell theory that the minimizing strategies never create large v(t)'s. As a matter of fact, let W 0 be the maximum payoff obtained by the strategy ϕ = 0 (after maximization in τ (•)) for any (u(0), v(0)) = (u 0 , v 0 ) ∈ R. Let a be a large number, chosen satisfying a > 2 exp(τ + T )[W 0 /(RC ) + 1], and S = aR. We claim the following: Proposition 2.1 Any nonanticipative strategy ϕ that causes |v| > S is dominated by the strategy ϕ = 0.

Proof Let ζ be a positive number, ζ < C exp(-τ + T )/4τ + . Any nonanticipative strategy ϕ may be challenged by the control function generated by the following rule:

If v(t -ζ) > S, choose τ (t) = τ -if v(t -ζ) < -S choose τ (t) = τ + .
Due to the small time delay ζ, it does generate an admissible control function τ (•) against a nonanticipative strategy ϕ. Notice that, because ϕ has to be non-anticipative, if a strategy ϕ is to cause |v| > S, it also does so against that response of τ , which comes after the fact.

It is easy to check that whether we reach v = S or v = -S, from |v 0 | ≤ R, the cost C ε ξ(t)dt is larger than C (e -τ + T S -R).

Consider, for instance, the case v(t -ζ) ≥ S, under the rule proposed to generate τ (•), after a delay ζ, we shall have either of three possibilities:

1. v(t) > 0, thus v(t)τ (t) < 0, 2.
-S < v(t) < 0, we are back in the "small v" domain, (and the minimizer has used twice the expense to produce large excursions in v),

3. v(t) ≤ -S, we start a new sequence of the same analysis, with the opposite sign.

In the three cases, the benefit accrued to the minimizer from this large |v| is not more than

t t-ζ Se τ + (s-t+ζ) ds = S(exp(τ + ζ) -1) < 2Sτ + ζ < C exp(-τ + T )S/2 .
Hence, any such excursion in v costs the minimizer at least

C (exp(-τ + T )a/2 -1)R > W 0 .
Since the terminal cost N (u(T ), v(T )) is itself non negative, that strategy ϕ does less well than ϕ = 0.

As a consequence, for initial states in R, we may, without modifying the Value, restrict the set of admissible strategies to strategies that keep |v| ≤ S. 2 With these strategies, the term τ (t)v(t) dt is bounded. And therefore, as useful strategies ϕ should keep C ε ξ(t) dt -τ (t)v(t) ≤ W 0 , we can furthermore, without modifying the Value, restrict the admissible strategies to be such that C ε ξ(t) dt be also bounded, and therefore also |ξ(t)| dt. (Say, by (W 0 + 1 + τ ST )/C .)

Let Φ b the set of admissible non anticipative strategies thus restricted. We now modify the original problem as follows: let

P [a,b] be the projection of R on [a, b] ⊂ R and, N b (u, v) = N (P [0,e τ + T R] (u), P [-S,S] (v)) , L b (v) = P [-S,S] (v) . (9)
We keep the same dynamics as we had, and define the pay-off as

W b (0, u 0 , v 0 ) = inf ϕ∈Φ b τ ∈Ψ N b (u(T ), v(T )) + T 0 -τ L b (v(t)) + C ε ξ(t) dt . ( 10
)
We have modified the problem only for states outside [0, e τ + T R] × [-S, S], never reached from initial states (u 0 , v 0 ) in R. Hence in R, the value of the modified game coincides with that of the original game:

W b | R = W | R . (11)
The next proposition follows from standard results (see e.g. [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations[END_REF][START_REF] Souganidis | Two-player, zero-sum differential games and viscosity solutions[END_REF]).

Proposition 2.2

The function W b is a bounded continuous viscosity solution of the modified DQVI:

∀(t, u, v) ∈ [0, T [×R 2 , 0 = max - ∂W b ∂t + min τ ∈[τ -,τ + ] τ - ∂W b ∂u u - ∂W b ∂v v + L b (v) , - ∂W b ∂v + C + , ∂W b ∂v + C -, ∀(u, v) ∈ R 2 , W b (T, u, v) = N b (u, v) . (12)
We prove in the next section the following technical result.

Theorem 2.2 The DQVI (12) admits a unique bounded continuous viscosity solution.

Now, in view of the equality [START_REF] El Farouq | Degenerate first-order quasi-variational inequalities: an approach to approximate the value function[END_REF] and the fact that R could be chosen at will, all uniqueness results we need for the full theory follow.

Proof of the uniqueness theorem

We now set to prove theorem (2.2). We omit all indices b, but it should be understood all along that we are dealing with the modified problem.

The Comparison Theorem

Modified DQVI's

We first apply a classical transformation to DQVI [START_REF] El Farouq | Deterministic Minimax Impulse Control[END_REF] introducing

Γ(t, u, v) := e t W b (t, u, v) ,
which is bounded if and only if W b is.

Proposition 3.1 The function W b is a viscosity solution of ( 12) if and only if Γ is a viscosity solution of the modified DQVI:

∀(t, u, v) ∈ [0, T ) × R 2 , 0 = max - ∂Γ ∂t + Γ(t, u, v) + min τ ∈[τ -,τ + ] τ - ∂Γ ∂u u - ∂Γ ∂v v + e t L(v) , - ∂Γ ∂v -e t C + , ∂Γ ∂v + e t C -, ∀(u, v) ∈ R 2 , Γ(T, u, v) = e T N (u, v) . ( 13
)
Theorem 3.1 If U is a bounded and upper semi-continuous viscosity subsolution of ( 13) and V is a bounded lower semi-continuous viscosity supersolution of ( 13), then

U ≤ V in [0, T ] × R 2 .
We need the following Lemma for the proof of this theorem. Let us recall the definition of the quantities τ * and C * in equations (1).

Lemma 3.1 Let U be an upper semi-continuous viscosity subsolution of ( 13), then for any µ ∈ (0, 1), α > 0 and K satisfying the following conditions,

α < 2(1 -µ)C * , K > 2C * τ * + e T τ * L ∞ , the function
Z(t, u, v) = µU (t, u, v) - α 2 ln(1 + u 2 ) - α 2 ln(1 + v 2 ) -K(1 -µ)
is a strict viscosity subsolution of [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations[END_REF].

Proof Observe first that Z is an upper semi-continuous function.

Let Φ ∈ C 1 [0, T ) × R 2 and ( t, ū, v) a global maximum point of Z -Φ.
We have that for all (t, u, v)

∈ [0, T ) × R 2 , U ( t, ū, v) - 1 µ α 2 ln(1 + ū2 ) + α 2 ln(1 + v2 ) + Φ( t, ū, v) ≥ U (t, u, v) - 1 µ α 2 ln(1 + u 2 ) + α 2 ln(1 + v 2 ) + Φ(t, u, v) .
This inequality means that ( t, ū, v) is a global maximal point of U -Φ 1 where

Φ 1 (t, u, v) = 1 µ α 2 ln(1 + u 2 ) + α 2 ln(1 + v 2 ) + Φ(t, u, v) , then at t, ū, v , max - ∂Φ 1 ∂t + U + min τ ∈[τ -,τ + ] τ - ∂Φ 1 ∂u ū - ∂Φ 1 ∂v v + e tL(v) , - ∂Φ 1 ∂v -e tC + , ∂Φ 1 ∂v + e tC -≤ 0 , that is - ∂Φ ∂t + µU + min τ ∈[τ -,τ + ] τ - ∂Φ ∂u ū - αū 2 1 + ū2 - ∂Φ ∂v v - αv 2 1 + v2 + µe tL(v) ≤ 0 , (14)
- ∂Φ ∂v - αv 1 + v2 -µe tC + ≤ 0 , (15)
∂Φ ∂v + αv 1 + v2 + µe tC -≤ 0 . (16) Inequality (15) implies
- ∂Φ ∂v -e tC + ≤ (µ -1)e tC + + αv 1 + v2 ≤ (µ -1)C + + α 2 ≤ (µ -1)C * + α 2 .
Inequality (16) implies

∂Φ ∂v + e tC -≤ - αv 1 + v2 + (1 -µ)e tC -≤ (1 -µ)C -+ α 2 ≤ (µ -1)C * + α 2 . Let δ = (µ -1)C * + α 2
. Then, by choosing 0 < α < 2(1 -µ)C * , we get δ < 0. Inequality ( 14) implies

- ∂Φ ∂t + Z( t, ū, v)+ min τ ∈[τ -,τ + ] τ - ∂Φ ∂u ū - αū 2 1 + ū2 - ∂Φ ∂v v - αv 2 1 + v2 + µe tL(v) ≤ K(µ -1) , then - ∂Φ ∂t + Z( t, ū, v) + min τ ∈[τ -,τ + ] τ - ∂Φ ∂u ū - ∂Φ ∂v v + e tL(v) ≤ - ∂Φ ∂t + Z( t, ū, v) + min τ ∈[τ -,τ + ] τ - ∂Φ ∂u ū - αū 2 1 + ū2 - ∂Φ ∂v v - αv 2 1 + v2 + µe tL(v) + max τ ∈[τ -,τ + ] τ αū 2 1 + ū2 + αv 2 1 + v2 + (1 -µ)e tL(v) ≤ K(µ -1) + max τ ∈[τ -,τ + ] τ αū 2 1 + ū2 + αv 2 1 + v2 + (1 -µ)e tL(v) ≤ 2ατ * + (µ -1)(K -e T τ * L ∞ ) ≤ (µ -1)(K -2C * τ * -e T τ * L ∞ ) . Let δ = (µ -1)(K -2C * τ * -e T τ * L ∞ ). Choose K > 2C * τ * + e T τ * L ∞ , to get δ < 0.
Now, let V be a viscosity supersolution of (13), and let

R = exp U ∞ + V ∞ α , R = [0, T ] × (-R, R) × (-R, R) ,
and for any function

X : R → R, X R = sup t,u,v∈R |X(t, u, v)|. Then, for all (t, u, v) / ∈ R, we have Z(t, u, v) ≤ V (t, u, v). Indeed, for all (t, u, v) such that max{|u|, |v|} ≥ R, Z(t, u, v) ≤ (µ -1) U ∞ -V ∞ ≤ V (t, u, v) .

It remains to prove this inequality for

(t, u, v) ∈ R. Let M = sup (t,u,v)∈[0,T ]×R×R [Z(t, u, v) -V (t, u, v)] . If M is non-positive, then the proof is finished. If M is positive, then M = sup (t,u,v)∈R (Z -V ). Let us consider the test-function Φ γ : R × R → R: Φ γ (t, u, v, t , u , v ) = Z(t, u, v) -V (t , u , v ) - |u -u | 2 γ 2 - |v -v | 2 γ 2 - |t -t | 2 γ 2 .
Let ( t, ū, v, t , ū , v ) be the maximum point of Φ γ . It exists since on the one hand, this is a bounded and upper semi-continuous function on a bounded set, and on the other hand, it is negative in a neighborhood of the boundary |u| = R or |v| = R, and, by hypothesis, positive for some (t, u, v) = (t , u , v ). Thus the search for the maximum may be restricted to a compact set [0, T ] × [-R + , R -] 2 . Let M γ = Φ γ ( t, ū, v, t , ū , v ). We need the following lemma for the proof of Theorem 3.1.

Lemma 3.2 Assume that M > 0. Then, for any β > 0, there exists γ 0 such that, for any γ ≤ γ 0 ,

|ū -ū | 2 γ 2 + |v -v | 2 γ 2 + | t -t | 2 γ 2 ≤ β .
Proof By definition, ( t, ū, v, t , ū , v ) satisfies, for all (t, u, v), (t , u , v ) ∈ R

Z( t, ū, v) -V ( t , ū , v ) - |ū -ū | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 ≥ Z(t, u, v) -V (t , u , v ) - |u -u | 2 γ 2 - |v -v | 2 γ 2 - |t -t | 2 γ 2 .
Let us take (t, u, v) = (t , u , v ), then

M γ ≥ Z(t, u, v) -V (t, u, v) .
We then obtain M ≤ M γ . Let r 2 = Z R + V R . We have

0 < M ≤ M γ ≤ r 2 - |ū -ū | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 . Hence |ū -ū | ≤ γr, |v -v | ≤ γr and | t -t | ≤ γr . (17)
We also have that

M ≤ M γ = Z( t, ū, v) -V ( t , ū , v ) - |ū -ū | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 . ( 18
)
The function Z is upper semi-continuous, then for any β > 0, there exists γ 0 , such that for any γ ≤ γ 0 , using (17), we get

M ≤ Z( t, ū, v) -V ( t , ū , v ) ≤ Z( t , ū , v ) -V ( t , ū , v ) + β ≤ M + β . (19)
Using (18), we get the result of the lemma.

Let us now prove the theorem. Proof Case t and t smaller than T We recall that ( t, ū, v, t , ū , v ) satisfies

∀(t, u, v) ∈ R, Z( t, ū, v) -V ( t , ū , v ) - |ū -ū | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 ≥ Z(t, u, v) -V ( t , ū , v ) - |u -ū | 2 γ 2 - |v -v | 2 γ 2 - |t -t | 2 γ 2 .
Let

Φ 1 (t, u, v) = V ( t , ū , v ) + |u -ū | 2 γ 2 + |v -v | 2 γ 2 + |t -t | 2 γ 2 , Φ 1 ∈ C 1 (R).
This inequality means that ( t, ū, v) is a maximal point of Z(t, u, v) -Φ 1 (t, u, v). We also have, ∀(t , u , v ) ∈ R,

Z( t, ū, v) -V ( t , ū , v ) - |ū -ū | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 ≥ Z( t, ū, v) -V (t , u , v ) - |ū -u | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 . Let Φ 2 (t , u , v ) = Z( t, ū, v) - |ū -u | 2 γ 2 - |v -v | 2 γ 2 - | t -t | 2 γ 2 . Φ 2 ∈ C 1 (R). This inequality means that ( t , ū , v ) is a minimal point of V (t , u , v ) -Φ 2 (t , u , v ). Then - ∂Φ 1 ∂t + Z( t, ū, v) + min τ ∈[τ -,τ + ] τ - ∂Φ 1 ∂u ū - ∂Φ 1 ∂v v + e tL(v) ≤ δ < 0 , - ∂Φ 1 ∂v -e tC + ≤ δ < 0 , ∂Φ 1 ∂v + e tC -≤ δ < 0 .
These inequalities can be written as

-2( t -t ) γ 2 + Z( t, ū, v)+ min τ ∈[τ -,τ + ] τ - 2(ū -ū ) γ 2 ū - 2(v -v ) γ 2 v + e tL(v) ≤ δ < 0 , (20)
- 2(v -v ) γ 2 -e tC + ≤ δ < 0 , (21) 2
(v -v ) γ 2 + e tC -≤ δ < 0 . (22)
We also have

max - 2( t -t ) γ 2 + V ( t , ū , v )+ min τ ∈[τ -,τ + ] τ - 2(ū -ū ) γ 2 ū - 2(v -v ) γ 2 v + e t L(v ) , -2(v -v ) γ 2 -e t C + , 2(v -v ) γ 2 + e t C -≥ 0 .
Let β = 1 in Lemma 3.2 and use the fact that exp function is uniformly continuous over [0, T ], then one can find 0 < γ 1 ≤ γ 0 such that for all γ ≤ γ 1 , we have |e t -e t | < -δ/2C + , and likewise, 0 < γ 2 ≤ γ 0 such that for all γ ≤ γ 2 , we have |e t -e t| < δ/2C -. Using ( 21) and ( 22), we get, for all γ ≤ min(γ 1 , γ 2 ),

- 2(v -v ) γ 2 -e t C + ≤ δ 2 < 0 , 2(v -v ) γ 2 + e t C -≤ δ 2 < 0 .
We then get

-2( t -t ) γ 2 + V ( t , ū , v )+ min τ ∈[τ -,τ + ] τ - 2(ū -ū ) γ 2 ū - 2(v -v ) γ 2 v + e t L(v ) ≥ 0 . (23)
We subtract (23) to (20) and we get

Z( t, ū, v) -V ( t , ū , v ) ≤ max τ ∈[τ -,τ + ] τ 2(ū -ū ) 2 γ 2 + 2(v -v ) 2 γ 2 + e t L(v ) -e tL(v) ≤ max τ ∈[τ -,τ + ] τ 2(ū -ū ) 2 γ 2 + 2(v -v ) 2 γ 2 + e t (L(v ) -L(v)) + (e t -e t)L(v) .
We use again as previously Lemma 3.2 with β = 1 and the fact that L and the exp function are uniformly continuous to get, for all ε > 0, one can find 0 < γ 3 ≤ min(γ 1 , γ 2 ) and 0 < γ 4 ≤ min(γ 1 , γ 2 ) such that for all γ ≤ min(γ 3 , γ 4 ), we have |L(v ) -L(v)| ≤ ε and |e t -e t| < ε. Finally, we use Lemma 3.2 with arbitrary β to find 0 < γ 5 ≤ min(γ 3 , γ 4 ) such that for all γ ≤ γ 5 , we have 2(ū -ū ) 2 /γ 2 + 2(v -v ) 2 /γ 2 ≤ 2β. Then, for all γ ≤ γ 5 , we get Z( t, ū, v) -V ( t , ū , v ) ≤ 2βτ * + ετ * e T + L ∞ .

As β and ε are arbitrary, we get Z( t, ū, v) -V ( t , ū , v ) ≤ 0. Using (19), we get M ≤ 0 and then Z(t, u, v) ≤ V (t, u, v), ∀(t, u, v) ∈ [0, T ) × R 2 . We let α → 0 and µ → 1 to get U (t, u, v) ≤ V (t, u, v) for all ∀(t, u, v) ∈ [0, T ) × R 2 .

Case t or t equal to T In the case where t or t are equal to T , QVI (13) cannot be used, because it assumes t < T . Because of the fact that U is a viscosity subsolution of ( 13) and V is a viscosity supersolution of (13), we have the following inequality ∀(u, v) ∈ R 2 , U (T, u, v) ≤ e T N (u, v) ≤ V (T, u, v) .

We then get the result of the theorem.

Theorem 3.2 The quasi-variational inequality (13) has a unique bounded continuous viscosity solution.

Proof Existence follows from propositions 2.2 and 3.1. Now, assume that it has two bounded continuous viscosity solutions Γ 1 and Γ 2 . We first use Γ 1 as a bounded upper semi-continuous subsolution, and Γ 2 as a bounded viscosity supersolution, and we use the result of Theorem 3.1. We change the role of Γ 1 and Γ 2 and then we get Γ 1 (t, u, v) = Γ 2 (t, u, v) for all (t, u, v) ∈ [0, T ] × R 2 .

Corollary 3.1

1. Theorem 2.2 is proved.

2. The Value function of the continuous trading problem is the unique viscosity solution of (8).

Proof

1. Theorem 2.2 is a direct consequence of theorem 3.2 and of proposition 3.1.

2. The DQVI (8) has a viscosity solution by theorem 2.1. It is unique, because if it had two solutions W 1 and W 2 , with (t, u, v) such that W 1 (t, u, v) = W 2 (t, u, v), by taking R large enough in the modified game, we could manage to get (u, v) ∈ R, and therefore a contradiction of theorem 2.2.

As mentioned at the beginning of paragraph 2.2.2, as a consequence, the other results in [START_REF] Bernhard | Robust control approach to option pricing: representation theorem and fast algorithm[END_REF] that depend on this uniqueness are proved, most noticeably convergence of the Value function of the discrete trading problem to that of the continuous trading one, and therefore convergence of the fast algorithm toward the Value function sought.

According to the above, the integral term in ξ includes the terms k C ε k ξ k

We may notice that from a modelling viewpoint, we might as well have introduced such a constraint a priori, from market liquidity considerations.

Acknowledgment

The uniqueness proof owes much to discussions with Guy Barles, of University Franc ¸ois Rabelais, Tours, France.