Artem Danilov

Gleb Tselikov

Fan Wu

Vasyl G Kravets

Igor Ozerov

Frederic Bedu

Alexander N Grigorenko
email: alexander.grigorenko@manchester.ac.uk

Andrei V Kabashin
email: kabashin@lp3.univ-mrs.fr

Ultra-narrow surface lattice resonances in plasmonic metamaterial arrays for biosensing applications

Keywords: plasmonic surface lattice resonances, diffraction-coupled localized plasmon resonances, biosensor, attenuated total reflection, phase sensitivity, plasmonic metamaterials for biosensing

When excited over a periodic metamaterial lattice of gold nanoparticles (~ 100 nm), localized plasmon resonances (LPR) can be coupled by a diffraction wave propagating along the array plane, which leads to a drastic narrowing of plasmon resonance lineshapes (down to a few nm full-widthat-half-maximum) and the generation of singularities of phase of reflected light. These phenomena look very promising for the improvement of performance of plasmonic biosensors, but conditions of implementation of such diffractively coupled plasmonic resonances, also referred to as plasmonic surface lattice resonances (PSLR), are not always compatible with biosensing arrangement implying the placement of the nanoparticles between a glass substrate and a sample medium (air, water). Here, we consider conditions of excitation and properties of PSLR over arrays of glass substrate-supported single and double Au nanoparticles (~100-200 nm), arranged in a periodic metamaterial lattice, in direct and Attenuated Total Reflection (ATR) geometries, and assess their sensitivities to variations of refractive index (RI) of the adjacent sample dielectric medium. First, we identify medium (PSLRair, PSLRwat for air and water, respectively) and substrate-related (PSLRsub) modes corresponding to the coupling of individual plasmon oscillations at medium-and substrate-related diffraction cut-off edges. We show that spectral sensitivity of medium-related modes to RI variations is determined by the lattice periodicity in both direct and ATR geometries (~ 320 nm per RIU change in our case), while substrate-related mode demonstrates much lower sensitivity. We finally show that phase sensitivity of PSLR can exceed 10 5 Deg. of phase shift per RIU change and thus outperform the relevant parameter for all other plasmonic sensor counterparts. Combining advantages of nanoscale architectures, including drastic concentration of electric field, possibility of manipulation at the nanoscale etc, and high phase and spectral sensitivities, PSLRs promise the advancement of current state-of-the-art plasmonic biosensing technology toward single molecule label-free detection.

Introduction

For last decade plasmonic biosensing has become a technology of choice for label-free characterization of biomolecular binding interactions between a target analyte (antigens, DNA, ligand etc.) and its corresponding receptor (e.g. an antibody, DNA capture, protein etc.) immobilized on the gold surface [START_REF] Liedberg | Surface plasmon resonance for gas detection and biosensing[END_REF][START_REF] Liedberg | Biosensing with surface plasmon resonance -how it all started[END_REF], Schasfort et al., 2008[START_REF] Homola | Surface Plasmon Resonance Based Sensors[END_REF]. Plasmonic biosensors take advantage of a resonant dependence of conditions of plasmon excitation on refractive index (RI) of the medium contacting the plasmon-supporting metal. Conventional plasmonic biosensors employ the phenomenon of Surface Plasmon Resonance (SPR), which is associated with the excitation of surface plasmon polaritons over a thin (~ 50 nm) gold film using Attenuated Reflection Geometry (Kretschmann-Raether arrangement). SPR effect leads to a dip in the reflected intensity at a defined combination of the angle of incidence and the wavelength, whose values are resonantly dependent on the RI of a thin layer near gold [START_REF] Liedberg | Surface plasmon resonance for gas detection and biosensing[END_REF]. Any change of the thickness of biological film on gold due to mass accumulation is accompanied by a RI change, which can be monitored by following spectral (Zhang et al., 1988), angular [START_REF] Liedberg | Surface plasmon resonance for gas detection and biosensing[END_REF] or phase [START_REF] Kabashin | Interferometer based on a surface-plasmon resonance for sensor applications[END_REF][START_REF] Kabashin | Phase-polarisation contrast for surface plasmon resonance biosensors1This paper was presented at the Fifth World Congress on Biosensors[END_REF][START_REF] Kabashin | Surface plasmon resonance bio-and chemical sensors with phase-polarisation contrast[END_REF] characteristics of reflected light. With the lower detection limit of 1 pg/mm 2 of biomaterial accumulated on the biosensor surface, SPR biosensors are efficient for studies of many biomolecular interactions (Schasfort et al., 2008), but they are hardly compatible with current trends in biosensing focusing on the development of compact (Patskovsky et al., 2004, Pilarik et al., 2003, Nemova et al., 2007) and nanoscale [START_REF] Anker | Biosensing with plasmonic nanosensors[END_REF], Kabashin et al., 2009[START_REF] Sreekanth | Extreme sensitivity biosensing platform based on hyperbolic metamaterials[END_REF] transducer implementations.

A huge potential for the upgrade of plasmonic biosensing technology is now expected from localized plasmon resonances (LPRs) excited over nanoscale structures [START_REF] Anker | Biosensing with plasmonic nanosensors[END_REF], and metamaterial arrays (Kabashin et al., 2009). Such biosensors are much better compatible with nanoscale biochemical architectures (beacons, hybrid structures etc.), as well as can offer novel functionalities such as size-based selectivity, possibility for a drastic localization and concentration of electric field to profit from SERS and other field-enhanced effects (Nie et al., 2008), resolution beyond the diffraction limit [START_REF] Kawata | Plasmonics for near-field nano-imaging and superlensing[END_REF], nanotweezing [START_REF] Grigorenko | Nanometric optical tweezers based on nanostructured substrates[END_REF], etc. Plasmonic Surface Lattice Resonances (PSLR) produced in conditions of diffraction coupling of LPR are of particular interest for biosensing applications due to their exceptional resonance quality [START_REF] Grigorenko | Nanometric optical tweezers based on nanostructured substrates[END_REF][START_REF] Auguie | Collective resonances in gold nanoparticle arrays[END_REF][START_REF] Chu | Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays[END_REF]. To excite PSLR, gold nanoparticles (nanodics, nanopillars etc.) having the size of about 100 nm are arranged in a 2D lattice in such a way that one of diffracted beams, appearing due to the periodicity of this structure, propagates over the array plane and can couple in far field localized plasmons over individual nanoparticles (Fig. 1a).

Such a coupling leads to a drastic narrowing of resonances in reflected and transmitted light down to 2-3 nm full width at half maximum (FWHM). In addition, the diffraction coupling gives rise to vanishing of light intensity in resonances, leading to the generation of singularities of light phase [START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF][START_REF] Kravets | Singular phase nano-optics in plasmonic metamaterials for label-free single-molecule detection[END_REF]. When used as a signal parameter to monitor refractive index variations, such phase singularities can be used to lower the detection limit of label-free plasmonic biosensing schemes down to single molecule level [START_REF] Kabashin | Interferometer based on a surface-plasmon resonance for sensor applications[END_REF](Kabashin et al., , 2009;;[START_REF] Huang | Phase-sensitive surface plasmon resonance biosensors: Methodology, instrumentation and applications[END_REF].

However, the excitation of diffraction-coupled PSLR is critically dependent on refractive index of the media surrounding a nanoparticle array. When the array is illuminated under normal incidence of light, the excitation of PSLR typically requires uniform surrounding, i.e. the match of refractive index of the substrate and that of the medium contacting the particles (Auguie et al., 2010), although certain narrowing of resonances is possible via a proper choice of size and geometry of nanoparticles [START_REF] Chu | Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays[END_REF], Trackray et al., 2014). On the other hand, the excitation of PSLR becomes possible under oblique incidence of light on the array structure and the monitoring of optical parameters in reflected light [START_REF] Kravets | Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles[END_REF][START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF]. In all cases, direct geometry of PSLR excitation is not fully compatible with biosensing experimental arrangement, as it implies light direction through a sample liquid leading to a dependence of PSLR parameters on bulk RI fluctuations inside the flow cell. Alternatively, PSLR can be excited in Attenuated Total Reflection (ATR) geometry [START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF], but conditions of excitation and sensitivity of such resonances

have not yet been systematically studied.

In this paper, we compare conditions of PSLR excitation in direct and ATR geometries and examine sensitivities of different PSLR modes under these conditions. Our analysis shows that PSLR can be excited in both geometries and exhibit distinct resonance features with the spectral width of a few nm FWHM and high resonance quality. We also show that the PSLR can provide spectral sensitivity correlating with the periodicity of the structures and exceptionally high phase sensitivity, which makes them very promising candidates for biosensing applications.

Methodology

Sample preparation

High-quality regular and homogenous square arrays of gold nanoparticles were produced by e-beam lithography (LEO-RAITH) on a clean microscopic glass substrate covered by a thin Cr sublayer. A double layered resist was used to improve lift-off (80 nm of 495 kD PMMA cast from a 3 wt% solution in anisole for the bottom resist layer and 50 nm of 95 kD PMMA cast from a 2 wt% solution in anisole for the top layer). After lithography we deposited 5nm of Cr (to improve adhesion) and 80-90nm Au by electron beam evaporation, which was followed by the lift-off procedure. The typical array size was 0.2*0.2 mm 2 . The samples on a clean glass substrate were obtained from the samples fabricated on a 5 nm Cr sublayer in which the Cr sublayer has been wet-etched after the fabrication procedure. Typical Scanning Electron Microscopy images of single and double nanodot arrays are shown in Fig. 1(b,c), respectively.

Optical measurement setup

In our experiments, we used commercially available M-2000 ellipsometer system (Woollam Inc, USA), which makes possible independent sample rotation and displacement, relative to illumination and collection light paths, thus granting the possibility for precise alignment of focal plane with respect to the examined area on a sample. Xenon wide spectrum lamp was used as a light source. The sample was illuminated with weakly focused light (about 0.5 mm spot diameter). Device output data represented in standard for ellipsometry Ψ-Δ values so that Ep/Es = tan(Ψ)exp(iΔ), where Ep and Es are the reflected field amplitudes for the incident light Ei of p and s polarizations and their values are Rp=|Ep/Ei| and Rs=|Es/Ei| of p and s polarizations, respectively.

A homemade liquid flow cell was designed for sensitivity measurements. The cell contained a rubber O-ring with input and output holes fixed between two cover slides for direct illumination measurements and a cover slide with 45-45-90 deg. glass prism (n=1.5) on the other side for attenuated total reflection (ATR) configuration. In ATR geometry, the light beam was passed through a prism and reflected from its opposite facet where nanoparticle arrays were deposited.

Results and discussions

Conditions of excitation and properties of PSLR in direct and ATR geometries

Under normal light illumination PSLR can be normally excited only under symmetric environment surrounding nanoparticle metamaterial arrays [START_REF] Auguie | Collective resonances in gold nanoparticle arrays[END_REF](Auguie et al., , 2010)), which prevents the implementation of biosensing schemes as the substrate and tested media typically have essentially different RI (nsub ~ 1.5 compared to nwat = 1.33 and nair = 1). However, this problem can be solved, e.g., by employing oblique light incidence [START_REF] Kravets | Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles[END_REF][START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF]. In this case, PSLR can be efficiently excited even under asymmetry of optical environment, but they split into two modes corresponding to the coupling of localized plasmons via diffracted waves propagating in the tested medium or the substrate (Fig. 1a). The reason of this splitting is straightforward. For any regular nanoparticle array, diffracted beam disappears when it crosses the boundary between ambient medium (air or water) and the substrate due to the impossibility of transition between media having essentially different light dispersion. As a result, the diffracted beam is cut-off at Rayleigh cut-off wavelength (diffraction edge). It is obvious that if a medium/substrate interface is present, there should be two cut-off wavelengths. One of them corresponds to the disappearance of the medium diffraction modes (when the diffraction wave crosses the sample boundary from air/water to the substrate) and other one is related to crossing the boundary in the opposite direction. Such two Rayleigh cut-off wavelengths for air/substrate interface can be presented as follows [START_REF] Kravets | Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles[END_REF]:

𝜆 𝑅 𝑎𝑖𝑟 = 𝑎/𝑚(𝑛 𝑎𝑖𝑟 ± 𝑠𝑖𝑛𝜃), 𝜆 𝑅 𝑠𝑢𝑏 = 𝑎/𝑚(𝑛 𝑠𝑢𝑏 ± 𝑠𝑖𝑛𝜃) (1)
where a is a period of the structure, m is integer, θ is the angle of incidence, nsub is substrate RI.

Similarly, for water/substrate interface Rayleigh cut-off wavelengths will be:

𝜆 𝑅 𝑤𝑎𝑡 = 𝑎/𝑚(𝑛 𝑤𝑎𝑡 ± 𝑠𝑖𝑛𝜃), 𝜆 𝑅 𝑠𝑢𝑏 = 𝑎/𝑚(𝑛 𝑠𝑢𝑏 ± 𝑠𝑖𝑛𝜃) (2)
The effect of splitting of resonances can be easily seen in experiments. PSLR typically appear in reflection as narrow Fano-shape dips over a positive background spectrum [START_REF] Kravets | Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles[END_REF][START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF], Nikitin et al., 2013), presenting the combination of reflected and scattered light from glass-supported nanoparticle array system. However, when measured by It is obvious that negative or positive polarity of PSLRs in Ψ(λ) spectrum is determined whether resonant dips take place for p-or s-polarized component of light, respectively, or whether the substrate reflection adds or subtracted from the reflection produced by the array. In general, the width and shape of PSLR critically depend on a variety of factors, including nanoparticle size, geometry of unit cell (single or double particles), refractive index of the environment. In addition, the emergence of different modes and their intensity strongly depend on the angle of light incidence.

Here, different structures appear to provide optimal resonance characteristics under different conditions. As an example, the single 163-nm particle array provides a very distinct medium-related mode under its contact with air medium (Fig. 2a), while the same mode is almost absent in the case of its contact with water ambience (Fig. 3a). The substrate-related mode for this sample demonstrates the opposite tendency, as it appears to be much more pronounced in the case of water medium. As another example, the double 134-nm particle array provides much more pronounced medium-related mode under its contact with water medium (Fig. 2b, Fig. 3b), while the substrate-related mode shows the opposite tendency. As shown in figure insets presenting some resonances at a magnified scale, the width of PSLR is also strongly dependent on array parameters and range from a few nm FWHM to 20-25 nm FWHM. In all cases, the implementation of ultra-narrow and high quality resonances requires the optimization of array structures and their adaptation for concrete biosensing architectures (air, water). Fig. 2c, 3c show optical energy dispersion E(kx), which is constructed on the basis of experimental data (combinations of the resonance angle of incidence and wavelength).

In fact, such dispersion summarizes conditions of excitation of medium-and substrate related modes in direct geometry for air and water environments, respectively. One can find that the dispersion curves E(kx) for both medium (PSLRair, PSLRwat) and substrate (PSLRsub) modes have a characteristic descending course, which is related to a delocalized nature of diffraction-coupled plasmon surface lattice resonances (Odom et al., 2011(Odom et al., , 2014)).

One of critically important characteristics of resonant phenomena is related to light intensity in the very minimum of the resonance (light darkness). If the intensity approaches to zero, it can lead to singularities of phase of reflected light [START_REF] Grigorenko | Phase jumps and interferometric surface plasmon resonance imaging[END_REF], which can be used to much improve the sensitivity of plasmonic biosensing [START_REF] Kabashin | Interferometer based on a surface-plasmon resonance for sensor applications[END_REF][START_REF] Kabashin | Phase-polarisation contrast for surface plasmon resonance biosensors1This paper was presented at the Fifth World Congress on Biosensors[END_REF](Kabashin et al., , 2009)). As follows from Fig. 2 and 3, all PSLRs are accompanied by phase jumps, shown by dashed curves, in the very minima/maxima of the resonance. For example, the generation of air mode PSLRair for single 163 nm particle array (Fig. 2a) and water mode PSLRwat for double 134 nm particle array (Fig. 2b), manifested as a dip and peak in Ψ(λ) dependence, respectively, is accompanied by extremely sharp phase jumps at the resonance points with a total phase variation around 180 Deg. Note that the generation of sharp phase jumps at resonant peaks and not dips does not contradict to our previous explanations of phase topological properties [START_REF] Grigorenko | Phase jumps and interferometric surface plasmon resonance imaging[END_REF], Kravets et al., 2014). Indeed, the appearance of PSLR as peaks in spectral dependence for Ψ(λ) is explained by a particular way of presentation of ellipsometric data, as reflectivity of s-polarized component finds itself in the equation denominator. In normal reflection, this resonance should obviously appear as a minimum for s-polarized component.

Our experiments showed that all phenomena related to the excitation of PSLR over essentially asymmetric substrate medium interface can be efficiently reproduced in ATR geometry implying the illumination of the nanoparticle arrays from the glass substrate side. Furthermore, characteristics of such PSLR can be advantageous compared to the direct geometry. Fig. 4 and5 show the excitation of PSLR in ATR geometry using the same arrays: single 163 nm particle array (panels a) and double 134 nm particle array (panels b), under their contact with air and water environments, respectively. Here, similarly to direct geometry, one can observe ultra-narrow PSLR modes near Rayleigh cut-off wavelengths, while the resonances equally appear as minima (e.g., PSLRair and PSLRwat for single 163 nm particle and double 134 nm particle arrays) or maxima (PSLRsub for the 163 nm particle array contacting air, and PSLRsub for the double 134 nm particle array contacting water). As in the case of direct geometry, PSLRs are generated exactly at Rayleigh cut-off wavelengths for single nanoparticle arrays (Fig. 4a, 5a), while for double nanoparticle arrays the position of PSLR can be much shifted (Fig. 5b). The width and quality of resonances also critically depends on nanoparticle size, geometry of unit cell (single or double particles), refractive index of the environment. As an example, the array of single 163-nm particles demonstrates a prominent medium related mode under its contact with air (Fig. 4a), while this resonance is very weak under the contact of this structure with water ambience (Fig. 5a). Based on statistics of PSLR excitation in ATR geometry using various samples, we can conclude that single nanoparticle arrays demonstrate much better medium resonances under their contact with air ambience, while double nanoparticle arrays can provide high quality medium-related resonances for both air and water ambiences. As shown in insets to Fig. 4 and Fig. 5, the widths of PSLR in the ATR geometry are in average smaller compared with this parameter in direct geometry and can range from a few nm to 15-20 nm FWHM. Similarly to the case of direct geometry, the generation of PSLRs is accompanied by sharp phase features and the sharpness of phase jump is determined by the depth of resonances.

As an example, very sharp phase features with total phase variation close to 180 Deg. are observed for medium related mode PSLRair using single 163 nm particle array (inset of Fig. 4a) and for PSLRair, PSLRwat using double 134 nm particle arrays (insets of Fig. 4b, Fig. 5b). As shown in Fig.

Sensitivity of PSLRs to local environment

Ultimate biosensing experiment implies the functionalization of the plasmonic nanoparticles and immobilization of a sensor recognition element on them. A target analyte (affinity partner) comes from aqueous ambience and is supposed to bind to the recognition element leading to an increase of refractive index of a thin layer around the nanoparticles, while a relevant change of conditions of PSLR excitation (spectral, phase) enables one to monitor the RI increase caused by the biological interaction. In this case, the resulting response of biosensor system is determined by both optical sensitivity of the plasmonic transducer to RI variations and the efficiency of the recognition element.

Since the main objective of this work is related to the development of optical transducer based on PSLR, we use a physical model to simulate RI changes in order to simplify the analysis and avoid artificial facts related to the performance of the recognition element. To simulate the increase of RI of water ambience, aqueous solutions of ethanol of different concentrations were pumped through a flow cell by a peristaltic pump and brought into contact with the nanoparticle arrays, while PSLR were excited in both direct and ATR geometries. A block containing the metamaterial slide with nanoparticle arrays and the flow cell was put onto the ellipsometer platform and examined.

Figure 6a shows spectral shifts of resonances due to changes of refractive index of the aqueous medium caused by the addition of different concentration of ethanol to water for an array of single 190-nm nanoparticles. It is visible that an increase of refractive index (RI) due to the ethanol addition leads to a red shift of PSLR and this shift is nearly linear. However, sensitivities of medium and substrate-related modes appear to be quite different. Here, the slope of the linear dependence for the medium mode PSLRwat evidences the sensitivity of 357 nm/RIU, while the sensitivity of substrate mode is much lower (200 nm/RIU). Statistical analysis of sensitivities of medium and substrate modes for various samples shows that this parameter does not depend whether single or double nanoparticle arrays are used and is in the range of 310-360 nm/RIU and 190-200 nm/RIU for PSLRwat and PSLRsub, respectively. In general, sensitivity of medium modes PSLRwat for different samples strongly correlates with the period of structures (320 nm), similarly to other transducers based on diffraction phenomena such as e.g., surface plasmon polaritons over nanohole thin film arrays [START_REF] Brolo | Surface plasmon sensor based on the enhanced light transmission through arrays of nanoholes in gold films[END_REF], while sensitivity of the substrate mode PSLRwat does not show any correlation with the array period and is much lower.

A more complicated situation takes place for sensitivities of medium and substrates modes in ATR geometry. Here, the value of sensitivity of the medium mode starts to depend whether its excitation takes place before or after the angle of total internal reflection (TIR) for glass/water interface (~ 63º Deg.) Indeed, as follows from Fig. 6b, sensitivity of PSLRwat before the TIR angle (~46º Deg.) is also of the order of the period of the structure (305 nm/RIU for double 134 nm particle arrays), while for angles of incidence after TIR (73º Deg.) this parameter can exceed 400 nm/RIU.

Note that such a difference of sensitivities was recorded for a largely dominating majority of samples of single and double nanodot arrays, independently of the geometry of the unit cell. Here, the samples typically demonstrate sensitivity in the range of 300-340 nm/RIU before the TIR angle and improved sensitivity of 400-420 nm/RIU after the TIR angle. On the other hand, as follows from 6b, substraterelated mode appears to be weakly sensitive to variations of RI (50-55 nm/RIU) independently of angle of incidence (before or after TIR).

It is clear that sensitivities of PSLR modes can be obtained within diffraction theory, but a detailed consideration of the phenomenon should take into account not only the dependence of diffraction edge position on RI of the environment, but also accompanying effects such as a certain mismatch of actual PSLR positions with that of the Rayleigh cut-off wavelengths and evanescent near-fields in the case of ATR geometry. However, in the first approximation, this sensitivity should correlate with the sensitivity of the diffraction edge. The position of this Rayleigh cut-off wavelength for water and substrate modes in direct geometry can be derived as follows:

±𝜆 𝑅_𝐷𝑖𝑟 𝑊𝑎𝑡 = 𝑎𝑛 𝑊𝑎𝑡 (𝑠𝑖𝑛𝜃 ± 1) ±𝜆 𝑅_𝐷𝑖𝑟 𝑆𝑢𝑏 = 𝑎(𝑛 𝑤𝑎𝑡 𝑠𝑖𝑛𝜃 ± 𝑛 𝑠𝑢𝑏 )
In direct geometry, sensitivities of medium PSLRwat and substrate PSLRsub modes to variations of refractive index of the aqueous medium can be expressed as follows:

𝑆 𝑑𝑖𝑟 (𝑃𝑆𝐿𝑅 𝑤𝑎𝑡 ) = 𝜆 𝑅_𝐷𝑖𝑟 𝑤𝑎𝑡 𝑛 =  𝑛 [𝑎𝑛(𝑠𝑖𝑛𝜃 ± 1)]= 𝑎 (𝑠𝑖𝑛𝜃 ± 1 + 𝛿𝑠𝑖𝑛 𝛿𝑛 ) 𝑆 𝑑𝑖𝑟 (𝑃𝑆𝐿𝑅 𝑠𝑢𝑏 ) = 𝜆 𝑅_𝐷𝑖𝑟 𝑆𝑢𝑏 𝑛 =  𝑛 [𝑎(𝑛𝑠𝑖𝑛𝜃 ± 𝑛 𝑠𝑢𝑏 )]= 𝑎 (𝑠𝑖𝑛𝜃 + 𝑛 𝛿𝑠𝑖𝑛 𝛿𝑛 )
where n is refractive index of ethanol/water mixture. Since the angle of light incidence at water/substrate interface θ can be derived through the angle of light incidence  at the air/water interface (this interface is located at the flow cell entrance window) by the Snell's law nairsin= nsin, two latter equations can be rewritten as:

𝑆 𝑑𝑖𝑟 (𝑃𝑆𝐿𝑅 𝑤𝑎𝑡 ) = 𝑎 (𝑠𝑖𝑛𝜃 ± 1 - 𝑛 𝑛 𝑎𝑖𝑟 𝑠𝑖𝑛 𝑛 2 ) = 𝑎(𝑠𝑖𝑛𝜃 ± 1 -𝑠𝑖𝑛𝜃) = ±𝑎 𝑆 𝑑𝑖𝑟 (𝑃𝑆𝐿𝑅 𝑠𝑢𝑏 ) = 𝑎 (𝑠𝑖𝑛𝜃 - 𝑛 𝑛 𝑎𝑖𝑟 𝑠𝑖𝑛 𝑛 2 ) = 𝑎(𝑠𝑖𝑛𝜃 -𝑠𝑖𝑛𝜃) = 0
Thus, according to simplified diffraction theory, sensitivity of the medium mode (PSLRwat)

to RI in direct geometry is supposed to be of the order of the period of the structure (a nm/RIU), while the substrate mode should be insensitive to RI variations.

Similarly, sensitivities of medium PSLRwat and substrate PSLRsub modes in ATR geometry are mainly determined by positions of corresponding Rayleigh cut-off wavelength:

±𝜆 𝑅_𝐴𝑇𝑅 𝑊𝑎𝑡 = 𝑎(𝑛 𝑠𝑢𝑏 𝑠𝑖𝑛𝜃 ± 𝑛 𝑊𝑎𝑡 ) ±𝜆 𝑅_𝐴𝑇𝑅 𝑆𝑢𝑏 = 𝑎𝑛 𝑠𝑢𝑏 (𝑠𝑖𝑛𝜃 ± 1)
Therefore, these sensitivities can be expressed as follows:

𝑆 𝐴𝑇𝑅 (𝑃𝑆𝐿𝑅 𝑤𝑎𝑡 ) = 𝜆 𝑅_𝐴𝑇𝑅 𝑤𝑎𝑡 𝑛 =  𝑛 [𝑎(𝑛 𝑠𝑢𝑏 𝑠𝑖𝑛𝜃 ± 𝑛)] = ±𝑎 𝑆 𝐴𝑇𝑅 (𝑃𝑆𝐿𝑅 𝑠𝑢𝑏 ) = 𝜆 𝑅_𝐴𝑇𝑅 𝑆𝑢𝑏 𝑛 =  𝑛 [𝑎𝑛 𝑠𝑢𝑏 (𝑠𝑖𝑛𝜃 ± 1)] = 0
Here, again, sensitivity of the medium mode (PSLRwat) in ATR appears to be linked to the period of the structure (a nm/RIU), while the substrate mode should be insensitive to these variations.

Our experimental assessment of sensitivity of PSLRwat mode in direct geometry (Fig. 6a) and ATR geometry before TIR angle (Fig. 6b) confirm these theoretical predictions, as these sensitivities are in the range of 310-360 nm/RIU and 340-340 nm/RIU, respectively. However, a higher sensitivity of this mode in ATR geometry (400-420 nm/RIU) under angles exceeding the TIR angle is a pleasant surprise taking into account that this sensitivity does not follow from diffraction theory considerations. We believe that this effect is related to the involvement of evanescent wave under TIR, which can additionally react on RI variations. Another interesting issue is related to the observation of some dependence of substrate related modes PSLRsub on RIU variations for direct and ATR geometries (190-200 nm/RIU and 55-55 nm/RIU, respectively), although diffraction theory can not predict these dependencies. We believe that such a discrepancy of experiment and theory is related to too simplified model used in our calculations. In this model, we considered the sensitivity of the diffraction edge, which is not far from actual resonances for oblique angles, but still does not match them. Nevertheless, even if the deviation from Rayleigh wavelength is small, it changes fast with n, angle, etc, as the observed PSLR are very dispersive (Fig. 2c, 3c, 4c, 5c), providing non-zero sensitivities for substrate modes. In addition, the proposed theory does not take into account the excitation of evanescent wave under illumination of arrays at angles larger than 63º Deg. using ATR geometry, which can explain unexpectedly high sensitivity of medium mode under these conditions (Fig. 6 b).

Let us now consider phase sensitivity of plasmonic surface lattice resonances. It should be noted that this parameter does not directly correlate with the spectral sensitivity. Indeed, as we showed above, spectral sensitivity of PSLRwat is directly linked to the period of used metamaterial arrays and does not depend on width of depth of generated On the other hand, phase sensitivity is independent of the period of the arrays, but is determined by the sharpness of phase jump, which in turn is critically dependent on the depth of resonances ("light darkness" in the resonant minimum) [START_REF] Grigorenko | Phase jumps and interferometric surface plasmon resonance imaging[END_REF]. As an example, one can expect high phase sensitivity for double 134-nm particle arrays, which provide very deep resonances under ATR geometry leading to extreme phase features (Fig. 4b and 5b, respectively).

As shown in Figure 6c, the replacement of 80% solution of ethanol (80% ethanol / 20% water) by 100% ethanol solution, corresponding to a change of RI by 4.4•10 -3 RIU, leads to a dramatic shift of phase curve (Fig. 6c) and a huge (~205º Deg.) change of phase at the optimal wavelength of about 899 nm (inset to Fig. 6c). Based on this data, we can now estimate phase sensitivity, which appears to be equal to 5.7•10 4 Deg. of phase per RIU. Taking into account the level of noises of phasesensitive detection schemes, we can then estimate the low detection limit (LOD), which characterizes minimal variation of RI detectable by an optical transducer. It is clear that LOD should be very high for ellipsometry, as commercial ellipsometers are typically designed for optical characterization of films and not for sensing tests, and thus have a high level of noises. However, as shown in our previous study (Kabashin et al., 2009), resolution of phase measurements using an advanced photoacoustic modulator-based scheme of phase detection and a thermally stabilized flow cell system can be better than 5x10 -3 Deg. Under this phase resolution, projected LOD is estimated to be lower than 8•10 -8 RIU, which is better than in commercial amplitude-sensitive SPR units by almost 2 orders of magnitude (Lindberg et al., 1995). Notice that the minimal variation of RI that could be introduced into our system using calibrated ethanol-water mixtures was of the order of 4•10 -3 RIU.

Smaller variations of RI could not be resolved due to too high level of noises, conditioned by liquid pumping system and flow cell. As shown in Fig. 6c, such a variation of RI already exceeded the dynamic range of phase measurements, which inevitably lowered our estimation of phase sensitivity.

In our previous study [START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF], we assessed phase sensitivity of similar nanoparticle array metamaterials using a gas model, which made possible the introduction of much small variations of RI of the medium by employing a calibrated mixture of gases with known difference of RI (Ar and N2). In this case, the system was in the optimal dynamic range for the whole set of phase measurements, while the recorded sensitivity exceeded 10 6 Deg. of phase shift per RIU, giving access to ultrasensitive measurements of RI with the detection limit lower than 10 -9 RIU.

Thus, ultra-narrow PSLRs over ordered nanodot array metamaterials can exhibit fairly good sensitivity (300-400 nm/RIU), conditioned by the periodicity of used structures and extremely high phase sensitivity, conditioned by a very low light intensity in the resonance minimum (nearly complete light darkness). It is clear that a much reduced spectral width of PSLRs (2-20 nm FWHM) compared to LPR (80-100 nm FWHM) and SPR (50 nm FWHM) can be used to improve the precision of biosensing tests using plasmonic biosensors. To take into account the sharpness of the resonance and thus examine system efficiency to sensitively measure small wavelength changes, one normally uses a characteristic "Figure of Merit" (FOM) parameter [START_REF] Mayer | Localized surface plasmon resonance sensors[END_REF]:

FOM=(/n)(1/), where  is the width of resonance at FWHM and  is the resonance shift for a n refractive-index change. The essence of FOM is straightforward: it adequately quantifies the sensing potential of plasmonic modes in configurations similar to those used in commercial instruments. Typical FOMs do not exceed 8 and 23 for the sensors based on LPR [START_REF] Anker | Biosensing with plasmonic nanosensors[END_REF] and SPR (Zhang et al., 1988), respectively. The employment of PSLRs enables one to increase these parameters up to 10, 100 and more. In particular, in our experiments FOM reached 183.8, which is in a good agreement with our earlier works there it reached ~ 200-250 [START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF]. As another illustration, it was recently shown [START_REF] Offermans | Universal scaling of the figure of merit of plasmonic sensors[END_REF] that PSLRs in ordered metamaterial arrays of gold nanoparticles can offer a sensing performance that is at least an order of magnitude improvement over uncoupled LPRs associated with disordered particle arrays. Other studies experimentally demonstrated a high-performance RI-sensitive sensor based on PSLR, while its FOM reached 38 normal incidence (Li et al., 2015;Shen et al., 2015). However, the main advantage offered by PSLR is related to their phase sensitivity, which can be extremely high (over 10 6 Deg. of phase shift per RIU change). In general, the combination of ultrahigh point sensitivity (provided by phase interrogation) and fairly high sensitivity for a wider dynamic range of measurements (provided by spectral interrogation) looks like a very attractive basis for the development of versatile biosensing platform based on PSLR for characterization of biomolecular interactions.

The reported spectral sensitivity of PSLRs (300-400 nm/RIU) is comparable with the best values of sensitivity reported for alternative nanoperiodic plasmonic transducers, including uncoupled plasmonic arrays (200-300 nm/RIU) [START_REF] Anker | Biosensing with plasmonic nanosensors[END_REF] and nanohole arrays (300-400 nm/RIU) [START_REF] Brolo | Surface plasmon sensor based on the enhanced light transmission through arrays of nanoholes in gold films[END_REF][START_REF] Im | Template-stripped smooth Ag nanohole arrays with silica shells for surface plasmon resonance biosensing[END_REF], but almost one order of magnitude lower than in the case of SPR using Kretschmann-Raether prism arrangement (~2000-4000 nm/RIU) (Zhang et al., 1988). However, this tendency is not valid for phase sensitivities, which depend on "light darkness" in resonances and manifest themselves quite differently. As an example, even after a rigorous optimization and the thickness of thin Au films, SPR does not render possible intensities lower than ~5% in the resonance minimum compared to off-resonance conditions (Kabashin et al., 2009). As a result, the jump of phase of the light is considerably smoothed out, such a phase smoothing effect limits phase sensitivity of this approach by (5-10)•10 4 Deg. of phase per RIU. The employment of diffractively coupled PSLR enables one to achieve extremely low light intensities (down to 10 -6 %) [START_REF] Kravets | Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles[END_REF][START_REF] Kravets | Sensitivity of collective plasmon modes of gold nanoresonators to local environment[END_REF][START_REF] Kravets | Singular phase nano-optics in plasmonic metamaterials for label-free single-molecule detection[END_REF] and thus obtain extreme phase sensitivities exceeding 10 6 Deg.

per RIU change. Thus, despite a considerable handicap of PSLR in terms of amplitude sensitivity, the use of phase sensitivity completely compensates for this. To the best of our knowledge, PSLRs outperform all plasmonic counterparts in terms of phase sensitivity.

CONCLUSIONS

We established conditions of excitation and properties of diffraction-coupled plasmonic surface lattice resonances over metamaterial arrays of single and double Au nanoparticles in direct and ATR geometries. We first identified the possibility for excitation of medium and substrate-related modes, corresponding to the coupling of individual plasmon oscillations at medium-and substraterelated diffraction cut-off edges. We also determined that spectral sensitivity of medium modes is conditioned by the periodicity of used structures and is of the order of 300-400 nm/RIU, while substrate-related modes have much lower sensitivity. Finally, we found that PSLR can exhibit

  Fig. 2 and Fig. 3 present spectral dependencies of ellipsometric parameters Ψ and Δ for light reflected from a 320-nm period metamaterial arrays: an array of single 163 nm gold particles (panels (a)) and an array of double 134-nm particles having ~50 nm separation (panels (b)), contacting with air and water media, respectively. It is visible that the resonances are indeed split into medium-and substrate-related modes (PSLRair/PSLRwat and PSLRsub, respectively) and correlate with Rayleigh cut-off wavelengths 𝜆 𝑅 𝑎𝑖𝑟 / 𝜆 𝑅 𝑤𝑎𝑡 and 𝜆 𝑅 𝑠𝑢𝑏 , which are indicated as vertical yellow lines on the spectra. It should be noted that for single nanoparticle arrays diffraction-coupled PSLRs match Rayleigh cut-off wavelengths (Fig. 2a, 3a), while for double nanoparticle arrays the position of PSLRs can remarkably shift from these wavelengths (Fig. 2b, 3b) due to near-field coupling in the nanoparticle dimer structure.

  ellipsometry and presented as a function of ellipsometric reflection Ψ, which describes a ratio of amplitudes of p-and s-components upon reflection tan 𝛹 = |𝑅 𝑝 | |𝑅 𝑠 | , the resonances can appear as both minima (PSLRair for single and double nanoparticle arrays, Fig. 2, a,b) and maxima (PSLRwat for double nanoparticle arrays, Fig. 3b; PSLRsub for single and double nanoparticle arrays, Fig. 2a, 3b).

  Fig. 4c, 5c, optical dispersion E(kx) curves for medium (PSLRair and PSLRwat) and substrate (PSLRsub) modes demonstrate the same descending trend as it was observed in direct geometry. It should be noted that arrays of single 190-nm particles reveal a negative dispersion for PSLRair mode, which could be related to too large size of gold nanoparticles and possible involvement of complex plasmon modes (e.g., quadruple ones). Thus, ATR geometry makes possible the excitation of ultra-narrow diffraction-coupled plasmonic surface lattice resonances over arrays of gold nanodots contacting both air and water ambiences, while characteristics of these resonances can be optimized by a proper selection of array parameters. Our analysis shows that PSLR excited under ATR can be even narrower than relevant resonances under direct geometry.

  extremely high phase sensitivity due to a very low light intensity in the resonance minimum (nearly complete light darkness). Combining ultrahigh point sensitivity (provided by phase interrogation) and fairly high sensitivity for a wider dynamic range of measurements (provided by spectral interrogation) and a series of advantages of nanoscale metamaterial architecture, the proposed sensor transducer based on PSLR can serve as a basis for the development of versatile platform for ultrasensitive characterization of biomolecular interactions.

Fig. 1

 1 Fig. 1 (a) Schematics of excitation of diffraction-coupled plasmonic surface lattice resonances (PSLR) over a metal nanoparticle array. When metal nanoparticles are arranged in a periodic lattice, they may scatter light to produce diffracted waves. If one of the diffracted waves then propagates in the plane of the array, it may couple localized plasmon resonances associated with individual nanoparticles leading to the generation of PSLR with a very narrow lineshape. In the presence of a substrate, this phenomenon takes place at Rayleigh cut-off frequencies for the medium (𝜆 𝑅 𝑎𝑖𝑟 or 𝜆 𝑅 𝑤𝑎𝑡 ) and substrate (𝜆 𝑅 𝑠𝑢𝑏 ); Typical Scanning electron Microscopy images of single (a) and double nanoparticle arrays used for the excitation of PSLRs.

Fig. 2

 2 Fig. 2 Spectral dependencies of ellipsometric reflectivity Ψ (solid) and phase Δ (dashed) for a 320-nm period arrays of glass-supported gold nanoparticles contacting with air medium under different angles of their illumination in direct geometry: (a) single 163-nm particle array; (b) double 134 nm particle array with the distance between the nanoparticles 50 nm. Vertical yellow lines show positions of Rayleigh cut-off wavelengths associated with diffraction edges for air and substrate media (𝜆 𝑅 𝑎𝑖𝑟 and 𝜆 𝑅 𝑠𝑢𝑏 , respectively). The insets show some resonances at magnified scale. (c) Dispersion curves describing conditions for the generation of resonances (combinations of the resonance angle of incidence and wavelength) constructed on the basis of experimental data for air-and substrate-related modes PSLRair (solid) and PSLRsub (dashed)

ACKNOWLEDGEMENTS

AD, GT, AVK acknowledge the support from A*MIDEX project (No. ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the ANR, and LASERNANOCANCER and GRAVITY projects of ITMO "Plan Cancer 2014-2019" INSERM program. rANG acknowledges the support from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 696656 "GrapheneCore1", the EPSRC (EP/K011022/1) and SONY research agreement. Thackray, B.D., Kravets, V.G., Schedin, F., Auton, G., Thomas, P.A., Grigorenko, A.N., 2014. Narrow Collective Plasmon Resonances in Nanostructure Arrays Observed at Normal Light Incidence for Simplified Sensing in Asymmetric Air and Water Environments. ACS Photonics 1, 1116-1126. doi:10.1021/ph5002186 Zhang, L.-M., Uttamchandani, D., 1988. Optical chemical sensing employing surface plasmon resonance. Electron. Lett. 24, 1469-1470. doi:10.1049/el:19881004 Zhou, W., Hua, Y., Huntington, M.D., Odom, T.W., 2012. Delocalized lattice plasmon resonances show dispersive quality factors. J. Phys. Chem. Lett. 3, 1381-1385. doi:10.1021/jz300318v

Zhou, W., Odom, T.W., 2011. Tunable subradiant lattice plasmons by out-of-plane dipolar interactions. Nat. Nanotechnol. 6, 423-427. doi:10.1038/nnano.2011.72


Avram Florin
email: florin.avram@univ-pau.fr

A PONTRYAGHIN MAXIMUM PRINCIPLE APPROACH FOR THE OPTIMIZATION OF DIVIDENDS/CONSUMPTION OF SPECTRALLY NEGATIVE MARKOV PROCESSES, UNTIL A GENERALIZED DRAW-DOWN TIME

Keywords: first passage, draw-down process, spectrally negative process, scale functions, dividends, dividends barrier optimization, de Finetti, optimal harvesting, variational problem

), this framework provides a natural unification of draw-down and classic first passage times.

We illustrate here the potential of this unified framework by solving a variation of the De Finetti problem of maximizing expected discounted cumulative dividends/consumption gained under a barrier policy, until an optimally chosen Azema-Yor time, with a general spectrally negative Markov model.

While previously studied cases of this problem [APP07, SLG84, AS98, AVZ17, AH18, WZ18] assumed either Lévy or diffusion models, and the draw-down function to be fixed, we describe, for a general spectrally negative Markov model, not only the optimal barrier but also the optimal draw-down function. This is achieved by solving a variational problem tackled by Pontryaghins maximum principle. As a by-product we show that in the Lévy case the classic first passage solution is indeed optimal; in the diffusion case, we obtain the optimality equations, but the existence of solutions improving the classic ones is left for future work.

1. A brief review of the classic spectrally negative first passage theory 2 2. Generalized draw-down stopping for processes without positive jumps 5 3. Optimal dividends barrier problem for spectrally negative Markov processes with generalized draw-down stopping 6 4. A variational problem for de Finetti's optimal dividends until a generalized draw-down time, with a bound on the initial and total draw-down/regret area 7 5. Solving the de Finetti Markovian variational problem by Pontryaghin's minimum principle 8 5.1. Optimality Without Area Constraints 8 5.2. Optimality With Area Constraints 10 6. Back to the Lévy case: De Finetti's solution is optimal 11 7. Optimal dividends for functions of a Lévy process 13 7.1. Optimal dividends for functions of Brownian motion with drift 13 7.2. Geometric (logarithmic) Brownian motion 14 References 15

1. A brief review of the classic spectrally negative first passage theory Control of dividends/optimal consumption and capital injections. Many control problems in risk theory concern versions of X t which are reflected/constrained/regulated at first passage times (below or above):

X [a t = X t + L t , X b] t = X t -U t . (1)
Here,

L t = L [a t = -(X t -a) -, X t = inf 0≤t ≤t X t , U t = U b] t = X t -b + , X t := sup 0≤t ≤t X t ,
are the minimal 'Skorohod regulators' constraining X t to be larger than a, and smaller than b, respectively, and we use the notation x + = max(x, 0) and x -= min(x, 0).

Financial management and other applications require also studying the running maximum and the process reflected at its maximum/drawdown

Y t = X t -X t , X t = sup 0≤t ≤t X s ,
as well as the running infimum and the process reflected from below/drawup

Y t = X t -X t , X t = inf 0≤t ≤t X s .
The first passage times of the reflected processes, called draw-down/regret time and draw-up time, respectively, are defined for d > 0 by

(2) τ d := inf{t ≥ 0 : X(t) -X(t) ≥ d}, τ d := inf{t ≥ 0 : X(t) -X(t) ≥ d}.

Such times turn out to be optimal in several stopping problems, in statistics [START_REF] Page | Continuous inspection schemes[END_REF] in mathematical finance/risk theory and in queueing. More specifically, they figure in risk theory problems involving dividends at a fixed barrier or capital injections, and in studying idle times until a buffer reaches capacity in queueing theory-see for example [Tay75, Leh77, SS93, AKP04, MP12, SXZ08, Car14, LLZ17a, LLZ17b]-for results and references to numerous applications of draw-downs and draw-ups.

Optimization of dividends. One important optimization problem, going back to Bruno de Finetti [dF57], is to estimate the maximal expected discounted cumulative dividends of a financial company until its ruin.

Solving this is nontrivial even for Lévy process without positive jumps (a "multi-band" continuation region may be necessary); therefore, given the usual data uncertainty inherent to real problems, it is reasonable to restrict to simpler dividends policies which distribute all surpluses above a fixed level b, called dividends barrier. For fixed b, we arrive then to the optimal dividends barrier problem with ruin stopping.

Note that the time of ruin of the "Skorohod regulated process" X b] t = X t -U t may be decomposed as:

T b] 0 := T 0,-1 T 0,-<T b,+ + τ b 1 T b,+ <T 0,-. (3)
The "de Finetti barrier problem" consists in maximizing over b the present value of all dividend payments at the barrier b, until the time

T b] 0 : (4) V b (x) = V b q (x) := E x T b] 0 0 e -qt d(X t -b) + .
In the case of a spectrally negative Lévy process X t , the value function (4) and many other results may be expressed in terms of the scale function W q [START_REF] Bertoin | Lévy processes[END_REF][START_REF] Kyprianou | Fluctuations of Lévy Processes with Applications: Introductory Lectures[END_REF]. In preparation for the spectrally negative Markov case, we will also express the value function in terms of the logarithmic derivative

ν q (x) := W q (x+) W q (x) (5)
since it became apparent in [START_REF] Landriault | A unified approach for drawdown (drawup) of timehomogeneous markov processes[END_REF][START_REF] Avram | A unified analysis of taxed draw-down spectrally negative markov processes[END_REF] that for spectrally negative Markov processes it is more convenient to introduce first a natural extension ν q (x, y) (defined via the limit (12)) of the logarithmic derivative than the corresponding extension W q (x, y).

The Lévy factorization of the "gambler's winning/survival" probability may also be written as

E x e -qT b,+ 1 {Tb,+<Ta,-} = W q (x -a) W q (b -a) = e -b x νq(s-a)ds . (6)
Applying now the strong Markov property in (4) yields

V b (x) = E x e -qT b,+ ; T b,+ < T 0,-E b T b] 0 0 e -qt d(X t -b) (7) = W q (x) W q (b) V b (b) = W q (x) W q (b) W q (b) W q (b) = e -b x νq(m)dm 1 ν q (b) ,
where we have used (6) and [START_REF] Kyprianou | Distributional study of de finetti's dividend problem for a general lévy insurance risk process[END_REF][START_REF] Albrecher | Linking dividends and capital injections-a probabilistic approach[END_REF]. To understand the last equality, note that the dividends starting from b equal the local time spent at the reflecting boundary b, and that the latter has an exponential law, the rate of which is ν q (b), since the function ν q (x) is the rate of downwards excursions strictly larger than x, and occurring before an exponential horizon of rate q [Ber98, Don05].

V b (b) = 1 ν q (b) cf. [Ber98,
We will make use below of the fact that ν q (x) is nonincreasing and that

ν q (x) ≥ Φ q (8)
where Φ q is the unique positive root of the Cramér-Lundberg equation [START_REF] Bertoin | Lévy processes[END_REF][START_REF] Kyprianou | Fluctuations of Lévy Processes with Applications: Introductory Lectures[END_REF]

κ(s) := Log E 0 [e sX 1 ] = q. (9)
Assumption 1. To be able to write below equations like ν q (x) = W q (x) Wq(x) and formulas like (39) we will assume throughout the paper that W q (x) is three times differentiable in the Lévy case. In the spectrally negative Markov case, we will assume the scale function W q (x, y) (see last section) to be three times differentiable in x, or, alternatively, ν q (x, y) := ∂Wq (x,y) ∂x Wq(x,y) will be assumed to be twice differentiable.

See [START_REF] Chan | Smoothness of scale functions for spectrally negative lévy processes[END_REF] for more information on the smoothness of scale functions for Lévy processes, and note this problem has not yet been studied for spectrally negative Markov processes.

In conclusion, the Lévy De-Finetti barrier objective has a simple expression in terms of either the W q scale function or of ν q :

(10)

V b (x) = Wq(x) W q (b) = e -b x νq(m)dm 1 νq(b) , x ≤ b, V b (x) = x -b + V b (b), x > b .
where the second line follows upon completing the barrier strategy by "reduce holdings to b when above". Maximizing over the reflecting barrier b is simply achieved by finding the roots of

W q (b * ) = 0 ⇔ ∂ ∂b 1 ν q (b * ) = 1. (11)
This is a smooth fit equation at b * (see (10)). ¶ Our paper replaces ruin in the de Finetti dividends barrier optimization by more general Azema-Yor /generalized draw-down stopping times, to be chosen optimally. Then, the appropriate tool is the calculus of variations/optimal control. Let us note a recent related paper using general drawdown stopping times [START_REF] Wang | General draw-down based de finetti optimization for spectrally negative Lévy risk processes[END_REF], who study optimality of barrier policies under a fixed prespecified draw-down function.

First passage theory for spectrally negative Markov processes. Prior to [START_REF] Landriault | A unified approach for drawdown (drawup) of timehomogeneous markov processes[END_REF], the classic and draw-down first passage literatures were restricted mostly to parallel analytic treatments of the two particular cases of diffusions and of spectrally negative Lévy processes. [START_REF] Landriault | A unified approach for drawdown (drawup) of timehomogeneous markov processes[END_REF] showed that a direct unified approach (inspired by [START_REF] John P Lehoczky | Formulas for stopped diffusion processes with stopping times based on the maximum[END_REF] in the case of diffusions) may achieve the same results for all time homogeneous Markov processes; the known results for diffusions and spectrally negative Lévy processes are just particular cases of general formulas, once expressed in terms of W, Z, or of the differential exit parameters ν, δ -see below.

Assume the existence of differential versions of the ruin and survival problems:

Assumption 2. For all q, θ ≥ 0 and u ≤ x fixed, assume that Ψ b q (x, a) and Ψ b q,θ (x, a) are differentiable in b at b = x, and in particular that the following limits exist:

(12) ν q (x, a) := lim ε↓0 1 -Ψ x+ε q
(x, a) ε (total infinitesimal hazard rate) and (13) δ q,θ (x, a) := lim ε↓0 Ψ x+ε q,θ (x, a) ε (infinitesimal spatial killing rate).

Remark 1. It turns out that everything reduces to the differentiability of the two-sided ruin and survival probabilities as functions of the upper limit. Informally, we may say that the pillar of first passage theory for spectrally negative Markov processes is proving the existence of ν, δ. § Later, the differential characteristics ν, δ were extended in [START_REF] Avram | A unified analysis of taxed draw-down spectrally negative markov processes[END_REF] to the case of generalized draw-down times, which unify classic first passage times and draw-down times.

Since results for spectrally negative Lévy processes (like the de Finetti problem considered here) require often not much more than the strong Markov property, it was natural to attempt to extend them to the spectrally negative strong Markov case. As expected, everything worked out almost smoothly for "Lévy -type cases" like random walks [START_REF] Avram | First passage problems for upwards skip-free random walks via the phi, w, z paradigm[END_REF], Markov additive processes ¶ The equivalence between the two de-Finetti optimality conditions may be checked by differentiating W q (b * ) = Wq(b * )νq(b * ), which yields 0 = W q (b * ) = Wq(b * )(ν q (b * ) + ν 2 q (b * )). § ν and δ capture the behavior of excursions of the process away from its running maximum.

[IP12], and Lévy processes with Ω state dependent killing [START_REF] Ivanovs | Occupation densities in solving exit problems for Markov additive processes and their reflections[END_REF].

However, diffusions and spectrally negative Lévy processes were always tackled by different methods until the pioneering work [START_REF] Landriault | A unified approach for drawdown (drawup) of timehomogeneous markov processes[END_REF], who showed that certain draw-down problems could be treated by a unified approach, inspired by [START_REF] John P Lehoczky | Formulas for stopped diffusion processes with stopping times based on the maximum[END_REF] in the case of diffusions, which can be extended to all time homogeneous Markov processes.

When switching to spectrally negative Markov processes, W q (x) must be replaced by a two variables function W q (x, y) (which reduces in the Lévy case to W q (x, y) = W q (x -y), with W q being the scale function of the Lévy process).

However, the existence of W , as well as that of the scale function Z, are not obvious in the non-Lévy case, and it becomes more convenient to replace them by differential versions ν and δ defined by( 13), (13) below.

Computing ν, δ, W, Z is still an open problem, even for simple classic processes like the Ornstein-Uhlenbeck and the Feller branching diffusion with jumps. However, one may cut through this Gordian node by restricting to processes for which the limits defining ν, δ exist, and leaving to the user the responsibility to check this for their process. With this caveat, the results of [START_REF] Landriault | A unified approach for drawdown (drawup) of timehomogeneous markov processes[END_REF][START_REF] Avram | A unified analysis of taxed draw-down spectrally negative markov processes[END_REF] provide a unifying umbrella for spectrally negative Lévy processes, diffusions, branching processes(including with immigration), logistic branching processes, etc. Surprisingly, all these processes which were traditionally studied separately, may be viewed as particular cases of a unified general first passage theory for spectrally negative strong Markov processes!

In this paper we illustrate the potential of this framework via one application, a variation of the de Finetti problem of maximizing expected discounted cumulative dividends, where we replace stopping at ruin by an optimally chosen Azema-Yor/generalized draw-down stopping time.

Contents. We start by reviewing in Section 1 the classic spectrally negative first passage theory, and in Section 2 the first passage theory with generalized draw-down /Azema-Yor stopping times. Section 3 introduces the de Finetti dividends optimization problem with generalized draw-down /Azema-Yor stopping times for spectrally negative Markov processes. Section 4 spells out the calculus of variations problem to be solved, and Section 5 offers its solution via a Pontryaghintype approach.

Section 6 presents a detailed analysis of the particular case of Lévy processes. Finally, Section 7 considers a more general class of diffusions (general functions of drifted Brownian motion with particular emphasis on logarithmic cases).

Generalized draw-down stopping for processes without positive jumps

Generalized draw-down times appear naturally in the Azema-Yor solution of the Skorokhod embedding problem [START_REF] Azéma | Une solution simple au probleme de skorokhod[END_REF], and in the Dubbins-Shepp-Shiryaev, Peskir and Hobson optimal stopping problems [START_REF] Lester E Dubins | Optimal stopping rules and maximal inequalities for bessel processes[END_REF][START_REF] Peskir | Optimal stopping of the maximum process: The maximality principle[END_REF][START_REF] Hobson | Optimal stopping of the maximum process: a converse to the results of peskir[END_REF]. Importantly, they allow a unified treatment of classic first passage and draw-down times (see also [START_REF] Avram | A unified analysis of taxed draw-down spectrally negative markov processes[END_REF] for a further generalization to taxed processes)-see [START_REF] Avram | On draw-down stopping times for taxed spectrally negative lévy processes[END_REF][START_REF] Li | General drawdown times[END_REF]. The idea is to replace the upper side of the rectangle by a parametrized curve

(x, y) = ( d(s), d(s)), d(s) = s -d(s),
where s = x + y represents the value of X t during the excursion which intersects the upper boundary at (x, y). See Figure 1, where we put d(s) := s -d(s).

Alternatively, parametrizing by

x yields y = h(x), h(x) = (l) -1 (x) -x (note Y t ≥ d(X t ) ⇔ Y t ≥ h(X t )).
Definition 2. [AY79, LVZ17] For any function d(s) > 0 such that d(s) = s-d(s) is nondecreasing, a generalized draw-down time is defined by

(14) τ d := inf{t ≥ 0 : Y t > d(X t )} = inf t ≥ 0 : X t < d(X t ) . Figure 1. Exit of (X, Y ) from a trapezoid with a = 0, d(x) = (1 -ξ)x + d = 1 3 x + 1 Introduce Y t := Y t -d(X t ), t ≥ 0
to be called draw-down type process. Note that we have Y 0 = -d(X 0 ) < 0, and that the process Y t is in general non-Markovian. However, it is Markovian during each negative excursion of X t , along one of the oblique lines in the geometric decomposition sketched in Figure 1.

Example 3. With affine functions

(15) d(x) = (1 -ξ)x + d ⇔ d(x) = ξx -d, ξ ∈ [0, 1] ⇔ h(x) = (1 -ξ)x + d ξ , ξ ∈ [0, 1], d ≥ 0,
we obtain the affine draw-down/regret times studied in [START_REF] Avram | On draw-down stopping times for taxed spectrally negative lévy processes[END_REF]. Affine draw-down times reduce to a classic draw-down time (2) when ξ = 1, d(x) = d, and to a ruin time when ξ = 0, d(x) = -d, d(x) = x + d. When ξ varies, we are dealing with the pencil of lines passing through (x, y) = (-d, d). In particular, for ξ = 1 we obtain an infinite strip, and for ξ = 0, d = 0, we obtain the positive quadrant (this case corresponds to the classic ruin time).

One of the merits of affine draw-down times is that they allow unifying the classic first passage theory with the draw-down theory [START_REF] Avram | On draw-down stopping times for taxed spectrally negative lévy processes[END_REF]. A second merit is that they intervene in the variational problem considered below.

Optimal dividends barrier problem for spectrally negative Markov processes with generalized draw-down stopping

Consider now the extension of de Finetti's optimal dividend problem (16)

V b (x) = V b q, d(.) (x) := E x τ b] d 0 e -qt d(X t -b) ,
where

τ b]
d denotes a generalized draw-down time for the process X b] t reflected at b. Note that V b depends now also on the "spatial killing function" d(.).

Remark 4. This definition assumes that the initial point satisfies X 0 = X 0 = x, i.e. that the starting point is on the x axis in figure 1.

The strong Markov property yields again an explicit decomposition formula

(17) V b (x) = E x e -qT b,+ 1 T b,+ <τ b] d V b (b).
Furthermore, by [START_REF] Avram | A unified analysis of taxed draw-down spectrally negative markov processes[END_REF]Thm1] it holds that (18)

E x e -qT b,+ 1 T b,+ <τ b] d = e -b x νq(z, d(z))dz ,
where ν q (x, d(x)) is defined in (12). Concerning the expectation of the dividends starting from the barrier v(b) = V b (b), one may show again via standard bounding arguments (see for example [CKLP18, Sec 4]) that

(19) v(b) = v q (b, d(b)) := E b T d 0 e -qt d(X t -b) = ν q (b, d(b)) -1 .
Note that in the Lévy case, using x -d(x) = d(x), the equations above simplify to:

V b (x) = W q (d(x)) W q (d(b)) ν q (d(b)) -1 , which checks with [WZ18, Lem. 3.1-3.2].

4.

A variational problem for de Finetti's optimal dividends until a generalized draw-down time, with a bound on the initial and total draw-down/regret area Let us consider now de Finetti's optimal dividends with draw-down stopping. Suppose X 0 = X 0 = a ≥ 0, and view the total trapezoidal area A(b) = b a √ 2d(s)ds between the green and blue lines, in which the bivariate process (X t , Y t ) is allowed to evolve, as a measure of risk. With no upper bounds on A(b), the optimum will be b = ∞. We set therefore an upper limit A(b) ≤ K √ 2, and also an upper limit d 0 = d(a) on the initial maximum regret. Using ( 17

ν q (b * , d(b * )) + ν q (b * , d(b * )) ν q (b * , d(b * )) = 0, (22)
which implies the classic smooth-fit equation (11). In the Lévy case, the optimal draw-down boundary turns out to be De Finetti's d(y) = const, and thus the smooth-fit equation at b * determines completely the solution.

Solving the de Finetti Markovian variational problem by Pontryaghin's minimum principle

As usual in modern calculus of variations, we let u(t) denote the derivative of d(t), and reformulate the problem as (23)

                                   V (a, d(a)) := inf u J b (a, d, u), where J b (a, d, u) := b a ν q (t, t -d(t, u)) + ∂ 1 ν q (t, t -d(t, u)) + ∂ 2 ν q (t, t -d(t, u)) (1 -u(t)) ν q (t, t -d(t, u)) dt + log (ν q (a, a -d(a))) s.t. b a d(t, u)dt ≤ K, ∂ t d(t, u) = u(t), d(a, u) = d(a) ∈ [0, d 0 ] u measurable , u ∈ [u * = 0, u * = 1] .
Remark 7. In the case of non-decreasing draw-down functions, requiring d(t) ≥ 0 amounts to imposing the initial condition d(a) ≥ 0. In absence of such assumptions, one deals with state constraints and Pontryagin's principle has a (slightly) different form This is the first step towards defining an associated Hamiltonian H(t, d, u, p) (24), where the costate p(t) satisfies the conjugate equation ( 25). Then, one may apply Pontryagin's maximum principle [START_REF] Semenovich | Mathematical theory of optimal processes[END_REF].

It is convenient here to break the solution in four cases, with free area constraint

H(t, d, u, p) :=pu + ν q (t, t -d) + ∂ 1 ν q (t, t -d) + ∂ 2 ν q (t, t -d) (1 -u) ν q (t, t -d) =ν q (t, t -d) + ∂ 1 ν q (t, t -d) + ∂ 2 ν q (t, t -d) ν q (t, t -d) + p - ∂ 2 ν q (t, t -d) ν q (t, t -d) u, (24)
with costate p(t) satisfying:

∂ t p(t) =-∂ d H(t, d opt , u opt , p) =∂ 2 ν q + ∂ 1 ν q + ∂ 2 ν q ν q t, t -d opt (t) -∂ 2 ∂ 2 ν q ν q t, t -d opt (t) u opt (t). ( 25
)
Remark 8. Note that due to linearity in u, optimizing the Hamiltonian H(t, d, u, p) yields optimal control policies u opt of bang-bang type, except on sets where p(t) = ∂ 2 νq(t,t-d(t)) νq(t,t-d(t)) .

The previous remark implies:

Lemma 9. The optimal draw-down function d may have three possible types of subintervals.

(1) On sets [α 1 , β 1 ] on which u opt = u * = 0, it follows from (23) that d opt (t) = d(α 1 ) is constant. On such sets, (25) yields

(26) p(t) = p(α 1 ) + t α 1 ∂ 2 ν q + ∂ 1 ν q + ∂ 2 ν q ν q (s, s -d (α 1 )) ds > ∂ 2 ν q (t, t -d (α 1 )) ν q (t, t -d (α 1 )) ,
for all t ∈ (α 1 , β 1 ) by noting that the coefficient of u in H should be positive. We either have α 1 = a or equality at t = α 1 in (26). Similar assertions hold true at t = β 1 .

(2) Sets [α 2 , β 2 ] on which the costate satisfies the structural equality p(t) = ∂ 2 νq(t,t-d(t)) νq(t,t-d(t)) . Recalling that ∂ t d(t) = u(t) and combining with (25), whenever the function ν q is regular enough (of class C 2 such that second-order mixed partial derivatives coincide), this leads to the following implicit "structural equation" satisfied by the optimal draw-down d opt :

(27) ∂ 2 ν q t, t -d opt (t) = 0.
(3) The third and last case to be taken into consideration leads to u opt = u * = 1. On sets (α 3 , β 3 ) corresponding to this case (note that here it holds that that t -d(t) = α 3 -d (α 3 )), one gets ii. The second condition is linked to the freedom of d(b)

(28)              d(t) = d(α 3 ) + (t -α 3 ) , p(t) = p (α 3 ) + t α 3 ∂ 2 ν q (s, α 3 -d (α 3 )) ds + ∂ 2 ν q ν q (t, α 3 -d (α 3 )) - ∂ 2 ν q ν q (α 3 , α 3 -d (α 3 )) ≤ ∂ 2 ν q (t, α 3 -d (α 3 )) ν q (t,
(30) p(b * ) = 0.
iii. If the initial position d(a) is not fixed (thus, one searches for 0 < d(a) < d 0 ), one further imposes

(31) p(a) = ∂ 2 ν q ν q (a, a -d(a)).
In other words, assuming optimality, either the optimal initial position satisfies 0 < d(a) < d 0 and, in such cases, (31) holds true, or, otherwise, d(a) ∈ {0, d 0 } (saturating this constraint for some a priori given d 0 ).

Arguments for

= ν q (t, t -d) + ∂ 1 ν q (t, t -d) + ∂ 2 ν q (t, t -d) ν q (t, t -d) + p - ∂ 2 ν q (t, t -d) ν q (t, t -d) u + rd. ( 33
)
The arguments are exactly the same but the equations of the costates are given here by (34)

           ∂ t r(t) = -∂ e H + (t, d, e, u, p, r) = 0=⇒ r(t) = r = const ∂ t p(t) = -∂ d H + (t, d opt , e opt , u opt , p, r) = -∂ d H(t, d opt , u opt , p) -r = ∂ 2 ν q + ∂ 1 ν q + ∂ 2 ν q ν q t, t -d opt (t) + (∂ 2 ν q ) 2 -∂ 2 22 ν q × ν q ν 2 q t, t -d opt (t) u opt (t) -r.
Cases are exactly the same as before. Formulas ( 26) and ( 28) are similar:

(35) p(t) = p(α 1 ) + t α 1 ∂ 2 ν q + ∂ 1 ν q + ∂ 2 ν q ν q (s, s -d (α 1 )) ds -r (t -α 1 ) > ∂ 2 ν q (t, t -d (α 1 )) ν q (t, t -d (α 1 )) , resp. ( 36
)                        d(t) = d(α 3 ) + (t -α 3 ) , p(t) = p (α 3 ) + t α 3 ∂ 2 (ν q (s, α 3 -d (α 3 ))) ds + ∂ 2 ν q ν q (t, α 3 -d (α 3 )) - ∂ 2 ν q ν q (α 3 , α 3 -d (α 3 )) -r (t -α 3 ) ≤ ∂ 2 ν q (t, α 3 -d (α 3 )) ν q (t, α 3 -d (α 3 ))
Finally, the structure equation (27

) becomes § (37) ∂ 2 ν q t, t -d opt (t) = r.
The transversality conditions (see Remark 13) are similar and allow to determine the optimal horizon b * . We have proven (2) In the absence of the area constraint, the equation (37) holds with r = 0. § Or, in a more symmetric form,

Wq ∂ 12 Wq -∂ 1 Wq ∂ 2 Wq W 2 q t, t -d opt (t) = r.
Remark 11. i. It can be easily shown that overoptimizing in the sense of allowing unbounded derivatives for d by setting u * = ∞ (and not respecting the contraint y-d(y) nondecreasing) leads to rather trivial results: either constant d or d(•) continuously evolving among the points of the critical set for ν q (usually void). ii. In the classical Lévy framework and for a certain class of diffusions, we will give reasonable (and rather general) conditions on ν q yielding affine optimal draw-down.

6. Back to the Lévy case: De Finetti's solution is optimal

Let us go back to the Lévy case where ν q (x, y) = ν q (x -y).

This case has the further particularity that (

∂ 1 + ∂ 2 ) (ν q ) = 0.
In the rest of this section we will drop the tilde in ν q (x). Recall that the one-variable functions ν q is non-increasing and non-zero. In this framework, the (time-homogeneous) extended Hamiltonian (for which we drop the superscript + ) is given by ( 38)

H(d, u, p, r) := ν q (d) + ν q (d) ν q (d) + p u+rd.
Therefore, like in any homogeneous setting, the Hamiltonian H is constant and by the transversality condition H(b * ) = 0 it must equal 0 along optimal trajectories. The costate (cf. (34)) satisfies (39)

∂ t p(t) = -ν q d opt (t) - ν q ν q d opt (t) u opt (t) -r
and the structural equation (37) for non-extremal solutions in the present setting writes down (40) ν q d opt (t) = -r.

We proceed to show now that only sets with u = 1 ⇔ d opt (t) = d(a)+t-a ⇔ d(t) = d(a) constant (the de Finetti solution) are possible in the Lévy case.

(1) Sets on which p(t) = -ν q νq d opt (t) cannot exist, and the optimal control is reduced to bang-bang 0/1, with or without area constraints. Indeed, in this case the Hamiltonian reduces to 0 = H d opt (t), u opt (t), p(t), r = ν q d opt (t) + rd opt (t) which is impossible since ν q > 0 and r ≥ 0 by (40) (since ν q is non-increasing).

(2) Sets [α 1 , β 1 ] on which u opt = 0 and d opt (t) = d opt (α 1 ) is constant cannot exist either.

Indeed, on such sets, one must have

(41) p(t) = p(α 1 ) -(t -α 1 ) ν q d opt (α 1 ) + r ≥ - ν q d opt (α 1 ) ν q (d opt (α 1 )) , ∀t ∈ [α 1 , β 1 ] ,
Moreover, since the Hamiltonian is null, it follows that ν q d opt (α 1 ) + rd opt (α 1 ) = 0.

• Without area constraints. Here r = 0, and the previous equality cannot hold. Thus, 0 control cannot be optimal. • With 0 initial datum d opt (α 1 ) = 0. Again, 0 control cannot be used.

• With area constraints and non-zero initial datum. r = 0 and it should be picked such that r = -νq(d opt (α 1 )) d opt (α 1 ) . Since r is negative and ν q is non-increasing, the inequality in (41) is strengthened as t ≥ α 1 increases, hence u opt = 0 and d opt (t) = d(α 1 ) is constant for all t ∈ (α 1 , b * ]. But, then, assuming that b * > α 1 , it follows that p(b * ) > -ν q (d(α 1 )) νq(d(α 1 )) ≥ 0 which contradicts the transversality condition (30). We have the following more precise result.

Lemma 12. Let a and the initial datum d(a) be given.

(1) Then the optimal draw-down (with or without area constraints) is d opt (t) = d(t) = d(a) + (t -a).

(2) Without state constraints, b * should satisfy the de Finetti 'smooth fit' equation Proof. The first assertion on the optimality of 1-slope d and the last assertion have been provided prior to the Lemma (recall that the optimal control is u opt = 1). One has (44)

ν q (d(b * )) + ν q (d(b * )) ν q (d(b * )) = 0 ⇔ ∂ b 1 ν q (d(b * )) = ∂ b V b * (b) = 1. ( 42
     d opt (t) = d(a) + t -a, p(t) = p 0 -ν q d opt (t) - ν q d opt (t) ν q (d opt (t)) -rt ≤ - ν q d opt (t) ν q (d opt (t)) .
Since the Hamiltonian should be equal to 0 (see again the transversality condition (29), it follows that p(t) = -ν q d opt (t) + ν q ν q d opt (t) + rd opt (t) .

We focus on the case without area constraints i.e. r = 0. The transversality condition (30) yields p (b * ) = 0 which, given the previous form for p yields the second assertion.

For the third assertion, we note that the presence of a further constraint (on the area) can only increase the value function. This area is given exactly by

ν q (d(b * )) + ν q (d(b * )) ν q (d(b * )) u * = 0 =⇒ ν q (d(b * )) ν 2 q (d(b * )) = - 1 u * ⇔ ∂ b 1 ν q (d(b * )) = 1 u * . ( 45
)
The solution in this case will still be to use u = u * , leading to the affine draw-down barriers already studied in [START_REF] Avram | On draw-down stopping times for taxed spectrally negative lévy processes[END_REF]

under the different parametrization d(y) = (1 -ξ) (y-a) + d ⇔ u * = 1 -ξ ∈ [0, 1].

Example 14 ([AVZ17]

). For Brownian motion with drift X t = σB t + µt, the scale function is

(46) W q (x) = 1 ∆ [e (-µ+∆)x/σ 2 -e -(µ+∆)x/σ 2 ] := 1 ∆ [e Φqx -e -ρqx ],
where ∆ = µ 2 + 2qσ 2 , and Φ q , -ρ q are the nonnegative and negative roots of σ 2 θ 2 +2µθ-2q = 0.

In the case of affine optimal profiles and with the extra restriction u * := 1 -ξ ≤ 1 as in Remark 13, (ii) the transversality conditions (45) yield

(47) W q W q -W q 2 W q 2 (d(b * )) = - 1 1 -ξ =⇒ W q W q (W q ) 2 (d(b * )) = - ξ 1 -ξ , a result already obtained in [AVZ17]. § 7.
Optimal dividends for functions of a Lévy process Consider a process implicitly defined by

F (X t ) -F (x 0 ) = Z t , Log E 0 [e sZt ] = tκ(s) (48)
for an arbitrary increasing function F (x) and x 0 = 0 (w.l.o.g.). This class of processes generalizes the geometric Brownian motion, obtained when F (x) = ln( α β x + 1), Z t = σB t + µt, σ > 0. 7.1. Optimal dividends for functions of Brownian motion with drift . The monotone harmonic functions are ϕ ± (x) = ϕ Z ± (F (x)), where ϕ Z ± are the monotone harmonic functions of Z t . The q-scale and excursions function may therefore be expressed in terms of the corresponding one-dimensional characteristics of Z t :

W q (x, y) = W Z q (F (x) -F (y)) := ω q (F (x) -F (y)), ν q (x, y) = F (x)ν Z q (F (x) -F (y)) := F (x)µ q (F (x) -F (y)). As well-known, ϕ Z ± (F (x)) = e r ± x
, where r ± are the positive /negative roots of σ 2 2 r 2 +rµ-q = 0, and the q-scale function is: ω q (x) := e r + x -e r -x . Recalling that ω q (x) satisfies the equation (49) σ 2 2 ω q + µω q -qω q = 0 ⇔ ω q = -2µ σ 2 ω q + 2q σ 2 ω q we find that

µ q = w q w q -µ 2 q = - 2µ σ 2 µ q + 2q σ 2 -µ 2 q . (50)
Lemma 15. Let X t be defined by (48), with F (x) strictly increasing, and set ∆ = F (x) -F (y). Then:

A) The structure equation (37) becomes

(51) µ 2 q ∆ opt (x) + 2µ σ 2 µ q ∆ opt (x) - 2q σ 2 = r F (x)F (F -1 (F (x) -∆ opt (x))) ,
for some r ≥ 0.

B) If F (x) is furthermore convex, then for fixed x the equation (51) admits exactly one solution ∆ opt (x).

Proof: A) The reader is invited to note that (52)

∂ 2 ν q (x, y) = -F (x)F (y)µ q (∆) = -F (x)F F -1 (F (x) -∆) µ q (∆)
The structure equation becomes therefore (recall the case r = 0 corresponds to the absence of area constraints). Dividing now (53) by F (y)F (x) and substituting (50) yields (51) leads to our first assertion.

(53) r = -F (y opt (x))F (x)µ q ∆ opt (x) = -F (x)F F -1 F (x) -∆ opt (x) µ q ∆ opt (x) § When ξ = d = 0,
For notation simplicity we drop the dependence ∆ opt (x) and write ∆ from now on.

B) For the uniqueness assertion, fix x and assume that ∆ satisfies (51). Note now that the applications

∆ → µ 2 q (∆)+ 2µ σ 2 µ q (∆) - 2q σ 2 and ∆ → r F (x)F (F -1 (F (x) -∆))
are decreasing with range [0, ∞) and increasing with range [ r (F (x)) 2 , r F (x)F (0) ], respectively. Indeed, the derivative of the first 2µ q (∆) µ q (∆)+ µ σ 2 is negative by the strict monotony of µ q . The values start from ∞ since the positivity of σ implies ν(0) = ∞, and their positivity follows from the well-known (8).

For the second term, note that besides the negative sign and the inversion, it consists of a composition of the increasing functions F (here convexity of F is used), F -1 and F § . This terms is thus increasing and the assertion follows. Remark 16. Note that F is concave and therefore Lemma 15 may not apply. Numeric experiments reveal however that a unique solution ∆(x) exists sometimes.

The monotone harmonic functions are ϕ ± (x) = (αx + β) r ± , where 2 ) -q = 0, appearing in the scale function ω q (x) := e r + x -e r -x associated to the drifted Brownian motion Z t = εαB t + (α -ε 2 α 2 2 )t. The two variables scale function satisfying W q (x, y) = 0 is W q (x, y) = ( αx + β αy + β ) r + -( αx + β αy + β ) r -= ω q ln αx + β αy + β , and its logarithmic derivative is ν q (x, y) = ∂ 1 W q (x, y) W q (x, y) = α αx + β µ q ln αx + β αy + β , § Equivalently, the function Remark 17. Note that if r = 0 (57) becomes r + e r + ∆ -r -e r -∆ = r + e r + ∆ -e r -∆ =⇒ r + = r -, which is impossible. With r > 0 how, if an adequate solution ∆ exists, it is a non-constant function of the position x.

r ± = 1 -2 ε 2 α ± 1 -2 ε 2 α 2 + 8q

In conclusion

Lemma 18. Assuming u ∈ [0, 1 -ξ], it holds that (1) Without area restrictions for geometric Brownian motions, the structure equation (for r = 0) has no solution and the optimal profile is still affine as before with the maximal slope 1 -ξ.

(2) For area restrictions, the structure equation is given by (57). The optimal draw-down belongs to the class of functions whose piecewise components either satisfy (57) (for some fixed r > 0) or are affine with the maximal slope 1 -ξ.

  )-(19), we arrive to the following Bolza problem d(y)≥0, d(y),y-d(y) nondecreasing V b (a) = max d(y)≥0,d(y),y-d(y) nondecreasing e -b a νq (y, d(y))dy νq(b, d(b)) A(b) ≤ K √ 2 b, d(b) free, d(a) ≤ d 0 After taking logarithms, (20) becomes: d(y)≥0,d(y),y-d(y) nondecreasing b a ν q (y, d(y))dy + log(ν q (b, d(b)) = min d(y)≥0,d(y),y-d(y) nondecreasing b a ν q (y, d(y)) + ν q (y, d(y)) νq(y, d(y)) dy + log(ν q (a, d(a)) b a d(y)dy ≤ K b, d(b) free, d(a) ≤ d 0 . Remark 5. Let us relate (21) to the classic de Finetti problem, which is the particular case obtained by imposing the additional constraint d(y) = d(a) ⇔ d(y) = d(a) + y -a =⇒ X t ≥ d(a), ∀t. Here the constraint d(a) = d 0 ⇔ d(a) = a -d 0 quantifies an imposed initial bankruptcy level, and the subsequent values d(y) quantify bankruptcy levels dependent on the attained maximum y. The area constraint is thus an acceptable "integrated bankruptcy risk". Remark 6. If we fix the draw-down boundary in (21), the optimality condition for the dividend barrier b * is

d

  opt (t)dt = (resp. <)K, respectively with free starting data d(a) > 0 and with fixed starting data d(a) = 0. 5.1. Optimality Without Area Constraints. 5.1.1. Arguments for Free Initial d(a) > 0. The associated Hamiltonian to be minimized is

  α 3 -d (α 3 )) Remark 10. One should add the following transversality conditions, taking into account the liberty to choose b and d(b), and the initial conditions d(a): i. The first condition is linked to the freedom of b (29) H(b * ) = 0.

V

  Null Initial Datum d(a) = 0. The program presented before still holds true but having fixed the initial datum d(a) = 0, the transversality conditions are reduced to (29, 30).5.2. Optimality With Area Constraints. Again, as before, we reason for d(a) > 0 (the restriction d(a) = 0 being taken into account by the absence of the transversality condition (31). To cope with the additional constraint b * a d(y)dy≤K, we use a classic trick and introduce a further variable in the control system. We deal now with (a, d(a), e(a)) := inf u J + b (a, d, e, u), where J + b (a, d, e, u) := J b (a, d, u), s.t. ∂ t d(t, u) = u(t), ∂ t e(t, u) = d(t, u), e(a, u) = e(a), d(a, u) = d(a), e(b, u) = K. u ∈ [0, 1] . The associated Hamiltonian is H + (t, d, e, u, p, r) := H(t, d, u, p) + rd

Theorem 1 .

 1 Assume that the problem (23) admits an optimal pair (d opt , b * ) such that d opt is smooth of class Lip u * ([0, b * ], non-decreasing, with t → t -d opt (t) non-decreasing.Then, (1) The equations (34)-(37) provide the three possible behaviors for d opt , while b * is determined from the transversality conditions (29), (30) written for the extended Hamiltonian H + instead of H to which one adds the area saturation d opt (b * ) = K;

) ( 3 )

 3 With area constraints, b * constr is the minimum between b * and b + , where b + ≥ a is the solution of the equation (43) b + a d opt (s)ds = b + -a 0 d(a) + y dy = d(a)(b + -a) + (b + -a) 2 /2 = K. (4) Without area restriction, the best value is obtained for d(a) extreme (i.e. d(a) ∈ {0, d 0 }).

d

  opt (s)ds ≤ K. If b * ≤ b + , then it satisfies the constraint, thus providing the best solution. Otherwise, one retains b + . (b) Without area restrictions, the optimal initial datum d(a) is either d 0 or 0 since if d(a) does not satisfy this restriction, then by the transversality condition (31) it follows that p(a) = -ν q νq (d(a)) and, thus H d opt (a), u opt (a), p(a), 0 = ν q d opt (a) > 0. This contradicts the previous assertion on H being 0. Remark 13. If, instead of searching for draw-down functions s.t. y → y -d(y) is non-decreasing (i.e. u * = 1) one searches for d opt with u * < 1-bounded derivative, then the condition (42) becomes

  we recover in the compound Poisson case the equation W q (b) = 0.

7. 2 .

 2 Geometric (logarithmic) Brownian motion . Consider the diffusion defined by the SDEdX t αX t + β = dt + εdB t ,(54)with coefficients µ(x) = αx + β, σ(x) = (αx + β), β > 0. By Ito's formula, this process may be represented as:X t = (x 0 + β α )e Ztβ α , Z t = εαB t + (α -ε 2 α 2 2 )t ⇔ (55) Z t = F (X t ), F (x) = ln( αx + β αx 0 + β ).

ε 2 α 2 2 are the positive /negative roots of 2 α 2 2

 22 r 2 + r(α -2 α 2

rF

  (x)F (y) is decreasing in y where µ q := ω q ωq . SinceF (x)F (y) = F (x)F F -1 (F (x)r + e r + ∆ -r -e r -∆e r + ∆ -e r -r + e r + ∆ -r -e r -∆ -e r -∆ .Finally (58) d opt (x) = x+ β α1 -e -∆(x) .

Acknowledgement. We thank Hongzhong Zhang for help in formulating the variational problem.