Evgeniya A Saverina

Daria Yu Zinchenko

Sofia D Farafonova

Alexey S Galushko

Andrei A Novikov

Maxim V Gorbachevskii

Valentine P Ananikov

Mikhail P Egorov

Viatcheslav V Jouikov
email: vjouikov@univ-rennes1.fr

Dr Mikhail A Syroeshkin
email: syroeshkin@ioc.ac.ru

Porous silicon preparation by electrochemical etching in ionic liquids

Keywords: porous silicon, electrochemical etching, HF-free, fluorescence, confocal microscopy, green chemistry

Anodic etching of n-type {111} silicon in ionic liquid (IL) systems ([RMIM][X], R = H, Bu; X = BF 4 -, PF 6 -), realized under galvanostatic conditions and room temperature, allowed the formation of porous silicon surfaces with different pore morphology depending on the etching time, current density and the IL used. The study of the effect of water content in IL on the etching process has shown water content of 1% to be optimal. The role of the anion on the etching process was elucidated using 1methylimidazolium tetrafluoroborate ([HMIM][BF 4 ]) and 1-methylimidazolium hexafluorophosphate ([HMIM][PF 6 ]) IL systems. [HMIM][BF 4 ] was found to be most efficient for the formation of silicon nanostructured array with a pore size of 30-80 nm.

The thus prepared porous silicon samples show fluorescence in blue light (475 nm).

The NMR spectra of [HMIM][BF 4 ] ionic liquid before and after etching does not show

Introduction

Silicon-based materials, in particular porous silicon (PSi), are of high demand in the materials science. Besides dominating solar cells market [START_REF] Shin | Effect of Layer Number and Metal-Chloride Dopant on Multiple Layers of Graphene/Porous Si Solar Cells[END_REF][START_REF] Mohammed | Reflectivity Effect of the PS on Solar Cell Efficiency[END_REF][START_REF] Praveenkumar | An Experimental Study of Optoelectronic Properties of Porous Silicon for Solar Cell Application[END_REF][4] silicon and its porous form are widely used in various research fields such as optoelectronics, [START_REF] Korotcenkov | How to Improve the Performance of Porous Silicon-Based Gas and Vapor Sensors? Approaches and Achievements[END_REF][START_REF] Wang | Metal Nanoparticles/Porous Silicon Microcavity Enhanced Surface Plasmon Resonance Fluorescence for the Detection of DNA[END_REF] biosensing [START_REF] Terracciano | Porous Silicon-Based Aptasensors: The Next Generation of Label-Free Devices for Health Monitoring[END_REF][START_REF] Ge | Optical Sensing Analysis of Bilayer Porous Silicon Nanostructure[END_REF][START_REF] Harraz | Porous Silicon Chemical Sensors and Biosensors: A Review[END_REF] and biomedical applications. [START_REF] Gopal | Porous Silicon Nanoneedles Modulate Endocytosis to Deliver Biological Payloads[END_REF][START_REF] Park | Photoluminescent and Biodegradable Porous Silicon Nanoparticles for Biomedical Imaging[END_REF][START_REF] Santos | Porous Silicon Nanoparticles for Nanomedicine: Preparation and Biomedical Applications[END_REF][START_REF] Tieu | Advances in Porous Silicon-Based Nanomaterials for Diagnostic and Therapeutic Applications[END_REF][START_REF] Zhang | Porous Silicon Nanomaterials: Recent Advances in Surface Engineering for Controlled Drug-Delivery Applications[END_REF][START_REF] Alhmoud | A MACEing Silicon: Towards single-step etching of defined porous nanostructures for biomedicine[END_REF][START_REF] Jones | Crystallisation Behaviour of Pharmaceutical Compounds Confined within Mesoporous Silicon[END_REF] The primary method of obtaining PSi is oxidative etching of a non-porous silicon precursor in hydrofluoric acid solutions. [START_REF] Quiroga-González | Porous Silicon: From Formation to Application[END_REF] In electrochemical version of silicon etching in fluoride solutions the process can be carried out both in potentiostatic and in galvanostatic mode; the latter option is often preferable because of easier control of the process. [START_REF] Harraz | Porous Silicon Chemical Sensors and Biosensors: A Review[END_REF] A c c e p t e d M a n u s c r i p t

Although the extreme safety hazards of hydrofluoric acid for nature and humans are well known, [START_REF] Peng | Calcium Gluconate Alleviates the Toxic Effect of Hydrofluoric Acid on Human Dermal Fibroblasts through the Wnt/Β-catenin Pathway[END_REF] using HF in PSi production seems unavoidable since it is needed for removal of native oxides on silicon surface and for binding silicon in etching products.

The search for different approaches for replacing HF acid by more environmentalfriendly agents is of great interest in the field of silicon etching [19,[START_REF] Zhao | An HF-Free Etching of SiO 2 for Soft Lithography[END_REF] and in other areas of materials science. [START_REF] Kumar | HF-Free Synthesis of K 2 SiF 6 and BaSiF 6 Nanoparticles by Thermal Decomposition[END_REF][START_REF] Teo | Solventand HF-Free Synthesis of Flexible Chromium-Based MIL-53 and MIL-88B[END_REF][START_REF] Fialho | Development of Stacked Porous Tantalum Oxide Layers by Anodization[END_REF] A method of silicon etching replacing HF with silicophilic reagents, in particular, salts of pyridine or N-aromatic bases with tetrafluoroboric acid in an organic solvent was previously proposed. [19] It should be noted that for etching silicon in such systems, it is necessary to use the salts solutions of high concentration. In this regard, etching systems based on protic ionic liquids, [START_REF] Greaves | Protic ionic liquids: properties and applications[END_REF] combining solvent and supporting salt in one compound [START_REF] Buzzeo | Non-haloaluminate roomtemperature ionic liquids in electrochemistry-a review[END_REF][START_REF] Silvester | Electrochemistry in Room Temperature Ionic Liquids: A Review and Some Possible Applications[END_REF] that were successfully used in surface electro-chemistry [START_REF] Liu | Ionic liquids in surface electrochemistry[END_REF] might be more promising.

A c c e p t e d M a n u s c r i p t

Besides entirely merited attention and wide use in almost all chemistry-related areas, [28- 31] ionic liquids permit to address and in many cases to improve very important ecological issues.

In this work, we propose a simple, environment-friendly, and efficient technique for preparing porous silicon by means of an HF-free electrochemical etching in the ionic liquid medium (primarily derivatives of 1-methylimidazolium, [HMIM][BF 4 ]). This method allows one to prepare a uniformly porous silicon surface with the possibility of controlling the pore size.

Results and discussion

The [HMIM][BF 4 ] ionic liquid was synthesized by stirring 1-methylimidazole with tetrafluoroboric acid in appr. 1:1 molar ratio with minimal excess of 1-methylimidazole to prevent the presence of HBF 4 in product. After that, the resulting solution was evaporated, washed with ethyl acetate and dried under vacuum according to [START_REF] Goc-Maciejewska | Diels-Alder reaction in protic ionic liquids[END_REF] . Several samples of [HMIM][BF 4 ] were then prepared with a water content of 1%, 5%, and 10%, controlled by Fischer titration. [START_REF] Azov | Solvent-in-Salt" Systems for Design of New Materials in Chemistry, Biology and Energy Research[END_REF] Synthesized in the similar way [

HMIM][PF 6 ]

A c c e p t e d M a n u s c r i p t was conditioned to water content of 2%. In contrast to dry ILs, such wet ionic liquids (w-IL) remain liquid at room temperature; this enables using them for silicon etching under the ambient conditions without heating which is otherwise needed to liquefy these ILs. These w-IL were then tested for etching silicon and revealing the influence of current density and etching time on the morphology of the obtained samples.

A polished n-type (111) oriented single-crystal silicon wafer was used (Fig. S1) for this purpose. The etching was carried out at room temperature in a galvanostatic mode, using a specially designed polytetrafluorethylene (PTFE) cell (Fig. 1). The cell was fitted with a PTFE nut with a 4 mm hole protecting the Si sample so that the etching area, limited by the hole, remains constant throughout all experiments. A platinum grid was used as a cathode. We found that etching in 1% [HMIM][BF 4 ] w-IL system at a current density j = 8 mA/cm 2 during 30 min results in the development of significant roughness of the surface of silicon wafer (Fig. 2 left). Further increase in etching time leads to the formation of a porous surface.

In particular, a uniform mesoporous surface with the irregularly shaped pores sized of 31±2 nm separated by 8±1 nm walls was obtained after a 5 hours etching (Fig. 2

right).

A B C D

A c c e p t e d M a n u s c r i p t When etching in HF-based systems, higher current densities are known to increase the pore size and depth. [START_REF] Ramizy | The Effect of Etching Current Density on Porous Silicon Fabricated by Electrochemical Etching Process[END_REF][START_REF] Al-Husseini | Influence of Current Density on Morphology of Electrochemically Formed Porous Silicon[END_REF] This trend was also observed during the etching of silicon in

[HMIM][BF 4 ]-based w-IL system. Fig. 3 shows an scanning electron microscope (SEM) picture of the sample obtained by etching a silicon wafer at j = 24 mA/cm 2 for 1 hour.

It can be seen that the pores at thus prepared homogeneous porous surface are larger than those observed after etching for 5 hours at j = 8 mA/cm 2 . Now, the pore size and During etching gas formation is observed, but the change in the amount of water cannot be determined within the measurement accuracy, and after the process the system remains liquid. It contain white amorphous solid.

The increase of water content in the IL significantly affects the nature of etching. So, at 5% water in IL the morphology of the formed layer changed now resembling to a A c c e p t e d M a n u s c r i p t network of bulges around deeper etching zones of commensurate size (Fig. S2). This case is intermediate between the cellular structure (Fig. 2, 3) and a spongy surface when etching is achieved with higher water content (10%) (Fig. S3).

Elucidating the role of anion in the etching process encounters the intrinsic difficulties of strongly varying electrochemical properties of ILs when changing the anion.

Alkylimidazolium ILs with PF 6 -anion have about ten times higher viscosity and ten-fold lower conductivity than with BF 4 -, [START_REF] Susan | Brønsted acid-base ionic liquids and their use as new materials for anhydrous proton conductors[END_REF] which dramatically alters etching efficiency with [BMIM][BF 4 ] (108.25 cP [37] and 3.53 mS/cm [START_REF] Stoppa | Conductivities of Binary Mixtures of Ionic Liquids with Polar Solvents[END_REF] ), whereas an intermediate regime was

observed with [BMIM][PF 6 ] (284.49 cP [37] and 1.65 mS/cm [START_REF] Nakamura | Systematic Dielectric and NMR Study of the Ionic Liquid 1-Alkyl-3-Methyl Imidazolium[END_REF] ) when mesoporous etching (Fig. 4 a) goes concomitantly with the formation of a sub 100 nm pattern (Fig. 4 c).

It should be noted that the formation of pores occurs without significant oxidation of the silicon surface, as the fraction of silicon oxide in the resulting material is less than 1.5%

(according energy dispersive X-ray spectroscopy (EDS) data).

One of the essential qualities of porous silicon is its strong fluorescence. [START_REF] Gu | In Vivo Time-Gated Fluorescence Imaging with Biodegradable Luminescent Porous Silicon Nanoparticles[END_REF][START_REF] Liu | Enhancement of QDs' Fluorescence Based on Porous Silicon Bragg Mirror[END_REF] The fluorescence spectrum of a porous silicon sample obtained by etching in 1%

[HMIM][BF 4 ] w-IL system at a current density of 8 mA/cm 2 for 6 hours (Fig. 5) has a maximum in blue light at 475 nm. A more detailed study of the optical properties of the material is the subject of further study.

A c c e p t e d M a n u s c r i p t Raman spectra of the studied samples (Fig. S19, S20) are quite similar, with the more pronounced 2TO mode of the etched silicon, especially with the 532 nm excitation.

The IL used for preparing the wafer as in Fig. 2 (right) was diluted with CD 3 CN, and a comprehensive range of 1 H, [START_REF] Tieu | Advances in Porous Silicon-Based Nanomaterials for Diagnostic and Therapeutic Applications[END_REF] C, [START_REF] Park | Photoluminescent and Biodegradable Porous Silicon Nanoparticles for Biomedical Imaging[END_REF] B, and 19 F NMR spectra were recorded. Comparison of NMR spectra of this IL before and after etching showed that it does not decompose during the etching process (all signals at all nuclei before and after etching are completely identical and only the acidic proton signal shifts to the region of the high fields, which indicates a some decrease in pH [42] ) and it can be considered as a potentially recyclable etching agent (see Fig. S7-S14). Of course, the interpretation that the ionic liquids will be recyclable because there is no degradation of the IL as determined by NMR spectroscopy is just a preliminary investigation; and, we will need to be examined more closely in the future including examination of the purity of the IL after several uses and using a more sensitive technique. Ionic liquid systems were washed with ethyl acetate (Acros).

The n-type (111) single-crystalline silicon wafer (d = 5 mm, thickness = 2 mm) was polished to a high gloss with chromium oxide GOI polishing paste and then degreased with isopropyl alcohol (SEM/EDS -see Fig. S1).

Synthesis of ionic liquids.

[HMIM][BF 4 ] was synthesized according to known literary technic. [START_REF] Goc-Maciejewska | Diels-Alder reaction in protic ionic liquids[END_REF] The synthesis of [HMIM][PF 6 ] was carried out similarly.

The purity and structure of prepared NMR spectra were recorded on Bruker AV600 spectrometer at ambient temperature.

Standards according to Bruker almanac [43] were used. Water content in the ILs was controlled by Fischer titration using a KAS-01 MD apparatus.

Electrochemical etching. The etching was carried out in a PTFE cell of the original design (see Fig. 1). It consisted of a PTFE rod fitted with a PTFE nut with a 4 mm hole that allowed the silicon wafer to be pressed against a stainless steel rod, which served as a current collector. A PTFE nut also allows us to limit the etching area, so it remains constant throughout all experiments. In parallel with the silicon wafer, a platinum grid A c c e p t e d M a n u s c r i p t cathode was fixed above the PTFE cell. The etching agent was placed on the surface of the silicon wafer using a micropipette. The etching process was carried out in galvanostatic mode at current density in the range from 8 mA/cm 2 to 24 mA/cm 2 using a PC-piloted digital potentiostat IPC-Pro-MF (Econix). Etching time was varied from 30 minutes to 6 hours. After that, the silicon wafer was washed with ethyl alcohol and stored in a high-purity argon atmosphere in a septum-sealed vial.

SEM/EDS. Morphology of pristine and etched silicon surface was studied using a Hitachi SU8000 field-emission scanning electron microscope (FE-SEM). The images were acquired in secondary electron mode at 20 kV accelerating voltage and the working distance 8-10 mm. Energy-dispersive X-ray spectroscopy (EDS) was carried out using an Oxford Instruments X-max 80 EDS system at 20 kV accelerating voltage and a working distance 15 mm. The samples were studied without metallization in order to avoid metal coating effects. [44] A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t the proposed technique suitable for scaling-up. Application of HF-free solvent-in-salt system for eco-friendly etching is a novel approach with excellent practical opportunities.

Conflicts of interest

There are no conflicts to declare.

ASSOCIATED CONTENT

Supporting Information.

Additional SEM data, NMR and Raman spectra are provided in the Supporting Information.

This file is available free of charge.

Figure 1 .

 1 Figure 1. Photograph of a silicon etching setup (A), the scheme of assembling the

Figure 2 .

 2 Figure 2. Scanning electron microscope (SEM) images of porous silicon obtained by

A c c e

  p t e d M a n u s c r i p t the thickness of the separating walls are increased up to 79±6 nm and 21±1 nm, respectively. It is noteworthy that the majority of the cells formed are 4 or 5-faceted.

Figure 3 .

 3 Figure 3. SEM image of porous silicon obtained by etching in 1% [HMIM][BF 4 ] w-IL

[

  HMIM][PF 6 ]. The process becomes non-uniform developing locally (Fig.S5, S6), though at the bottom of the affected zones the dilution of the IL with etching products makes the conditions closer to those with less viscous [HMIM][BF 4 ] and leads to the formation of a similar cellular structure (Fig.S6). Thus, changing the counterion to PF 6 - has significant impact on the surface structure and the viscosity effects point at a diffusion-based process.When the difference in viscosity/conductivity is smaller, ca. 2 times, finer tuning of etching properties is seen. An isotropic electropolishing occurs (Fig.S4) with Page 10 of 34 ACS Paragon Plus Environment ACS Sustainable Chemistry & Engineering A c c e p t e d M a n u s c r i p t

Figure 4 .Figure 5 .

 45 Figure 4. SEM images of a silicon samples etched in w-ILs with [PF 6 -] anion: (a)

A c c e p t e d M a n u s c r i p t

  ExperimentalReagents, solvents, and materials. [BMIM][BF 4 ] (99 %), used for etching, was supplied by Sigma Aldrich. [BMIM][PF 6 ] (99%) was supplied by ABCR. [HMIM][BF 4 ] and [HMIM][PF 6 ] were prepared from 1-methylimidazol, 99 % (Acros) and fluoroboric acid, 48 wt % (Acros) or hexafluorophosphoric acid, 55 wt % (Sigma Aldrich) respectively.

Page 31 of 34 ACS

 34 s c r i p t shifts of all other protons remain the same after etching,[START_REF] Park | Photoluminescent and Biodegradable Porous Silicon Nanoparticles for Biomedical Imaging[END_REF] B and19 F spectra are also identical), can cause a significant high field shift of the acidic proton. In support of this explanation, we have taken a cyclic voltammetry measurement of [HMIM][BF 4 ] under the same conditions as were during etching but in the presence of 1-methylimidazole. It was revealed that its oxidation begins at rather low potentials, namely +0.8 V vs. Ag/AgCl (the wave does not have a pronounced maximum because of high viscosity of the system), which is significantly lower than the real etching potential (> 2 V vs. Ag/AgCl). Also 1 Н NMR measurement of [HMIM][BF 4 ] in acetonitrile showed that the presence of 3 mol% HBF 4 (a minimal amount that can be added to a NMR tube with reasonable accuracy) causes a high field shift of the acidic NH-proton signal by 3 ppm. 43. Bruker corporation, Almanac, 2010, ISBN 978-3-929431-24-7, Chemical Shift Ranges and Standards for Selected Nuclei, https://www.pascalman.com/pulseprogram/Almanac2010.pdf, accessed 10.06.2020

  [HMIM][BF 4 ], [HMIM][PF 6 ] were confirmed by 1 H,[START_REF] Tieu | Advances in Porous Silicon-Based Nanomaterials for Diagnostic and Therapeutic Applications[END_REF] C,[START_REF] Park | Photoluminescent and Biodegradable Porous Silicon Nanoparticles for Biomedical Imaging[END_REF] B, 19 F and 31 P NMR analysis (see SI). The NMR spectra of [HMIM][BF 4 ] in CD 12.05 (s, N + -H). 13 C NMR (150 MHz): 34.2, 116.9, 121.9, 136.8. 11 B NMR (193 MHz): -1.06 (s). 19 F NMR (565 MHz): -150.99 (s, 10 BF 4 ) and -151.04 (s, 11 BF 4 ) in 1:4 intensity ratio, which corresponding to the natural abundance of 10 B and 11 B isotopes in ca. 20% and ca. 80%, respectively. The NMR spectra of [HMIM][PF 6 ] in D 2 O: 1 H NMR (600

	M MHz): 3.65 (s, 3H, CH 3 ), 7.16 (s, 2H), 8.37 (s, H). 13 C NMR (150 MHz): 35.3, 119.3, 122.8, 134.8. 31 P NMR (243 MHz): -145.3 (sep, J = 709.17 Hz) 19 F NMR (565 MHz): -a n u s c r i p t 72.85 (d, J = 709.14 Hz)
	A c c e p t e d

3 CN: 1 H NMR (600 MHz): 3.84 (s, 3H, CH 3 ), 7.30 (s, H), 7.32 (s, H), 8.26 (s, H),

ABBREVIATIONS [BMIM][BF4], 1-Butyl-3-methylimidazolium tetrafluoroborate; EDS, energy dispersive Xray spectroscopy; EMIMF•2.3HF, 1-ethyl-3-methylimidazolium oligofluorohydrogenate; IL, ionic liquid; [HMIM][BF4], 1-Methylimidazolium tetrafluoroborate; [HMIM][PF6], 1-Methylimidazolium hexafluorophosphate; NMR, nuclear magnetic-resonance; PTFE, polytetrafluorethylene; PSi, porous silicon; SEM, scanning electron microscope; w-IL, wet ionic liquids.

Funding Sources ACKNOWLEDGMENT (E.A.S., M.P.E., M.A.S. acknowledge the support by Russian Science Foundation Grant 17-73-20281. Electron microscopy and NMR spectra characterization were performed at the Department of Structural Studies of Zelinsky Institute of Organic Chemistry, Moscow. Raman spectroscopy and confocal microscopy were performed by A.A.N. and M.V.G., and were funded by the Ministry of Science and Higher Education of the Russian Federation within the State Assignment (project 0768-2020-0007).

Raman spectroscopy. Raman spectra were acquired with a BWS415 spectrometer (BWTEK, Germany) with 785 nm laser, and with EnSpectr532 spectrometer (Enspectr, Russia), integrated with 532 nm laser and CX41 microscope (Olympus, Japan).

Confocal microscopy.

Confocal microscopy was performed with Nikon A1 confocal microscope equipped with 401 nm and 514 nm lasers and spectral detector (Nikon, Japan).

Summary

High-purity porous silicon with uniform and size-controlled pores was produced using wet 1-methylimidazolium tetrafluoroborate as the etching solution. Such a non-volatile, non-flammable, and chemically and electrochemically stable system is a good alternative to commonly used aggressive, toxic and dangerous HF. [START_REF]It should be noted that a technique of etching silicon using ionic liquid 1-ethyl-3methylimidazolium oligofluorohydrogenate (EMIMF•2.3HF) has been published in the literature. However, HF is also actually present in such a system[END_REF] 1-Methylimidazolium tetrafluoroborate is easily synthesized and does not require special storage conditions. Moreover, it can be considered as a potentially recyclable etching agent. Besides, the etching process is very simple, secure and efficient with no need of additional heating, control of potential or any other special precautions. All this makes

AUTHOR INFORMATION

Corresponding Author


the time spent below a level by Brownian motion with drift 6.8 Another class of past-future martingales . . . . . . . . . . . .

A rough description of Part B of the course

In Part B of the Course, we further our discussion of relations bewteen Black-Scholes type formulae and distributions of last times, begun in Notes 1 to 5 of Part A ( [START_REF] Bentata | From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon[END_REF]).

In Note 6, following a suggestion by J. Akahori, we consider, instead of the last passage times G K , which "look into the filtration up to time ∞", the times : g = l}1 which present the advantage, although they are not stopping times, to look into the filtration only up to the finite horizon t. Computing the associated Azéma's supermartingale : Z s,t = P g (ν) l (t) > s|F s , for s < t, leads us to study (super-)martingales with respect to the two-parameter filtration F s,t = σ{B

(ν) u , u ≤ s, B (ν) v , v ≥ t}, 0 < s < t < ∞.
In Note 7, we no longer make the assumption that (M t ) is continuous, and we discuss which of the results from the previous notes still hold, or how they may be modified. In particular, some of our discussion applies to the Esscher martingales : E ψ t = exp (λX t -tψ(λ)), t ≥ 0, associated with a Lévy process (X t ) with some exponential moments.

In Note 8, we discuss a recent result by P. Carr, C. Ewald and Y. Xiao asserting that E 1 T A T -K + is increasing in T , when A t = t 0 ds exp B s -s 2 ; clearly, our previous discussion does not apply there.

In Note 9, we examine how careful one should be when changing "puts" into "calls", and examining the modifications one should make in our previous discussions.

Note 10 concludes : a number of our results may be stated in quite some generality, at the cost of introducing certain conditional expectations, which, in the Markovian context, are easily computed, but in the general martingale framework, bring some fundamental "unknowns" into the discussion. Thus, this case study may be seen as a further illustration of the passage from a Markovian discussion to a "general theory of stochastic processes" type discussion.

Finally, an appendix, presents a number of developments/ solutions for some of the exercises suggested in the different notes of both Part A and Part B.

Throughout the text, we may refer to some of the results of Part A; when we do so we write e.g : A -(65) for the equation (65) in Part A.

Note : Working under a finite horizon

This note aims to counterbalance the difficulty of bringing into the picture quantities such as G K = sup{t, M t = K} which take into account the whole history of M (or (F t )).

Thus, we are now interested in finding formulae up to finite maturity, which may replace our previous formula :

P (G K ≤ t|F t ) = 1 - M t K + , (1)
when M ∈ M + 0 , or even the more general2 :

E (G K ≤ t) ; (K -M ∞ ) + ) = E (K -M t ) + (2)
The following developments are most recent 3 , therefore we shall study only the particular case where : M t = E t = exp B t -t 2 , and G K is replaced by G K (t) = sup{s ≤ t, E s = K}.

Past-future (sub)martingales

Rather than starting with some "explicit" computation about G K (t), it seems of interest (see below !) to introduce the past-future filtration of Brownian motion :

F s,t = σ{B u , u ≤ s, B v , v ≥ t}, indexed by pairs (s, t), with 0 ≤ s ≤ t < ∞. Note that, for s = t, F s,s = F ∞ , and that, if [s, t] ⊂ [s ′ , t ′ ], then :

F s,t ⊇ F s ′ ,t ′ .
In this note, we shall naturally deal with :

i. (F s,t ) positive submartingales, i.e : R + valued, (F s,t ) adapted processes (Σ s,t ) such that :

E [Σ s,t |F s ′ ,t ′ ] ≥ Σ s ′ ,t ′ ; (3)
ii. (F s,t ) positive martingales, i.e :

E [M s,t |F s ′ ,t ′ ] = M s ′ ,t ′ . (4)
Here are some particular examples of such processes : let Γ be a Borel set in R, and define :

Σ Γ s,t = P ({∀u ∈ (s, t), B u ∈ Γ} |F s,t ) , (5)
and the sets : A Γ s,t = {∀u ∈ (s, t), B u ∈ Γ}. They satisfy : A Γ s,t ⊃ A Γ s ′ ,t ′ , which implies that Σ Γ s,t satisfies (3). Moreover, from the Markov property of Brownian motion, there exists a function f Γ (s, t; x, y) such that :

Σ Γ s,t = f Γ (s, t; B s , B t ). (6)
We shall call such functions f (depending on the 4 arguments) past-future subharmonic functions (P F S-functions). Naturally, we may also be interested in past-future harmonic functions h(s, t; x, y) (P F H-functions) such that : Σ h s,t = h(s, t; B s , B t ) is a (F s,t )-martingale. Thanks to basic properties of Brownian motion, i.e : stability by time inversion, that is : if (B t , t ≥ 0) is a Brownian motion starting from 0, then :

( Bt = tB 1/t , t > 0)
is also a Brownian motion (which extends to 0, by continuity), and so on, the following Proposition is easily obtained : Proposition 6.1.

i. If f (≡ f (s, t; x, y)) is P F H, resp P F S, then f is also P F H, resp P F S, where :

f (s, t; x, y) = f ( 1 t , 1 s , y t , x s ) (0 < s < t; x, y ∈ R) (7)
We also note that the application f → f is involutive, i.e :

f = f.
ii. If f is P F H, then, for any ν, λ ∈ R :

f ν, l (s, t; x, y) ≡ f (s, t; x + νs + l, y + νt + l) (8)
is P F H.

iii. If f is P F H, then for any a > 0,

f (a) (s, t; x, y) ≡ f (a 2 s, a 2 t; ax; ay) (9)
is also P F H

Proof. We leave the proof to the reader.

6.2

The case of Brownian motion with drift 6.2.1

Here are some particular examples of the previous discussion, which are closely connected with our random times :

g (ν) l (t) = sup{s ≤ t, B (ν) s = l},
where :

B (ν) s = B s + νs. Indeed, for s ≤ t : g (ν) l (t) ≤ s = A + s,t + A - s,t , (10)
where :

A + s,t = {∀u ∈ (s, t), B (ν) u > l} A - s,t = {∀u ∈ (s, t), B (ν) u < l}. (11) Therefore, Σ (l, ν) s,t def = P g (ν) l (t) ≤ s|F s,t = Σ + s,t + Σ - s,t , with Σ + s,t = P A + s,t |F s,t Σ - s,t = P A - s,t |F s,t . (12)
Our main result here is :

Theorem 6.1. Σ (l, ν) s,t = 1 -exp - 2 t -s (B (ν) s -l)(B (ν) t -l) + ( 13
)
Thus the function :

f (l, ν) (s, t; x, y) = 1 -exp - 2 t -s (x + νs -l)(y + νt -l) + = 1 -exp - 2 t -s ((x + νs -l)(y + νt -l)) + (14)
is P F S. This leads us naturally to the next theorem : Theorem 6.2. The function :

h (l,ν) (s, t; x, y) = exp - 2 t -s (x + νs -l)(y + νt -l) (15)
is P F H.

The sequel of this note consists in proving both theorems.

As an illustration of both theorems, and of Proposition 6.1, an elementary computation shows that :

h (l, ν) = h (ν, l) ; f (l, ν) = f (ν, l) (16)

A connection with Section 4.3 (Part A)

We now interpret our result [START_REF] Doob | Classical Potential Theory and Its Probabilistic Counterpart[END_REF] in terms of the martingale : (E u , u ≤ t), which terminates at E t > 0. Now, with the same notation as in Section 4.3 (Part A), the Azéma supermartingale : (Z K s , s ≤ t) is :

Z K s = 1 -Σ (l, ν) s,t ,
with l = log (K) and ν = -1 2 . Hence :

Z K s = exp - 2 (t -s) (B (-1/2) s -l)(B (-1/2) t -l) + .
We consider the easiest case : K = 1, hence : l = 0, so that :

G 1 (t) = sup{s ≤ t, B s - s 2 = 0},
and we wish to compute :

E f exp (B t - t 2 ) |G 1 (t) = E f (exp (B t )) exp (-Bt 2 )|g t E exp (-Bt 2 )|g t , (17)
where : g t = sup{u ≤ t, B u = 0}. Now, we use the fact that : B t = B gt+(t-gt) = √ t -g t m1 where m1 = ǫ √ 2e, and g t , ǫ, e are independent. ǫ is symmetric Bernoulli, and e is a standard exponential variable. Since :

P ( m1 ∈ dµ) = 1 2 |µ|e -µ 2 /2 dµ, (µ ∈ R),
we obtain with λ = √ t -g t :

E [f (E t ) |G 1 (t)] = ∞ -∞ dµ 2 |µ|e -µ 2 /2 e -λµ/2 f (e λµ ) ∞ -∞ dµ 2 |µ|e -µ 2 /2 e -λµ/2 , ( 18
)
which provides us with an explicit form of the law ν 1 (dm) defined in P artA-(125). From there, all quantities found in Subsection A -4.3 can be made explicit.

6.

h ′ s (s, t; x, y) + y -x t -s h ′ x (s, t; x, y) + 1 2 h ′′ x 2 (s, t; x, y) = 0 (-) -h ′ t (s, t; x, y) - y -x t -s h ′ y (s, t; x, y) + 1 2 h ′′ y 2 (s, t; x, y) = 0 (+)
• The proof of this proposition hinges on the following lemma : Lemma 6.1. Let M f s,t = f (s, t; B s , B t ), for f a regular function; it is a past-future martingale if and only if :

a) for fixed t, {(M f s,t ) s , s < t} is a (F (t)
s , s < t)-martingale, where :

F (t) s = F s ∨ σ(B t ), b) for fixed s, {(M f s,t ) t , t > s} is a ( (s) F (+)
t , t > s)-martingale where :

(s) F (+) t = σ(B s ) ∨ F (+) t , and
F (+) t = σ(B v , v ≥ t) Proof. Note that (M f s,t ) is an (F s,t )-martingale if and only if : for every φ s ′ ∈ b(F s ′ ), ψ t ′ ∈ b(F (+) t ′ ), with s ′ < s < t < t ′ , one has : E φ s ′ M f s,t ψ t ′ = E φ s ′ M f s ′ ,t ′ ψ t ′ . ( 19
)
We now prove : We wish to prove [START_REF] Kaji | The tail estimation of the quadratic variation of a quasi left continuous local martingale[END_REF]. From the Markov property, the LHS of [START_REF] Kaji | The tail estimation of the quadratic variation of a quasi left continuous local martingale[END_REF], say l, is equal to :

l = E φ s ′ M f s,t γ(t, B t ) , where γ(t, B t ) = E [ψ t ′ |F t ] = E [ψ t ′ |B t ] .
From a), we now get :

l = E φ s ′ M f s,t γ(t, B t ) = E φ s ′ M f s ′ ,t ψ t ′ .
Again, from the Markov property, we get :

l = E β(s ′ , B s ′ )M f s ′ ,t ψ t ′ ,
where :

β(s ′ , B s ′ ) = E [φ s ′ |B s ′ ] .
Now, using b), we get :

l = E β(s ′ , B s ′ )M f s ′ ,t ψ t ′ = E φ s ′ M f s ′ ,t ψ t ′ .
The lemma is proven4 .

• We now give a proof of Proposition 6.2 :

Proof. For fixed t > 0, (B s , s ≤ t) is a (F (t)
s , s ≤ t) semimartingale, with :

B s = β (t) s + s 0 du B t -B u t -u , (20)
and (β

(t) s , s ≤ t) a (F (t)
s , s ≤ t) Brownian motion (see [START_REF] Jeulin | Inégalité de Hardy, semimartingales et faux-amis[END_REF]; also [START_REF] Yor | Some Aspects of Brownian Motion[END_REF]).

We now apply Itô's formula to (M f s,t , s ≤ t) in the filtration (F

(t)
s , s ≤ t). We obtain for u < s < t :

M s,t = M u,t + s u dh f ′ s (h, t; B h , B t ) + s u (dβ (t) h + dh B t -B h t -h ) f ′ x (h, t; B h , B t ) + 1 2 s u dh f ′′ x 2 (h, t; B h , B t ). (21)
Hence, the martingale property holds if and only if :

f ′ s (s, t; x, y) + y -x t -s f ′ x (s, t; x, y) + 1 2 f ′′ x 2 (s, t; x, y) = 0. ( 22
)
Thus, we have obtained (-). We might obtain (+) using some similar arguments; however a time inversion argument is much quicker and reduces the obtention of (+) to (-).

• Elementary computations now show that h (l, ν) is P F H; that is we have proven Theorem 6.2.

Exercise 6.1. Consider h (l, ν) as a generating function, depending on the 2 parameters l, ν. We write :

h (l, ν) (s, t; x, y) = exp - 2 t -s (x + νs -l)(y + νt -l) = ∞ p=0 ∞ q=0
l p ν q H p,q (s, t; x, y).

Prove that for any p, q, H p,q is P F H, and give an expression for H p,q .

Exercise 6.2. Introduce the 5 variables Hermite polynomials H p,q (a, b, c, d, f ) defined by the generating function :

exp (al + bν - cν 2 2 - dl 2 2 + f lν) = ∞ p=0, q=0 l p ν q H p, q (a, b, c, d, f ).
Write H p, q in terms of H p, q .

Solutions to Exercises 6.1 and 6.2 are given in the Appendix.

Exercise 6.3. (A second family of P F H functions). Check that, for any a ∈ R,

h (a) (s, t; x, y) = exp a y -x t -s - a 2 2 1 t -s (23)
is a P F H function. Hint: recall that, if

B [s,t] = B t -B s t -s , then : E B [s,t] |F s ′ ,t ′ = B [s ′ ,t ′ ] .
This property of (B t ) is the Harness property, which holds more generally for any integrable Lévy process. More generally, see ) on Harnesses. We refer to [START_REF] Chaumont | Exercises in Probability[END_REF] (exercise 6.19), [START_REF] Jacod | Time reversal of Lévy processes[END_REF], [START_REF] Williams | Some basic theorems on harnesses. Stochastic Analysis (a tribute to the memory of Rollo Davidson[END_REF], [START_REF] Williams | Brownian motion as a harness[END_REF].

Exercise 6.4. Give other examples of P F H functions for other Lévy processes.

An excellent candidate is the Gamma process (γ t , t ≥ 0), since the Harness property in this case :

E γ [s,t] |F s ′ ,t ′ = γ [s ′ ,t ′ ] ,
is reinforced as follows :

γ t -γ s γ t ′ -γ s ′ is independent from F s ′ ,t ′ ,
and is distributed as :

β(t -s; (t ′ -s ′ ) -(t -s))
where β(a; b) is the beta variable with parameters a and b. Thus, we have, for any m > 0 :

E γ t -γ s γ t ′ -γ s ′ m |F s ′ ,t ′ = E [(γ t -γ s ) m ] E [(γ t ′ -γ s ′ ) m ] ,
ie : for any m > 0 :

(γ t -γ s ) m E [(γ t -γ s ) m ] = (γ t -γ s ) m Γ(m + (t -s))/Γ(t -s) is a (F s,t ) martingale.
For more references about the nice properties of the Gamma process, see, e.g. [START_REF] Yor | Some remarkable properties of Gamma processes[END_REF].

Exercise 6.5. Give a version of Proposition 6.2, i.e : the differential system ( (-) and (+) ), when Brownian motion now takes values in R n . (Solution : y-x t-s h ′ x is replaced by : y-x t-s • ∇ x h, and h ′′ x 2 is replaced by : ∆ x h, and so on...)

Proof of Theorem 6.1

We have :

Z (+) s,t = 1 {B (ν) s >l} P inf {s≤u≤t} B (ν) u -B (ν) s > l -B (ν) s |B (ν) t -B (ν) s . (24)
Thus, we need to compute, for l, and λ = l -B (ν) s

:

P inf {s≤u≤t} B (ν) u -B (ν) s > l -B (ν) s |B (ν) t -B (ν) s = m = P T λ > (t -s)|B (t-s) = m , (25) since :
i. the quantity in [START_REF] Mansuy | Harnesses, Lévy bridges and Monsieur Jourdain[END_REF] clearly depends only on (t -s) and not on the pair (s, t),

ii. the law of the Bridge from a to b, over the time interval [0, α] of a Brownian motion with drift ν does not depend on ν.

Thus, from [START_REF] Mansuy | Harnesses, Lévy bridges and Monsieur Jourdain[END_REF], we need to compute :

P T λ > (t -s)|B (t-s) = m = 1 -P T λ < (t -s)|B (t-s) = m . (26)
Let us denote by

P (u)
0→m the law of the Brownian bridge of length u, starting at 0, ending at m. Then, there is the result :

P (u) 0→m (T λ < u) = exp - (2λ(λ -m)) + u (27)
Comment : We learnt from G.Pagès [START_REF] Pagès | Introduction to Numerical Probability for Finance[END_REF] that the Brownian Bridge method to estimate the error due to discretization for path dependent options relies upon formula [START_REF] Pagès | Introduction to Numerical Probability for Finance[END_REF], which is also proven in Section 7.7 of [START_REF] Pagès | Introduction to Numerical Probability for Finance[END_REF].

We give two proofs of ( 27) :

• A first proof via the reflection principle :

(a) By scaling, we can restrict ourselves to u = 1;

(b) In [START_REF] Yor | Some remarks about the joint law of Brownian motion and its supremum[END_REF], it is remarked, following [START_REF] Seshadri | Exponential models, Brownian motion and independence[END_REF], that if S t = sup u≤t B u , then 2S 1 (S 1 -B 1 ) is independent from B 1 , and satisfies :

2S 1 (S 1 -B 1 ) law = e,
where e is a standard exponential variable. Thus :

P (1) 0→m (T λ < 1) = P (1) 0→m (S 1 > λ) = P (1) 0→m (2S 1 (S 1 -B 1 ) > 2λ(λ -m)) = exp (-2λ(λ -m))
(here, we have assumed a priori : λ(λ -m) > 0.)

• A second proof via Doob's maximal identity : In order to show :

p λ, m ≡ P sup u≤1 B u > λ|B 1 = m = exp (-2λ(λ -m)), (28)
(λ > m, λ > 0), we use time inversion :

p λ, m = P sup t≥1 B 1/t > λ|B 1 = m = P sup t≥1 Bt t > λ| B1 = m = P ∃t ≥ 1, Bt -λt > 0| B1 = m = P ∃u ≥ 0, B1+u -λ(1 + u) > 0| B1 = m = P ∃u ≥ 0, ( B1+u -B1 ) -λu > (λ -m) = P sup u≥0 (B u -λu) > (λ -m) = P sup u≥0 exp (2λ(B u -λu)) > exp (2λ(λ -m)) = P 1 U > exp (2λ(λ -m)) = P [U < exp (-2λ(λ -m))] = exp (-2λ(λ -m)).
We note that since we derive formula (28) from Doob's maximal identity, we can recover the joint law of (S 1 , B 1 ) from this identity; traditionally, this joint law is obtained from the reflection principle.

Exercise 6.6. a) Prove [START_REF] Pagès | Introduction to Numerical Probability for Finance[END_REF] with the help of the absolute continuity relationship between P (u) 0→m and P 0 on F t , for any t < u.

b) Compute the law of T λ under P

(u)
0→m . The answer is :

P (u) 0→m (T λ ∈ dt) = dt λ t p t (0, λ) p u-t (λ, m) p u (0, m) .
Exercise 6.7. Compute :

P sup t≤1 B t a + bt > λ|B 1 = m .
The previous method yields :

P sup t≤1 B t a + bt > λ|B 1 = m = exp (-2λa(λ(a + b) -m)),
for λ > m a+b , λ > 0.

Towards an integral representation of the positive P F H functions

So far, we have discovered two families of P F H functions, i.e : h (l, ν) on one hand and h (a) on the other hand. It would be nice to exhibit a unifying formula for all these P F H functions.

In fact, we are interested in the following question : is there an integral representation of all positive P F H functions? We were motivated to raise this question from the following classical result, due to D. Widder [START_REF] Widder | The Heat Equation[END_REF], see also J.L. Doob [START_REF] Doob | Classical Potential Theory and Its Probabilistic Counterpart[END_REF], concerning positive space-time harmonic functions (of Brownian motion) : h ≥ 0 is space-time harmonic, if, by definition, {h(B s , s), s ≥ 0} is a Brownian martingale. Thus, there exists a positive finite measure µ(dλ) such that :

h(x, s) = ∞ -∞ dµ(λ) exp λx - λ 2 s 2 . ( 29
)
Comment on (29):

In chapter I of [START_REF] Yor | Some Aspects of Brownian Motion[END_REF], a probabilistic proof of Widder's result is obtained with the help of the non canonical Brownian motion :

B ′ t = B t - t 0 ds s B s , t ≥ 0.
It would be most interesting to obtain a representation of the P F H functions by constructing a two-parameter process, with a past-future filtration F ′ s,t such that :

F s,t = F ′ s,t ∨ σ(B s , B t ),
and F ′ s,t is independent from the pair (B s , B t ).Such a filtration is exhibited in the next exercise : Exercise 6.8. The past-future Bridge filtration.

i. Let us consider the following two-parameter filtration :

F ′ s,t = σ{B u - u s B s , u ≤ s; B t+h -B t , h ≥ 0}.
Note that the σ-field F s,t satisfies :

F s,t = F ′ s,t ∨ σ(B s , B t ),
and that

F ′ s,t is independent from σ(B s , B t ), in fact from F [s,t] = σ{B u , s ≤ u ≤ t}.
ii. Prove that F ′ s,t 0≤s<t<∞ is the natural two-parameter filtration associated with :

X s,t = B s - s 0 du u B u + 1 t B t - ∞ t dv v 2 B v ,
that is :

F ′ s,t = Ξ s,t ≡ σ{X u,v ; u ≤ s, v ≥ t}. Hint : simply show that : E[X s,t B s ] = E[X s,t B t ] = 0,
and proceed from there.

iii. Note that :

X s,t = X (-) s + X (+) t , where : X (-) s = β(B) s ; X (+) t = β( B) 1/t , and : β(B) s = B s - s O du u B u
is the Brownian motion introduced in Chapter I of Yor, Zürich [START_REF] Yor | Some Aspects of Brownian Motion[END_REF]. Furthermore : Bu = uB 1/u . Question 6.1. It may also be of interest to consider the family of intervalσ-field : F [s,t] = σ{B u , s ≤ u ≤ t}, and to look for the "generation" of

F ′ [s,t] , a sub σ-fields of F [s,t] : F ′ [s,t] = σ{ B u -B v u -v - B t -B s t -s ; s ≤ u ≤ v ≤ t}.
Can one give a simpler description of F ′ [s,t] ? Note that, if I 1 , . . . , I k are disjoint intervals then

F ′ I 1 , . . . , F ′ I k are independent
Independently from the previous exercise, we have obtained the following partial answers presented in the next two propositions. Proposition 6.3. Let h(s, t; x, y) = φ( 1 t-s ; y-x t-s ) for a function φ(u, z). Then, h is P F H if and only if φ is space-time harmonic, i.e :

φ ′ u + 1 2 φ ′′ z 2 = 0. ( 30
)
Proof. The LHS of the first equation in (-) writes, in this case : (with :

u = 1/(t -s); z = (y -x)/(t -s)) 1 (t -s) 2 φ ′ u (u, z) - y -x (t -s) 2 φ ′ z (u, z) + y -x (t -s) 2 φ ′ z (u, z) + 1 2(t -s) 2 φ ′′ z 2 . ( 31
)
Hence, the result.

Proposition 6.4. Let h(s, t; x, y) = F ( x √ t-s , y √ t-s ) for a function F . Then, h is P F H if and only if : aF ′ a + (2a -b)F ′ b + F ′′ b 2 = 0 (2b -a)F ′ a + bF ′ b + F ′′ a 2 = 0 (32)
Proof.

h ′ s = 1 2 x (t -s) 3/2 F ′ a + 1 2 y (t -s) 3/2 F ′ b ; h ′ x = 1 √ t -s F ′ a ; h ′′ x 2 = 1 t -s F ′′ a 2 .
We must have :

1 2 1 t-s x √ t-s F ′ a + y √ t-s + 1 t -s y -x √ t -s F ′ a + 1 2(t -s) F ′′ a 2 = 0 Hence : 1 2 (aF ′ a + bF ′ b ) + (b -a)F ′ a + 1 2
F ′′ a 2 = 0. We leave the end of the proof to the reader.

Note that F , solution of [START_REF] Widder | The Heat Equation[END_REF], is harmonic for the 2-dimensional Ornstein-Uhlenbeck process (X, Y ) defined by :

X t = β t + t 0 Y s ds; Y t = γ t + t 0 X s ds,
where β and γ is a 2-dimensional Brownian motion.

A question : We dispose of the equations (-) and (+). We would like to find a more general family of P F H functions, than the ones we obtained in Proposition 6.3 and Proposition 6.4. More precisely, we consider the functions defined by : h(s, t; x, y) = exp (a(s, t)xy + b(s, t)x + c(s, t)y + d(s, t)), associated to the four time dependent functions a, b, c, d.

• h satisfies formula (-) if and only if :

(after computation of h ′ s , h ′ x , h ′ x 2
and identification of the terms in xy, y 2 , x, y and the constant parts)

                 (1) ∂a ∂s = 1 t-s a(s, t) (2) a(s, t) 1 t-s + 1 2 a(s, t) = 0 (3) ∂b ∂s = 1 t-s b(s, t) = 0 (4) ∂c ∂s + b(s, t) 1 t-s + a(s, t) = 0 (5) ∂d ∂s + 1 2 b(s, t) 2 = 0 (33)
(2) gives : a(s, t) = 0, or a(s, t) = -2 t -s .

(3) gives :

b(s, t) = β(t) t -s .
Let us take a(s, t) = -2 t-s (leaving the case a(s, t) = 0 aside for the moment). Then (4) gives :

c(s, t) = β(t) t -s + γ(t).
(5) gives :

d(s, t) = - 1 2
β(t) 2 t -s + δ(t).
• h satisfies formula (+) if and only if :

(after computation of h ′ t , h ′ y , h ′ y 2
and identification of the terms in xy, y 2 , x, y and the constant parts)

                 (1 ′ ) -∂a ∂t = 1 t-s a(s, t) (2 ′ ) a(s, t) 1 t-s + 1 2 a(s, t) = 0 (3 ′ ) -∂b ∂t + a(s, t)c(s, t) + c(s,t) t-s = 0 (4 ′ ) ∂c ∂t 1 t-s + c(s, t) = 0 (5 ′ ) ∂d ∂t + c(s, t) 2 = 0 (34)
Again (2 ′ ) gives :

a(s, t) = 0, or a(s, t) = - 2 t -s . ( 4
′ ) gives : c(s, t) = γ(s)(t -s). ( 3
′ ) gives : b(s, t) = -γ(s)t + β(s).
(5 ′ ) gives :

d(s, t) = γ(s) 2 (t -s) 3 3 + δ(s).
We now have to compare the different expressions we have obtained for a, b, c, d and determine 6 functions of one variable, i.e : β, γ, δ on one hand and β, γ, δ on the other hand. H t,y (s, x) = h(s, t; x, y)p t-s (x, y)is space -time harmonic in(s, x).

We shall note : (s, x)-space-time harmonic. Hence, h(s, t; x, y) = 1 p t-s (x, y) H t,y (s, x).

Moreover, thanks5 to Proposition 6.1, we have also :

h( 1 t , 1 s ; y t , x s ) = 1 p t-s (x, y) K t,y (s, x), (36)
where K t,y is a (s, x)-space-time harmonic function. Let us rewrite [START_REF] Yor | Some Aspects of Brownian Motion[END_REF] with

u = 1 t , v = 1 s , x ′ = y t , y ′ = x s : h(u, v; x ′ , y ′ ) = 1 p 1 u -1 v (sy ′ , tx ′ ) K 1 u ,tx ′ ( 1 v , sy ′ ) = 1 p 1 u -1 v ( y ′ v , x ′ u ) K 1 u , x ′ u ( 1 v , y ′ v ). (37)
Thus, H and K need to satisfy :

h(s, t; x, y) = 1 p t-s (x, y) H t,y (s, x) = 1 p 1 s -1 t ( y t , x s ) K 1 s , x s ( 1 t , y t ). ( 38
)
After some elementary computation, we have :

1 p 1 s -1 t ( y t , x s ) = √ st 2π(t -s) exp - 1 2 (sy -xt) 2 st(t -s) . (39)
Hence, [START_REF] Yor | Some remarks about the joint law of Brownian motion and its supremum[END_REF] becomes :

H t,y (s, x) = 1 √ st exp (sy -xt) 2 st(t -s) - (y -x) 2 t -s K 1 s , x s ( 1 t , y t ) = 1 √ st exp - 1 2
y 2 t - x 2 s K 1 s , x s ( 1 t , y t ). (40)
If we assume that :

K 1 s , x s ( 1 t , y t ) = ∞ -∞ dη K 1 s , x s (η) exp η y t - η 2 2t ,
i.e : the representing measure of K a,z (s, x)is absolutely continuous, it "is likely" that :

1 √ s exp - x 2 2s K 1 s , x s (η),
is (s, x)-space-time harmonic, i.e :

1 √ s exp - x 2 2s K 1 s , x s (η) = τ η (dλ) exp λx - λ 2 s 2 . ( 41
)
Using again [START_REF] Yor | Some remarks about the joint law of Brownian motion and its supremum[END_REF], and taking t = 1 u , y t = uy = z, we have :

K 1 s , x s (u, z) = u s exp 1 2 x 2 s + z 2 t H 1 u , z u (s, x) (42)
If we assume that :

H t,y (s, x) = ∞ -∞ dλ H t,y (λ) exp λx - λ 2 s 2 ,
it "is likely" that :

1 √ u exp z 2 u H 1 u , z u (λ), is (u, z)-space-time harmonic, i.e : 1 √ u exp z 2 u H 1 u , z u (λ) = θ λ (dρ) exp ρz - ρ 2 u 2 . ( 43
)
It now remains to find a correspondence between τ η and θ λ ....... Exercise 6.9. Show from the above arguments that :

e (t) s ≡ exp - 2B s B t t -s is a (F s,t , s < t) martingale.
More generally, if (B u ) is a n-dimensional Brownian motion, show that :

exp - 2B s • B t t -s is a PF martingale.
Solution : We need to show in particular that :

exp - 2B s m t -s p t-s (B s , m) is a (F s , s < t) martingale.
In fact, this expression is : p t-s (B s , -m), which is of course a (F s , s < t) martingale!! Exercise 6.10. ( : T. Fujita, M. Yor [START_REF] Fujita | Some past-future martingales for Symmetric Random Walk and Symmetric Lévy processes[END_REF]).

i. Assume (X t ) is a symmetric Lévy process, with semigroup density : p t (x, y) ≡ q t (|x -y|), for q t a function on R + . Prove that, for any

a ∈ R, p t-s (X s + a, -(X t + a)) p t-s (X s , X t ) , is a F s,t martingale.
ii. Application : for the Cauchy process X t = C t , one gets : Based on this, we can state the following : Proposition 6.5. h is P F H ≥ 0 if and only if : a) ∀(t, y), there exists a space-time harmonic function for Brownian motion :

(t -s) 2 + (C t -C s ) 2 (t -s) 2 + (C t + C s + 2a) 2 is a (F s,
K (+) (t,y) such that : h(s, t; x, y) = K (+) (t,y) s t -s ; xt -ys √ t(t -s) (44)
b) ∀(s, x), there exists a space-time harmonic function for Brownian motion :

K (-) ( 1 s , x s ) such that : h(s, t; x, y) = K (-) ( 1 s , x s ) s t -s ; √ s(y -x) t -s (45)
To help the reader with these slightly complicated formulae, let us compute K (+) and K (-) for :

h(s, t; x, y) = exp - 2xy t -s ,
in the following exercise : Exercise 6.13. a) Fix t and m; find K (+) a space-time harmonic function such that :

exp - 2xm t -s = K (+) s t -s ; xt -ms √ t(t -s) .
Answer :

K (+) (u, ξ) = exp -2m ′ ξ -2m ′2 u ,
where m ′ = m √ t . (Note the remarkable fact that K (+) depends only on m ′ ). b) Fix s and x; find K (-) a space-time harmonic function such that :

exp - 2xy t -s = K (-) s t -s ; √ s(y -x)
t -s .

Answer :

K (-) (u, ξ) = exp (-2x ′ (ξ + x ′ u)),
where :

x ′ = x/ √ s.
(Note the remarkable fact that K (-) depends only on x ′ ).

Proof. (of Proposition 6.5) As explained just before stating that Proposition, the proof consists in expressing the generic ≥ 0, space-time harmonic function h

(t) m (s, x) for the Brownian bridge BB (t)
0→m in terms of a space-time harmonic function for Brownian motion. The result is :

h (t) m (s; x) = K s t -s ; xt -ms √ t(t -s) . ( 46
)
we denote by H

h (t) m (s; x) = h (1) m/ √ t s t ; x √ t . ( 47
) 2. Let h ∈ H (1)
m ; then, we shall show that there exists K ∈ H such that :

h 1 τ + 1 ; m + y τ + 1 = K 1 τ ; y τ . (48)
Then, we obtain (46) from ( 47) and (48).

It now remains to prove (48).

In order to state properties in Brownian terms, and not in terms of Brownian bridges, we use time inversion (in a first instance) :

B ′ v = vB 1/v .
Then, the martingale property for h writes : (s < s ′ < 1)

E F uB ′ 1/u , 1 u ≥ 1 s ′ h s; sB ′ 1/s |B ′ 1 = m = E F uB ′ 1/u , 1 u ≥ 1 s ′ h s ′ ; s ′ B ′ 1/s ′ |B ′ 1 = m .
We use : σ = 1/s; σ ′ = 1/s ′ , then :

E F (B ′ v , v ≥ σ ′ ) h 1 σ ; B ′ σ σ |B ′ 1 = m = E F (B ′ v , v ≥ σ ′ ) h 1 σ ′ ; B ′ σ ′ σ ′ |B ′ 1 = m .
We now shift time by 1 : σ = τ + 1; σ ′ = τ ′ + 1; and we introduce :

Bu = B ′ (u+1) -B ′ 1
, a new Brownian motion. Then :

E F Bu , u ≥ τ ′ h 1 τ + 1 ; m + Bτ τ + 1 = E F Bu , u ≥ τ ′ h 1 τ ′ + 1 ; m + Bτ ′ τ ′ + 1 .
It is advantageous to introduce the notation :

H(τ, x) = h 1 τ + 1 ; m + x τ + 1 .
The previous equality writes :

E F Bu , u ≥ τ ′ H τ ; Bτ = E F Bu , u ≥ τ ′ H τ ′ ; Bτ ′ .
Finally, we shall bring time "back in order" by introducing β such that : Bu = uβ 1/u , (time inversion : second instance), and :

1 τ ′ = θ ′ < 1 τ = θ. Hence : E F β 1/k , 1 k ≤ θ ′ H 1 θ ; 1 θ β θ = E F β 1/k , 1 k ≤ θ ′ H 1 θ ′ ; 1 θ ′ β θ ′ . (49)
Consequently, K(θ; x) = H 1 θ ; x θ belongs to H. We have obtained (48).

An explicit expression for the law of g (ν)

x (t)

In Note 2, we discussed the Black-Scholes formula in relation with the law of G K . Similarly, here in our finite horizon framework, we would like to give an explicit expression for the law of :

g (ν)
x (t) = sup{s ≤ t, B (ν) s = x}.

Already, in the case ν = 0, the following result has been obtained (see [START_REF] Yor | The distribution of Brownian quantiles[END_REF] and alsoo [START_REF] Jeanblanc | Mathematical Methods for financial Markets[END_REF] Proposition 4.3.3.3) :

P g (0) x (t) ∈ du = du π u(t -u) exp - x 2 2u (0 < u < t). (50)
Note that this is a sub-probability, since

du π √ u(t-u) (0 < u < t) is the arcsin distribution; indeed, P g (0)
x (t) = 0 = P (T x ≥ t)

= P |N| ≤ |x| √ t .
We are now interested in the general case : ν = 0; we shall compute, for 0 ≤ s ≤ t : γ

(ν)
x, t (s)

def = P(0 < g (ν) x (t) ≤ s). γ (ν) x, t (s) = E 1 {0<g (0) x (t)≤s} exp νB t - ν 2 t 2 . ( 51
)
On {0 < g (0)

x (t) ≤ s}, we have : T x ≤ s, and introducing :

Bu = B Tx+u -x,
a new Brownian motion, independent from F Tx , we have :

g (0) x (t) = T x + g(t -T x ),
where g(h) = sup{s < h, Bs = 0}. We have :

γ (ν)
x, t (s) = E 1 {Tx≤s} 1 {g(t-Tx)≤s-Tx} exp ν(x + Bt-Tx )

ν 2 t 2 = exp νx - ν 2 t 2 E 1 {Tx+g(t-Tx)≤s} exp ν (t -T x ) -g(t -T x ) m1 ,
where we have used the factorisation :

Bt-Tx = (t -T x ) -g(t -T x ) m1 , with m1 = ǫm 1 independent from (T x , g(t -T x )), ǫ is Bernoulli, m 1 law = √ 2e. We introduce the function :

φ(λ) = E[exp (λ m1 )] = E[cosh (λm 1 )] = ∞ 0 dt e -t cosh (λ √ 2t). (52)
An integration by parts shows that :

φ(λ) = 1 + e λ 2 /2 |λ| √ 2πP (|N| ≤ |λ|) . (53)
See [START_REF] Azéma | Etude d'une martingale remarquable[END_REF] for similar computations. Consequently, we obtain :

γ (ν) x, t (s) = P 1 {0<gx(t)≤s} e νx-ν 2 t 2 φ(ν t -g x (t)) , (54)
which from (50), gives :

P(g (ν) x (t) ∈ du) = du π u(t -u) exp - x 2 2u φ(ν √ t -u). ( 55
)
Very close computations may be found in Chapter 4, subsection 4.3.9 in [START_REF] Jeanblanc | Mathematical Methods for financial Markets[END_REF].

On the time spent below a level by Brownian motion with drift

Consider :

A (ν) x (t) = t 0 ds 1 {B (ν)
s ≤x} , x > 0. Thus, it is shown in [START_REF] Yor | The distribution of Brownian quantiles[END_REF], that, for ν = 0 :

A (0) x (t) law = g (0) x (t). ( 56
)
Question 6.2. We may ask whether this identity in law also holds with (B (ν) ) instead of (B).

Question 6.3. Another question consists in studying :

P A (ν)
x (t) ≤ s|F s . (Maybe already studied by Dassios [START_REF] Dassios | The distribution of the quantile of a Brownian motion with drift and the pricing related path-dependent options[END_REF], and Embrechts, Rogers and Yor [START_REF] Embrechts | A proof of Dassios' representation of the α-quantile of Brownian motion with drift[END_REF])

Another class of past-future martingales

(To appear in second edition of Chaumont-Yor, [START_REF] Chaumont | Exercises in Probability[END_REF], Chapter 7). We are interested in constructing (F s,t ) martingales from the simple recipe : if F ∈ L 1 (W, F ∞ ), then :

M s,t (F ) def = E [F |F s,t ] is a (F s,t )-martingale.
For a number of functionals F , we are able to compute M s,t (F ). Example 6.1.

F = exp ∞ 0 f (u)dB u - 1 2 ∞ 0 f 2 (u)du , ( 57
)
where f ∈ L 2 ([0, ∞), du). Then, E [F |F s,t ] = exp s 0 f (u)dB u + ∞ t f (u)dB u - 1 2 ∞ 0 f 2 (u)du × E exp t s f (u)dB u |F s,t .
We know that :

E t s f (u)dB u |F s,t = 1 t -s t s f (u)du (B t -B s ) = t s f (u)du B [s,t] .
Hence :

E exp t s f (u)dB u |F s,t = exp t s f (u)du B [s,t] exp 1 2 t s f 2 (du)du - 1 t -s t s f (u)du 2 .
Finally, we have obtained :

E [F |F s,t ] = F (1) s F (2) s,t F (3) t , (58)
where :

         F (1) s = exp s 0 f (u)dB u -1 2 s 0 f 2 (u)du F (2) s,t = exp t s f (u)du B [s,t] -1 2(t-s) t s f (u)du 2 F (3) t = exp ∞ t f (u)dB u -1 2 ∞ t f 2 (u)du . ( 59
)
Wishful thinking : Recall that the computation of E[F |F t ], for F in (57), and the fact that these functionals are total in L 1 (W, F ∞ ) easily leads to the representation result of Brownian martingales as stochastic integrals with respect to Brownian motion. Could there be an analogous result for the (F s,t ) filtration? What might be the P F martingale of reference, perhaps : B [s,t] ? Exercise 6.14. Compute :

E ∞ 0 du e -λu f (B u )|F s,t .
A first attempt :

E ∞ 0 du e -λu f (B u )|F s,t = s 0 du e -λu f (B u ) + ∞ t du e -λu f (B u ) + t s du e -λu E [f (B u )|F s,t ] .
We observe that, for s < u < t :

B u = B s + u -s t -s (B t -B s ) + R, with R independent of (B s , B t -B s ) and E [R 2 ] ≡ r 2 = (u-s)(t-u) (t-s)
. Hence :

E [f (B u )|F s,t ] = 1 √ 2πr 2 ∞ -∞ dx e -x 2 2r 2 f (x + B s + (u -s)B [s,t] ). ( 60
)
We also note that we have obtained :

∀y ∈ R, 1 √ 2πr 2 e -( y-(Bs+(u-s)B [s,t] ) ) 2 2r 2
is a (F s,t ) martingale for s < u < t, but we need to compute :

t s du e -λu E [f (B u )|F s,t ]
with the help of (60)

sup s≥0 M s law = 1 U ; b) if G K (M) = sup{s ≥ 0, M s ≥ K}, with K > 0, then, for any stopping time T : P (G K (M) > T |F T ) = M T K ∧ 1. ( 61
)
Remark : In the same vein, we mention that the so-called "Azéma-Yor martingales" F (S t , M t ), with S t = sup s≤t M s admit an extension (with no formal change in the formula) to the family of martingales with no positive jumps. See L. Nguyen-M. Yor [START_REF] Nguyen | Some martingales associated to reflected Lévy processes[END_REF].

A formula in the Lévy process framework

We would like to apply the preceeding to M t = exp (X t -tψ(1)), t ≥ 0, where (X t , t ≥ 0) is a Lévy process with no Brownian component.

A general version of Tanaka's formula ( see [START_REF] Yor | Rappels et Préliminaires généraux-Temps Locaux[END_REF]) is :

(M t -K) + = (M 0 -K) + + t 0 1 {M s ->K} dM s + Σ (K) t , (62)
where :

Σ (K) t = s≤t 1 {M s ->K} (M s -K) -+ 1 {M s -≤K} (M s -K) + . (63)
Let ν(dx) denote the Lévy measure of X. Then, Lévy's compensation formula

gives :

E Σ (K) t = E t 0 1 {M s ->K} ds ν(dx) (M s -e x -K) - + E t 0 1 {M s -≤K} ds ν(dx) (M s -e x -K) + = t 0 ds φ(s, K), (64)
with :

φ(s, K) = 1 0 du µ(]0, u[)E M s 1 {K<Ms< K u } + ∞ 1 du µ([u, ∞[)E M s 1 { K u <Ms<K} , (65)
where µ is the image of ν by the exponential application : x → exp (x). Finally, we have obtained :

E (M t -K) + = (M 0 -K) + + t 0 ds φ(s, K),
which plays the same role as Tanaka's formula in the continuous framework.

8 Note 8 : Other option prices which increase with maturity

The contents of this note are taken entirely from a preprint by Carr-Ewald-Xiao [START_REF] Carr | On the qualitative effect of volatility and duration on prices of Asian options[END_REF], to whom we are grateful for free access.

The main result of that paper is the following :

Theorem 8.1. Let g be a continuous convex function; then, the function :

A (t) g = E g 1 t t 0 ds E s
is an increasing function of t.

Exercise 8.1. Compute, for any n ≥ 1, the function :

a n (t) = E 1 t t 0 ds E s n ≡ ãn (t) t n .
Partial solution and hints :

• a 1 (t) = 1;

• a 2 (t) = 2 t 2 (e t -1 -t); • In order to compute a n (t), it may be useful to first prove the following formula : for α sufficiently large,

∞ 0 dt e -αt ãn (t) = n! α 2 (α -1)(α -3) . . . (α -n(n-1)

2

) ,

where

ãn (t) = E t 0 ds E s n ≡ t n a n (t);
• More directly, we obtain :

a n (t) = n! t n E t 0 ds 1 t s 1 ds 2 . . . t s n-1 ds n exp (B s 1 + • • • + B sn ) - 1 2 (s 1 + • • • + s n ) = n! t n t 0 ds 1 t s 1 ds 2 . . . t s n-1 ds n exp 1 2 C(s 1 , . . . , s n )
where :

C(s 1 , . . . , s n ) = E (B s 1 + • • • + B sn ) 2 -(s 1 + • • • + s n ).
and we find :

E (B s 1 + • • • + B sn ) 2 = n 2 s 1 + (n -1) 2 (s 2 -s 1 ) + • • • + (s n -s n-1 ).
Hence :

C(s 1 , . . . , s n ) ≡ n-1 j=0 (n -j) 2 -(n -j) (s j+1 -s j ) ≡ n-1 j=0 ((n -j)(n -(j + 1))) (s j+1 -s j )
• Finally, it may be useful in order to solve this question to use the "Asian Option identity in law" :

A (ν) T λ = T λ 0 ds exp 2(B s + νs) law = β 1,a 2γ b ,
where T λ is an independent exponential time with parameter λ, and β 1,a and γ b are 2 independent random variables, respectively distributed as beta(1, a) and gamma(b), with a = µ+ν 2 , b = µ-ν 2 , and µ = √ 2λ + ν 2 . For a compendium of results/papers on this topic, see Yor [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF]. 9 Note 9 : From puts to calls : more care is needed! 9.1 Why should one be careful?

In pricing financial options , the LHS of Part A-( 12) arises very naturally in terms of put options, i.e. when considering :

E (K -M t ) + (66)
On the other hand, the price of a call option is :

E (M t -K) + (67)
A most common argument to "reduce" (67) to (66) is to invoke "call-put parity" and/or "change of numéraire". Mathematically, this means that we consider the new probability Q defined via :

Q| Ft = M t • P| Ft , (68)
and the martingale ( 1 Mt , t ≥ 0) under Q, since :

E P (M t -K) + = E Q 1 - K M t + = K E Q 1 K - 1 M t + . (69)
However, two difficulties arise in order to perform these operations rigorously :

i. in order that Q, as defined via (68), be a probability, we need that (M t , t ≥ 0) is a true martingale under P, i.e : it satisfies in particular

E P (M t ) = 1;
ii. some care is needed also concerning (69); in particular M t might take the value 0 on some F t -set of positive P-probability.

To summarize, (68) and (69) are correct if (M t , t ≥ 0) is a strictly positive true martingale under P.

Formally, this may be stated as :

Proposition 9.1. If (M t
) is a strictly positive true continuous martingale under P, define P M via :

P M | Ft = M t • P| Ft . Denote : g (1) ∞ = sup{t ≥ 0, M t = 1}. Then : i. E P F t (M t -1) + = E M F t 1 {g (1)
∞ ≤t} , for every F t ∈ F t ;

ii.

E P [F t |M t -1|] = E P F t 1 {g (1) ∞ ≤t} + E M F t 1 {g (1) ∞ ≤t} ; iii. g (1)
∞ has the same distribution under P and under P M .

Proof.

i. We write :

E P F t (M t -1) + = E M F t 1 - 1 M t + = E M F t 1 {g (1) ∞ ≤t} ,
from Theorem 1.1 (Part A), since 1 Mt , t ≥ 0 is, under P M , a martingale which converges to 0 as t → ∞.

ii.

E P [F t |M t -1|] = E P F t (M t -1) + + E P F t (1 -M t ) + ,
and we apply both the previous result and Theorem 1.1 (Part A).

iii. Taking F t = 1 in i., we obtain :

P M g (1) ∞ ≤ t = E P (M t -1) + = E P [(M t -1)] + E P (M t -1) - = E P (1 -M t ) + = E P g (1) ∞ ≤ t , from Theorem 1.1 (Part A).
We note that Proposition 9.1 extends in the case when (M t ) is a true continuous martingale, taking values in R + , but it may vanish, i.e : P (T 0 < ∞) > 0. Indeed, under P M , T 0 = ∞ a.s., and the previous arguments are still valid.

In order to obtain some analogue of Theorem 1.1 (Part A) for :

E F t (M t -K) +
in the general case when (M t ) is a local martingale belonging to L (+) 0 , we shall proceed directly.

The "common" Bessel example

Before discussing in a general framework, we write some version of the identity Part A-( 12) for (M t = 1 Rt , t ≥ 0), where R is a BES(3), starting from 1. We denote by W a and P

(3) a (a > 0) the respective laws of Brownian motion and BES(3) starting at a > 0, on the canonical space C(R + , R), with X t (ω) = ω(t), and F t = σ{X s , s ≤ t}; then, there is the well-known Doob's h-transform relationship :

P (3) a | Ft = X t∧T 0 a • W a | Ft . (70)
Proposition 9.2. There is the identity :

E (3) 1 F t 1 X t -1 + = W 1 F t 1 {γ≤t≤T 0 } , (71)
for every F t ∈ F t , and γ = sup{t < T 0 , X t = 1}.

Proof. Thanks to (70), the LHS of (71) equals :

W 1 F t (1 -X t∧T 0 ) + 1 {t≤T 0 } ,
which is equal to the RHS of (71) thanks to formula Part A-( 12), applied with

F ′ t = F t 1 {t≤T 0 } . Remarks.
a) We understand that the present discussion is close to recent work by S. Pal and P. Protter motivated by financial bubbles. In fact, we had acces to their preprint [START_REF] Pal | Strict local martingales, bubbles and no early exercise[END_REF], at the beginning of April 2008 and after writing this Note 9.

b) It is worth noting that, as a consequence of (71), there is the identity

E (3) 1 1 X t -1 + = W 1 (γ ≤ t ≤ T 0 ) ,
which shows clearly that the LHS is not an increasing function of t; indeed, the RHS converges to 0 as t → ∞, as a consequence of Lebesgue's dominated convergence theorem.

In fact, we can compute explicitly this RHS, which equals :

r(t) ≡ W 1 (T 0 ≥ t) -W 1 (γ ≥ t) ≡ (1 -W 1 (T 0 ≤ t)) -(1 -W 1 (γ ≤ t)) ≡ W 1 (γ ≤ t) -W 1 (T 0 ≤ t) .
Recall that, under W 1 :

T 0 law = 1 B 2 1 and γ law = U 2 [0,2] B 2 1
; thus :

r(t) = P |B 1 | ≤ 1 √ t -P |B 1 | ≤ U [0,2] √ t , (72)
that is :

r(t) = 2 π 1 √ t 0 dx e -x 2 /2 - 2 π ∞ 0 dx e -x 2 /2 1 - √ tx 2 + ( 73
)
In particular, it easily follows from (72) that :

r(t) ∼ t→∞ 2 πt 3 1 6 . (74)
It is also easily proven directly that :

r(t) ∼ t→0 t 2π (75)
Formulae ( 74), (75) are proven in the Appendix.

Exercise 9.1. Prove the formula :

∞ 0 λe -λt dt r(t) = 1 -e -2 √ 2λ 2 √ 2λ -e - √ 2λ .
b) There is an extension of formula (71) for pairs of Bessel processes with respective dimensions δ ∈ (0, 2), and 4 -δ; formula (70) generalizes as :

P (4-δ) 1 F t 1 X 2-δ t -1 + = P (δ) 1 F t 1 {γ≤t≤T 0 } ,
for every F t ∈ F t , and γ = sup{t < T 0 , X t = 1}. Proposition 9.2 leads us easily to a general statement, whose proof simply mimicks that of Proposition 9.2, hence it is left to the reader. Proposition 9.3. Let Q| Ft = M t •P| Ft , with M 0 = 1, and let us assume that T 0 = inf{t, M t = 0} < ∞ P a.s.

Then,

M ′ t = 1 {t<T 0 } Mt
is well defined and strictly positive under Q;

M ′ t ∈ L (+)
0 ; moreover it is a strict local martingale under Q. Finally :

E Q F t (M ′ t -1) + = E P F t 1 {γ≤t<T 0 } , (76)
where γ = sup{t < T 0 , M t = 1}.

Note that, as for Proposition 9.2, the RHS of (76), for F t ≡ 1, is no longer an increasing function of t.

The result (76) should be compared with the general expression given in [START_REF] Madan | Itô's integrated formula for strict local martingales[END_REF], Proposition 2, p.160, for

E Q (M ′ t -K) + .
The contruction of (M ′ t ) in Proposition 9.3 is very close to the discussion by Delbaen-Schachermayer [START_REF] Delbaen | Arbitrage probabilities in Bessel processes and their relations to local martingales[END_REF] of arbitrage within the Bessel processes framework.

A general discussion

This is based on [START_REF] Madan | Itô's integrated formula for strict local martingales[END_REF]; extensions of that discussion to discontinuous local martingales have been done by Chybiryakov [START_REF] Chybiryakov | Itô's Integrated Formula for Strict Local Martingales with Jumps[END_REF], Kaji [START_REF] Kaji | The tail estimation of the quadratic variation of a quasi left continuous local martingale[END_REF].

Let (M t , t ≥ 0) denote an R + -valued continuous local martingale, with M 0 = a > 0. We denote by (L K t , t ≥ 0) the local time at level K of (M t , t ≥ 0).

The following theorem shows how careful one should be when "integrating Tanaka's formula" when (M t∧τ , t ≥ 0) is not a uniformly integrable martingale, with τ a stopping time. This care should be taken in particular for fixed time τ when (M t ) is a strict local martingale.

E (M τ -K) + = (a -K) + + 1 2 E L K τ -c M (τ ), (77)
where the "correction term" c M (τ ) does not depend on K, and is equal to either of the four following quantities :

i

. E [a -M τ ]; ii. lim K→∞ 1 2 E L K τ ; iii. lim σ→∞ σ P sup u≤τ M u ≥ σ ; iv. lim q→∞ π 2 q P M τ ≥ q .
See [START_REF] Madan | Itô's integrated formula for strict local martingales[END_REF] for a discussion of the financial meaning of (77). Let us insist, once again, about the formula different, from (77), which one obtains when considering the put quantity : E (K -M τ ) + , instead of the call quantity :

E (M τ -K) + : E (K -M τ ) + = (K -a) + + 1 2 E L K τ . (78)
Clearly, there is agreement between the identities (77),(78) and the expression (i) of c M (τ ).

Dealing with last passage times :

Throughout both Part A and Part B, last passage times (of local martingales) play a crucial role. It has been comforting that in private discussions around these lectures we did not experience. The reluctance about these "non-stopping times", which existed during the seventies, as K. L. Chung [START_REF] Chung | Probabilistic approach in potential theory to the equilibrium problem[END_REF] describes very well : "For some reason, the notion of a last exit time would not be dealt with openly and directly. This may be due to the fact that such a time is not an "optional", or "stopping time", does not belong to the standard equipment, and so must be evaded at all costs."

The positive attitude (which we experienced in these private discussions) may be partly be due to the fact that the main results of enlargments of filtrations theory are nowadays reasonably well-known.

On the other hand, one should not deny that, when dealing with last passage times, difficulties occur, due to the fact that these times look forever into the future. Undoubtedly, J. Akahori's suggestion (Note 6) originated from these diffculties. From the strong Markov property :

G (ν) a = sup{t, B (ν) t = a} = T (ν) a + sup{u, B (ν)
T (ν) a +u -a = 0} = T (ν) a + G(ν) 0 ,
where : G(ν) 0 = sup{u, Bu + νu = 0}, and ( Bu , u ≥ 0) is a Brownian motion starting from 0 and independent from (B t ). Now, the law of G(ν) 0 has a simple expression, since :

G(ν) 0 = sup{u, Bu = -νu} law = 1 T (ν) ( B) (: time inversion) law = 1 ν 2 T1 (: scaling) law = B2 1 ν 2 (: classical result).
Hence :

T (ν) a , G (ν) a law = T (ν) a , T (ν) a + B2 1 ν 2 , with B1 independent from T (ν)
a .

• Solution to Exercise 1.2 :

We first have thanks to Theorem A -1.1 :

E (E t -K) + = KP 0 < G (-1/2) ln K ≤ t .
The laws of G (ν) a and G (-ν) a are related via the Cameron-Martin formula :

W (-ν) | F Ga ∩(Ga>0) = exp (-2νa)W (ν) | F Ga . (79)
Hence :

P G (-ν) a ∈ dt|G (-ν) a > 0 = P G (ν)
a ∈ dt . Moreover thanks to (79) :

P G (-ν) a > 0 = exp (-2νa).
So the result :

E (E t -K) + = P G (1/2)
ln K ≤ t , follows. We leave the end of the proof to the reader (take derivative in t of the RHS of (A -5) and recover the density of the law of G (1/2) ln K given in (A -22).

• Solution to Exercise 1.3 :

We recall that :

E (E t -K) ± = (1 -K) ± + 1 2 E L K t (E) .
From the time occupation formula :

E t 0 f (E s ) d E s = E t 0 f (E s ) E 2 s ds = E ∞ 0 f (K)L K t (E) dK ,
and the change of variable e y-s 2 = K, we deduce :

t 0 ds √ 2πs ∞ 0 f (e y-s 2 ) e 2y-s e -y 2 2s dy = t 0 ds √ 2πs ∞ 0 f (K) K e -1 2s (log K+ s 2 )
2 dK, so that :

E L K t (E) = t 0 ds √ 2πs K e -1 2s (log K+ s 2 ) 2 = √ K t 0 ds √ 2πs e -(log K) 2 2s -s 8 = 2 √ KE 1 {4B 2 1 ≤t} exp - (log K) 2 8B 2 1
since the density of the r.v B where (R u , u ≥ 0) is a BES 0 (3) and G 1 (R) = inf{u ≥ 0, R u = 1}. Hence :

T 0 -G 1 (M) law = T K (R),
where on the LHS, T 0 and G 1 (M) are independent and on the RHS, T K (R) = inf{u ≥ 0, R u = K}. Taking the Laplace transform in ( λ 2 2 ) of both sides, one obtains :

e -λ = λK sinh λK E e -λ 2 2 G K (M ) , (80)
since : E e -λ 2 2 T 0 = e -λ ; E e -λ 2 2 G K (R) = λK sinh λK .

Formula (80) becomes :

E e -λ 2 2 G K (M ) = e -λ(1-K) -e -λ(1+K) 2λK = 1 2K 1+K 1-K exp (-λx) dx = 1 2K 1+K 1-K E exp - λ 2 2 T x ,
where T x is the first hitting time at level x of a Brownian motion (β t , t ≥ 0) starting from 0; thus :

E e -λ 2 2 G K (M ) = 1 2K 1+K 1-K E exp - λ 2 x 2 2β 2 1 = E exp - λ 2 2 U 2 K β 2 1
, since :

T x law = x 2 β 2 1 .
• Discussion of Example 2.1 :

We observe that :

sup{t ≥ 0, 1 R t = K} = sup{t ≥ 0, R t = 1 K }.
We consider the process R as obtained by time reversal from a Brownian motion (B t , t ≥ 0) starting from 1 K and killed when it first hits 0. Hence, with the same notation as before, we have :

T 0 law = G K (M) + T 1 (R), (with independence).
Thus, taking once again the Laplace transform in ( λ 2

2 ) of both sides, one obtains :

e -λ 1 K = λ sinh λ E e -λ 2 2 G K (M ) , E e -λ 2 2 G K (M ) = 1 2λ exp -λ( 1 K -1) -exp -λ( 1 K + 1) = 1 2 1 K +1 1 K -1
e -λx dx.

We leave the end of the proof to the reader (conclude as above).

• Solution to Exercise 2.4 : From Itô's formula, one obtains :

σ t = sinh (B t )e -t 2 .
Hence : σ 2 t = M 2 t -e -t ; therefore, using Theorem A -2.1 :

P (G K ∈ dt) = 1 - 1 K + ǫ 0 (dt) + 1 {t>0} 2K θ t (K) m t (K) dt.
We leave the computation of m t (K) to the reader.

Note 6

• Solution to Exercise 6. l p ν q H p,q (a, b, c, d, f

) = ∞ k,j,n=0 l k+m ν j+m k!j!m! H k (a, d)H j (b, c)f m = ∞ m=0 f m m! ∞ p=m l p (p -m)! H p-m (a, d) ∞ q=m ν q (q -m)! H q-m (b, c) = ∞ p,q=0
l p ν q m≤p,q f m m! H p-m (a, d)H q-m (b, c) 1 (p -m)!(q -m)! Consequently, we have : H p,q (a, b, c, d, f ) = m≤p,q H p-m (a, d)H q-m (b, c) 1 (p -m)!(q -m)! .

• Solution to Exercise 6.1 : First, we need to write : From the representation (72) of r(t) :

- 2 (t -s) (x + νs -l)(y + νt -l) = - 2
r(t) = P |B 1 | ≤ 1 √ t -P |B 1 | ≤ 2U √ t ,
we can show :

r(t) = 2 πt ∞ n=1
r n t n , with :

r n = (-1) n n!(2n + 1) 1 2 n - 2 n n + 1
.

(Note : r 1 = 1 6 , r 2 = -13 120 ). c) We recall that : r(t) ≡ W 1 (T 0 ≥ t) -W 1 (γ ≥ t) . From (72), it is easy enough to obtain :

r(t) = 2 π ρ( 1 √ t ),
with :

ρ(u) = 1 2u 1 -e -2u 2 - 2u u dx e -x 2 2 .
We now show the graph of ρ, for which we thank G. Pagès.

e) Other properties of r :

m

  functions, space-time harmonic functions for the Brownian bridge BB (t) 0→m , and by H the set of the space-time harmonic functions for Brownian motion.

1 .

 1 Clearly, by scaling,

Theorem 9 . 1 .

 91 [[20],Theorem 1] With the previous notation, and hypotheses, there is the formula :

11 1 1 •

 11 Appendix : Developments of some particular points-solutions to the exercises for both Part A and Part B This appendix is decomposed into sections 11-1, 11-2, ... which correspond to the Notes 1, 2, ...11.Note Solution to Exercise 1.1 :

11. 2 2 •

 22 Note Discussion of Example 2.1 : Thanks to time reversal, one has :(B T 0 -u , u ≤ T 0 ) law = (R u , u ≤ G 1 (R)),

2 :l

 2 We recall the generating function expansion of the Hermite polynomials: k+m ν j+m k!j!m! H k (a, d)H j (b, c)f m .Now, we may write :∞ p,q=0

Figure 1 :

 1 Figure 1: Graph of ρ

  3 Proof of Theorem 6.2

It will follow from the next proposition Proposition 6.2. A regular function h(s, t; x, y) is P F H if and only if it satisfies the following system :

  , y)p t-s (B s , y) is a martingale, (but only until t) for Brownian motion, i.e (with obvious notations) :

	(t) 0→y , that
	is : h(s, t; B

6.5.1 A first attempt to represent P F H ≥ 0 functions :

Let h(s, t; B s , B t ) be an F s,t -martingale. Then, for fixed t,

s → h(s, t; B s , B t ) (s < t) is an F (t)

s martingale.

Hence : for fixed (t, y), h(s, t; x, y) is space-time harmonic for P s

  t ) martingale.Question : does there exist an adequate extension of this result to nonsymmetric Lévy processes? Exercise 6.11. Define R u = |B u |, the n-dimensional Bessel process associated with B a Brownian motion in R n ; compute the projection of e Recall again that : h(s, t; x, y) is P F H if and only if : i. ∀(t, y), (s, x) → h(s, t; x, y) is harmonic under P

	(t) s on
	R s,t .
	(t) Exercise 6.12. Show that : e Identify the law of G (t) 1 = sup{s < t, e (t) s = 1}, and more generally the law of (t) G K
	6.5.2 A second attempt to represent P F H ≥ 0 functions :
	(t) 0→y ;
	ii. h(s, t; x, y) ≡ k( 1 t , 1 s ; y t , x

s → 0 when s → t. s ), where k is P F H; and we shall apply (i) to k, for fixed (s, x).

  In this section, (M t , t ≥ 0) denotes a càdlàg R + -valued local martingale, such that : M 0 = 1, lim t→∞ M t = 0, and (M t ) has no positive jumps.Then, our previous main results still hold; precisely :

	7 Note 7 : Which results still hold for dis-
	continuous martingales?
	7.1 No positive jumps
	Proposition 7.1. Under the previous hypotheses,
	a)

  t -s (x + νs)(y + νt) -l(x + y + ν(s + t)) + l 2 = -2 t -s xy + ν(sy + tx) + ν 2 (st) -l(x + y) -lν(s + t) + l 2

	b) (t → ∞) : From (72), we obtain :
	r(t) = P |B 1 | ≤	1 √	t		-P |B 1 | ≤	2U √ t
	=		2 π	0	1 √ t	dx e -x 2 2 -E	0	2U √ t	dx e -x 2 2
	∼		2 πt	0	1	dy e -y 2 2t -E 2U	0	1	dy e -2 y 2 U 2 t
	=		2 πt	0	1	dy e -y 2 2t -1 -E 2U	0	1	dy e -2 y 2 U 2 t	-1
	∼		2 πt	0	1	dy	-y 2 2t	-E (2U)	0	1	dy -	2y 2 U 2 t
	Hence :
		r(t) ∼				2 πt		E	4U 3 t	0	1	dy y 2 -	1 2t	1 3
			=				2 πt 3 E 4U 3 1 3	-	1 6
			=				2 πt 3	1 6
	≡ al -	dl 2 2	+ bν -		cν 2 2	+ f lν.

In general, we use capital letters such as K for "geometric type" objects, and small letters such as l for items related to Brownian motion with drift, with the correspondence : l = ln K.

We often write E [Λ; X] for E [1 Λ X]

They were very much motivated by some computations of Akahori-Imamura-Yano[START_REF] Akahori | On the pricing of options written on the last exit time[END_REF], which Madan-Roynette-Yor[START_REF] Madan | Unifying Black-Scholes type formulae which involve last passage times up to finite horizon[END_REF] have later shown to match formula (13) below.

We remark that these arguments only make use of the Markov property and not of the Brownian framework. A further discussion is provided in[START_REF] Bentata | Further developments stemming from the last passage times viewpoint on Black-Scholes formula[END_REF].

Here, our treatment is not totally rigorous since we assume that an (s, x)-space-time harmonic function, for s < t, may be extended to an (s, x)-space-time harmonic, for all s > 0. Can one characterize such t-space-time harmonic functions? Our second attempt (see

6.5.2) does not suffer from this abuse.

Acknowledgments

What started it all is M. Qian's question (15/08/2007). M. Yor is also very grateful to D. Madan and B. Roynette for several attempts to solve various questions, rewriting, and summarizing... J. Akahori gave the stimulation for Section 6. C. Ewald sent the preprint [6] very early on. The graph of ρ (see A-9) is due to G. Pagès. Lecturing in Osaka and Ritsumeikan (October 2007), Melbourne and Sydney (December 2007), then finally at the Bachelier Séminaire (February 2008) has been a great help. We gave further lectures in Oxford and at Imperial College, London, both in May 2008 and in Amsterdam at VU University in June 2008.

10 Note 10 : A temporary conclusion [START_REF] Dang-Ngoc | Champs markoviens et mesures de Gibbs sur R[END_REF].1 What have we learnt? a) Clearly, the knowledge of the call/put quantities :

for all strikes and maturities, is equivalent to the knowledge of the 1dimensional marginals of M. We have learnt that it is also equivalent to the knowledge of the laws of last passage times G K . b) In our discussion, some fundamental quantities are certain conditional expectations, namely :

] and E σ 2 s |M s , which would certainly deserve more study. However, once introduced, they allow to develop our discussion in a purely martingale framework instead of the more usual Markovian framework.

Where is all this going?

This course seems to open a number of fairly general questions : a) (Note 1). May the universal character of the law of sup u≥t M u be extended to a study involving :

). Can one develop a stochastic calculus with respect to the two-parameter filtration (F s,t )? Creating a two-parameter Itô stochastic integration has already been done, in the context of the study of double points of 2-and 3-dimensional Brownian motion by J. Rosen, M. Yor [START_REF] Rosen | Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane[END_REF]. We would also like to connect this study with the fairly well developed Markovian Fields theory. (see, e.g, Dang-Ngoc-Yor [START_REF] Dang-Ngoc | Champs markoviens et mesures de Gibbs sur R[END_REF], Royer-Yor [START_REF] Royer | Représentation intégrale de certaines mesures quasi-invariantes sur C(R); mesures extrémales et propriété de Markov[END_REF]). c) (Note 8 )The Black-Scholes formula has been the first fundamental option pricing formula, and may be considered to have fathered many other such formulae, e.g : for quantile options, Asian options, and so on... Could one develop a last time or more generally a past-future viewpoint for these options? d) We plan to develop answers to some of these questions in [START_REF] Bentata | Further developments stemming from the last passage times viewpoint on Black-Scholes formula[END_REF].

Comparing the two developments as polynomials in (l, ν), we obtain :

Consequently, we obtain :

11.9 Note 9

• Asymptotic study of r(t), t → 0, and t → ∞.

a) (t → 0) : We use

(1 -X t ) + . Now, using the decomposition :

with (β t ) a one-dimensional Brownian motion starting from 0, one obtains :

and, it is now easily seen that :

-We note that (r(t), t ≥ 0) is integrable over R + , since : r(t) ∼ C t 3/2 , as t → ∞. More precisely, we have :

-Moreover,

-We note that r(t) = 3r(t), is a probability density, and we denote by R a random variable such that:

The Laplace transform of R is given by :

Since :

W 1 e -λ(T 0 -γ) = E Can we exhibit a "natural" Brownian variable which is distributed as R?